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Bifurcation of positive periodic solutions of first-order impulsive differential equations

Ruyun Ma*, Bianxia Yang and Zhenyan Wang

Author Affiliations

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. China

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Boundary Value Problems 2012, 2012:83  doi:10.1186/1687-2770-2012-83

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/83


Received:18 May 2012
Accepted:20 July 2012
Published:1 August 2012

© 2012 Ma 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

We give a global description of the branches of positive solutions of first-order impulsive boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M1">View MathML</a>

which is not necessarily linearizable. Where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M2">View MathML</a> is a parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M3">View MathML</a> are given impulsive points. Our approach is based on the Krein-Rutman theorem, topological degree, and global bifurcation techniques.

MSC: 34B10, 34B15, 34K15, 34K10, 34C25, 92D25.

Keywords:
Krein-Rutman theorem; topological degree; bifurcation from interval; impulsive boundary value problem; existence and multiplicity

1 Introduction

Some evolution processes are distinguished by the circumstance that at certain instants their evolution is subjected to a rapid change, that is, a jump in their states. Mathematically, this leads to an impulsive dynamical system. Differential equations involving impulsive effects occur in many applications: physics, population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, optimal control, etc. Therefore, the study of this class of impulsive differential equations has gained prominence and it is a rapidly growing field. See [1-9] and the references therein.

Let us consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M4">View MathML</a>

(1.1)

subjected to the impulsive boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M5">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M2">View MathML</a> is a real parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M3">View MathML</a> are given impulsive points. We make the following assumptions:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M9">View MathML</a> is a 1-periodic function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M10">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M13">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M14">View MathML</a>, there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M15">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M16">View MathML</a>

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M17">View MathML</a> is 1-periodic function with respect to the first variable, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M19">View MathML</a> exist, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M20">View MathML</a>. Moreover, there exist functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M21">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M22">View MathML</a> in any subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M24">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M25">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M26">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M27">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M28">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M29">View MathML</a>), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M30">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M31">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M32">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M33">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M28">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M35">View MathML</a>);

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M37">View MathML</a>;

(H5) there exists function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M39">View MathML</a> in any subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M41">View MathML</a>

Some special cases of (1.1), (1.2) have been investigated. For example, Nieto [3] considered the (1.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M43">View MathML</a>. By using Schaeffer’s theorem, some sufficient conditions for existence of solutions of the IBVP (1.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M43">View MathML</a> were obtained.

Li, Nieto, and Shen [4] studied the existence of at least one positive periodic solutions of (1.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M47">View MathML</a> (m is a constant). By using Schaeffer’s fixed-point theorem, they got the solvability under f satisfied at most linear growth and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M48">View MathML</a> is bounded or f is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M48">View MathML</a> satisfied at most linear growth.

Liu [7] studied the existence and multiplicity of (1.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M44">View MathML</a>, by using the fixed- point theorem in cones, and he proved the following:

Theorem A ([7], Theorem 3.1.1])

Let (H1) hold. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M53">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M54">View MathML</a>

(1.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M55">View MathML</a>

(1.4)

Then the problem (1.1), (1.2) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M44">View MathML</a>has at least one positive solution where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M57">View MathML</a>will be defined in (2.2) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M58">View MathML</a>

(1.5)

Theorem B ([7], Theorem 3.1.2])

Let (H1) hold. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M53">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M62">View MathML</a>

(1.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M63">View MathML</a>

(1.7)

Then the problem (1.1), (1.2) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M44">View MathML</a>has at least one positive solution whereW, wdefined as (1.5) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M65">View MathML</a>

(1.8)

It is worth remarking that the [3,4,7] only get the existence of solutions, and there is not any information of global structure of positive periodic solutions.

By using global bifurcation techniques, we obtain a complete description of the global structure of positive solutions for (1.1), (1.2) under weaker conditions. More precisely, our main result is the following theorem.

Theorem 1.1Let (H1), (H2), and (H3) hold. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M12">View MathML</a>. Then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M70">View MathML</a>is a bifurcation interval of positive solutions from infinity for (1.1), (1.2), and there exists no bifurcation interval of positive solutions from infinity which is disjoint with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M70">View MathML</a>. More precisely, there exists a component<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72">View MathML</a>of positive solutions of (1.1), (1.2) which meets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M73">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M75">View MathML</a>will be defined in Section 2;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M76">View MathML</a>is a bifurcation interval of positive solutions from the trivial solutions for (1.1), (1.2), and there exists no bifurcation interval of positive solutions from the trivial solutions which is disjoint with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M76">View MathML</a>. More precisely, there exists a component<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78">View MathML</a>of positive solutions of (1.1), (1.2) which meets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M79">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M81">View MathML</a>will be defined in Section 4;

(iii) If (H4) and (H5) also hold, then there is a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M82">View MathML</a>such that problem (1.1), (1.2) admits no positive solution with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M83">View MathML</a>. In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M84">View MathML</a>.

Remark 1.1 There is no paper except [9] studying impulsive differential equations using bifurcation ideas. However, in [9], they only dealt with the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M85">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M87">View MathML</a> do exist. Where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M88">View MathML</a>

From (H3), it is easy to see that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M87">View MathML</a> may be not exist, the method used in [9] is not helpful any more in this case.

Remark 1.2 From (iii) of Theorem 1.1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72">View MathML</a> are involved in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M93">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M70">View MathML</a> is a unique bifurcation interval of positive solutions from infinity for (1.1), (1.2), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M76">View MathML</a> is a unique bifurcation interval of positive solutions from the trivial solutions for (1.1), (1.2). Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78">View MathML</a> must be intersected with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M73">View MathML</a>.

Remark 1.3 Obviously, (H3) is more general than (1.5), (1.8). Moreover, if we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M99">View MathML</a>, under conditions (1.3), (1.4), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M101">View MathML</a>, respectively. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78">View MathML</a> cross the hyperplane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M103">View MathML</a>. Therefore, Theorem 3.1.1 of [7] is the corollary of Theorems 1.1 even in the special case.

Remark 1.4 Similar, if we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M105">View MathML</a>, only under condition (1.6), we can obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M106">View MathML</a>. From Proposition 3.1, we will know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72">View MathML</a> is unbounded in λ direction, so, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72">View MathML</a> cross the hyperplane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M109">View MathML</a>. Therefore, Theorem 3.1.2 of [7] is the corollary of Theorems 1.1 even in the special case and weaker condition.

Remark 1.5 There are many papers which get the positive solutions using bifurcation from the interval. For example, see [10,11]. However, in those papers, the linear operator corresponding problem is self-adjoint. It is easy to see that the linear operator corresponding (1.1), (1.2) is not self-adjoint. So, the method used in [9,10] is not helpful in this case.

Remark 1.6 Condition (H3) means that f is not necessarily linearizable near 0 and infinity. So, we will apply the following global bifurcation theorems for mappings which are not necessarily smooth to get a global description of the branches of positive solutions of (1.1), (1.2), and then, we obtain the existence and multiplicity of positive solutions of (1.1), (1.2).

Theorem C (K. Schmitt, R. C. Thompson [12])

LetVbe a real reflexive Banach space. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M110">View MathML</a>be completely continuous such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M112">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M113">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M114">View MathML</a>) be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M115">View MathML</a>is an isolated solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M116">View MathML</a>

(1.9)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M117">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M118">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M120">View MathML</a>are not bifurcation points of (1.9). Furthermore, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M121">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M122">View MathML</a>is an isolating neighborhood of the trivial solution. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M123">View MathML</a>

Then there exists a connected component<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M124">View MathML</a>ofcontaining<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M125">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M126">View MathML</a>, and either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M124">View MathML</a>is unbounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M126">View MathML</a>, or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M129">View MathML</a>.

Theorem D (K. Schmitt [13])

LetVbe a real reflexive Banach space. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M110">View MathML</a>be completely continuous, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M113">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M114">View MathML</a>) be such that the solution of (1.9) are, a priori, bounded inVfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M117">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M118">View MathML</a>, i.e., there exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M135">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M136">View MathML</a>

for alluwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M137">View MathML</a>. Furthermore, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M138">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M135">View MathML</a>large. Then there exists a closed connected set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M124">View MathML</a>of solutions of (1.9) that is unbounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M141">View MathML</a>, and either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M124">View MathML</a>is unbounded inλdirection, or

(ii) there exist an interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M143">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M144">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M124">View MathML</a>bifurcates from infinity in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M146">View MathML</a>.

The rest of the paper is organized as follows: In Section 2, we state some notations and preliminary results. Sections 3 and 4 are devoted to study the bifurcation from infinity and from the trivial solution for a nonlinear problem which are not necessarily linearizable, respectively. Finally, in Section 5, we consider the intertwining of the branches bifurcating from infinity and from the trivial solution.

2 Preliminaries

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M147">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148">View MathML</a> is a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M149">View MathML</a>.

By a positive solution of the problem (1.1), (1.2), we mean a pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M150">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M2">View MathML</a> and u is a solution of (1.1), (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M14">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M153">View MathML</a> be the closure of the set of positive solutions of (1.1), (1.2), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M154">View MathML</a>.

Lemma 2.1 ([14], Theorem 6.26])

The spectrum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M155">View MathML</a>of compact linear operatorThas following properties:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M155">View MathML</a>is a countable set with no accumulation point which is different from zero;

(ii) each nonzero<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M157">View MathML</a>is an eigenvalue ofTwith finite multiplicity, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M158">View MathML</a>is an eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M159">View MathML</a>with the same multiplicity, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M158">View MathML</a>denote the conjugate ofλ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M159">View MathML</a>denote the conjugate operator ofT.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M162">View MathML</a>, with inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M163">View MathML</a> and norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M164">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M166">View MathML</a> in any subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23">View MathML</a>. Further define the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M168">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M169">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M170">View MathML</a> as defined in (H2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M57">View MathML</a> is the Green’s function of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M172">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M173">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M174">View MathML</a>, it is easy to see that (H1) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M175">View MathML</a>.

By virtue of Krein-Rutman theorems (see [15]), we have the following lemma.

Lemma 2.2Suppose that (H1) holds, then for the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M176">View MathML</a>defined by (2.1), has a unique characteristic value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M177">View MathML</a>, which is positive, real, simple, and the corresponding eigenfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M178">View MathML</a>is of one sign, i.e., we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M179">View MathML</a>.

Proof It is a direct consequence of the Krein-Rutman theorem [15], Theorem 19.3]. □

Remark 2.1 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M177">View MathML</a> is real number, so from Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M177">View MathML</a> is also the characteristic value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M182">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M183">View MathML</a> denote the nonnegative eigenfunction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M182">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M177">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M182">View MathML</a> denote the conjugate operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M176">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M188">View MathML</a>

We extend the function f to function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M189">View MathML</a>, defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M190">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M191">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M192">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M190">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M194">View MathML</a>, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M195">View MathML</a>

(2.3)

is equivalent to the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M196">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M197">View MathML</a>

Remark 2.2 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M2">View MathML</a>, if u is a nontrivial solution of (2.3), from the positivity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M189">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M201">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23">View MathML</a>, so u is a nontrivial solution of (1.1), (1.2). Therefore, the closure of the set of nontrivial solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M150">View MathML</a> of (2.3) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M204">View MathML</a> is exactly Σ.

The problem (2.3) is now equivalent to the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M205">View MathML</a>

(2.4)

In the following, we shall apply the Leray-Schauder degree theory, mainly to the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M206">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M207">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M135">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M209">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M210">View MathML</a> denote the degree of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M211">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M212">View MathML</a> with respect to 0.

3 Bifurcation from infinity

In this section, we are devoted to study the bifurcation from infinity.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M213">View MathML</a>be a compact interval with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M214">View MathML</a>. Then there exists a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M215">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M216">View MathML</a>

Proof Suppose on the contrary that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M217">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M218">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M219">View MathML</a>), such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M220">View MathML</a>. We may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M221">View MathML</a>. By Remark 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M222">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M224">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M225">View MathML</a>

From (H2), (H3), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M226">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M228">View MathML</a> is a relatively compact set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M230">View MathML</a> is bounded and continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M231">View MathML</a>. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M232">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M235">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23">View MathML</a>.

Now, from condition (H2), we know that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M237">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M238">View MathML</a>

From (H3), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M239">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M240">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M241">View MathML</a>

accordingly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M242">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M243">View MathML</a>

(3.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M245">View MathML</a> denote the nonnegative eigenfunctions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M247">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M74">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M75">View MathML</a>, respectively. Then we have from the (3.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M250">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M219">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M252">View MathML</a>

we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M253">View MathML</a>

and consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M254">View MathML</a>

Similarly, we deduce from (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M255">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M256">View MathML</a>. This contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M257">View MathML</a>. □

Corollary 3.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M258">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M259">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M260">View MathML</a>.

Proof Lemma 3.1, applied to the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M261">View MathML</a>, guarantees the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M215">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M259">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M264">View MathML</a>

Hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M259">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M266">View MathML</a>

which implies the assertion. □

On the other hand, we have

Lemma 3.2Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M267">View MathML</a>. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M268">View MathML</a>with the property that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M269">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M270">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M271">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M272">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M273">View MathML</a>is the nonnegative eigenfunction of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M274">View MathML</a>corresponding to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M75">View MathML</a>.

Proof Let us assume that for some sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M276">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M218">View MathML</a> and numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M279">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M280">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M281">View MathML</a>

and we conclude from Remark 2.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M222">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23">View MathML</a>. So we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M284">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M285">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M286">View MathML</a>

(3.3)

By (H3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M287">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M288">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M218">View MathML</a>, then exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M290">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M291">View MathML</a>

and consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M292">View MathML</a>

(3.4)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M293">View MathML</a>, applying (3.4), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M294">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M295">View MathML</a>

this contradicts (3.3). □

Corollary 3.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M267">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M297">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M298">View MathML</a>.

Proof By Lemma 3.2, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M268">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M300">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M301">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M297">View MathML</a>. The assertion follows. □

We are now ready to prove

Proposition 3.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M303">View MathML</a>is a bifurcation interval of positive solutions from infinity for the problem (2.4). There exists an unbounded component<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72">View MathML</a>of positive solutions of (2.4) which meets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M305">View MathML</a>, and is unbounded inλdirection. Moreover, there exists no bifurcation interval of positive solutions from infinity which is disjointed with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M306">View MathML</a>.

Proof For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M307">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M308">View MathML</a>, let us take that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M310">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M311">View MathML</a>. It is easy to check that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M312">View MathML</a>, all of the conditions of Theorem D are satisfied. So, there exists a closed connected set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M313">View MathML</a> of solutions of (2.4) that is unbounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M314">View MathML</a>, and either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M313">View MathML</a> is unbounded in λ direction, or else

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M316">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M317">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M313">View MathML</a> bifurcates from infinity in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M319">View MathML</a>.

By Lemma 3.1, the case (ii) cannot occur. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M313">View MathML</a> bifurcates from infinity in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M314">View MathML</a> and is unbounded in λ direction. Furthermore, we have from Lemma 3.1 that for any closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M322">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M323">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M313">View MathML</a> must be bifurcated from infinity in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M326">View MathML</a> and is unbounded in λ direction. □

Assertion (i) of Theorem 1.1 follows directly.

4 Bifurcation from the trivial solutions

In this section, we shall study the bifurcation from the trivial solution for a nonlinear problem which is not necessarily linearizable near 0 and infinity.

As in Section 2, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M166">View MathML</a> in any subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23">View MathML</a>. Further define the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M330">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M331">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M332">View MathML</a> is defined in (H2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M57">View MathML</a> is defined in (2.2).

Similar as Lemma 2.2, we have the following lemma.

Lemma 4.1Suppose that (H1) holds, then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M334">View MathML</a>has a unique characteristic value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M335">View MathML</a>, which is positive, real, simple, and the corresponding eigenfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M336">View MathML</a>is of one sign, i.e., we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M337">View MathML</a>.

Remark 4.1 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M335">View MathML</a> is real number, so from Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M335">View MathML</a> is also the characteristic value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M340">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M340">View MathML</a> denote the conjugate operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M334">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M343">View MathML</a> denote the nonnegative eigenfunction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M340">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M335">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M346">View MathML</a>

Lemma 4.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M213">View MathML</a>be a compact interval with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M348">View MathML</a>. Then there exists a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M349">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M350">View MathML</a>

Proof Suppose on the contrary that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M217">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M352">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M219">View MathML</a>), such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M220">View MathML</a>. We may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M221">View MathML</a>. By Remark 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M222">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M224">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M359">View MathML</a>

From (H2), (H3), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M226">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148">View MathML</a>, so we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M362">View MathML</a> is a relatively compact set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148">View MathML</a>, hence (for a subsequence) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M232">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M235">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M234">View MathML</a>.

Now, from condition (H2), we know that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M368">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M369">View MathML</a>

From (H3), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M370">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M371">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M372">View MathML</a>

accordingly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M373">View MathML</a>

(4.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M374">View MathML</a>

(4.3)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M375">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M376">View MathML</a> denote the nonnegative eigenfunctions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M377">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M378">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M80">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M81">View MathML</a>, respectively. Then we have from the (4.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M381">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M219">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M383">View MathML</a>

we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M384">View MathML</a>

and consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M385">View MathML</a>

Similarly, we deduce from (4.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M386">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M387">View MathML</a>. This contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M257">View MathML</a>. □

Corollary 4.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M389">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M390">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M391">View MathML</a>.

On the other hand, we have

Lemma 4.3Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M392">View MathML</a>. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M393">View MathML</a>with the property that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M269">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M395">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M396">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M397">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M398">View MathML</a>is the nonnegative eigenfunction of the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M399">View MathML</a>corresponding to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M81">View MathML</a>.

Proof We assume again on the contrary that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M279">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M402">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M403">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M404">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M148">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M406">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M307">View MathML</a>.

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M408">View MathML</a>

and we conclude from Remark 2.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M222">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M23">View MathML</a>. So, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M411">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M285">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M413">View MathML</a>

(4.4)

By (H3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M414">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M415">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M352">View MathML</a>, then exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M290">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M418">View MathML</a>

and consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M419">View MathML</a>

(4.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M420">View MathML</a>, applying (4.5), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M421">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M422">View MathML</a>

this contradicts with (4.4). □

Corollary 4.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M392">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M424">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M425">View MathML</a>.

Proof By Lemma 4.3, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M393">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M427">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M428">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M429">View MathML</a>. Then the assertion follows. □

Now, using Theorem C and the similar method to prove Proposition 3.1 with obvious changes, we may prove the following proposition.

Proposition 4.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M430">View MathML</a>is a bifurcation interval of positive solutions from the trivial solution for the problem (2.4). There exists an unbounded component<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78">View MathML</a>of positive solutions of (2.4) which meets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M432">View MathML</a>. Moreover, there exists no bifurcation interval of positive solutions from the trivial solution which is disjointed with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M433">View MathML</a>.

This is exactly the assertion (ii) of Theorem 1.1.

5 Global behavior of the component of positive solutions

In this section, we consider the intertwining of the branches bifurcating from infinity and from the trivial solution.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M434">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M12">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M436">View MathML</a>. From (H2), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M437">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M12">View MathML</a>.

Define the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M439">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M440">View MathML</a>

(5.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M441">View MathML</a> is defined in (H5), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M57">View MathML</a> is defined in (2.2).

Similar as Lemma 2.2, we have the following lemma.

Lemma 5.1The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M443">View MathML</a>has a unique characteristic value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M444">View MathML</a>, which is positive, real, simple, and the corresponding eigenfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M445">View MathML</a>is of one sign, i.e., we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M446">View MathML</a>.

Remark 5.1 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M444">View MathML</a> is real number, so from Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M444">View MathML</a> is also the characteristic value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M449">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M449">View MathML</a> denote the conjugate operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M443">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M452">View MathML</a> denote the nonnegative eigenfunction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M449">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M444">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M455">View MathML</a>

Lemma 5.2Let (H1)-(H5) hold. Then there exists a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M82">View MathML</a>such that there is no positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M150">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M458">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M83">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M150">View MathML</a> be a positive solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M458">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M462">View MathML</a>

From (H5) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M463">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M464">View MathML</a>

(5.2)

From (5.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M465">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M466">View MathML</a>

 □

The assertion that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M467">View MathML</a> in Theorem 1.1(iii) now easily follows. For, in the case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M72">View MathML</a> are contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M470">View MathML</a>. Moreover, there exists no bifurcation interval of positive solution from infinity which is disjointed with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M471">View MathML</a>, there exists no bifurcation interval of positive solution from the trivial solution which is disjointed with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M472">View MathML</a>. In Theorem 1.1(iii), the unbounded component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M78">View MathML</a> has to meet <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/83/mathml/M474">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

RM completed the main study and carried out the results of this article. BY drafted the manuscript. ZW checked the proofs and verified the calculation. All the authors read and approved the final manuscript.

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11061030), NSFC (No. 11126296), and the Fundamental Research Funds for the Gansu Universities.

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