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This article is part of the series A Tribute to Professor Ivan Kiguradze.

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On the Fučík spectrum of the scalar p-Laplacian with indefinite integrable weights

Wei Chen1, Jifeng Chu1, Ping Yan2* and Meirong Zhang2

Author Affiliations

1 Department of Mathematics, College of Science, Hohai University, Nanjing, 210098, People’s Republic of China

2 Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

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Boundary Value Problems 2014, 2014:10  doi:10.1186/1687-2770-2014-10

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/10


Received:18 November 2013
Accepted:11 December 2013
Published:9 January 2014

© 2014 Chen et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we study the structure of the Fučík spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1">View MathML</a> of Dirichlet and Neumann problems for the scalar p-Laplacian with indefinite weights <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>. Besides the trivial horizontal lines and vertical lines, it will be shown that, confined to each quadrant of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1">View MathML</a> is made up of zero, an odd number of, or a double sequence of hyperbolic like curves. These hyperbolic like curves are continuous and strictly monotonic, and they have horizontal and vertical asymptotic lines. The number of the hyperbolic like curves is determined by the Dirichlet and Neumann half-eigenvalues of the p-Laplacian with weights a and b. The asymptotic lines will be estimated by using Sturm-Liouville eigenvalues of the p-Laplacian with a weight a or b.

MSC: 34B09, 34B15, 34L05.

Keywords:
indefinite weights; p-Laplacian; Fučík spectrum; spectral structure

1 Introduction

Fučík spectrum was first introduced for the Laplacian on a bounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M6">View MathML</a>, by Dancer [1] and by Fučík [2] in the 1970s, in connection with the study of semilinear elliptic boundary value problems with jumping nonlinearities. Thereafter this important concept was generalized to the p-Laplacian, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M7">View MathML</a>. See [3] and references therein.

In this paper, we are concerned with the Fučík spectrum of the scalar p-Laplacian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M8">View MathML</a>

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, taking the notations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M10">View MathML</a>, let us consider the ODE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M11">View MathML</a>

(1.1)

in which a, b are called potentials, and the ODE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M12">View MathML</a>

(1.2)

in which a, b are called weights. For a pair of potentials a and b, the Fučík spectra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M14">View MathML</a> are defined as the sets of those <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15">View MathML</a> such that equation (1.1) has non-trivial solutions satisfying the Dirichlet boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M16">View MathML</a>

(1.3)

and the Neumann boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M17">View MathML</a>

(1.4)

respectively. Similarly, for a pair of weights a and b, the Fučík spectra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19">View MathML</a> are defined as the sets of those <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15">View MathML</a> such that equation (1.2) has non-trivial solutions satisfying the corresponding boundary conditions (1.3) and (1.4), respectively.

The Fučík spectra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M14">View MathML</a> have been comprehensively understood in [4]: each of them is composed of one horizontal line, one vertical line and a double sequence of differentiable, strictly decreasing, hyperbolic like curves; asymptotic lines of these hyperbolic like curves are given by using (Sturm-Liouville) eigenvalues of the p-Laplacian with a potential; moreover, these curves have a strong continuous dependence on the potentials.

Compared with potentials, indefinite weights will add difficulties to the study of the Fučík spectra. Alif [5] studied <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19">View MathML</a> by means of ‘zero functions’, where the weights a and b were assumed to be sign-changing (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M26">View MathML</a>) continuous functions without ‘singular points’ (which is a technical hypothesis). Their main results are as follows. Besides the trivial horizontal lines and vertical lines, confined to each quadrant of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1">View MathML</a> consists of an odd number of or infinitely many hyperbolic like curves. The asymptotic behavior of the first non-trivial curves in each quadrant was also studied. It was observed that for instance the first curve of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30">View MathML</a> is not asymptotic on any side to the trivial horizontal and vertical lines. In other words, there are always gaps between its asymptotic lines and the trivial horizontal and vertical lines. However, the exact asymptotic lines were not found in that paper.

In this paper, we are interested in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M32">View MathML</a>, where the weights <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a> are assumed to be indefinite (i.e., a and b may or may not change sign). In this case, since the weights are integrable, the method employed in [5] does not work anymore. Using the Prüfer transformation, we convert the second-order ODE (1.2) into a system of first-order ODEs (3.2) and (3.3), for the argument θ and the radius r, respectively. The ODE (3.2) for θ turns to be independent of r, and the boundary conditions (1.3) and (1.4) can be characterized by the solutions of equation (3.2), therefore the Fučík spectra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19">View MathML</a> are completely determined by this first-order ODE (3.2). The solutions of equation (3.2) admit (strong) continuity and Fréchet differentiability in the weights. Based on these properties, we will finally reveal the structure of the Fučík spectra. Our main results are as follows.

(i) Besides at most two vertical lines and two horizontal lines, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1">View MathML</a> confined to each quadrant of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M3">View MathML</a> is made up of zero, an odd number of, or a double sequence of continuous, strictly monotonic, hyperbolic like curves.

(ii) The number of those trivial lines in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1">View MathML</a> is determined by the Dirichlet and Neumann eigenvalues of the p-Laplacian.

(iii) The number of the hyperbolic like curves in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1">View MathML</a> is determined by the Dirichlet and Neumann half-eigenvalues of the p-Laplacian.

(iv) All the hyperbolic like curves have vertical and horizontal asymptotic lines, and these asymptotic lines will be estimated by using (Sturm-Liouville) eigenvalues of the p-Laplacian.

(v) If the weights a and b are positive, the structure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M40">View MathML</a> is comparable with that of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M41">View MathML</a>, the case with potentials. More precisely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1">View MathML</a> is composed of one horizontal line, one vertical line and a double sequence of differentiable, strictly decreasing, hyperbolic like curves in the quadrant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30">View MathML</a>. And all asymptotic lines of these hyperbolic like curves will be given by using (Sturm-Liouville) eigenvalues of the p-Laplacian.

The paper is organized as follows. In Section 2, we will give some preliminary results. Section 3 is devoted to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a>. We first decompose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a> in Section 3.1, according to the number of zeroes of the eigenfunctions. Sections 3.2 and 3.3 are devoted to eigenvalues and half-eigenvalues of the p-Laplacian, respectively. The results in these two subsections enables us to finally determine the structure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a> in Section 3.4. For a pair of positive weights a and b, we can get more information on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a> and the results are given in Section 3.5. The Fučík spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19">View MathML</a> can be studied by similar arguments and we just list the results in Section 4.

2 Preliminary results

Given an exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M49">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M50">View MathML</a> the conjugate number of p, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M51">View MathML</a>. The initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M52">View MathML</a>

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M54">View MathML</a>. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M56">View MathML</a> are the so-called p-cosine and p-sine because they possess properties similar to those of the standard cosine and sine, as shown in the following lemma.

Lemma 2.1 ([6,7])

Thep-cosine andp-sine have the following properties.

(i) Both<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M55">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M56">View MathML</a>are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M59">View MathML</a>-periodic, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M60">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M55">View MathML</a>is even intand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M56">View MathML</a>is odd int;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M63">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M64">View MathML</a>for allt;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M65">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M67">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M68">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M67">View MathML</a>;

(v) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M71">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M72">View MathML</a>; and

(vi) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M73">View MathML</a>.

Remark 2.1 For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M7">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M75">View MathML</a>. In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M76">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M77">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M78">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M79">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M80">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M81">View MathML</a>

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M83">View MathML</a>

(2.1)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M84">View MathML</a>. Via the p-polar coordinates (or Prüfer transformation)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M85">View MathML</a>

(2.2)

we can transform equation (2.1) into the following equations for r and θ:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M86">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M87">View MathML</a>

(2.4)

Note that equation (2.3) for θ is independent of r. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M89">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M91">View MathML</a>, the unique solution of system (2.3)-(2.4) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M92">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M93">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M94">View MathML</a>

The p-polar coordinates (2.2), one can verify that equation (2.1) has a non-trivial solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M95">View MathML</a>

(2.5)

One basic observation on equation (2.3) is that the vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M96">View MathML</a> at those θ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M97">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M99">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M101">View MathML</a> are only integrable, the derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M102">View MathML</a> at any specific t is meaningless. However, one can still use such an observation to obtain the following property, called quasi-monotonicity. We refer the readers to [[7], Lemma 2.3] for a detailed proof.

Lemma 2.2Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M104">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M89">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M106">View MathML</a>be the solution of equation (2.3). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M107">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M99">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M109">View MathML</a>

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M110">View MathML</a> the weak topology in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M111">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M112">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M113">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M114">View MathML</a>, we mean that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M115">View MathML</a>

Some important properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M119">View MathML</a> are collected in the following theorem.

Theorem 2.1 ([8])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M88">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M89">View MathML</a>be fixed. We have the following results.

(i) As mappings from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M122">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M117">View MathML</a>are continuous. More precisely, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M126">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M127">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M128">View MathML</a>

as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M129">View MathML</a>.

(ii) The functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M131">View MathML</a>is continuous. More precisely, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M126">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M127">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M134">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M129">View MathML</a>.

(iii) The functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M131">View MathML</a>is continuously differentiable in the sense of Fréchet. The differentials of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M118">View MathML</a>ataandb, denoted, respectively, by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M139">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M140">View MathML</a>, are the following mappings:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M141">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M142">View MathML</a>

(2.7)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M143">View MathML</a>is the dual space of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M144">View MathML</a>. Moreover, as mappings from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M122">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M146">View MathML</a>, both<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M147">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M140">View MathML</a>are continuous.

Remark 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M151">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M153">View MathML</a>, then it follows from formulations (2.6) and (2.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M154">View MathML</a>

3 Fučík spectrum for Dirichlet problems: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M155">View MathML</a>

3.1 Decomposition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a>

Given a pair of weights <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, the (Dirichlet type) Fučík spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a> is defined as the set of those <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15">View MathML</a> such that system (1.2)-(1.3) has non-trivial solutions. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M160">View MathML</a>

(3.1)

In the p-polar coordinates (2.2), equation (1.2) is equivalent to the following two equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M161">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M162">View MathML</a>

(3.3)

Compared with equations (2.3) and (2.4), the pair of weights a and b are now replaced by λa and μb, respectively. Since the right-hand side of equation (3.2) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M163">View MathML</a>-periodic in θ, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M164">View MathML</a>

(3.4)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M167">View MathML</a>. One can also check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M168">View MathML</a>

(3.5)

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169">View MathML</a> is an eigenfunction of system (1.2)-(1.3) associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M170">View MathML</a>. By equation (2.2), the corresponding solution of equation (3.2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M171">View MathML</a>, satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M172">View MathML</a>

(3.6)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M173">View MathML</a>. Due to equation (3.4), we may restrict <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M174">View MathML</a>. In other words, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M175">View MathML</a> and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M176">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M177">View MathML</a>. Moreover, it follows from the quasi-monotonicity result in Lemma 2.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M178">View MathML</a>. We distinguish two cases: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M179">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M180">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M179">View MathML</a>, then it follows from equation (3.1) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M182">View MathML</a>. By equation (2.2), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M183">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M184">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M185">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M186">View MathML</a>

(3.7)

Now equation (3.6) tells us that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M187">View MathML</a>. In fact, the subscript k is related to the number of zeroes of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189">View MathML</a>. By Lemma 2.2, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M190">View MathML</a>

has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M191">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M192">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M193">View MathML</a>

By equation (2.2), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169">View MathML</a> has exactly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M195">View MathML</a> zeroes in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189">View MathML</a>. Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169">View MathML</a> has exactly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M195">View MathML</a> zeroes in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M203">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M204">View MathML</a>

(3.8)

Till now, we have proved that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M205">View MathML</a>

Conversely, let us show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M206">View MathML</a>

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M187">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M209">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M210">View MathML</a>

For this specific <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211">View MathML</a>, take a non-trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M212">View MathML</a> of equation (3.3). Then we can construct a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M213">View MathML</a>, which is a solution of equation (1.2) with exactly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M195">View MathML</a> zeroes on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189">View MathML</a>. Particularly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M216">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M217">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M218">View MathML</a>. Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M219">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M220">View MathML</a>

Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M203">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M170">View MathML</a> and any associating eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M180">View MathML</a> and has exactly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M195">View MathML</a> zeroes in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189">View MathML</a>.

Combining the previous arguments, we can conclude that.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>. The Fučík spectrum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a>can be decomposed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M230">View MathML</a>

Moreover, the following characterization on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M231">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M232">View MathML</a>holds.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>any eigenfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169">View MathML</a>associated with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M236">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M238">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169">View MathML</a>has precisely<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M240">View MathML</a>zeroes in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M241">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>any eigenfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169">View MathML</a>associated with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M236">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M238">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M169">View MathML</a>and has precisely<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M240">View MathML</a>zeroes in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M241">View MathML</a>.

By equation (3.5), the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M251">View MathML</a> defined as in equation (3.8) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M252">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M253">View MathML</a>

(3.9)

In other words, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M251">View MathML</a> is symmetric to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M255">View MathML</a> about the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M256">View MathML</a>. For this reason, essentially we need only to characterize those sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M231">View MathML</a>.

In Section 3.4, we will see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M258">View MathML</a> is made up of straight lines which are in connection with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M260">View MathML</a>, the Dirichlet eigenvalues of p-Laplacian with the weight a. See Theorem 3.2.

For those sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, it is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M263">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M264">View MathML</a>

(3.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M265">View MathML</a>

(3.12)

Therefore we need only to focus our study on the subset

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M266">View MathML</a>

(3.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M267">View MathML</a>. In Section 3.4, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a> we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a> is either an empty set or a continuous, strictly decreasing, hyperbolic like curve with a horizontal asymptotic line and a vertical asymptotic line.

With the help of half-eigenvalues of the p-Laplacian with a pair of weights, we can determined whether <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a> is an empty set or not. Using eigenvalues of the p-Laplacian with a weight, we can roughly locate the hyperbolic like curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a>. For these reasons, we will give in the successive two subsections some useful characterization on eigenvalues and half-eigenvalues of the p-Laplacian with weights.

3.2 Eigenvalues of p-Laplacian with an indefinite weight

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M273">View MathML</a> the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M274">View MathML</a>

(3.14)

satisfying the initial value condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M275">View MathML</a>. Particularly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M276">View MathML</a>, it follows from Lemma 2.1(vi) that equation (3.14) turns to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M277">View MathML</a>, and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M278">View MathML</a>

(3.15)

Because the right-hand side of equation (3.14) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M279">View MathML</a>-periodic in θ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M280">View MathML</a>

(3.16)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M282">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M283">View MathML</a>. Since equation (3.14) can also be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M284">View MathML</a>

using the notations in Section 2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M285">View MathML</a>

By Lemma 2.2, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M286">View MathML</a> is also quasi-monotonic in t.

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M290">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M291">View MathML</a> the sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M292">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M293">View MathML</a>

(3.17)

has a non-trivial solution satisfying the Dirichlet boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M216">View MathML</a>, the Neumann boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M295">View MathML</a>, the Dirichlet-Neumann boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M296">View MathML</a> and the Neumann-Dirichlet boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M297">View MathML</a>, respectively. Similar arguments as in Section 3.1 show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M298">View MathML</a>

(3.18)

These spectra have been studied in [9]. Consider the function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M292">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M300">View MathML</a>

It follows from formulations (2.6) and (2.7) in Theorem 2.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M301">View MathML</a>

(3.19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M302">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M303">View MathML</a>

(3.20)

See equation (2.5) for the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M304">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M304">View MathML</a> is also a non-trivial solution of equation (3.17). Multiplying equation (3.20) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M304">View MathML</a> and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M308">View MathML</a>

Substituting this into equation (3.19), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M309">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M310">View MathML</a>

(3.21)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M311">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M304">View MathML</a> becomes the associated eigenfunction of equation (3.17) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M313">View MathML</a>. In this case, the first item on the right-hand side of equation (3.21) equals 0, and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M314">View MathML</a>

(3.22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M315">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M316">View MathML</a>. Similarly, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M317">View MathML</a>

(3.23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M318">View MathML</a>

(3.24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M319">View MathML</a>

(3.25)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, it follows from equations (3.18) and (3.22) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M321">View MathML</a>

has at most one positive solution and one negative solution, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M322">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M323">View MathML</a>, respectively, if they exist. In other words, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M324">View MathML</a>

(3.26)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M325">View MathML</a>

(3.27)

It has been proved in [9] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M326">View MathML</a> has at most one nonzero solution, called the principal Neumann eigenvalue and denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M327">View MathML</a>, if it exists. By equation (3.23) and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M328">View MathML</a>, we can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M329">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, has at most one positive solution and one negative solution, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M331">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M332">View MathML</a> respectively, if they exist. In other words, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M333">View MathML</a>

(3.28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M334">View MathML</a>

(3.29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M335">View MathML</a>

(3.30)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272">View MathML</a>, use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M338">View MathML</a> for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M340">View MathML</a> on a subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189">View MathML</a> of positive measure. Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M343">View MathML</a>.

Lemma 3.1 ([9])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272">View MathML</a>. Then it is necessary that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M345">View MathML</a>.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M288">View MathML</a>contains no negative eigenvalues, and it consists of a sequence of positive eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M348">View MathML</a>

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M288">View MathML</a>contains no positive eigenvalues, and it consists of a sequence of negative eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M351">View MathML</a>

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M352">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M353">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M288">View MathML</a>contains both positive and negative eigenvalues, and it consists of a double sequence of eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M355">View MathML</a>

Lemma 3.2 ([9])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272">View MathML</a>. Then it is necessary that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M357">View MathML</a>.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M289">View MathML</a>contains no negative eigenvalues, and it consists of a sequence of non-negative eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M360">View MathML</a>

The principal eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M361">View MathML</a>does not exist in this case.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M289">View MathML</a>contains no positive eigenvalues, and it consists of a sequence of non-positive eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M364">View MathML</a>

The principal eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M361">View MathML</a>does not exist in this case.

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M366">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M367">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M289">View MathML</a>contains both positive and negative eigenvalues, and it consists of a double sequence of eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M369">View MathML</a>

The principal eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M361">View MathML</a>is positive in this case.

(iv) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M366">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M372">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M289">View MathML</a>consists of a double sequence of eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M374">View MathML</a>

The principal eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M361">View MathML</a>is negative in this case.

(v) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M366">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M377">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M378">View MathML</a>consists of a double sequence of eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M379">View MathML</a>

The principal eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M361">View MathML</a>does not exist in this case.

The following lemma reveals, to some extent, the essential reason of the existence of positive eigenvalues.

Lemma 3.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M382">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M383">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M384">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M283">View MathML</a>. Denote the indicator function of the subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M386">View MathML</a>of the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189">View MathML</a>by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M388">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M389">View MathML</a>

(3.31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M390">View MathML</a>

(3.32)

Proof We only prove equation (3.32), and equation (3.31) can be proved similarly.

Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M391">View MathML</a> for simplicity.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M392">View MathML</a>, by similar arguments as in [[9], Lemma 3.4] (see also Lemma 3.5) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M393">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M394">View MathML</a> in equation (3.2) and we get the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M395">View MathML</a>

which has equilibria <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M396">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M283">View MathML</a>. Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M398">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M399">View MathML</a>

Therefore there must exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M401">View MathML</a>.

On the other hand, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M402">View MathML</a>, namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M403">View MathML</a> for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M404">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M405">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M406">View MathML</a> for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M404">View MathML</a>. Now it follows from the comparison theorem, equation (3.15), and Remark 2.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M408">View MathML</a>

completing the proof of equation (3.32). □

In the rest of this subsection, we aim to reveal some quasi-monotonicity property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M409">View MathML</a> in λ, which will play an important role in analyzing the structure of the Fučík spectra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1">View MathML</a>.

Using equation (3.18), the characterization on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M411">View MathML</a>, we can rewritten equation (3.22) more precisely as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M412">View MathML</a>

Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M413">View MathML</a>

(3.33)

because it follows from equation (3.16) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M414">View MathML</a>

Though we have always been considering equations on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M91">View MathML</a>, similar results as in Theorem 2.1 still hold when the interval is replaced by any general interval. Thus equation (3.33) can also be generalized. In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M416">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M417">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M418">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M419">View MathML</a>

(3.34)

Similar arguments can be applied to (3.23)-(3.25) to obtain results analogous to equation (3.34). We skip the proof and collect these results in the following lemma, which can be understood as the quasi-monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M420">View MathML</a> in λ.

Lemma 3.4Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M416">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M417">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M418">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M425">View MathML</a>

(i) If there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400">View MathML</a>and an integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M427">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M428">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M429">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M430">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M431">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M432">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M433">View MathML</a>.

(ii) If there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400">View MathML</a>and an integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M435">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M436">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M429">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M438">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M431">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M440">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M441">View MathML</a>.

(iii) If there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400">View MathML</a>and an integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M435">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M444">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M445">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M446">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M431">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M448">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M441">View MathML</a>.

(iv) If there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400">View MathML</a>and an integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M427">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M452">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M445">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M454">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M431">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M456">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M433">View MathML</a>.

3.3 Half-eigenvalues of p-Laplacian with a pair of indefinite weights

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M458">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M459">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M460">View MathML</a> the sets of half-eigenvalues of the scalar p-Laplacian, namely, the sets of those <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M292">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M462">View MathML</a>

has a non-trivial solution satisfying the boundary conditions (1.3) and (1.4), respectively.

Based on the p-polar transformation (2.2) and the quasi-monotonicity results in Lemma 2.2, by similar arguments as in Section 3.1 we can show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M463">View MathML</a>

Applying the differentiability results (2.6) and (2.7) in Theorem 2.1, together with the Dirichlet boundary condition (1.3), by similar arguments as in Section 3.2 we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M464">View MathML</a> is also quasi-monotonic in λ. More precisely, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M465">View MathML</a>

(3.35)

We also know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M466">View MathML</a>, because the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M467">View MathML</a>

has equilibria <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M396">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M283">View MathML</a>. Combining the quasi-monotonicity results in Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M470">View MathML</a>

(3.36)

and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M471">View MathML</a>. It follows from equations (3.35) and (3.36) that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M473">View MathML</a>

has at most one positive solution and one negative solution, denoted, respectively, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M475">View MathML</a>, if they exist. More precisely, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M476">View MathML</a>

(3.37)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M477">View MathML</a>

(3.38)

By equations (3.26), (3.27), and Lemma 2.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M478">View MathML</a>. Some immediate results are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M479">View MathML</a>

(3.39)

Similarly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M480">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M481">View MathML</a>

Thus the Neumann type half-eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M482">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, are defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M484">View MathML</a>

And the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M485">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, implies the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M487">View MathML</a>. By Lemma 2.2, the solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M488">View MathML</a> is also that of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M489">View MathML</a>. Thus there may exist at most one principal Neumann half-eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M490">View MathML</a>, which is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M491">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M492">View MathML</a> by equation (3.28). Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M493">View MathML</a> may not exist even if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M494">View MathML</a> exist.

It is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M495">View MathML</a>

Essentially we need only to concern <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M496">View MathML</a> and those positive half-eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M497">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>. Now a natural question arises: for what kind of weights a and b do there exist no, finitely many, or infinitely many positive Dirichlet or Neumann type half-eigenvalues?

Lemma 3.5Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M382">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M501">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M502">View MathML</a>

(3.40)

uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M282">View MathML</a>.

Proof This lemma can be proved by similar argument as in the proof of Lemma 2.3 in [10], thus we skip the details. □

Lemma 3.6Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M505">View MathML</a>, and there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M506">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M507">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M508">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M509">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M510">View MathML</a>

(3.41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M511">View MathML</a>

(3.42)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M512">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M513">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M514">View MathML</a>

(3.43)

and inequality (3.43) becomes an equality if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M515">View MathML</a>

Proof Let us write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M516">View MathML</a> for simplicity.

Claim I: there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M517">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M518">View MathML</a>. If this is false, then it follows from Lemma 2.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M519">View MathML</a>

(3.44)

and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M520">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M521">View MathML</a>. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211">View MathML</a> satisfies the ODE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M523">View MathML</a>

Then we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211">View MathML</a> also satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M525">View MathML</a>

(3.45)

on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M526">View MathML</a>. Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M527">View MathML</a>

(3.46)

Particularly, we get from equations (3.44) and (3.46)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M528">View MathML</a>

(3.47)

On the other hand, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M529">View MathML</a> in equation (3.41), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M530">View MathML</a>

(3.48)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M531">View MathML</a>, it follows from Lemma 3.4(i) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M532">View MathML</a>

a contradiction to equation (3.47). Thus there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M533">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M534">View MathML</a>, proving Claim I.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M535">View MathML</a>, we aim to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M536">View MathML</a>. If this is not true, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M537">View MathML</a>, and one can check that equation (3.46) is still true. It follows from equations (3.46), (3.48), Lemma 3.4(i), and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M535">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M539">View MathML</a>

a contradiction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M534">View MathML</a>.

Claim II: there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M541">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M542">View MathML</a>. If this is not true, then the quasi-monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211">View MathML</a> in t shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M544">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M546">View MathML</a>

(3.49)

on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M547">View MathML</a>, and the initial value condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M534">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M549">View MathML</a>

(3.50)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M537">View MathML</a>, similar arguments as in the proof of Claim I show the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M541">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M552">View MathML</a>, and Claim II is proved. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M553">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M554">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M555">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M556">View MathML</a> by Lemma 2.2. We can improve the result in Claim II as the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M557">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M542">View MathML</a>. If this result is not true, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M559">View MathML</a>

and we still have equations (3.49)-(3.50). Now both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M561">View MathML</a> satisfy the same ODE (3.49) on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M562">View MathML</a>, while the initial values satisfy the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M563">View MathML</a>

By case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M529">View MathML</a> in equation (3.42) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M565">View MathML</a>. Now the existence and uniqueness theorem for the first-order ODEs shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M566">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M534">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M557">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M542">View MathML</a>.

Now we can conclude that Claim II is true. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M570">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M571">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M554">View MathML</a>.

Inductively, we can show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M573">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M574">View MathML</a>. Moreover, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M575">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M576">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M577">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M578">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M579">View MathML</a>. And if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M575">View MathML</a>, it follows from Lemma 2.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M581">View MathML</a>.

Finally, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M582">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M583">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M584">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M585">View MathML</a>, then it follows from equations (3.41) and (3.42) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M586">View MathML</a>, completing the proof of the lemma. □

Property 3.1Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, we have the following results:

(i) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342">View MathML</a>, then any positive half-eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M589">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, does not exist;

(ii) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M591">View MathML</a>, then all positive half-eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, exist;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M594">View MathML</a>exists, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>both<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597">View MathML</a>exist;

(iv) both<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597">View MathML</a>exist, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M594">View MathML</a>exists or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M602">View MathML</a>exists.

Proof (i) Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342">View MathML</a>. If there exists a positive half-eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M604">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M605">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M606">View MathML</a>

and hence there must exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M607">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M608">View MathML</a>

It follows from Lemma 2.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M609">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M610">View MathML</a>

Particularly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M611">View MathML</a>

By Lemma 3.3, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M612">View MathML</a>, a contradiction to the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342">View MathML</a>. Consequently, there is not any positive half-eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M589">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342">View MathML</a>.

(ii) This result follows immediately from equations (3.35), (3.36), and Lemma 3.5.

(iii) Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M617">View MathML</a> exists and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>. The existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474">View MathML</a> has been given in equation (3.39). We need only to prove the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597">View MathML</a>. Take the following notations for simplicity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M621">View MathML</a>

By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M594">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M623">View MathML</a>. By Lemma 2.2, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M624">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M625">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M626">View MathML</a>

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M627">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M628">View MathML</a> and i is even (odd). Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M629">View MathML</a>

(3.51)

We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M630">View MathML</a>

(3.52)

In fact, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M631">View MathML</a> by Lemma 2.2, proving equation (3.52) for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M632">View MathML</a>. To prove the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M633">View MathML</a>, we assume on the contrary that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M634">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M635">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M636">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M632">View MathML</a> in equation (3.51), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M639">View MathML</a> satisfy the same ODE,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M640">View MathML</a>

(3.53)

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M641">View MathML</a> is also a solution of equation (3.53). Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M642">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M643">View MathML</a>

thus the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M634">View MathML</a> is false, proving equation (3.52) for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M633">View MathML</a>. Inductively, we can prove equation (3.52).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M646">View MathML</a> in equation (3.52); we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M647">View MathML</a>

Combining with equation (3.36), we conclude that there must exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M648">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M649">View MathML</a>

(iv) Suppose that both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M650">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M651">View MathML</a> exist and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M653">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M654">View MathML</a>. The case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M655">View MathML</a> can be proved similarly.

Let us take the notations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M656">View MathML</a>

By the definition of half-eigenvalues in equation (3.37) and Lemma 2.2, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M657">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M658">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M659">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M660">View MathML</a>

Particularly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M661">View MathML</a>

(3.54)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M662">View MathML</a>

(3.55)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M663">View MathML</a>

(3.56)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M664">View MathML</a>

(3.57)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M665">View MathML</a> in equations (3.54) and (3.56). We get from Lemma 3.3

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M666">View MathML</a>

(3.58)

Without loss of generality, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M667">View MathML</a>. We need only to distinguish three cases in the following.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M668">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M669">View MathML</a>. In this case, it follows from Lemma 3.3 that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M670">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M671">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M672">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M673">View MathML</a>. Combining equations (3.54), (3.55), and the above two conditions, we get from Lemma 3.6

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M674">View MathML</a>

and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M594">View MathML</a> exists.

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M676">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M677">View MathML</a>. In this case, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M591">View MathML</a>. And it follows from the result (ii) in this property that both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M679">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M680">View MathML</a> exist.

Case 3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M676">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M682">View MathML</a>. In this case, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M676">View MathML</a>, the condition (3.58) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M684">View MathML</a>

Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M685">View MathML</a>, there must exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M686">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M687">View MathML</a>

(3.59)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M688">View MathML</a>

(3.60)

Then we can apply similar arguments as in Case 1 to get the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M689">View MathML</a> if equation (3.59) holds, and the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M690','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M690">View MathML</a> if equation (3.59) holds. □

Corollary 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>and denote<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M315">View MathML</a>. One of the following three cases must occur.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M693">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M694">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M695">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M696','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M696">View MathML</a>.

(iii) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M605">View MathML</a>such that either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M698">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M699">View MathML</a>

Applying Lemmas 3.3 and 3.6, one can verify the following three examples.

Example 3.1 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M701">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M702">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M703','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M703">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M704">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M705">View MathML</a>.

Example 3.2 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M707">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M708">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M709">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M710">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M711">View MathML</a>.

Example 3.3 Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M713">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M714','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M714">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M715">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M716">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M717">View MathML</a>.

The following property can be proved by similar arguments as used for Property 3.1.

Property 3.2Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, we have the following results:

(i) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342">View MathML</a>, then any positive half-eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M720">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M721">View MathML</a>, does not exist, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M722','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M722">View MathML</a>

(ii) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M591">View MathML</a>, then those positive half-eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M724">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, exist, but the existence of a positive principal half-eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M726','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M726">View MathML</a>is indefinite;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M727','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M727">View MathML</a>exists, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>both<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M724">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M730">View MathML</a>exist;

(iv) both<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M724">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M730">View MathML</a>exist, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M727','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M727">View MathML</a>exists or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M735','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M735">View MathML</a>exists.

3.4 Structure of the Fučík spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M155">View MathML</a>

In this subsection, we always use the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M737','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M737">View MathML</a>

for simplicity if there is no confusion. By Theorem 2.1, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15">View MathML</a>.

The following lemma tells us that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738">View MathML</a> is quasi-monotonic in λ and in μ. This property is crucial for us to characterize the structure of the Fučík spectra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a>.

Lemma 3.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>. The following results hold:

(i) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M744">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M745">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M747','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M747">View MathML</a>

(ii) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M744">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M745">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M752','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M752">View MathML</a>

Proof We only prove (i), and (ii) can be proved by similar arguments.

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M744">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M755','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M755">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>. It follows from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M231">View MathML</a> in equation (3.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M758','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M758">View MathML</a>

In the following, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M759','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M759">View MathML</a>

for simplicity. By Lemma 2.2, there exist (0=) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M760','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M760">View MathML</a> (=1), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M761','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M761">View MathML</a>

Furthermore, we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M762','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M762">View MathML</a>

Particularly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M763','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M763">View MathML</a>

Then it follows from Lemma 3.6 that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M764','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M764">View MathML</a> (>0), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M765','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M765">View MathML</a>

To complete the proof of (i), we need only to prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M766','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M766">View MathML</a>

If this is not true, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M767','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M767">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M768','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M768">View MathML</a>. Write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M769','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M769">View MathML</a>

for simplicity. Then there exist (0=) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M770','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M770">View MathML</a> (≤1) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M771','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M771">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M772','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M772">View MathML</a>, we can use similar arguments as in the previous paragraph to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M773','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M773">View MathML</a>

Finally, it follows from the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M774','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M774">View MathML</a> and Lemma 2.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M775','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M775">View MathML</a>

a contradiction to the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M776','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M776">View MathML</a>. □

By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M258">View MathML</a> in equation (3.7), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M778','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M778">View MathML</a>

By the quasi-monotonicity results in Lemma 2.2, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M779','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M779">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M780','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M780">View MathML</a>

It follows from Lemma 3.1 that the solutions to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M781','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M781">View MathML</a>

are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M782','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M782">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M366">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M784','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M784">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M786','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M786">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M258">View MathML</a> can be characterized as in the following theorem.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>and assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M790','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M790">View MathML</a>.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M352">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M353">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M793','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M793">View MathML</a>.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M795','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M795">View MathML</a>.

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M342">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M797','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M797">View MathML</a>.

Remark 3.1 It follows from equation (3.9) and the above theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M798','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M798">View MathML</a> is made up of two horizontal lines <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M799','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M799">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M800','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M800">View MathML</a>. If the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M801','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M801">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M802','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M802">View MathML</a> do not exist, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M799','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M799">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M804','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M804">View MathML</a> should be understood as empty sets, respectively.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M261">View MathML</a> is more complicated than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M258">View MathML</a>. As analyzed in Section 3.1, essentially we need only to discuss the subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a> as defined in equation (3.13). Finally we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a> is either an empty set, or a hyperbolic like curve. The following property helps us to locate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a> roughly.

Property 3.3 (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M811','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M811">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M813','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M813">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M814','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M814">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M815','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M815">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M816','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M816">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M817','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M817">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M814','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M814">View MathML</a>.

Proof We will only prove (i), and (ii) can be proved similarly.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M811','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M811">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M820','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M820">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M821','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M821">View MathML</a>, and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M822','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M822">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M823','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M823">View MathML</a>

satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M824','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M824">View MathML</a>

(3.61)

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M211">View MathML</a> satisfies equation (3.45) on each interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M826','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M826">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M827','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M827">View MathML</a>, and it satisfies equation (3.49) on each interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M828','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M828">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M827','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M827">View MathML</a>. In other words, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M830','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M830">View MathML</a>

(3.62)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M831','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M831">View MathML</a>

(3.63)

Take the following notations for simplicity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M832','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M832">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M632">View MathML</a> and take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M834','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M834">View MathML</a> in equation (3.62). Then it follows from equation (3.61) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M835','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M835">View MathML</a>

By the quasi-monotonicity results in Lemma 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M836','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M836">View MathML</a>

(3.64)

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M837','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M837">View MathML</a>

Inductively, we can show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M838','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M838">View MathML</a>

(3.65)

Similarly, applying equations (3.16), (3.61), (3.63), and Lemma 2.2, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M839','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M839">View MathML</a>

(3.66)

The case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M840','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M840">View MathML</a> (≥1) in equation (3.65) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M841','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M841">View MathML</a>

By equation (3.36), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M842','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M842">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M400">View MathML</a>. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M844','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M844">View MathML</a> as in equation (3.26), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M845','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M845">View MathML</a>

Then it follows from Lemma 3.4(i) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M846','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M846">View MathML</a> (>0).

Similarly, it can be deduced from equations (3.66), (3.26) and Lemma 3.4 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M847','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M847">View MathML</a>. □

Theorem 3.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M848','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M848">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M850','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M850">View MathML</a>is a continuous, strictly decreasing, hyperbolic like curve

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M851','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M851">View MathML</a>

with the horizontal asymptotic line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M852','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M852">View MathML</a>and the vertical asymptotic line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M853','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M853">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M854','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M854">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M855','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M855">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M856','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M856">View MathML</a>is the floor function.

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M848','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M848">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>. We prove the theorem by seven steps.

Step 1. We aim to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a> is not a single-point set. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M860','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M860">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M861','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M861">View MathML</a>

By Property 3.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M862','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M862">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M863','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M863">View MathML</a>. Then it follows from Lemma 3.7 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M864','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M864">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15">View MathML</a>, there must exist a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M867','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M867">View MathML</a> in the open line segment with endpoints <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M868','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M868">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M869','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M869">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M870','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M870">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M871','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M871">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M872','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M872">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M873','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M873">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M874','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M874">View MathML</a>, proving the claim. Furthermore, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M875','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M875">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M876','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M876">View MathML</a> is not a single-point set, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M877','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M877">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M878','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M878">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M879','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M879">View MathML</a>.

Step 2. It follows from Lemma 3.7 that every horizontal line intersects <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a> at one time at most, so does every vertical line. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a> is the image of some function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M882','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M882">View MathML</a>

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M883','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M883">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M884','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M884">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M885','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M885">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M151">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M887','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M887">View MathML</a>, then it follows from Lemma 3.7 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M888','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M888">View MathML</a>

which is a contradiction. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M889','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M889">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M890','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M890">View MathML</a> is a strictly decreasing function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M891','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M891">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M892','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M892">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M893','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M893">View MathML</a>

By Property 3.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M894','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M894">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M895','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M895">View MathML</a>.

Step 3. We claim that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M883','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M883">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M897','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M897">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M878','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M878">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M899','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M899">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M885','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M885">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M151">View MathML</a>. By the monotonicity result proved in Step 2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M889','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M889">View MathML</a>. By Lemma 3.7 again, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M903','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M903">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M904','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M904">View MathML</a>

Then the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15">View MathML</a> implies the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M907','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M907">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M908','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M908">View MathML</a>

and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M860','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M860">View MathML</a>, completing the proof of the claim.

Step 4. We aim to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M910','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M910">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M911','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M911">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M912','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M912">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M913','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M913">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M914','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M914">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M915','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M915">View MathML</a>. Similar arguments as in Step 1 show that there exists some point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M916','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M916">View MathML</a> in the open line segment with endpoints <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M917','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M917">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M918','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M918">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M860','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M860">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M920','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M920">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M878','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M878">View MathML</a>, a contradiction. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M922','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M922">View MathML</a>. Similarly, we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M923','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M923">View MathML</a>. Combining the results in Step 3, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M910','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M910">View MathML</a>.

Step 5. We aim to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M925','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M925">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M910','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M910">View MathML</a> and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M927','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M927">View MathML</a> is strictly decreasing, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M928','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M928">View MathML</a> is not a single-point set. Then similar arguments as in Step 3 and Step 4 show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M929','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M929">View MathML</a>.

Step 6. We aim to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M890','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M890">View MathML</a> is a continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M910','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M910">View MathML</a>. In fact, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M932','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M932">View MathML</a>

(3.67)

Fix any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M933','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M933">View MathML</a>. By the monotonicity results proved in Step 2, we see that both

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M934','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M934">View MathML</a>

exist, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M935','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M935">View MathML</a>. Furthermore, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M936','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M936">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M937','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M937">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M938','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M938">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M939','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M939">View MathML</a> in equation (3.67), then the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15">View MathML</a> guarantees that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M942','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M942">View MathML</a>

Similarly, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M943','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M943">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M944','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M944">View MathML</a> in equation (3.67), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M945','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M945">View MathML</a>

Now Lemma 3.7 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M946','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M946">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M890','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M890">View MathML</a> is continuous at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M948','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M948">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M933','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M933">View MathML</a> can be chosen arbitrarily, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M890','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M890">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M951','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M951">View MathML</a>.

Step 7. In the above six steps, we have shown that the continuous and strictly decreasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M952','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M952">View MathML</a> maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M953','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M953">View MathML</a> onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M954','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M954">View MathML</a>. Then it is necessary that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M955','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M955">View MathML</a>

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M952','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M952">View MathML</a> is a hyperbolic like curve with the horizontal asymptotic line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M852','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M852">View MathML</a> and the vertical asymptotic line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M853','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M853">View MathML</a>. □

When the weight a or b is positive, we can improve the results about the asymptotic lines in Theorem 3.3.

Theorem 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>. The following results hold.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M352">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M961','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M961">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M962','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M962">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, and the vertical asymptotic line of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M965','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M965">View MathML</a>.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M337">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M967','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M967">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M962','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M962">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, and the horizontal asymptotic line of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M971','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M971">View MathML</a>.

Proof We only prove (i) for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M972','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M972">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M973','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M973">View MathML</a>. Other cases can be proved similarly.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M974','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M974">View MathML</a>, it follows from Lemma 3.1 that the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M975','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M975">View MathML</a> exists. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M975','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M975">View MathML</a> as in equation (3.26), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M977','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M977">View MathML</a>. By Lemma 2.2, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M978','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M978">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M979','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M979">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M980','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M980">View MathML</a>

By equation (3.16), we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M981','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M981">View MathML</a>

Given any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M982','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M982">View MathML</a>, it follows from Lemma 3.4(i) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M983','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M983">View MathML</a>

and hence there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M984','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M984">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M985','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M985">View MathML</a>

(3.68)

We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M986','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M986">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M961','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M961">View MathML</a>. By Lemma 3.3, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M508">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M989','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M989">View MathML</a>

(3.69)

Define a weight q on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M991','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M991">View MathML</a>

Then it follows from equations (3.68) and (3.69) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M992','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M992">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M993','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M993">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M994','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M994">View MathML</a>. One has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M995','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M995">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M189">View MathML</a>. Now Remark 2.2 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M997','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M997">View MathML</a>

By the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M738">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M236">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1000','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1000">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M744">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1002','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1002">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1003','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1003">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1004','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1004">View MathML</a>.

Now it follows from Theorem 3.3 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1005','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1005">View MathML</a> is a hyperbolic like curve and its vertical asymptotic line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1006','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1006">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1007','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1007">View MathML</a>. On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1008','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1008">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1009','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1009">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1010','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1010">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1011','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1011">View MathML</a> can be chosen arbitrarily. Therefore the vertical asymptotic line of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1012','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1012">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1013','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1013">View MathML</a>. □

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1014','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1014">View MathML</a> exists and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, then it follows from equation (3.37) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1016','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1016">View MathML</a>

and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1017','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1017">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M850','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M850">View MathML</a> is a hyperbolic like curve by Theorem 3.3. On the other hand, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M850','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M850">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, is a hyperbolic like curve. Theorem 3.3 tells us it has a horizontal asymptotic line and a vertical one. Then it must intersect the diagonal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1021','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1021">View MathML</a> at a unique point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1022','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1022">View MathML</a>. Furthermore, we can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1023','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1023">View MathML</a>. In conclusion, we have the following property.

Property 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>. The following results hold.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1026','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1026">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M589">View MathML</a>does not exist.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a>is a hyperbolic like curve if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474">View MathML</a>exists.

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a>is either an empty set or a hyperbolic like curve in the quadrant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30">View MathML</a>emanating from the point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1032','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1032">View MathML</a>.

Remark 3.2 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1033','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1033">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, then it follows from Properties 3.1 and 3.4 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M962','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M962">View MathML</a>. Furthermore, we can deduce by Lemma 3.7 that the higher-order hyperbolic like curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1036','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1036">View MathML</a> always lies above the lower-order curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1037','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1037">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1039','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1039">View MathML</a> and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1040','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1040">View MathML</a>

By equations (3.9) and (3.13), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1041','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1041">View MathML</a> is asymmetric to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1042','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1042">View MathML</a> about the diagonal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1021','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1021">View MathML</a>. Namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1044','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1044">View MathML</a>

Then we get can the following results immediately from Property 3.4.

Property 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1047','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1047">View MathML</a>is either an empty set or a hyperbolic like curve in the quadrant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30">View MathML</a>emanating from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1049','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1049">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1050','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1050">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1051','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1051">View MathML</a>exists.

By Properties 3.4 and 3.5, we see that the existence of those hyperbolic like curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1041','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1041">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, is determined by the existence of those half-eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597">View MathML</a>, respectively. By Corollary 3.1, we can conclude that besides those trivial lines, the Fučík spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a> confined to the quadrant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30">View MathML</a> is an empty set, or made up of an odd number of hyperbolic like curves, or made up of a double sequence of hyperbolic like curves. Taking the relations (3.10)-(3.12) into consideration, we obtain the following theorem.

Theorem 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1060','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1060">View MathML</a>

If one of the half-eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1061','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1061">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1062','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1062">View MathML</a>does not exist, the corresponding straight line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1063','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1063">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1064','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1064">View MathML</a>should be understood as an empty set. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1065','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1065">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1066','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1066">View MathML</a>consists of zero, an odd number of, or a double sequence of hyperbolic like curves:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1067','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1067">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597">View MathML</a>does not exist, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1037','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1037">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1041','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1041">View MathML</a>should be understood as an empty set, respectively. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M589">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597">View MathML</a>exist, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1041','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1041">View MathML</a>are continuous, strictly decreasing, hyperbolic like curves. Moreover,

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1076','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1076">View MathML</a>is asymmetric to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1077','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1077">View MathML</a>about the vertical line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M394">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1079','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1079">View MathML</a>is asymmetric to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1080','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1080">View MathML</a>about the origin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1081','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1081">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1082','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1082">View MathML</a>is asymmetric to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1083','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1083">View MathML</a>about the horizontal line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1084','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1084">View MathML</a>.

3.5 Fučík spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a> with positive weights

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1087','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1087">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M961','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M961">View MathML</a>. Then it follows from Lemma 3.1 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M801','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M801">View MathML</a> exist, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M802','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M802">View MathML</a> do not exist. By Property 3.1 and Example 3.1, all half-eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M474">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M597">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, exist; but none of the half-eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1096','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1096">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1097','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1097">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1098','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1098">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1099','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1099">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, exist. Then we have the following theorem.

Theorem 3.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1087','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1087">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M961','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M961">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M18">View MathML</a>is made up of one vertical line, one horizontal line and a double sequence of differentiable, strictly decreasing, hyperbolic like curves in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1108">View MathML</a>

For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a>has the vertical asymptotic line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M965','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M965">View MathML</a>and the horizontal asymptotic line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1112">View MathML</a>.

Proof We need only to prove the differentiability of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a>. Recall from Theorem 3.3 that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, the curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M952','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M952">View MathML</a> is determined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1117">View MathML</a>

(3.70)

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M860','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M860">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>, denote the associated eigenfunction by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M304">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1121">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M262">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1123">View MathML</a> for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M91">View MathML</a>, it follows from formulation (2.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1125">View MathML</a>

Thus the Implicit Function Theorem can be applied to equation (3.70), and we see that the hyperbolic like curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M269">View MathML</a> is differentiable and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1127">View MathML</a>

 □

4 Fučík spectrum for Neumann problems

Given a pair of indefinite weights <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, the (Neumann type) Fučík spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19">View MathML</a> is defined as the set of those <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M15">View MathML</a> such that system (1.2)-(1.4) has non-trivial solutions.

Via similar arguments as in the previous sections, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19">View MathML</a> can also be characterized. We list the results in the following but omit the detailed proof.

Theorem 4.1The Neumann Fučík spectrum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19">View MathML</a>can be decomposed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1133">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1134">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1135">View MathML</a>are defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1136">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1137">View MathML</a>

Moreover, for any set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M721">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1140">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1141">View MathML</a>

(4.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1142">View MathML</a>

(4.3)

By the relations (4.1)-(4.3), we need only to consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1144">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1145">View MathML</a>

Theorem 4.2The set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1143">View MathML</a>is made up of two vertical lines<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1147">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1148">View MathML</a>. And<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1144">View MathML</a>is made up of two horizontal lines<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1150">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1151">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1152">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1153">View MathML</a>does not exist, the corresponding straight line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1148">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1151">View MathML</a>should be understood as an empty set.

Theorem 4.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1158">View MathML</a>is a continuous, strictly decreasing, hyperbolic like curve

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1159">View MathML</a>

with the horizontal asymptotic line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M852','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M852">View MathML</a>and the vertical asymptotic line<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M853','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M853">View MathML</a>. Moreover, the lower bound of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1162">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1163">View MathML</a>can be estimated as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1164">View MathML</a>

Theorem 4.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>. Then:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1156">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M720">View MathML</a>exists. And<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1168">View MathML</a>is either an empty set or a hyperbolic like curve emanating from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1169">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1170">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M730">View MathML</a>exists. And<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1172">View MathML</a>is either an empty set or a hyperbolic like curve emanating from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1173">View MathML</a>.

From the relation between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M724">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1175">View MathML</a> as stated in Property 3.2, we obtain the following spectral structure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1176">View MathML</a>.

Theorem 4.5One of the following three cases must occur.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1177">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1178">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1156">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1181">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M208">View MathML</a>.

(iii) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M605">View MathML</a>such that either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1184">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1185">View MathML</a>

From the relations (4.1)-(4.3) and the above theorems, the structure of the Neumann Fučík spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1176">View MathML</a> becomes clear.

Theorem 4.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1176">View MathML</a>is composed of (at most) four trivial lines<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1192">View MathML</a> (if one of the involved principal eigenvalues does not exist, the corresponding straight line is understood as an empty set), and in each quadrant of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M3">View MathML</a>zero, a finite odd number of, or a double sequence of hyperbolic like curves.

Finally, if the weights a and b are positive, then neither <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M327">View MathML</a> nor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1153">View MathML</a> exists, and we have the following results.

Theorem 4.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1087','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1087">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M961','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M961">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M19">View MathML</a>is made up of one vertical line, one horizontal line and a double sequence of differentiable, strictly decreasing, hyperbolic like curves in the quadrant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M30">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1201">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M972','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M972">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M973','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M973">View MathML</a>, the hyperbolic like curve<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1168">View MathML</a>has the asymptotic lines

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1205">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M973','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M973">View MathML</a>, the hyperbolic like curve<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1168">View MathML</a>has the asymptotic lines

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/10/mathml/M1209">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

PY gave the idea of this article and drafted the manuscript. All authors discussed the methods of proving the main results. All authors read and approved the final manuscript.

Acknowledgements

Jifeng Chu was supported by the National Natural Science Foundation of China (Grant No. 11171090, No. 11271078 and No. 11271333), China Postdoctoral Science Foundation funded project (Grant No. 2012T50431) and the Alexander von Humboldt Foundation of Germany. Ping Yan was supported by the National Natural Science Foundation of China (Grant No. 10901089, No. 11171090 and No. 11371213). Meirong Zhang was supported by the National Natural Science Foundation of China (Grant No. 1123001) and the National 111 Project of China (Station No. 111-2-01).

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