Existence of Solutions for Fractional Differential Inclusions with Antiperiodic Boundary Conditions

  • Bashir Ahmad1 and

    Affiliated with

    • Victoria Otero-Espinar2Email author

      Affiliated with

      Boundary Value Problems20092009:625347

      DOI: 10.1155/2009/625347

      Received: 21 January 2009

      Accepted: 18 March 2009

      Published: 4 May 2009

      Abstract

      We study the existence of solutions for a class of fractional differential inclusions with anti-periodic boundary conditions. The main result of the paper is based on Bohnenblust- Karlins fixed point theorem. Some applications of the main result are also discussed.

      1. Introduction

      In some cases and real world problems, fractional-order models are found to be more adequate than integer-order models as fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electro dynamics of complex medium, polymer rheology, and so forth, involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For details and examples, see [114] and the references therein.

      Antiperiodic boundary value problems have recently received considerable attention as antiperiodic boundary conditions appear in numerous situations, for instance, see [1522].

      Differential inclusions arise in the mathematical modelling of certain problems in economics, optimal control, and so forth. and are widely studied by many authors, see [2327] and the references therein. For some recent development on differential inclusions, we refer the reader to the references [2832].

      Chang and Nieto [33] discussed the existence of solutions for the fractional boundary value problem:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ1_HTML.gif
      (1.1)

      In this paper, we consider the following fractional differential inclusions with antiperiodic boundary conditions

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ2_HTML.gif
      (1.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq1_HTML.gif denotes the Caputo fractional derivative of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq3_HTML.gif Bohnenblust-Karlin fixed point theorem is applied to prove the existence of solutions of (1.2).

      2. Preliminaries

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq4_HTML.gif denote a Banach space of continuous functions from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq5_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq6_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq7_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq8_HTML.gif be the Banach space of functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq9_HTML.gif which are Lebesgue integrable and normed by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq10_HTML.gif

      Now we recall some basic definitions on multivalued maps [34, 35].

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq11_HTML.gif be a Banach space. Then a multivalued map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq12_HTML.gif is convex (closed) valued if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq13_HTML.gif is convex (closed) for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq14_HTML.gif The map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq15_HTML.gif is bounded on bounded sets if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq16_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq17_HTML.gif for any bounded set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq18_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq19_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq20_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq21_HTML.gif is called upper semicontinuous (u.s.c.) on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq22_HTML.gif if for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq23_HTML.gif the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq24_HTML.gif is a nonempty closed subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq25_HTML.gif , and if for each open set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq26_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq27_HTML.gif containing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq28_HTML.gif there exists an open neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq29_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq30_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq31_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq32_HTML.gif is said to be completely continuous if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq33_HTML.gif is relatively compact for every bounded subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq34_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq35_HTML.gif If the multivalued map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq36_HTML.gif is completely continuous with nonempty compact values, then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq37_HTML.gif is u.s.c. if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq38_HTML.gif has a closed graph, that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq39_HTML.gif imply http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq40_HTML.gif In the following study, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq41_HTML.gif denotes the set of all nonempty bounded, closed, and convex subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq42_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq43_HTML.gif has a fixed point if there is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq44_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq45_HTML.gif

      Let us record some definitions on fractional calculus [8, 11, 13].

      Definition 2.1.

      For a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq46_HTML.gif the Caputo derivative of fractional order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq47_HTML.gif is defined as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ3_HTML.gif
      (2.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq48_HTML.gif denotes the integer part of the real number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq50_HTML.gif denotes the gamma function.

      Definition 2.2.

      The Riemann-Liouville fractional integral of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq51_HTML.gif for a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq52_HTML.gif is defined as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ4_HTML.gif
      (2.2)

      provided the right-hand side is pointwise defined on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq53_HTML.gif

      Definition 2.3.

      The Riemann-Liouville fractional derivative of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq54_HTML.gif for a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq55_HTML.gif is defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ5_HTML.gif
      (2.3)

      provided the right-hand side is pointwise defined on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq56_HTML.gif

      In passing, we remark that the Caputo derivative becomes the conventional http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq57_HTML.gif derivative of the function as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq58_HTML.gif and the initial conditions for fractional differential equations retain the same form as that of ordinary differential equations with integer derivatives. On the other hand, the Riemann-Liouville fractional derivative could hardly produce the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations (the same applies to the boundary value problems of fractional differential equations). Moreover, the Caputo derivative for a constant is zero while the Riemann-Liouville fractional derivative of a constant is nonzero. For more details, see [13].

      For the forthcoming analysis, we need the following assumptions:

      (A1)let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq59_HTML.gif be measurable with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq60_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq61_HTML.gif , u.s.c. with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq62_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq63_HTML.gif , and for each fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq64_HTML.gif the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq65_HTML.gif is nonempty,

      (A2)for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq66_HTML.gif there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq67_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq68_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq69_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq70_HTML.gif , and

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ6_HTML.gif
      (2.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq71_HTML.gif depends on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq72_HTML.gif For example, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq73_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq74_HTML.gif and hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq75_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq76_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq77_HTML.gif is not finite.

      Definition 2.4 ([16, 33]).

      A function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq78_HTML.gif is a solution of the problem (1.2) if there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq79_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq80_HTML.gif a.e. on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq81_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ7_HTML.gif
      (2.5)
      which, in terms of Green's function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq82_HTML.gif , can be expressed as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ8_HTML.gif
      (2.6)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ9_HTML.gif
      (2.7)
      Here, we remark that the Green's function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq83_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq84_HTML.gif takes the form (see [22])
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ10_HTML.gif
      (2.8)

      Now we state the following lemmas which are necessary to establish the main result of the paper.

      Lemma 2.5 (Bohnenblust-Karlin [36]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq85_HTML.gif be a nonempty subset of a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq86_HTML.gif , which is bounded, closed, and convex. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq87_HTML.gif is u.s.c. with closed, convex values such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq88_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq89_HTML.gif is compact. Then G has a fixed point.

      Lemma 2.6 ([37]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq90_HTML.gif be a compact real interval. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq91_HTML.gif be a multivalued map satisfying http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq92_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq93_HTML.gif be linear continuous from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq94_HTML.gif then the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq95_HTML.gif is a closed graph operator in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq96_HTML.gif

      3. Main Result

      Theorem 3.1.

      Suppose that the assumptions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq97_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq98_HTML.gif are satisfied, and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ11_HTML.gif
      (3.1)

      Then the antiperiodic problem (1.2) has at least one solution on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq99_HTML.gif

      Proof.

      To transform the problem (1.2) into a fixed point problem, we define a multivalued map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq100_HTML.gif as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ12_HTML.gif
      (3.2)
      Now we prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq101_HTML.gif satisfies all the assumptions of Lemma 2.6, and thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq102_HTML.gif has a fixed point which is a solution of the problem (1.2). As a first step, we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq103_HTML.gif is convex for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq104_HTML.gif For that, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq105_HTML.gif Then there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq106_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq107_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ13_HTML.gif
      (3.3)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq108_HTML.gif Then, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq109_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ14_HTML.gif
      (3.4)

      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq110_HTML.gif is convex ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq111_HTML.gif has convex values), therefore it follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq112_HTML.gif

      In order to show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq113_HTML.gif is closed for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq114_HTML.gif let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq115_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq116_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq117_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq118_HTML.gif and there exists a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq119_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ15_HTML.gif
      (3.5)
      As http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq120_HTML.gif has compact values, we pass onto a subsequence to obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq121_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq122_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq123_HTML.gif Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq124_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ16_HTML.gif
      (3.6)

      Hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq125_HTML.gif

      Next we show that there exists a positive number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq126_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq127_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq128_HTML.gif Clearly http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq129_HTML.gif is a bounded closed convex set in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq130_HTML.gif for each positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq131_HTML.gif If it is not true, then for each positive number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq132_HTML.gif , there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq133_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq134_HTML.gif and

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ17_HTML.gif
      (3.7)
      On the other hand, in view of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq135_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ18_HTML.gif
      (3.8)
      where we have used the fact that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ19_HTML.gif
      (3.9)

      Dividing both sides of (3.8) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq136_HTML.gif and taking the lower limit as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq137_HTML.gif , we find that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq138_HTML.gif which contradicts (3.1). Hence there exists a positive number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq139_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq140_HTML.gif

      Now we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq141_HTML.gif is equicontinuous. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq142_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq143_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq144_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq145_HTML.gif then there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq146_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq147_HTML.gif we have

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ20_HTML.gif
      (3.10)
      Using (3.8), we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ21_HTML.gif
      (3.11)

      Obviously the right-hand side of the above inequality tends to zero independently of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq148_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq149_HTML.gif Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq150_HTML.gif is equicontinuous.

      As http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq151_HTML.gif satisfies the above assumptions, therefore it follows by Ascoli-Arzela theorem that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq152_HTML.gif is a compact multivalued map.

      Finally, we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq153_HTML.gif has a closed graph. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq154_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq155_HTML.gif We will show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq156_HTML.gif By the relation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq157_HTML.gif we mean that there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq158_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq159_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ22_HTML.gif
      (3.12)
      Thus we need to show that there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq160_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq161_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ23_HTML.gif
      (3.13)
      Let us consider the continuous linear operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq162_HTML.gif so that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ24_HTML.gif
      (3.14)
      Observe that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ25_HTML.gif
      (3.15)
      Thus, it follows by Lemma 2.6 that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq163_HTML.gif is a closed graph operator. Further, we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq164_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq165_HTML.gif therefore, Lemma 2.6 yields
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ26_HTML.gif
      (3.16)

      Hence, we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq166_HTML.gif is a compact multivalued map, u.s.c. with convex closed values. Thus, all the assumptions of Lemma 2.6 are satisfied and so by the conclusion of Lemma 2.6, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq167_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq168_HTML.gif which is a solution of the problem (1.2).

      Remark 3.2.

      If we take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq169_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq170_HTML.gif is a continuous function, then our results correspond to a single-valued problem (a new result).

      Applications

      As an application of Theorem 3.1, we discuss two cases in relation to the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq171_HTML.gif in (1.2), namely, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq172_HTML.gif has (a) sublinear growth in its second variable (b) linear growth in its second variable (state variable). In case of sublinear growth, there exist functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq173_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq174_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq175_HTML.gif In this case, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq176_HTML.gif For the linear growth, the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq177_HTML.gif satisfies the relation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq178_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq179_HTML.gif In this case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq180_HTML.gif and the condition (3.1) modifies to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq181_HTML.gif In both the cases, the antiperiodic problem (1.2) has at least one solution on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq182_HTML.gif

      Examples
      1. (a)

        We consider http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq183_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq184_HTML.gif in (1.2). Here, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq185_HTML.gif Clearly http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq186_HTML.gif satisfies the assumptions of Theorem 3.1 with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq187_HTML.gif (condition (3.1). Thus, by the conclusion of Theorem 3.1, the antiperiodic problem (1.2) has at least one solution on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq188_HTML.gif

         
      2. (b)

        As a second example, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq189_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq190_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq191_HTML.gif in (1.2). In this case, (3.1) takes the form http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq192_HTML.gif As all the assumptions of Theorem 3.1 are satisfied, the antiperiodic problem (1.2) has at least one solution on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq193_HTML.gif

         

      Declarations

      Acknowledgments

      The authors are grateful to the referees for their valuable suggestions that led to the improvement of the original manuscript. The research of V. Otero-Espinar 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)
      Department of Mathematics, Faculty of Science, King Abdulaziz University
      (2)
      Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela

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