Open Access

Second-Order Boundary Value Problem with Integral Boundary Conditions

Boundary Value Problems20102011:260309

DOI: 10.1155/2011/260309

Received: 28 May 2010

Accepted: 1 October 2010

Published: 13 October 2010

Abstract

The nonlinear alternative of the Leray Schauder type and the Banach contraction principle are used to investigate the existence of solutions for second-order differential equations with integral boundary conditions. The compactness of solutions set is also investigated.

1. Introduction

This paper is concerned with the existence of solutions for the second-order boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq1_HTML.gif is a given function and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq2_HTML.gif is an integrable function.

Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers [19] and the references therein. Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for example [1014]. The goal of this paper is to give existence and uniqueness results for the problem (1.1). Our approach here is based on the Banach contraction principle and the Leray-Schauder alternative [15].

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq3_HTML.gif be the space of differentiable functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq4_HTML.gif whose first derivative, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq5_HTML.gif , is absolutely continuous.

We take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq6_HTML.gif to be the Banach space of all continuous functions from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq7_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq8_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ2_HTML.gif
(2.1)
and we let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq9_HTML.gif denote the Banach space of functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq10_HTML.gif that are Lebesgue integrable with norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ3_HTML.gif
(2.2)

Definition 2.1.

A map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq11_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq12_HTML.gif -Carathéodory if

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq13_HTML.gif is measurable for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq14_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq15_HTML.gif is continuous for almost each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq16_HTML.gif

(iii)for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq17_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq18_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ4_HTML.gif
(2.3)

3. Existence and Uniqueness Results

Definition 3.1.

A function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq19_HTML.gif is said to be a solution of (1.1) if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq20_HTML.gif satisfies (1.1).

In what follows one assumes that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq21_HTML.gif One needs the following auxiliary result.

Lemma 3.2.

. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq22_HTML.gif . Then the function defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ5_HTML.gif
(3.1)
is the unique solution of the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ6_HTML.gif
(3.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ7_HTML.gif
(3.3)

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq23_HTML.gif be a solution of the problem (3.2). Then integratingly, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ8_HTML.gif
(3.4)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ9_HTML.gif
(3.5)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ10_HTML.gif
(3.6)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ11_HTML.gif
(3.7)
Now, multiply (3.6) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq24_HTML.gif and integrate over https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq25_HTML.gif , to get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ12_HTML.gif
(3.8)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ13_HTML.gif
(3.9)
Substituting in (3.6) we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ14_HTML.gif
(3.10)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ15_HTML.gif
(3.11)
Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq26_HTML.gif Note that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ16_HTML.gif
(3.12)

Our first result reads

Theorem 3.3.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq27_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq28_HTML.gif -Carathéodory function and the following hypothesis

(A1) There exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq30_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ17_HTML.gif
(3.13)
holds. If
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ18_HTML.gif
(3.14)

then the BVP (1.1) has a unique solution.

Proof.

Transform problem (1.1) into a fixed-point problem. Consider the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq31_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ19_HTML.gif
(3.15)
We will show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq32_HTML.gif is a contraction. Indeed, consider https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq33_HTML.gif Then we have for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq34_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ20_HTML.gif
(3.16)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ21_HTML.gif
(3.17)

showing that, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq35_HTML.gif is a contraction and hence it has a unique fixed point which is a solution to (1.1). The proof is completed.

We now present an existence result for problem (1.1).

Theorem 3.4.

Suppose that hypotheses

(H1) The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq37_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq38_HTML.gif -Carathéodory,

(H2) There exist functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq40_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq41_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ22_HTML.gif
(3.18)
are satisfied. Then the BVP (1.1) has at least one solution. Moreover the solution set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ23_HTML.gif
(3.19)

is compact.

Proof.

Transform the BVP (1.1) into a fixed-point problem. Consider the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq42_HTML.gif as defined in Theorem 3.3. We will show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq43_HTML.gif satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps.

Step 1 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq44_HTML.gif is continuous).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq45_HTML.gif be a sequence such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq46_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq47_HTML.gif Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ24_HTML.gif
(3.20)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq48_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq49_HTML.gif -Carathéodory and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq50_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ25_HTML.gif
(3.21)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ26_HTML.gif
(3.22)

Step 2 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq51_HTML.gif maps bounded sets into bounded sets in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq52_HTML.gif ).

Indeed, it is enough to show that there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq53_HTML.gif such that for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq54_HTML.gif one has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq55_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq56_HTML.gif . Then for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq57_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ27_HTML.gif
(3.23)
By (H2) we have for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq58_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ28_HTML.gif
(3.24)
Then for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq59_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ29_HTML.gif
(3.25)

Step 3 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq60_HTML.gif maps bounded set into equicontinuous sets of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq61_HTML.gif ).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq64_HTML.gif be a bounded set of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq65_HTML.gif as in Step 2. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq67_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ30_HTML.gif
(3.26)

As https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq68_HTML.gif the right-hand side of the above inequality tends to zero. Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq69_HTML.gif is equicontinuous. As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem we can conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq70_HTML.gif is completely continuous.

Step 4 (A priori bounds on solutions).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq71_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq72_HTML.gif . This implies by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq73_HTML.gif that for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq74_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ31_HTML.gif
(3.27)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ32_HTML.gif
(3.28)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq75_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ33_HTML.gif
(3.29)
Thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ34_HTML.gif
(3.30)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ35_HTML.gif
(3.31)
Set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ36_HTML.gif
(3.32)

and consider the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq76_HTML.gif From the choice of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq77_HTML.gif , there is no https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq78_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq79_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq80_HTML.gif As a consequence of the nonlinear alternative of Leray-Schauder type [15], we deduce that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq81_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq82_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq83_HTML.gif which is a solution of the problem (1.1).

Now, prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq84_HTML.gif is compact. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq85_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq86_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ37_HTML.gif
(3.33)
As in Steps 3 and 4 we can easily prove that there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq87_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ38_HTML.gif
(3.34)
and the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq88_HTML.gif is equicontinuous in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq89_HTML.gif hence by Arzela-Ascoli theorem we can conclude that there exists a subsequence of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq90_HTML.gif converging to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq91_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq92_HTML.gif Using that fast that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq93_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq94_HTML.gif -Carathédory we can prove that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ39_HTML.gif
(3.35)

Thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq95_HTML.gif is compact.

4. Examples

We present some examples to illustrate the applicability of our results.

Example 4.1.

Consider the following BVP
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ40_HTML.gif
(4.1)
Set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ41_HTML.gif
(4.2)
We can easily show that conditions (A1), (3.14) are satisfied with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ42_HTML.gif
(4.3)

Hence, by Theorem 3.3, the BVP (4.1) has a unique solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq96_HTML.gif .

Example 4.2.

Consider the following BVP
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ43_HTML.gif
(4.4)
Set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ44_HTML.gif
(4.5)
We can easily show that conditions (H1), (H2) are satisfied with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ45_HTML.gif
(4.6)

Hence, by Theorem 3.4, the BVP (4.4) has at least one solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq97_HTML.gif . Moreover, its solutions set is compact.

Declarations

Acknowledgment

The authors are grateful to the referees for their remarks.

Authors’ Affiliations

(1)
Department of Mathematics, University of Sidi Bel Abbes
(2)
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela

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© Mouffak Benchohra et al. 2011

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