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Half-linear eigenvalue problem: Limit behavior of the first eigenvalue for shrinking interval

Gabriella Bognár1 and Ondřej Došlý2*

Author Affiliations

1 University of Miskolc, Egyetemváros, Miskolc, 3515, Hungary

2 Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, Brno, 611 37, Czech Republic

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Boundary Value Problems 2013, 2013:221  doi:10.1186/1687-2770-2013-221

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/221


Received:27 June 2013
Accepted:4 September 2013
Published:7 November 2013

© 2013 Bognár and Do¿lý; 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

We investigate the limit behavior of the first eigenvalue of the half-linear eigenvalue problem when the length of the interval tends to zero. We show that the important role is played by the limit behavior of ratios of primitive functions of coefficients in the investigated half-linear equation.

MSC: 34C10.

1 Introduction

We consider the eigenvalue problem associated with the half-linear second order differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M1">View MathML</a>

(1)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M5">View MathML</a> being the conjugate exponent of p, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M7">View MathML</a>. Equation (1) can be written as the first order system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M8">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M9">View MathML</a> being the inverse function of Φ, and the integrability assumption on the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M10">View MathML</a>, c, w implies the unique solvability of this system. The original paper of Elbert [1], where the existence and uniqueness results are proved via the half-linear version of the Prüfer transformation, deals with continuous functions in (2), but the idea of the proof applies without change to integrable coefficients when, as a solution x, u, absolutely continuous functions are considered (which satisfy (2) a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M11">View MathML</a>).

Along with (1), we consider the separated boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M12">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M14">View MathML</a> are the half-linear goniometric functions, which will be recalled in the next section. Motivated by the paper [2], where the linear Sturm-Liouville differential equation (which is the special case of (1)) is considered, we investigate the limit behavior (as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15">View MathML</a>) of the first eigenvalue of (1), (3) in dependence on α, β. We show that this limit behavior is, in a certain sense, the same as for an eigenvalue problem when boundary conditions (3) are associated with an equation with constant coefficients.

The investigation of half-linear eigenvalue problems is motivated, among others, by the fact that the partial differential equation with the p-Laplacian (which models, e.g., the flow of non-Newtonian fluids, while the linear case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M16">View MathML</a> corresponds to the Newtonian fluid)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M17">View MathML</a>

and the spherically symmetric potential c, can be reduced to an equation of the form (1). For this reason, motivated also by the linear case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M16">View MathML</a>, the problem of dependence of eigenvalues of (1), (3) on the functions r, c, w and the boundary data a, b, α, β was a subject of the investigation in several recent papers. We refer to [3-6] and the references therein.

The paper is organized as follows. In the next section, we recall essentials of the qualitative theory of half-linear differential equations. Section 3 deals with the eigenvalue problem for an equation with constant coefficients. The main results of the paper, limit formulas for the first eigenvalue of (1), (3), are given in Section 4.

2 Preliminaries

First, we recall the concepts of half-linear goniometric functions. These functions, in the form presented here, appeared for the first time in [1]. In a modified form, they can also be found in other papers, e.g., in [7].

The half-linear sine function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M13">View MathML</a> is defined as the solution of the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M20">View MathML</a>

(4)

given by the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M22">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M13">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M24">View MathML</a> anti-periodic (and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M25">View MathML</a> periodic) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M26">View MathML</a>. The derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M27">View MathML</a> defines the half-linear cosine function. These functions satisfy the half-linear Pythagorean identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M28">View MathML</a>

(5)

The half-linear functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M30">View MathML</a> are defined in a natural way as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M31">View MathML</a>

The inverse functions to these functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M32">View MathML</a> resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M33">View MathML</a> are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M35">View MathML</a>. By a direct computation, using (5) and the fact that (4) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M36">View MathML</a>

one can verify the formulas

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M37">View MathML</a>

The original proof of the unique solvability of (1) (and hence of (2), see [1]) is based on the half-linear version of the Prüfer transformation. Let x be a nontrivial solution of (1), put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M38">View MathML</a>

(6)

Then the Prüfer angle φ solves the first order differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M39">View MathML</a>

(7)

The right-hand side of (7) is a Lipschitzian function with respect to φ, hence the standard existence, uniqueness, and continuous dependence on the initial data theory applies to this equation, and these results carry over via (6) to (2) and (1). Observe at this place that the right-hand side of (2) is not Lipschitzian, so this theory cannot be directly applied to (2).

The Prüfer transformation is closely associated with the Riccati-type differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M40">View MathML</a>

(8)

which is related to (1) by the Riccati substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M41">View MathML</a>. The fact that we have in disposal a Riccati-type differential equation and the generalized Prüfer transformation implies that the linear oscillation theory extends almost verbatim to (1). In particular, similarly to the linear case, the eigenvalues of (1), (3) form an increasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M42">View MathML</a>, and the nth eigenfunction has exactly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M43">View MathML</a> zeros in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M11">View MathML</a>, see [8] and also [[9], Section 5.7]. For some recent references in this area, we refer to [3] and the references therein.

3 Equation with constant coefficients

In this section, as a motivation, we consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M45">View MathML</a>

(9)

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M48">View MathML</a> in (1). The Riccati equation associated with (9) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M49">View MathML</a>

(10)

First, consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M50">View MathML</a> in (3). Then the first eigenfunction of (9), (3) corresponds to the solution of (10) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M51">View MathML</a>

(11)

To underline the dependence of eigenvalues of our eigenvalue problem on b, α, β, we denote the first eigenvalue by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M52">View MathML</a>. Also, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M53">View MathML</a>, we use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M54">View MathML</a>.

Obviously, the solution of (10) satisfying (11) is a constant solution when

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M55">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M56">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M57">View MathML</a>, we need v to be decreasing. When the length of the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M11">View MathML</a> tends to zero, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M59">View MathML</a> cannot be bounded from below in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M11">View MathML</a>, and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M61">View MathML</a>

In the opposite case, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M62">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M63">View MathML</a>, we need v to be increasing, and using the same argument as before, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M64">View MathML</a>. Finally, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M66">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M67">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M68">View MathML</a>, similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M70">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M71">View MathML</a>, and hence also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M72">View MathML</a>.

The previous simple considerations are summarized in the next theorem.

Theorem 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M52">View MathML</a>denote the first eigenvalue of (9), (3) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M75">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M76">View MathML</a>

Remark 1 In this section, for the sake of simplicity, we have considered a constant coefficients equation in the form (9), i.e., with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M78">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M79">View MathML</a>. If the functions r, c, w in (1) are constants equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M80">View MathML</a>, a slight modification of the previous considerations shows that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M53">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M82">View MathML</a>

(actually, this least eigenvalue does not depend on the endpoints a, b).

4 Limit behavior of the first eigenvalue

In this section, we consider general half-linear eigenvalue problem (1), (3). We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M83">View MathML</a>

In the next theorems, we discuss various asymptotic behavior of ratios of the functions C, R, W for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M84">View MathML</a>, which implies various limit behavior of the first eigenvalue. The behavior of the higher eigenvalues is described at the end of this section.

In the proofs of the next theorems, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M86">View MathML</a>, denotes the Prüfer angle of a solution x of (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M87">View MathML</a> satisfying (3) at a, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M88">View MathML</a>.

We start with the most interesting case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M53">View MathML</a> in (3).

Theorem 2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M90">View MathML</a>. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M91">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M92">View MathML</a>

(12)

Proof We will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M93">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M94">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M95">View MathML</a> is sufficiently close to a. Since the eigenfunction corresponding to the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M96">View MathML</a> has no zero on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M97">View MathML</a> (see, e.g., [10]), we can use the Riccati equation (8) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M98">View MathML</a> instead of equation (7) for φ. Using the mean-value theorem for Lebesgue integrals in computing the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M99">View MathML</a>, integration of (8) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M100">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M101">View MathML</a>. Hence, for b sufficiently close to a,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M102">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M103">View MathML</a>. Thus, since φ was the Prüfer angle corresponding to a solution of (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M87">View MathML</a> satisfying (3) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M105">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M93">View MathML</a> for the first eigenvalue when b is in a sufficiently small right neighborhood of a (since we need <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M107">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M108">View MathML</a>). Therefore, (12) holds. □

Theorem 3Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M109">View MathML</a>

and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M110">View MathML</a>

Then we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M111">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M113">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M114">View MathML</a> be fixed, and take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M115">View MathML</a> so small that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M116">View MathML</a>

(13)

Such a positive δ exists according to the definition of the number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M117">View MathML</a>. Formula (13) implies that for τ sufficiently close to α, we have the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M118">View MathML</a>

when t is sufficiently close to a. Again, integration of (8), together with the mean value theorem applied to the last integral on the right-hand side of (8), gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M119">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M120">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M93">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M85">View MathML</a>, and thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M123">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M124">View MathML</a>. □

Theorem 4Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M125">View MathML</a>

(14)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M126">View MathML</a>

(15)

Then we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M127">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M124">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M129">View MathML</a>.

Proof The first formula in (14) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M130">View MathML</a> for t in a right neighborhood of a. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M131">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M115">View MathML</a> be so small that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M133">View MathML</a>

(16)

Again, such <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M115">View MathML</a> exists according to the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M117">View MathML</a>. Hence, for τ sufficiently close to α, from (16), we have for t close to a that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M136">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M85">View MathML</a> be arbitrary. Similarly as in the proof of the previous theorem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M138">View MathML</a>

if b is sufficiently close to a, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M139">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M140">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M141">View MathML</a>. □

Theorem 5Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M142">View MathML</a>

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M143">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M144">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M85">View MathML</a> be arbitrary. Similarly as in the previous theorems,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M146">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M147">View MathML</a>

(18)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M148">View MathML</a>, the expression in line (17) tends to −∞ as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15">View MathML</a> while remaining terms on the right-hand side of the previous formula are bounded. Hence the expression on the right-hand side is negative for b close to a, which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M120">View MathML</a> for these b. However, this implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M151">View MathML</a> in a right neighborhood of a, and since Λ was arbitrary, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M152">View MathML</a>. The same arguments imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M153">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M154">View MathML</a>.

Finally, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M155">View MathML</a>, take first <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M156">View MathML</a>. Since the last term in (18) tends to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15">View MathML</a>, we obtain using the same argument as in the previous part of the proof that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M158">View MathML</a> for b sufficiently close to a. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M159">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M160">View MathML</a> for b in a right neighborhood of a, and this completes the proof. □

Theorem 6Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M161">View MathML</a>

(19)

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M91">View MathML</a>. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M163">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M164">View MathML</a>

(20)

Proof Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M165">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M166">View MathML</a> be the Prüfer angle of the solution x of (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M167">View MathML</a> satisfying (3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M53">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M169">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M170">View MathML</a>

(22)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M171">View MathML</a>. Therefore, from (21), (22), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M172">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15">View MathML</a> and using that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M174">View MathML</a>

(which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M175">View MathML</a> is bounded in a neighborhood of a), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M176">View MathML</a>

and (20) is proved. □

Theorem 7If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M177">View MathML</a>has a finite value for two different values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M178">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M179">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M180">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M181">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M182">View MathML</a>

exist finite, and for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M183">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M184">View MathML</a>

(23)

Proof First of all, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M185">View MathML</a>

(24)

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M186">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M187">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M188">View MathML</a> be the Prüfer angle of the solution of (1), (3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M53">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M191">View MathML</a>, we have from (8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M192">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M193">View MathML</a>. Hence, one gets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M194">View MathML</a>

(25)

Subtracting these two equations and using the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M195">View MathML</a>

for b sufficiently close to a (this follows from (24)), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M196">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M198">View MathML</a> exists finite, and hence from (25), the same holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M199">View MathML</a>, and the conclusion follows from Theorem 5. □

We finish the paper with a brief treatment of the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M200">View MathML</a> in (3). We show that the situation is similar as in the case of a constant coefficients equation treated in Section 3.

Theorem 8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M75">View MathML</a>.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M203">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M204">View MathML</a>

(26)

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M205">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M206">View MathML</a>

Proof We will prove the part (i) only, the proof of (ii) is analogical. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M85">View MathML</a> be arbitrary, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M208">View MathML</a> be the Prüfer angle of the solution x of (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M209">View MathML</a> satisfying (3) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M105">View MathML</a>. Since φ is a continuous function of t, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M211">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M212">View MathML</a> if b is sufficiently close to a. Hence, using the same argument as before, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M213">View MathML</a>, which implies (26). □

Remark 2 Until now, we have considered the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M214">View MathML</a> only. Concerning the asymptotic behavior of higher eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M215">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M216">View MathML</a>

(27)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M75">View MathML</a>. This formula follows in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M203">View MathML</a> from the general theory of half-linear eigenvalues problem (see [8,10]), which says that the eigenvalues form an increasing sequence tending to ∞, and from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M220">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M205">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M222">View MathML</a>, but for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M223">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M224">View MathML</a> and higher eigenvalues correspond to the situation when the Prüfer angle of a solution of (1) satisfying the first condition in (3) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M225">View MathML</a>. Hence, the growth of φ must be unbounded when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M15">View MathML</a>, and hence (27) holds also for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/221/mathml/M205">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally and read and approved the final version of the manuscript.

Acknowledgements

The research of the first author was carried out as a part of the TAMOP-4.2.2/B-10/1-2010-0008 project in the framework of the New Hungarian Development Plan. The realization of this project is supported by the European Union and co-financed by the European Social Fund. The second author was supported by the Grant GAP 201/11/0768 of the Czech Grant Agency.

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