Open Access

Positive Solutions of Singular Initial-Boundary Value Problems to Second-Order Functional Differential Equations

Boundary Value Problems20082008:457028

DOI: 10.1155/2008/457028

Received: 23 August 2007

Accepted: 5 August 2008

Published: 26 August 2008

Abstract

Positive solutions to the singular initial-boundary value problems https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq1_HTML.gif are obtained by applying the Schauder fixed-point theorem, where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq2_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq4_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq6_HTML.gif . As an application, an example is given to demonstrate our result.

1. Introduction

Recently, in [14], Erbe, Kong, Jiang, Wang, and Weng considered the following singular functional differential equations:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq7_HTML.gif and the existence of positive solutions to (1.1) is obtained. When https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq8_HTML.gif in (1.1), Agarwal and O'Regan in [5], Lin and Xu in [6] discussed the existence of positive solutions to (1.1) also. We notice that the nonlinearities https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq9_HTML.gif in all the above-mentioned references depend on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq10_HTML.gif .

The more difficult case is that the term https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq11_HTML.gif depends on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq12_HTML.gif for second-order functional differential equations with delay. When https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq13_HTML.gif has no singularity at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq14_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq15_HTML.gif , there are many results on the following (1.2) (see [79] and references therein). Up to now, to our knowledge, there are fewer results on (1.2) when the term https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq16_HTML.gif is allowed to possess singularity for the term https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq17_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq19_HTML.gif , which is of more actual significance.

In this paper, motivated by above results, we consider the second-order initial-boundary value problems:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq20_HTML.gif . By Leray-Schauder fixed-point theorem, the existence of positive solutions to (1.2) is obtained when https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq21_HTML.gif is singular at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq23_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq25_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq26_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq27_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq28_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq29_HTML.gif are Banach spaces. Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq31_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq32_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq33_HTML.gif are cones in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq35_HTML.gif respectively. Now, we give a new definition.

Deffinition.

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq36_HTML.gif is said to be singular at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq37_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq38_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq39_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq40_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq42_HTML.gif is said to be singular at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq43_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq44_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq45_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq46_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq47_HTML.gif .

And one defines some functions which one has to use in this paper.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq48_HTML.gif is a Green's function. It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq49_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq51_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq52_HTML.gif

We now introduce the definition of a solution to IBVP(1.2).

Deffinition.

A function https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq53_HTML.gif is said to be a solution to IBVP(1.2) if it satisfies the following conditions:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq54_HTML.gif is continuous and nonnegative on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq55_HTML.gif ;

(2) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq56_HTML.gif ;

(3) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq58_HTML.gif exist on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq59_HTML.gif ;

(4) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq60_HTML.gif is Lebesgue integrable on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq61_HTML.gif ;

(5) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq62_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq63_HTML.gif .

Furthermore, a solution https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq64_HTML.gif is said to be positive if https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq65_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq66_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq67_HTML.gif be a solution to IBVP(1.2). Then, it can be represented as
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ4_HTML.gif
(1.4)
It is clear that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ5_HTML.gif
(1.5)

for all solutions, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq68_HTML.gif , to IBVP(1.2), where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq69_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq70_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq71_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq72_HTML.gif throughout this paper. Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq74_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq75_HTML.gif .

Throughout this paper, we assume the following hypotheses hold.

(H1) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq76_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq77_HTML.gif .

(H2) There exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq78_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ6_HTML.gif
(1.6)

Lemma 1.3.

Assume that (H 1 )-(H 2 ) hold, then there exists a , such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ7_HTML.gif
(1.7)

for all solutions, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq80_HTML.gif , to (1.2).

Proof.

Suppose that the claim is false. (1.5) guarantees that there exists a sequence https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq81_HTML.gif of solutions to IBVP(1.2) such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ8_HTML.gif
(1.8)
Without loss of generality, we may assume that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ9_HTML.gif
(1.9)
From https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq82_HTML.gif and (1.5), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ10_HTML.gif
(1.10)

which contradicts the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq83_HTML.gif and hence the claim is true provided https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq84_HTML.gif is suitably small.

Remark 1.4.

The following inequality
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ11_HTML.gif
(1.11)

holds provided that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq85_HTML.gif is sufficiently small, where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq86_HTML.gif is in Lemma 1.3.

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq87_HTML.gif There exist a nonnegative continuous function https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq88_HTML.gif defined on (0,1) and two nonnegative continuous functions https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq89_HTML.gif defined on, respectively, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq90_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ12_HTML.gif
(1.12)
where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq91_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq92_HTML.gif ) satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ13_HTML.gif
(1.13)
Furthermore, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq93_HTML.gif is nonincreasing and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq94_HTML.gif is nondecreasing, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ14_HTML.gif
(1.14)

Lemma 1.5 (see [7]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq95_HTML.gif be the Banach space and let X be any nonempty, convex, closed, and bounded subset of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq96_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq97_HTML.gif is a continuous mapping of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq98_HTML.gif into itself and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq99_HTML.gif is relatively compact, then the mapping https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq100_HTML.gif has at least one fixed point (i.e., there exists an https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq101_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq102_HTML.gif ) .

Using Lemma 1.5, we present the existence of at least one positive solution to (1.2) when https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq103_HTML.gif is singular at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq105_HTML.gif (notice the new Definition 1.1). To some extent, our paper complements and generalizes these in [16, 810].

2. Main Results

Theorem 2.1.

Assume that (H 1 )–(H 3 ) hold. Then, the IBVP( 1.2 ) has at least one positive solution.

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq106_HTML.gif , we can choose an https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq107_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ15_HTML.gif
(2.1)
where the positive number https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq108_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ16_HTML.gif
(2.2)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ17_HTML.gif
(2.3)
For each https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq109_HTML.gif , we define https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq110_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ18_HTML.gif
(2.4)

It is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq111_HTML.gif satisfies the hypotheses https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq112_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq113_HTML.gif .

We now consider the modified initial-boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ19_HTML.gif
(2.5)
We claim that for all solutions, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq114_HTML.gif , to IBVP(2.5),
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ20_HTML.gif
(2.6)
Suppose that the claim is false. Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq115_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ21_HTML.gif
(2.7)

Since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq116_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq117_HTML.gif , there are the following three cases.

Case 1.

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq118_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq119_HTML.gif .

The solution of IBVP(2.5) can be represented as (notice https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq120_HTML.gif Remark 1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ22_HTML.gif
(2.8)

which contradicts (2.7).

Case 2.

There exists a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq121_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq122_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq123_HTML.gif .

In this case, we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ23_HTML.gif
(2.9)

which contradicts (2.7).

Case 3.

There exists a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq124_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq126_HTML.gif .

From (1.5), we get
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ24_HTML.gif
(2.10)

which contradicts (2.7).

So we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ25_HTML.gif
(2.11)

To prove the existence of positive solutions to IBVP(2.5), we seek to transform (2.5) into an integral equation via the use of Green's function and then find a positive solution by using Lemma 1.5.

Define a nonempty convex and closed subset of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq127_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ26_HTML.gif
(2.12)
Then, we define an operator https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq128_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ27_HTML.gif
(2.13)
From https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq129_HTML.gif and the definition of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq130_HTML.gif , we have, for every https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq131_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ28_HTML.gif
(2.14)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ29_HTML.gif
(2.15)

Together with the definition of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq132_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq133_HTML.gif .

Also,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ30_HTML.gif
(2.16)
is continuous in (0,1), and
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ31_HTML.gif
(2.17)
From https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq134_HTML.gif and (2.15), we can get
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ32_HTML.gif
(2.18)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq135_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq136_HTML.gif .

Now, we claim that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq137_HTML.gif is equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq138_HTML.gif . We will prove the claim. For any https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq139_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ33_HTML.gif
(2.19)
Since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq140_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq142_HTML.gif , then for any https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq143_HTML.gif , there is a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq144_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ34_HTML.gif
(2.20)
By (2.6), we have, for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq145_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ35_HTML.gif
(2.21)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq146_HTML.gif is a constant number.

Put https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq147_HTML.gif , then for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq148_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ36_HTML.gif
(2.22)
Set https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq149_HTML.gif . Then for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq150_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ37_HTML.gif
(2.23)

Since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq151_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq152_HTML.gif , the above inequality holds for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq153_HTML.gif .

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq154_HTML.gif is a relative compact subset of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq155_HTML.gif . That is, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq156_HTML.gif is a compact operator.

We are now going to prove that the mapping https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq157_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq158_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq159_HTML.gif be arbitrarily chosen and let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq160_HTML.gif converge to https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq161_HTML.gif uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq162_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq163_HTML.gif . Now, we claim that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq164_HTML.gif converge to https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq165_HTML.gif uniformly as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq166_HTML.gif . From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq167_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ38_HTML.gif
(2.24)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ39_HTML.gif
(2.25)

that is, the claim is true.

Since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq168_HTML.gif is continuous with respect to https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq169_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq170_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ40_HTML.gif
(2.26)
for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq171_HTML.gif . From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq173_HTML.gif , we know that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ41_HTML.gif
(2.27)
and hence
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ42_HTML.gif
(2.28)
where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq174_HTML.gif is a Lebesgue integrable function defined on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq175_HTML.gif because of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq176_HTML.gif . Consequently, we apply the dominated convergence theorem to get
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ43_HTML.gif
(2.29)

which shows that the mapping https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq177_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq178_HTML.gif .

Then from Lemma 1.5, we get that there exists at least one positive solution, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq179_HTML.gif , to IBVP(2.5) in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq180_HTML.gif . The solution can be represented by (1.4), where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq181_HTML.gif is replaced with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq182_HTML.gif . So, (2.6) holds. Furthermore, from the definition of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq183_HTML.gif , we can get
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ44_HTML.gif
(2.30)

Thus, the solution of IBVP(2.5) is also the one of (1.2). The proof is complete.

3. Application

Example 3.1.

Consider the singular IBVP(3.1):
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ45_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq184_HTML.gif .

4. Conclusion

Equation (3.1) has at least one positive solution.

Now, we will check that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq185_HTML.gif hold in (3.1).

In IBVP(3.1), https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq186_HTML.gif . It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq187_HTML.gif is continuous and singular at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq188_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq189_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq190_HTML.gif we choose
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ46_HTML.gif
(4.1)
when https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq191_HTML.gif ; by simple computation, we can get
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ47_HTML.gif
(4.2)

It is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq192_HTML.gif is nonincreasing and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq193_HTML.gif is nondecreasing.

Now, we check https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq194_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq195_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq196_HTML.gif (notice the definition of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq197_HTML.gif ), we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ48_HTML.gif
(4.3)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ49_HTML.gif
(4.4)
We define
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ50_HTML.gif
(4.5)

Now, we will prove that there exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq198_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq199_HTML.gif is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq200_HTML.gif .

Obviously,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ51_HTML.gif
(4.6)
Put https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq201_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ52_HTML.gif
(4.7)

From the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq202_HTML.gif , we can find https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq203_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq204_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq205_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq206_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq207_HTML.gif . That is, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq208_HTML.gif is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq209_HTML.gif .

Furthermore, we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ53_HTML.gif
(4.8)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ54_HTML.gif
(4.9)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq210_HTML.gif holds.

So, from Theorem 2.1, IBVP(3.1) has at least one positive solution.

Declarations

Acknowledgments

The research was supported by NNSF of China (10571111) and the fund of Shandong Education Committee (J07WH08).

Authors’ Affiliations

(1)
School of Mathematics Sciences, Shandong Normal University

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© F. Jin and B. Yan. 2008

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