Existence of Solutions for Fourth-Order Four-Point Boundary Value Problem on Time Scales

  • Dandan Yang1,

    Affiliated with

    • Gang Li1 and

      Affiliated with

      • Chuanzhi Bai2Email author

        Affiliated with

        Boundary Value Problems20092009:491952

        DOI: 10.1155/2009/491952

        Received: 11 April 2009

        Accepted: 28 July 2009

        Published: 19 August 2009

        Abstract

        We present an existence result for fourth-order four-point boundary value problem on time scales. Our analysis is based on a fixed point theorem due to Krasnoselskii and Zabreiko.

        1. Introduction

        Very recently, Karaca [1] investigated the following fourth-order four-point boundary value problem on time scales:

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ1_HTML.gif
        (1.1)

        for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq1_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq2_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq3_HTML.gif And the author made the following assumptions:

        (A1) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq6_HTML.gif

        (A2) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq8_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq9_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq10_HTML.gif

        The following key lemma is provided in [1].

        Lemma 1.1 (see [1, Lemma  2.5]).

        Assume that conditions ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq11_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq12_HTML.gif ) are satisfied. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq13_HTML.gif then the boundary value problem
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ2_HTML.gif
        (1.2)
        has a unique solution
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ3_HTML.gif
        (1.3)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ4_HTML.gif
        (1.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ5_HTML.gif
        (1.5)
        Here http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq14_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq15_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq16_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq17_HTML.gif are given as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ6_HTML.gif
        (1.6)
        Unfortunately, this lemma is wrong. Without considering the whole interval http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq18_HTML.gif the author only considers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq19_HTML.gif in the Green's function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq20_HTML.gif Thus, the expression of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq21_HTML.gif (1.3) which is a solution to BVP (1.2) is incorrect. In fact, if one takes http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq22_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq23_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq24_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq25_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq26_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq27_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq28_HTML.gif then (1.1) reduces to the following boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ7_HTML.gif
        (1.7)
        The counterexample is given by [2], from which one can see clearly that [1, Lemma  2.5] is wrong. If one takes http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq29_HTML.gif , here http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq30_HTML.gif is a constant, then (1.1) reduces to the following fourth-order four-point boundary value problem on time scales:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ8_HTML.gif
        (1.8)

        The purpose of this paper is to establish some existence criteria of solution for BVP (1.8) which is a special case of (1.1). The paper is organized as follows. In Section 2, some basic time-scale definitions are presented and several preliminary results are given. In Section 3, by employing a fixed point theorem due to Krasnoselskii and Zabreiko, we establish existence of solutions criteria for BVP (1.8). Section 4 is devoted to an example illustrating our main result.

        2. Preliminaries

        The study of dynamic equations on time scales goes back to its founder Hilger [3] and it is a new area of still fairly theoretical exploration in mathematics. In the recent years boundary value problem on time scales has received considerable attention [46]. And an increasing interest in studying the existence of solutions to dynamic equations on time scales is observed, for example, see [716].

        For convenience, we first recall some definitions and calculus on time scales, so that the paper is self-contained. For the further details concerning the time scales, please see [1719] which are excellent works for the calculus of time scales.

        A time scale http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq31_HTML.gif is an arbitrary nonempty closed subset of real numbers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq32_HTML.gif . The operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq34_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq35_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq36_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ9_HTML.gif
        (2.1)

        are called the forward jump operator and the backward jump operator, respectively.

        For all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq37_HTML.gif we assume throughout that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq38_HTML.gif has the topology that it inherits from the standard topology on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq39_HTML.gif The notations http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq40_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq41_HTML.gif and so on, will denote time-scale intervals

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ10_HTML.gif
        (2.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq42_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq43_HTML.gif

        Definition 2.1.

        Fix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq44_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq45_HTML.gif Then we define http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq46_HTML.gif to be the number (if it exists) with the property that given http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq47_HTML.gif there is a neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq48_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq49_HTML.gif with
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ11_HTML.gif
        (2.3)

        Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq50_HTML.gif is called derivative of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq51_HTML.gif

        Definition 2.2.

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq52_HTML.gif then we define the integral by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ12_HTML.gif
        (2.4)
        We say that a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq53_HTML.gif is regressive provided
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ13_HTML.gif
        (2.5)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq54_HTML.gif which is called graininess function. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq55_HTML.gif is a regressive function, then the generalized exponential function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq56_HTML.gif is defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ14_HTML.gif
        (2.6)
        for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq57_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq58_HTML.gif is the cylinder transformation, which is defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ15_HTML.gif
        (2.7)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq59_HTML.gif be two regressive functions, then define
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ16_HTML.gif
        (2.8)

        The generalized function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq60_HTML.gif has then the following properties.

        Lemma 2.3 (see [18]).

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq61_HTML.gif are two regressive functions, then

        (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq62_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq63_HTML.gif

        (ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq64_HTML.gif

        (iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq65_HTML.gif

        (iv) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq66_HTML.gif

        (v) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq67_HTML.gif

        (vi) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq68_HTML.gif

        (vii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq69_HTML.gif

        The following well-known fixed point theorem will play a very important role in proving our main result.

        Theorem 2.4 (see [20]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq70_HTML.gif be a Banach space, and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq71_HTML.gif be completely continuous. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq72_HTML.gif is a bounded linear operator such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq73_HTML.gif is not an eigenvalue of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq74_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ17_HTML.gif
        (2.9)

        Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq75_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq76_HTML.gif

        Throughout this paper, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq77_HTML.gif be endowed with the norm by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ18_HTML.gif
        (2.10)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq78_HTML.gif And we make the following assumptions:

        () http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq80_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq81_HTML.gif

        () http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq84_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq85_HTML.gif

        () http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq87_HTML.gif

        Set

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ19_HTML.gif
        (2.11)

        For convenience, we denote

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ20_HTML.gif
        (2.12)

        First, we present two lemmas about the calculus on Green functions which are crucial in our main results.

        Lemma 2.5.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq88_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq89_HTML.gif are satisfied. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq90_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq91_HTML.gif is a solution of the following boundary value problem (BVP):
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ21_HTML.gif
        (2.13)
        if and only if
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ22_HTML.gif
        (2.14)
        where the Green's function of (2.13) is as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ23_HTML.gif
        (2.15)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq92_HTML.gif are given as (2.12), respectively.

        Proof.

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq93_HTML.gif is a solution of (2.13), setting
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ24_HTML.gif
        (2.16)
        then it follows from the first equation of (2.13) that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ25_HTML.gif
        (2.17)
        Multiplying (2.17) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq94_HTML.gif and integrating from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq95_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq96_HTML.gif we get
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ26_HTML.gif
        (2.18)
        Similarly, by (2.18), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ27_HTML.gif
        (2.19)
        Then substituting (2.18) into (2.19), we get for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq97_HTML.gif that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ28_HTML.gif
        (2.20)
        Substituting this expression for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq98_HTML.gif into the boundary conditions of (2.13). By some calculations, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ29_HTML.gif
        (2.21)
        Then substituting (2.21) into (2.20), we get
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ30_HTML.gif
        (2.22)
        By interchanging the order of integration and some rearrangement of (2.22), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ31_HTML.gif
        (2.23)

        Thus, we obtain (2.14) consequently.

        On the other hand, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq99_HTML.gif satisfies (2.14), then direct differentiation of (2.14) yields
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ32_HTML.gif
        (2.24)

        And it is easy to know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq101_HTML.gif satisfies (2.13).

        Corollary 2.6.

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq102_HTML.gif then BVP (2.13) reduces to the following problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ33_HTML.gif
        (2.25)
        From Lemma 2.5, BVP (2.25) has a unique solution
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ34_HTML.gif
        (2.26)
        where the Green's function of (2.25) is as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ35_HTML.gif
        (2.27)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ36_HTML.gif
        (2.28)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ37_HTML.gif
        (2.29)

        Proof.

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq103_HTML.gif is a solution of (2.25), take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq104_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq105_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq106_HTML.gif Hence, from (2.20) we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ38_HTML.gif
        (2.30)
        Substituting this expression for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq107_HTML.gif into the boundary conditions of (2.25). By some calculations, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ39_HTML.gif
        (2.31)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq108_HTML.gif is given as (2.28). Then substituting (2.31) into (2.30), we get
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ40_HTML.gif
        (2.32)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq109_HTML.gif are as in (2.29), respectively. By some rearrangement of (2.32), we obtain (2.26) consequently.

        From the proof of Corollary 2.6, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq110_HTML.gif take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq111_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq112_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq113_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq114_HTML.gif we get the following result.

        Corollary 2.7.

        The following boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ41_HTML.gif
        (2.33)
        has a unique solution
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ42_HTML.gif
        (2.34)
        where the Green's function of (2.33) is as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ43_HTML.gif
        (2.35)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ44_HTML.gif
        (2.36)
        After some rearrangement of (2.35), one obtains
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ45_HTML.gif
        (2.37)

        Remark 2.8.

        Green function (2.37) associated with BVP (2.33) which is a special case of (2.13) is coincident with part of [21, Lemma  1].

        Lemma 2.9.

        Assume that conditions ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq115_HTML.gif )–( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq116_HTML.gif ) are satisfied. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq117_HTML.gif then boundary value problem
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ46_HTML.gif
        (2.38)
        has a unique solution
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ47_HTML.gif
        (2.39)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ48_HTML.gif
        (2.40)

        and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq118_HTML.gif is given in Lemma 2.5.

        Proof.

        Consider the following boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ49_HTML.gif
        (2.41)

        The Green's function associated with the BVP (2.41) is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq119_HTML.gif . This completes the proof.

        Remark 2.10.

        In [1, Lemma  2.5], the solution of (1.2) is defined as
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ50_HTML.gif
        (2.42)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq120_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq121_HTML.gif are given as (1.4) and (1.5), respectively. In fact, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq122_HTML.gif is incorrect. Thus, we give the right form of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq123_HTML.gif as the special case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq124_HTML.gif in our Lemma 2.9.

        3. Main Results

        Theorem 3.1.

        Assume ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq125_HTML.gif )–( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq126_HTML.gif ) are satisfied. Moreover, suppose that the following condition is satisfied:

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq127_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq128_HTML.gif are continuous, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq129_HTML.gif with
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ51_HTML.gif
        (3.1)
        and there exists a continuous nonnegative function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq130_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq131_HTML.gif If
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ52_HTML.gif
        (3.2)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ53_HTML.gif
        (3.3)

        then BVP (1.8) has a solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq132_HTML.gif .

        Proof.

        Define an operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq133_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ54_HTML.gif
        (3.4)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq134_HTML.gif is given by (2.40). Then by Lemmas 2.5 and 2.9, it is clear that the fixed points of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq135_HTML.gif are the solutions to the boundary value problem (1.8). First of all, we claim that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq136_HTML.gif is a completely continuous operator, which is divided into 3 steps.

        Step 1 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq137_HTML.gif is continuous).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq138_HTML.gif be a sequence such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq139_HTML.gif then we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ55_HTML.gif
        (3.5)

        Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq140_HTML.gif are continuous, we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq141_HTML.gif which yields http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq142_HTML.gif That is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq143_HTML.gif is continuous.

        Step 2 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq144_HTML.gif maps bounded sets into bounded sets in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq145_HTML.gif ).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq146_HTML.gif be a bounded set. Then, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq147_HTML.gif and any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq148_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ56_HTML.gif
        (3.6)

        By virtue of the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq149_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq150_HTML.gif , we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq151_HTML.gif is bounded uniformly, and so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq152_HTML.gif is a bounded set.

        Step 3 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq153_HTML.gif maps bounded sets into equicontinuous sets of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq154_HTML.gif ).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq155_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq156_HTML.gif then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ57_HTML.gif
        (3.7)

        The right hand side tends to uniformly zero as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq157_HTML.gif Consequently, Steps 1–3 together with the Arzela-Ascoli theorem show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq158_HTML.gif is completely continuous.

        Now we consider the following boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ58_HTML.gif
        (3.8)
        Define
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ59_HTML.gif
        (3.9)

        Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq159_HTML.gif is a completely continuous bounded linear operator. Moreover, the fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq160_HTML.gif is a solution of the BVP (3.8) and conversely.

        We are now in the position to claim that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq161_HTML.gif is not an eigenvalue of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq162_HTML.gif

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq163_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq164_HTML.gif then (3.8) has no nontrivial solution.

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq165_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq166_HTML.gif suppose that the BVP (3.8) has a nontrivial solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq167_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq168_HTML.gif then we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ60_HTML.gif
        (3.10)
        which yields
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ61_HTML.gif
        (3.11)
        On the other hand, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ62_HTML.gif
        (3.12)

        From the above discussion (3.11) and (3.12), we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq169_HTML.gif . This contradiction implies that boundary value problem (3.8) has no trivial solution. Hence, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq170_HTML.gif is not an eigenvalue of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq171_HTML.gif

        At last, we show that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ63_HTML.gif
        (3.13)
        By http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq172_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq173_HTML.gif then for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq174_HTML.gif there exist a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq175_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ64_HTML.gif
        (3.14)

        Set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq176_HTML.gif and select http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq177_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq178_HTML.gif

        Denote
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ65_HTML.gif
        (3.15)
        Thus for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq179_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq180_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq181_HTML.gif it follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ66_HTML.gif
        (3.16)
        In a similar way, we also conclude that for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq182_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ67_HTML.gif
        (3.17)
        Therefore,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ68_HTML.gif
        (3.18)
        On the other hand, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ69_HTML.gif
        (3.19)
        Combining (3.18) with (3.19), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ70_HTML.gif
        (3.20)

        Theorem 2.4 guarantees that boundary value problem (1.8) has a solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq183_HTML.gif It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq184_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq185_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq186_HTML.gif In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq187_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq188_HTML.gif will lead to a contradiction, which completes the proof.

        4. Application

        We give an example to illustrate our result.

        Example 4.1.

        Consider the fourth-order four-pint boundary value problem
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ71_HTML.gif
        (4.1)
        Notice that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq189_HTML.gif To show that (4.1) has at least one nontrivial solution we apply Theorem 3.1 with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq190_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq191_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq192_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq193_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq194_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq195_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq196_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq197_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq198_HTML.gif Obviously, ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq199_HTML.gif )–( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq200_HTML.gif ) are satisfied. And
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ72_HTML.gif
        (4.2)

        Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq201_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq202_HTML.gif we have the following.

        By simple calculation we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ73_HTML.gif
        (4.3)
        On the other hand, we notice that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ74_HTML.gif
        (4.4)
        Hence,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ75_HTML.gif
        (4.5)

        That is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq203_HTML.gif is satisfied. Thus, Theorem 3.1 guarantees that (4.1) has at least one nontrivial solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq204_HTML.gif

        Declarations

        Acknowledgments

        The authors would like to thank the referees for their valuable suggestions and comments. This work is supported by the National Natural Science Foundation of China (10771212) and the Natural Science Foundation of Jiangsu Education Office (06KJB110010).

        Authors’ Affiliations

        (1)
        School of Mathematical Science, Yangzhou University
        (2)
        Department of Mathematics, Huaiyin Normal University

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        © Dandan Yang et al. 2009

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