Open Access

Existence of Solutions for Fractional Differential Inclusions with Antiperiodic Boundary Conditions

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:

https://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

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq2_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq5_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq6_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq7_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq8_HTML.gif be the Banach space of functions https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq12_HTML.gif is convex (closed) valued if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq13_HTML.gif is convex (closed) for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq14_HTML.gif The map https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq15_HTML.gif is bounded on bounded sets if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq16_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq17_HTML.gif for any bounded set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq18_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq19_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq20_HTML.gif . https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq22_HTML.gif if for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq23_HTML.gif the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq24_HTML.gif is a nonempty closed subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq25_HTML.gif , and if for each open set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq26_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq27_HTML.gif containing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq28_HTML.gif there exists an open neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq29_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq30_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq31_HTML.gif . https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq34_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq35_HTML.gif If the multivalued map https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq38_HTML.gif has a closed graph, that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq39_HTML.gif imply https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq40_HTML.gif In the following study, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq42_HTML.gif . https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq44_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq46_HTML.gif the Caputo derivative of fractional order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq47_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ3_HTML.gif
(2.1)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq49_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq51_HTML.gif for a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq52_HTML.gif is defined as
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq54_HTML.gif for a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq55_HTML.gif is defined by
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq57_HTML.gif derivative of the function as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq59_HTML.gif be measurable with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq60_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq61_HTML.gif , u.s.c. with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq62_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq63_HTML.gif , and for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq64_HTML.gif the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq65_HTML.gif is nonempty,

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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq71_HTML.gif depends on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq72_HTML.gif For example, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq73_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq74_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq75_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq76_HTML.gif then https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq79_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq80_HTML.gif a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq81_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq82_HTML.gif , can be expressed as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ8_HTML.gif
(2.6)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq83_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq84_HTML.gif takes the form (see [22])
https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq88_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq90_HTML.gif be a compact real interval. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq91_HTML.gif be a multivalued map satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq92_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq93_HTML.gif be linear continuous from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq94_HTML.gif then the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq95_HTML.gif is a closed graph operator in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq98_HTML.gif are satisfied, and
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq100_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ12_HTML.gif
(3.2)
Now we prove that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq103_HTML.gif is convex for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq104_HTML.gif For that, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq105_HTML.gif Then there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq106_HTML.gif such that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq107_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ13_HTML.gif
(3.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq108_HTML.gif Then, for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq109_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ14_HTML.gif
(3.4)

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq110_HTML.gif is convex ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq112_HTML.gif

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ15_HTML.gif
(3.5)
As https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq121_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq122_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq123_HTML.gif Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq124_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ16_HTML.gif
(3.6)

Hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq126_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq127_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq128_HTML.gif Clearly https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq130_HTML.gif for each positive constant https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq132_HTML.gif , there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq133_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq134_HTML.gif and

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq135_HTML.gif , we have
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq136_HTML.gif and taking the lower limit as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq137_HTML.gif , we find that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq139_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq140_HTML.gif

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

https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq148_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq149_HTML.gif Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq150_HTML.gif is equicontinuous.

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq160_HTML.gif such that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq161_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq162_HTML.gif so that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ24_HTML.gif
(3.14)
Observe that
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq164_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq165_HTML.gif therefore, Lemma 2.6 yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_Equ26_HTML.gif
(3.16)

Hence, we conclude that https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq167_HTML.gif has a fixed point https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq169_HTML.gif where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq171_HTML.gif in (1.2), namely, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq173_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq174_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq175_HTML.gif In this case, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq176_HTML.gif For the linear growth, the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq177_HTML.gif satisfies the relation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq178_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq179_HTML.gif In this case https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq182_HTML.gif

Examples
  1. (a)

    We consider https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq183_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq184_HTML.gif in (1.2). Here, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq185_HTML.gif Clearly https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq189_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F625347/MediaObjects/13661_2009_Article_865_IEq190_HTML.gif and https://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 https://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 https://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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© B. Ahmad and V. Otero-Espinar. 2009

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