Existence of Positive Solutions of a Singular Nonlinear Boundary Value Problem

Boundary Value Problems20102010:458015

DOI: 10.1155/2010/458015

Received: 21 May 2010

Accepted: 11 August 2010

Published: 18 August 2010

Abstract

We are concerned with the existence of positive solutions of singular second-order boundary value problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq3_HTML.gif , which is not necessarily linearizable. Here, nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq4_HTML.gif is allowed to have singularities at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq5_HTML.gif . The proof of our main result is based upon topological degree theory and global bifurcation techniques.

1. Introduction

Existence and multiplicity of solutions of singular problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ1_HTML.gif
(1.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq6_HTML.gif is allowed to have singularities at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq8_HTML.gif , have been studied by several authors, see Asakawa [1], Agarwal and O http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq9_HTML.gif Regan [2], O http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq10_HTML.gif Regan [3], Habets and Zanolin [4], Xu and Ma [5], Yang [6], and the references therein. The main tools in [16] are the method of lower and upper solutions, Leray-Schauder continuation theorem, and the fixed point index theory in cones. Recently, Ma [7] studied the existence of nodal solutions of the singular boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ2_HTML.gif
(1.2)

by applying Rabinowitz's global bifurcation theorem, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq11_HTML.gif is allowed to have singularities at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq13_HTML.gif is linearizable at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq14_HTML.gif as well as at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq15_HTML.gif . It is the purpose of this paper to study the existence of positive solutions of (1.1), which is not necessarily linearizable.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq16_HTML.gif be Banach space defined by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ3_HTML.gif
(1.3)
with the norm
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ4_HTML.gif
(1.4)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ5_HTML.gif
(1.5)

Definition 1.1.

A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq17_HTML.gif is said to be an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq18_HTML.gif -Carathéodory function if it satisfies the following:

(i)for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq20_HTML.gif is measurable;

(ii)for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq21_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq22_HTML.gif is continuous;

(iii)for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq23_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq24_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ6_HTML.gif
(1.6)

In this paper, we will prove the existence of positive solutions of (1.1) by using the global bifurcation techniques under the following assumptions.

(H1) Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq26_HTML.gif be an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq27_HTML.gif -Carathéodory function and there exist functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq30_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq31_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ7_HTML.gif
(1.7)
for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq32_HTML.gif -Carathéodory functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq33_HTML.gif defined on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq34_HTML.gif with
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ8_HTML.gif
(1.8)
uniformly for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq35_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ9_HTML.gif
(1.9)
for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq36_HTML.gif -Carathéodory functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq37_HTML.gif defined on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq38_HTML.gif with
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ10_HTML.gif
(1.10)

uniformly for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq39_HTML.gif .

(H2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq40_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq42_HTML.gif .

(H3) There exists function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq43_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ11_HTML.gif
(1.11)

Remark 1.2.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq46_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq47_HTML.gif , then (1.8) implies that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ12_HTML.gif
(1.12)
and (1.10) implies that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ13_HTML.gif
(1.13)

The main tool we will use is the following global bifurcation theorem for problem which is not necessarily linearizable.

Theorem A (Rabinowitz, [8]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq48_HTML.gif be a real reflexive Banach space. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq49_HTML.gif be completely continuous, such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq51_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq52_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq53_HTML.gif is an isolated solution of the following equation:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ14_HTML.gif
(1.14)
for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq55_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq56_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq57_HTML.gif are not bifurcation points of (1.14). Furthermore, assume that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ15_HTML.gif
(1.15)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq58_HTML.gif is an isolating neighborhood of the trivial solution. Let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ16_HTML.gif
(1.16)

then there exists a continuum (i.e., a closed connected set) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq59_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq60_HTML.gif containing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq61_HTML.gif , and either

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq62_HTML.gif is unbounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq63_HTML.gif , or

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq64_HTML.gif .

To state our main results, we need the following.

Lemma 1.3 (see [1, Proposition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq65_HTML.gif ]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq66_HTML.gif , then the eigenvalue problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ17_HTML.gif
(1.17)
has a sequence of eigenvalues as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ18_HTML.gif
(1.18)

Moreover, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq67_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq68_HTML.gif is simple and its eigenfunction http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq69_HTML.gif has exactly http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq70_HTML.gif zeros in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq71_HTML.gif .

Remark 1.4.

Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq73_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq74_HTML.gif . Therefore, there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq75_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ19_HTML.gif
(1.19)

Our main result is the following.

Theorem 1.5.

Let (H1)–(H3) hold. Assume that either
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ20_HTML.gif
(1.20)
or
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ21_HTML.gif
(1.21)

then (1.1) has at least one positive solution.

Remark 1.6.

For other references related to this topic, see [914] and the references therein.

2. Preliminary Results

Lemma 2.1 (see [15, Proposition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq76_HTML.gif ]).

For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq77_HTML.gif , the linear problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ22_HTML.gif
(2.1)
has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq79_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ23_HTML.gif
(2.2)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ24_HTML.gif
(2.3)
Furthermore, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq80_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ25_HTML.gif
(2.4)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq81_HTML.gif be the Banach space with the norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq82_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ26_HTML.gif
(2.5)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq83_HTML.gif be an operator defined by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ27_HTML.gif
(2.6)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ28_HTML.gif
(2.7)

Then, from Lemma 2.1, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq84_HTML.gif is well defined.

Lemma 2.2.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq86_HTML.gif be the first eigenfunction of (1.17). Then for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq87_HTML.gif , one has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ29_HTML.gif
(2.8)

Proof.

For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq88_HTML.gif , integrating by parts, we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ30_HTML.gif
(2.9)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq90_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ31_HTML.gif
(2.10)
Therefore, we only need to prove that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ32_HTML.gif
(2.11)
Let us deal with the first equality, the second one can be treated by the same way. Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq91_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ33_HTML.gif
(2.12)
which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq92_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq93_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq94_HTML.gif . Now, we claim that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ34_HTML.gif
(2.13)
Suppose on the contrary that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq95_HTML.gif , then for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq96_HTML.gif small enough, we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ35_HTML.gif
(2.14)
Therefore,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ36_HTML.gif
(2.15)
which is a contradiction. Combining (1.19) with (2.13), we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ37_HTML.gif
(2.16)

This completes the proof.

Remark 2.3.

Under the conditions of Lemma 2.2, for the later convenience, (2.8) is equivalent to
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ38_HTML.gif
(2.17)

Lemma 2.4 (see [1, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq97_HTML.gif ]).

For every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq98_HTML.gif , the subset http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq99_HTML.gif defined by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ39_HTML.gif
(2.18)

is precompact in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq100_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq101_HTML.gif be the closure of the set of positive solutions of the problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ40_HTML.gif
(2.19)
We extend the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq102_HTML.gif to an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq103_HTML.gif -Carathéodory function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq104_HTML.gif defined on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq105_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ41_HTML.gif
(2.20)
Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq106_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq107_HTML.gif and a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq108_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq109_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq110_HTML.gif be an arbitrary solution of the problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ42_HTML.gif
(2.21)

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq111_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq112_HTML.gif , Lemma 2.2 yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq113_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq114_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq115_HTML.gif is a nonnegative solution of (2.19), and the closure of the set of nontrivial solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq116_HTML.gif of (2.21) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq117_HTML.gif is exactly http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq118_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq119_HTML.gif be an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq120_HTML.gif -Carathéodory function. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq121_HTML.gif be the Nemytskii operator associated with the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq122_HTML.gif as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ43_HTML.gif
(2.22)

Lemma 2.5.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq123_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq124_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq125_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq126_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq127_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq128_HTML.gif . Then,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ44_HTML.gif
(2.23)

Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq129_HTML.gif , whenever http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq130_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq131_HTML.gif be the Nemytskii operator associated with the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq132_HTML.gif as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ45_HTML.gif
(2.24)
Then (2.21), with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq133_HTML.gif , is equivalent to the operator equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ46_HTML.gif
(2.25)
that is,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ47_HTML.gif
(2.26)

Lemma 2.6.

Let (H1) and (H2) hold. Then the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq134_HTML.gif is completely continuous.

Proof.

From (1.10) in (H1), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq135_HTML.gif , such that, for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq136_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq137_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ48_HTML.gif
(2.27)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq138_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq139_HTML.gif -Carathéodory function, then there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq140_HTML.gif , such that, for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq141_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq142_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq143_HTML.gif . Therefore, for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq144_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq145_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ49_HTML.gif
(2.28)
For convenience, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq146_HTML.gif . We first show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq147_HTML.gif is continuous. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq148_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq149_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq150_HTML.gif . Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq151_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq152_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq153_HTML.gif and there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq154_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq155_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq156_HTML.gif . It is easy to see that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ50_HTML.gif
(2.29)

By the Lebesgue dominated convergence theorem, we have that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq157_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq158_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq159_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq160_HTML.gif is continuous.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq161_HTML.gif be a bounded set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq162_HTML.gif . Lemma 2.4 together with (2.28) shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq163_HTML.gif is precompact in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq164_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq165_HTML.gif is completely continuous.

In the following, we will apply the Leray-Schauder degree theory mainly to the mapping http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq166_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ51_HTML.gif
(2.30)

For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq167_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq168_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq169_HTML.gif denote the degree of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq170_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq171_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq172_HTML.gif .

Lemma 2.7.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq173_HTML.gif be a compact interval with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq174_HTML.gif , then there exists a number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq175_HTML.gif with the property
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ52_HTML.gif
(2.31)

Proof.

Suppose to the contrary that there exist sequences http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq176_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq177_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq178_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq179_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq180_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq181_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq182_HTML.gif , then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq183_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq184_HTML.gif .

Set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq185_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq186_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq187_HTML.gif . Now, from condition (H1), we have the following:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ53_HTML.gif
(2.32)
and accordingly
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ54_HTML.gif
(2.33)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq188_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq189_HTML.gif denote the nonnegative eigenfunctions corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq190_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq191_HTML.gif , respectively, then we have from the first inequality in (2.33) that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ55_HTML.gif
(2.34)
From Lemma 2.2, we have that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ56_HTML.gif
(2.35)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq192_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq193_HTML.gif , from (1.12), we have that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ57_HTML.gif
(2.36)
By the fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq194_HTML.gif , we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq195_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq196_HTML.gif . Thus,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ58_HTML.gif
(2.37)
Combining this and (2.35) and letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq197_HTML.gif in (2.34), it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ59_HTML.gif
(2.38)
and consequently
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ60_HTML.gif
(2.39)
Similarly, we deduce from second inequality in (2.33) that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ61_HTML.gif
(2.40)

Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq198_HTML.gif . This contradicts http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq199_HTML.gif .

Corollary 2.8.

For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq200_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq201_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq202_HTML.gif .

Proof.

Lemma 2.7, applied to the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq203_HTML.gif , guarantees the existence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq204_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq205_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ62_HTML.gif
(2.41)
This together with Lemma 2.6 implies that for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq206_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ63_HTML.gif
(2.42)

which ends the proof.

Lemma 2.9.

Suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq207_HTML.gif , then there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq208_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq209_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq210_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq211_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ64_HTML.gif
(2.43)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq212_HTML.gif is the nonnegative eigenfunction corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq213_HTML.gif .

Proof.

Suppose on the contrary that there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq214_HTML.gif and a sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq215_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq216_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq217_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq218_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq219_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq220_HTML.gif . As
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ65_HTML.gif
(2.44)
and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq221_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq222_HTML.gif , it concludes from Lemma 2.2 that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ66_HTML.gif
(2.45)
Notice that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq223_HTML.gif has a unique decomposition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ67_HTML.gif
(2.46)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq224_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq225_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq226_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq227_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq228_HTML.gif , we have from (2.46) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq229_HTML.gif .

Choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq230_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ68_HTML.gif
(2.47)
By (H1), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq231_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ69_HTML.gif
(2.48)
Therefore, for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq232_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ70_HTML.gif
(2.49)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq233_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq234_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ71_HTML.gif
(2.50)
and consequently
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ72_HTML.gif
(2.51)
Applying (2.51), it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ73_HTML.gif
(2.52)
Thus,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ74_HTML.gif
(2.53)

This contradicts (2.47).

Corollary 2.10.

For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq235_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq236_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq237_HTML.gif .

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq238_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq239_HTML.gif is the number asserted in Lemma 2.9. As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq240_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq241_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq242_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq243_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq244_HTML.gif . By Lemma 2.9, one has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ75_HTML.gif
(2.54)
This together with Lemma 2.6 implies that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ76_HTML.gif
(2.55)

Now, using Theorem A, we may prove the following.

Proposition 2.11.

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq245_HTML.gif is a bifurcation interval from the trivial solution for (2.30). There exists an unbounded component http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq246_HTML.gif of positive solutions of (2.30) which meets http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq247_HTML.gif . Moreover,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ77_HTML.gif
(2.56)

Proof.

For fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq248_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq249_HTML.gif , let us take that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq250_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq251_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq252_HTML.gif . It is easy to check that, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq253_HTML.gif , all of the conditions of Theorem A are satisfied. So there exists a connected component http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq254_HTML.gif of solutions of (2.30) containing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq255_HTML.gif , and either

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq256_HTML.gif is unbounded, or

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq257_HTML.gif .

By Lemma 2.7, the case (ii) can not occur. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq258_HTML.gif is unbounded bifurcated from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq259_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq260_HTML.gif . Furthermore, we have from Lemma 2.7 that for any closed interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq261_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq262_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq263_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq264_HTML.gif is impossible. So http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq265_HTML.gif must be bifurcated from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq266_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq267_HTML.gif .

3. Proof of the Main Results

Proof of Theorem 1.5.

It is clear that any solution of (2.30) of the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq268_HTML.gif yields solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq269_HTML.gif of (1.1). We will show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq270_HTML.gif crosses the hyperplane http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq271_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq272_HTML.gif . To do this, it is enough to show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq273_HTML.gif joins http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq274_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq275_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq276_HTML.gif satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ78_HTML.gif
(3.1)

We note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq277_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq278_HTML.gif since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq279_HTML.gif is the only solution of (2.30) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq280_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq281_HTML.gif .

Case 1.

consider the following:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ79_HTML.gif
(3.2)
In this case, we show that the interval
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ80_HTML.gif
(3.3)

We divide the proof into two steps.

Step 1.

We show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq282_HTML.gif is bounded.

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq283_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq284_HTML.gif . From (H3), we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ81_HTML.gif
(3.4)

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq285_HTML.gif denote the nonnegative eigenfunction corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq286_HTML.gif .

From (3.4), we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ82_HTML.gif
(3.5)
By Lemma 2.2, we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ83_HTML.gif
(3.6)
Thus,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ84_HTML.gif
(3.7)

Step 2.

We show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq287_HTML.gif joins http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq288_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq289_HTML.gif .

From (3.1) and (3.7), we have that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq290_HTML.gif . Notice that (2.30) is equivalent to the integral equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ85_HTML.gif
(3.8)
which implies that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ86_HTML.gif
(3.9)
We divide the both sides of (3.9) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq291_HTML.gif and set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq292_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq293_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq294_HTML.gif , there exist a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq295_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq296_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq297_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq298_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq299_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ87_HTML.gif
(3.10)
relabeling if necessary. Thus, (3.9) yields that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ88_HTML.gif
(3.11)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq300_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq301_HTML.gif denote the nonnegative eigenfunctions corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq302_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq303_HTML.gif , respectively, then it follows from the second inequality in (3.11) that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ89_HTML.gif
(3.12)
and consequently
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ90_HTML.gif
(3.13)
Similarly, we deduce from the first inequality in (3.11) that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ91_HTML.gif
(3.14)
Thus,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ92_HTML.gif
(3.15)

So http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq304_HTML.gif joins http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq305_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq306_HTML.gif .

Case 2.

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq307_HTML.gif .

In this case, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq308_HTML.gif is such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ93_HTML.gif
(3.16)
then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ94_HTML.gif
(3.17)
and moreover,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ95_HTML.gif
(3.18)
Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq309_HTML.gif is bounded, applying a similar argument to that used in Step 2 of Case 1, after taking a subsequence and relabeling if necessary, it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ96_HTML.gif
(3.19)

Again http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq310_HTML.gif joins http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq311_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq312_HTML.gif and the result follows.

Remark 3.1.

Lomtatidze [13, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq313_HTML.gif ] proved the existence of solutions of singular two-point boundary value problems as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ97_HTML.gif
(3.20)

under the following assumptions:

(A1)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ98_HTML.gif
(3.21)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq314_HTML.gif satisfies the following condition:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ99_HTML.gif
(3.22)
?(A2) For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq315_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq316_HTML.gif be the solution of singular IVPs
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ100_HTML.gif
(3.23)

satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq317_HTML.gif has at least one zero in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq318_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq319_HTML.gif has no zeros in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq320_HTML.gif .

It is worth remarking that (A1)-(A2) imply Condition (1.21) in Theorem 1.5. However, Condition (1.21) is easier to be verified than (A1)-(A2) since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq321_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq322_HTML.gif are easily estimated by Rayleigh's Quotient.

The language of eigenvalue of singular linear eigenvalue problem did not occur until Asakawa [1] in 2001. The first part of Theorem 1.5 is new.

Declarations

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC 11061030, the Fundamental Research Funds for the Gansu Universities.

Authors’ Affiliations

(1)
College of Mathematics and Information Science, Northwest Normal University
(2)
The School of Mathematics, Physics & Software Engineering, Lanzhou Jiaotong University

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© R. Ma and J. Li. 2010

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