Existence of Periodic Solution for a Nonlinear Fractional Differential Equation

  • Mohammed Belmekki1,

    Affiliated with

    • JuanJ Nieto2 and

      Affiliated with

      • Rosana Rodríguez-López2Email author

        Affiliated with

        Boundary Value Problems20092009:324561

        DOI: 10.1155/2009/324561

        Received: 2 February 2009

        Accepted: 4 June 2009

        Published: 14 July 2009

        Abstract

        We study the existence of solutions for a class of fractional differential equations. Due to the singularity of the possible solutions, we introduce a new and proper concept of periodic boundary value conditions. We present Green's function and give some existence results for the linear case and then we study the nonlinear problem.

        1. Introduction

        Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. The subject is as old as the differential calculus, and goes back to time when Leibnitz and Newton invented differential calculus. The idea of fractional calculus has been a subject of interest not only among mathematicians but also among physicists and engineers. See, for instance, [16].

        Fractional-order models are more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a "memory" term in a model. This memory term insures the history and its impact to the present and future. For more details, see [7].

        Fractional calculus appears in rheology, viscoelasticity, electrochemistry, electromagnetism, and so forth. For details, see the monographs of Kilbas et al. [8], Kiryakova [9], Miller and Ross [10], Podlubny [11], Oldham and Spanier [12], and Samko et al. [13], and the papers of Diethelm et al. [1416], Mainardi [17], Metzler et al. [18], Podlubny et al. [19], and the references therein. For some recent advances on fractional calculus and differential equations, see [1, 3, 2024].

        In this paper we consider the following nonlinear fractional differential equation of the form
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq1_HTML.gif is the standard Riemann-Liouville fractional derivative, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq2_HTML.gif is continuous, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq3_HTML.gif .

        This paper is organized as follows. in Section 2 we recall some definitions of fractional integral and derivative and related basic properties which will be used in the sequel. In Section 3, we deal with the linear case where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq4_HTML.gif is a continuous function. Section 4 is devoted to the nonlinear case.

        2. Preliminary Results

        In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq5_HTML.gif the Banach space of all continuous real functions defined on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq6_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq7_HTML.gif Define for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq9_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq11_HTML.gif be the space of all functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq12_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq13_HTML.gif which turn out to be a Banach space when endowed with the norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq14_HTML.gif

        By http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq15_HTML.gif we denote the space of all real functions defined on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq16_HTML.gif which are Lebesgue integrable.

        Obviously http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq17_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq18_HTML.gif

        Definition 2.1 (see [11, 13]).

        The Riemann-Liouville fractional primitive of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq19_HTML.gif of a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq20_HTML.gif is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ2_HTML.gif
        (2.1)

        provided the right side is pointwise defined on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq21_HTML.gif , and where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq22_HTML.gif is the gamma function.

        For instance, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq23_HTML.gif exists for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq24_HTML.gif , when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq25_HTML.gif ; note also that when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq26_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq27_HTML.gif and moreover http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq28_HTML.gif

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq29_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq30_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq31_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq32_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq33_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq34_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq35_HTML.gif is bounded at the origin, whereas if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq36_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq37_HTML.gif , then we may expect http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq38_HTML.gif to be unbounded at the origin.

        Recall that the law of composition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq39_HTML.gif holds for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq40_HTML.gif

        Definition 2.2 (see [11, 13]).

        The Riemann-Liouville fractional derivative of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq41_HTML.gif of a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq42_HTML.gif is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ3_HTML.gif
        (2.2)

        We have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq43_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq44_HTML.gif .

        Lemma 2.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq45_HTML.gif . If one assumes http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq46_HTML.gif , then the fractional differential equation
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ4_HTML.gif
        (2.3)

        has http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq47_HTML.gif , as unique solutions.

        From this lemma we deduce the following law of composition.

        Proposition 2.4.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq48_HTML.gif is in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq49_HTML.gif with a fractional derivative of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq50_HTML.gif that belongs to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq51_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ5_HTML.gif
        (2.4)

        for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq52_HTML.gif .

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq53_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq55_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq56_HTML.gif .

        3. Linear Problem

        In this section, we will be concerned with the following linear fractional differential equation:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ6_HTML.gif
        (3.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq57_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq58_HTML.gif is a continuous function.

        Before stating our main results for this section, we study the equation
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ7_HTML.gif
        (3.2)
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ8_HTML.gif
        (3.3)

        for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq59_HTML.gif .

        Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq61_HTML.gif . However, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq62_HTML.gif since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq63_HTML.gif has a singularity at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq64_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq65_HTML.gif

        It is easy to show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq66_HTML.gif . Hence we should look for solutions, not in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq67_HTML.gif but in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq68_HTML.gif . We cannot consider the usual initial condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq69_HTML.gif , but http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq70_HTML.gif Hence, to study the periodic boundary value problem, one has to consider the following boundary condition of periodic type
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ9_HTML.gif
        (3.4)
        From (3.3), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ10_HTML.gif
        (3.5)

        that leads to the following.

        Theorem 3.1.

        The periodic boundary value problem (3.2)–(3.4) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq71_HTML.gif if and only if
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ11_HTML.gif
        (3.6)
        The previous result remains true even if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq72_HTML.gif . In this case, (3.2) is reduced to the ordinary differential equation
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ12_HTML.gif
        (3.7)
        with the periodic boundary condition
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ13_HTML.gif
        (3.8)
        and the condition (3.6) is reduced to the classical one:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ14_HTML.gif
        (3.9)
        Now, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq73_HTML.gif different from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq74_HTML.gif , consider the homogenous linear equation
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ15_HTML.gif
        (3.10)
        The solution is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ16_HTML.gif
        (3.11)
        Indeed, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ17_HTML.gif
        (3.12)

        since the series representing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq75_HTML.gif is absolutely convergent.

        Using the identities
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ18_HTML.gif
        (3.13)
        we get
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ19_HTML.gif
        (3.14)
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ20_HTML.gif
        (3.15)
        Note that the solution can be expressed by means of the classical Mittag-Leffler special functions [8]. Indeed
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ21_HTML.gif
        (3.16)
        The previous formula remains valid for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq76_HTML.gif . In this case,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ22_HTML.gif
        (3.17)
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ23_HTML.gif
        (3.18)
        which is the classical solution to the homogeneous linear differential equation
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ24_HTML.gif
        (3.19)
        Now, consider the nonhomogeneous problem (3.1). We seek the particular solution in the following form:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ25_HTML.gif
        (3.20)
        It suffices to show that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ26_HTML.gif
        (3.21)
        Indeed
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ27_HTML.gif
        (3.22)

        Using the change of variable

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ28_HTML.gif
        (3.23)

        we get

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ29_HTML.gif
        (3.24)
        Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ30_HTML.gif
        (3.25)
        Hence, the general solution of the nonhomogeneous equation (3.1) takes the form
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ31_HTML.gif
        (3.26)
        Now, consider the periodic boundary value problem (3.1)–(3.4). Its unique solution is given by (3.26) for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq77_HTML.gif . Also http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq78_HTML.gif is in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq79_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ32_HTML.gif
        (3.27)
        From (3.26), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ33_HTML.gif
        (3.28)
        which leads to
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ34_HTML.gif
        (3.29)
        since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq80_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq81_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ35_HTML.gif
        (3.30)
        Then the solution of the problem (3.1)–(3.4) is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ36_HTML.gif
        (3.31)

        Thus we have the following result.

        Theorem 3.2.

        The periodic boundary value problem (3.1)–(3.4) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq82_HTML.gif given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ37_HTML.gif
        (3.32)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ38_HTML.gif
        (3.33)

        For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq83_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq84_HTML.gif given, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq85_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq86_HTML.gif .

        For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq87_HTML.gif , (3.1) is
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ39_HTML.gif
        (3.34)
        and the boundary condition (3.4) is
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ40_HTML.gif
        (3.35)
        In this situation Green's function is
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ41_HTML.gif
        (3.36)

        which is precisely Green's function for the periodic boundary value problem considered in [25, 26].

        4. Nonlinear Problem

        In this section we will be concerned with the existence and uniqueness of solution to the nonlinear problem (1.1)–(3.4). To this end, we need the following fixed point theorem of Schaeffer.

        Theorem 4.1.

        Assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq88_HTML.gif to be a normed linear space, and let operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq89_HTML.gif be compact. Then either

        (i)the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq90_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq91_HTML.gif , or

        (ii)the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq92_HTML.gif is unbounded.

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq93_HTML.gif is a solution of problem (1.1)–(3.4), then it is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ42_HTML.gif
        (4.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq94_HTML.gif is Green's function defined in Theorem 3.2.

        Define the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq95_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ43_HTML.gif
        (4.2)

        Then the problem (1.1)–(3.4) has solutions if and only if the operator equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq96_HTML.gif has fixed points.

        Lemma 4.2.

        Suppose that the following hold:

        (i)there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq97_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ44_HTML.gif
        (4.3)
        (ii)there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq98_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ45_HTML.gif
        (4.4)

        Then the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq99_HTML.gif is well defined, continuous, and compact.

        Proof.
        1. (a)
          We check, using hypothesis (4.3), that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq100_HTML.gif , for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq101_HTML.gif . Indeed, for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq102_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq103_HTML.gif , we have
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ46_HTML.gif
          (4.5)
           
        From the previous expression, we deduce that, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq104_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ47_HTML.gif
        (4.6)
        Indeed, note that the integral http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq105_HTML.gif is bounded by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ48_HTML.gif
        (4.7)
        A similar argument is useful to study the behavior of the last three terms of the long inequality above. On the other hand, if we denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq106_HTML.gif the second term in the right-hand side of that inequality, then it is satisfied that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ49_HTML.gif
        (4.8)
        Note that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ50_HTML.gif
        (4.9)
        and, concerning http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq107_HTML.gif , we distinguish two cases. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq108_HTML.gif is such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq109_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ51_HTML.gif
        (4.10)
        and, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq110_HTML.gif is such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq111_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ52_HTML.gif
        (4.11)
        In consequence,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ53_HTML.gif
        (4.12)
        The first term in the right-hand side of the previous inequality clearly tends to zero as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq112_HTML.gif . On the other hand, denoting by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq113_HTML.gif the integer part function, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ54_HTML.gif
        (4.13)
        The finite sum obviously has limit zero as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq114_HTML.gif . The infinite sum is equal to
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ55_HTML.gif
        (4.14)

        and its limit as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq115_HTML.gif is zero. Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq116_HTML.gif is bounded above by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq117_HTML.gif .

        The previous calculus shows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq118_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq119_HTML.gif , hence we can define http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq120_HTML.gif .
        1. (b)

          Next, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq121_HTML.gif is continuous.

           
        Note that, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq122_HTML.gif and for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq123_HTML.gif , we have, using hypothesis (4.4),
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ56_HTML.gif
        (4.15)
        Using the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq124_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ57_HTML.gif
        (4.16)
        Moreover,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ58_HTML.gif
        (4.17)
        Using that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq125_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq127_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq128_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq129_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq130_HTML.gif we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ59_HTML.gif
        (4.18)
        Note that the Beta function, also called the Euler integral of the first kind,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ60_HTML.gif
        (4.19)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq131_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq132_HTML.gif , satisfies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq133_HTML.gif . In particular, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq134_HTML.gif . On the other hand, using the change of variable http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq135_HTML.gif , we deduce that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ61_HTML.gif
        (4.20)
        This proves that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ62_HTML.gif
        (4.21)
        Hence,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ63_HTML.gif
        (4.22)
        In consequence,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ64_HTML.gif
        (4.23)
        Finally, we check that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq136_HTML.gif is compact. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq137_HTML.gif be a bounded set in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq138_HTML.gif .
        1. (i)

          First, we check that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq139_HTML.gif is a bounded set in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq140_HTML.gif .

           
        Indeed,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ65_HTML.gif
        (4.24)
        Hence
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ66_HTML.gif
        (4.25)
        and then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ67_HTML.gif
        (4.26)
        which implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq141_HTML.gif is a bounded set in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq142_HTML.gif .
        1. (ii)

          Now, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq143_HTML.gif is an equicontinuous set in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq144_HTML.gif . Following the calculus in (a), we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq145_HTML.gif tends to zero as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq146_HTML.gif .

           

        Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq147_HTML.gif is equicontinuous in the space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq148_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq149_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq150_HTML.gif .

        As a consequence of (i) and (ii), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq151_HTML.gif is a bounded and equicontinuous set in the space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq152_HTML.gif .

        Hence, for a sequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq153_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq155_HTML.gif has a subsequence converging to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq156_HTML.gif , that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ68_HTML.gif
        (4.27)
        Taking http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq157_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ69_HTML.gif
        (4.28)

        which means that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq158_HTML.gif , which proves that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq159_HTML.gif is compact.

        Theorem 4.3.

        Assume that (4.3) and (4.4) hold. Then the problem (1.1)–(3.4) has at least one solution in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq160_HTML.gif

        Proof.

        Consider the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq161_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq162_HTML.gif be any element of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq163_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq164_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq165_HTML.gif . Thus for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq166_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ70_HTML.gif
        (4.29)
        As in Lemma 4.2, (i), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ71_HTML.gif
        (4.30)

        which implies that the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq167_HTML.gif is bounded independently of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq168_HTML.gif . Using Lemma 4.2 and Theorem 4.1, we obtain that the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq169_HTML.gif has at least a fixed point.

        Remark 4.4.

        In Lemma 4.2, condition (4.3) is used to prove that the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq170_HTML.gif is continuous. Hence, in Lemma 4.2 and, in consequence, in Theorem 4.3, we can assume the weaker condition.

        (i)For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq171_HTML.gif fixed, there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq172_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ72_HTML.gif
        (4.31)

        instead of (4.3).

        However, to prove the existence and uniqueness of solution given in the following theorem, we need to assume the Lipschitzian character of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq173_HTML.gif (condition (4.3).

        Theorem 4.5.

        Assume that (4.4) holds. Then the problem (1.1)–(3.4) has a unique solution in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq174_HTML.gif provided that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ73_HTML.gif
        (4.32)

        Proof.

        We use the Banach contraction principle to prove that the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq175_HTML.gif has a unique fixed point.

        Using the calculus in (b) Lemma 4.2, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq176_HTML.gif is a contraction by condition (4.32). As a consequence of Banach fixed point theorem, we deduce that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq177_HTML.gif has a unique fixed point which gives rise to a unique solution of problem (1.1)–(3.4).

        Remark 4.6.

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq178_HTML.gif , condition (4.32) is reduced to
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ74_HTML.gif
        (4.33)

        Declarations

        Acknowledgment

        The research of J. J. Nieto and R. Rodríguez-López has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PR.

        Authors’ Affiliations

        (1)
        Département de Mathématiques, Université de Saïda
        (2)
        Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela

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        © Mohammed Belmekki et al. 2009

        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.