Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq2_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq4_HTML.gif is allowed to have singularities at https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq6_HTML.gif is allowed to have singularities at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq7_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq9_HTML.gif Regan [2], O https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq11_HTML.gif is allowed to have singularities at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq13_HTML.gif is linearizable at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq14_HTML.gif as well as at https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq16_HTML.gif be Banach space defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ3_HTML.gif
(1.3)
with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ4_HTML.gif
(1.4)
Let
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq17_HTML.gif is said to be an https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq19_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq20_HTML.gif is measurable;

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

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

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

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

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

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

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

Remark 1.2.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq46_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq47_HTML.gif , then (1.8) implies that
https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq49_HTML.gif be completely continuous, such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq50_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq51_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq52_HTML.gif , such that https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ14_HTML.gif
(1.14)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq55_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq56_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ15_HTML.gif
(1.15)
where https://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
https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq59_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq60_HTML.gif containing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq61_HTML.gif , and either

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

(ii) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq65_HTML.gif ]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq66_HTML.gif , then the eigenvalue problem
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ18_HTML.gif
(1.18)

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

Remark 1.4.

Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq73_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq74_HTML.gif . Therefore, there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq75_HTML.gif , such that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ20_HTML.gif
(1.20)
or
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq76_HTML.gif ]).

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq77_HTML.gif , the linear problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ22_HTML.gif
(2.1)
has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq79_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ23_HTML.gif
(2.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ24_HTML.gif
(2.3)
Furthermore, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq80_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ25_HTML.gif
(2.4)
Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq82_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ26_HTML.gif
(2.5)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq83_HTML.gif be an operator defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ27_HTML.gif
(2.6)
where
https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq85_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq87_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ29_HTML.gif
(2.8)

Proof.

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq88_HTML.gif , integrating by parts, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ30_HTML.gif
(2.9)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq89_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq90_HTML.gif , then
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq91_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ33_HTML.gif
(2.12)
which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq92_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq93_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq94_HTML.gif . Now, we claim that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq95_HTML.gif , then for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq96_HTML.gif small enough, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ35_HTML.gif
(2.14)
Therefore,
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq97_HTML.gif ]).

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

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

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

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq111_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq112_HTML.gif , Lemma 2.2 yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq113_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq114_HTML.gif . Thus, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq116_HTML.gif of (2.21) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq117_HTML.gif is exactly https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq118_HTML.gif .

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

Lemma 2.5.

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

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

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq132_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ45_HTML.gif
(2.24)
Then (2.21), with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq133_HTML.gif , is equivalent to the operator equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ46_HTML.gif
(2.25)
that is,
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq135_HTML.gif , such that, for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq137_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ48_HTML.gif
(2.27)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq138_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq139_HTML.gif -Carathéodory function, then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq140_HTML.gif , such that, for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq142_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq143_HTML.gif . Therefore, for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq145_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ49_HTML.gif
(2.28)
For convenience, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq146_HTML.gif . We first show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq147_HTML.gif is continuous. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq148_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq149_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq150_HTML.gif . Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq151_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq152_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq153_HTML.gif and there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq154_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq155_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq156_HTML.gif . It is easy to see that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq157_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq158_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq159_HTML.gif . Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq160_HTML.gif is continuous.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq161_HTML.gif be a bounded set in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq163_HTML.gif is precompact in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq164_HTML.gif . Therefore, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq166_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ51_HTML.gif
(2.30)

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

Lemma 2.7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq173_HTML.gif be a compact interval with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq174_HTML.gif , then there exists a number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq175_HTML.gif with the property
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq176_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq177_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq178_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq179_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq180_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq181_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq182_HTML.gif , then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq183_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq184_HTML.gif .

Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq185_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq186_HTML.gif and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ53_HTML.gif
(2.32)
and accordingly
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ54_HTML.gif
(2.33)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq188_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq189_HTML.gif denote the nonnegative eigenfunctions corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq190_HTML.gif and https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ56_HTML.gif
(2.35)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq192_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq193_HTML.gif , from (1.12), we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ57_HTML.gif
(2.36)
By the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq194_HTML.gif , we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq195_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq196_HTML.gif . Thus,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq197_HTML.gif in (2.34), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ59_HTML.gif
(2.38)
and consequently
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ61_HTML.gif
(2.40)

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

Corollary 2.8.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq200_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq201_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq203_HTML.gif , guarantees the existence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq204_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq205_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq206_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq207_HTML.gif , then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq208_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq209_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq210_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq211_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ64_HTML.gif
(2.43)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq212_HTML.gif is the nonnegative eigenfunction corresponding to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq214_HTML.gif and a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq215_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq217_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq218_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq219_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq220_HTML.gif . As
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ65_HTML.gif
(2.44)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq221_HTML.gif in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ66_HTML.gif
(2.45)
Notice that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq223_HTML.gif has a unique decomposition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ67_HTML.gif
(2.46)

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

Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq230_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ68_HTML.gif
(2.47)
By (H1), there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq231_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ69_HTML.gif
(2.48)
Therefore, for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq232_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ70_HTML.gif
(2.49)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq233_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq234_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ71_HTML.gif
(2.50)
and consequently
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ73_HTML.gif
(2.52)
Thus,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq235_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq236_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq237_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq238_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq240_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq241_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq242_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq243_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq244_HTML.gif . By Lemma 2.9, one has
https://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
https://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.

https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq247_HTML.gif . Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ77_HTML.gif
(2.56)

Proof.

For fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq248_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq249_HTML.gif , let us take that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq250_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq251_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq254_HTML.gif of solutions of (2.30) containing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq255_HTML.gif , and either

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

(ii) https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq258_HTML.gif is unbounded bifurcated from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq259_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq261_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq262_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq263_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq264_HTML.gif is impossible. So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq265_HTML.gif must be bifurcated from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq266_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq268_HTML.gif yields solutions https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq270_HTML.gif crosses the hyperplane https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq271_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq273_HTML.gif joins https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq274_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq275_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq276_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ78_HTML.gif
(3.1)

We note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq277_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq278_HTML.gif since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq280_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq281_HTML.gif .

Case 1.

consider the following:
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq282_HTML.gif is bounded.

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

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

From (3.4), we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ83_HTML.gif
(3.6)
Thus,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq287_HTML.gif joins https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq288_HTML.gif to https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ85_HTML.gif
(3.8)
which implies that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq291_HTML.gif and set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq292_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq293_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq294_HTML.gif , there exist a subsequence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq295_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq296_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq297_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq298_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq299_HTML.gif , such that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ88_HTML.gif
(3.11)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq300_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq301_HTML.gif denote the nonnegative eigenfunctions corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq302_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ89_HTML.gif
(3.12)
and consequently
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ91_HTML.gif
(3.14)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ92_HTML.gif
(3.15)

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

Case 2.

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

In this case, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq308_HTML.gif is such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ93_HTML.gif
(3.16)
then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ94_HTML.gif
(3.17)
and moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ95_HTML.gif
(3.18)
Assume that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ96_HTML.gif
(3.19)

Again https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq310_HTML.gif joins https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq311_HTML.gif to https://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 https://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:
https://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)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ98_HTML.gif
(3.21)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq314_HTML.gif satisfies the following condition:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ99_HTML.gif
(3.22)
?(A2) For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq315_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq316_HTML.gif be the solution of singular IVPs
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_Equ100_HTML.gif
(3.23)

satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq317_HTML.gif has at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq318_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq319_HTML.gif has no zeros in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F458015/MediaObjects/13661_2010_Article_925_IEq321_HTML.gif and https://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

References

  1. Asakawa H: Nonresonant singular two-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2001,44(6):791-809. 10.1016/S0362-546X(99)00308-9MathSciNetView ArticleMATHGoogle Scholar
  2. Agarwal RP, O'Regan D: Singular Differential and Integral Equations with Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:xii+402.View ArticleMATHGoogle Scholar
  3. O'Regan D: Theory of Singular Boundary Value Problems. World Scientific, River Edge, NJ, USA; 1994:xii+154.View ArticleMATHGoogle Scholar
  4. Habets P, Zanolin F: Upper and lower solutions for a generalized Emden-Fowler equation. Journal of Mathematical Analysis and Applications 1994,181(3):684-700. 10.1006/jmaa.1994.1052MathSciNetView ArticleMATHGoogle Scholar
  5. Xu X, Ma J: A note on singular nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 2004,293(1):108-124. 10.1016/j.jmaa.2003.12.017MathSciNetView ArticleMATHGoogle Scholar
  6. Yang X: Positive solutions for nonlinear singular boundary value problems. Applied Mathematics and Computation 2002,130(2-3):225-234. 10.1016/S0096-3003(01)00046-7MathSciNetView ArticleMATHGoogle Scholar
  7. Ma R: Nodal solutions for singular nonlinear eigenvalue problems. Nonlinear Analysis: Theory, Methods & Applications 2007,66(6):1417-1427. 10.1016/j.na.2006.01.028MathSciNetView ArticleMATHGoogle Scholar
  8. Rabinowitz PH: Some aspects of nonlinear eigenvalue problems. The Rocky Mountain Journal of Mathematics 1973, 3: 161-202. 10.1216/RMJ-1973-3-2-161MathSciNetView ArticleMATHGoogle Scholar
  9. Agarwal RP, O'Regan D: An Introduction to Ordinary Differential Equations, Universitext. Springer, New York, NY, USA; 2008:xii+321.View ArticleMATHGoogle Scholar
  10. Agarwal RP, O'Regan D: Ordinary and Partial Differential Equations, Universitext. Springer, New York, NY, USA; 2009:xiv+410.MATHGoogle Scholar
  11. Ghergu M, Radulescu VD: Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications. Volume 37. Oxford University Press, Oxford, UK; 2008:xvi+298.MATHGoogle Scholar
  12. Kristály A, Radulescu V, Varga C: Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and Its Applications, no. 136. Cambridge University Press, Cambridge, UK; 2010.View ArticleMATHGoogle Scholar
  13. Lomtatidze AG: Positive solutions of boundary value problems for second-order ordinary differential equations with singularities. Differentsial'nye Uravneniya 1987,23(10):1685-1692.MathSciNetGoogle Scholar
  14. Kiguradze IT, Shekhter BL: Singular boundary-value problems for ordinary second-order differential equations. Journal of Soviet Mathematics 1988,43(2):2340-2417. translation from: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh., vol. 30, pp. 105–201, 1987 10.1007/BF01100361View ArticleMATHGoogle Scholar
  15. Coster CD, Habets P: Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering. 2006., 205:MATHGoogle Scholar

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© R. Ma and J. Li. 2010

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