Second-Order Boundary Value Problem with Integral Boundary Conditions

  • Mouffak Benchohra1Email author,

    Affiliated with

    • JuanJ Nieto2 and

      Affiliated with

      • Abdelghani Ouahab1

        Affiliated with

        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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq1_HTML.gif is a given function and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq3_HTML.gif be the space of differentiable functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq4_HTML.gif whose first derivative, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq5_HTML.gif , is absolutely continuous.

        We take http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq7_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq8_HTML.gif with the norm
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ2_HTML.gif
        (2.1)
        and we let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq9_HTML.gif denote the Banach space of functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq10_HTML.gif that are Lebesgue integrable with norm
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq11_HTML.gif is said to be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq12_HTML.gif -Carathéodory if

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

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

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

        http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq22_HTML.gif . Then the function defined by
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ6_HTML.gif
        (3.2)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ7_HTML.gif
        (3.3)

        Proof.

        Let http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ8_HTML.gif
        (3.4)
        Hence
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ9_HTML.gif
        (3.5)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ10_HTML.gif
        (3.6)
        where
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq24_HTML.gif and integrate over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq25_HTML.gif , to get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ12_HTML.gif
        (3.8)
        Thus,
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ14_HTML.gif
        (3.10)
        Therefore
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ15_HTML.gif
        (3.11)
        Set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq26_HTML.gif Note that
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq27_HTML.gif is an http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq30_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ17_HTML.gif
        (3.13)
        holds. If
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq31_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ19_HTML.gif
        (3.15)
        We will show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq32_HTML.gif is a contraction. Indeed, consider http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq33_HTML.gif Then we have for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq34_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ20_HTML.gif
        (3.16)
        Therefore
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ21_HTML.gif
        (3.17)

        showing that, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq37_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq38_HTML.gif -Carathéodory,

        (H2) There exist functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq40_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq41_HTML.gif such that
        http://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
        http://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 http://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 http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq44_HTML.gif is continuous).

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

        Step 2 ( http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq53_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq54_HTML.gif one has http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq55_HTML.gif .

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

        Step 3 ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq61_HTML.gif ).

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

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

        and consider the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq76_HTML.gif From the choice of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq77_HTML.gif , there is no http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq78_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq79_HTML.gif for some http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq81_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq82_HTML.gif in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq84_HTML.gif is compact. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq85_HTML.gif be a sequence in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq86_HTML.gif , then
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq87_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ38_HTML.gif
        (3.34)
        and the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq88_HTML.gif is equicontinuous in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq90_HTML.gif converging to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq91_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq92_HTML.gif Using that fast that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq93_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_IEq94_HTML.gif -Carathédory we can prove that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ39_HTML.gif
        (3.35)

        Thus http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ40_HTML.gif
        (4.1)
        Set
        http://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
        http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F260309/MediaObjects/13661_2010_Article_31_Equ43_HTML.gif
        (4.4)
        Set
        http://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
        http://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 http://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

        This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.