Boundary Value Problems

, 2009:625347 | Cite as

Existence of Solutions for Fractional Differential Inclusions with Antiperiodic Boundary Conditions

Open Access
Research Article
Part of the following topical collections:
  1. Singular Boundary Value Problems for Ordinary Differential Equations

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.

Keywords

Fractional Order Fractional Derivative Fractional Calculus Fractional Differential Equation Differential Inclusion 

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 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and the references therein.

Antiperiodic boundary value problems have recently received considerable attention as antiperiodic boundary conditions appear in numerous situations, for instance, see [15, 16, 17, 18, 19, 20, 21, 22].

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 [23, 24, 25, 26, 27] and the references therein. For some recent development on differential inclusions, we refer the reader to the references [28, 29, 30, 31, 32].

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

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

where Open image in new window denotes the Caputo fractional derivative of order Open image in new window , Open image in new window Bohnenblust-Karlin fixed point theorem is applied to prove the existence of solutions of (1.2).

2. Preliminaries

Let Open image in new window denote a Banach space of continuous functions from Open image in new window into Open image in new window with the norm Open image in new window Let Open image in new window be the Banach space of functions Open image in new window which are Lebesgue integrable and normed by Open image in new window

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

Let Open image in new window be a Banach space. Then a multivalued map Open image in new window is convex (closed) valued if Open image in new window is convex (closed) for all Open image in new window The map Open image in new window is bounded on bounded sets if Open image in new window is bounded in Open image in new window for any bounded set Open image in new window of Open image in new window (i.e., Open image in new window . Open image in new window is called upper semicontinuous (u.s.c.) on Open image in new window if for each Open image in new window the set Open image in new window is a nonempty closed subset of Open image in new window , and if for each open set Open image in new window of Open image in new window containing Open image in new window there exists an open neighborhood Open image in new window of Open image in new window such that Open image in new window . Open image in new window is said to be completely continuous if Open image in new window is relatively compact for every bounded subset Open image in new window of Open image in new window If the multivalued map Open image in new window is completely continuous with nonempty compact values, then Open image in new window is u.s.c. if and only if Open image in new window has a closed graph, that is, Open image in new window imply Open image in new window In the following study, Open image in new window denotes the set of all nonempty bounded, closed, and convex subset of Open image in new window . Open image in new window has a fixed point if there is Open image in new window such that Open image in new window

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

Definition 2.1.

For a function Open image in new window the Caputo derivative of fractional order Open image in new window is defined as

where Open image in new window denotes the integer part of the real number Open image in new window and Open image in new window denotes the gamma function.

Definition 2.2.

The Riemann-Liouville fractional integral of order Open image in new window for a function Open image in new window is defined as

provided the right-hand side is pointwise defined on Open image in new window

Definition 2.3.

The Riemann-Liouville fractional derivative of order Open image in new window for a function Open image in new window is defined by

provided the right-hand side is pointwise defined on Open image in new window

In passing, we remark that the Caputo derivative becomes the conventional Open image in new window derivative of the function as Open image in new window 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 Open image in new window be measurable with respect to Open image in new window for each Open image in new window , u.s.c. with respect to Open image in new window for a.e. Open image in new window , and for each fixed Open image in new window the set Open image in new window is nonempty,

(A2)for each Open image in new window there exists a function Open image in new window such that Open image in new window for each Open image in new window with Open image in new window , and

where Open image in new window depends on Open image in new window For example, for Open image in new window we have Open image in new window and hence Open image in new window If Open image in new window then Open image in new window is not finite.

Definition 2.4 ([16, 33]).

A function Open image in new window is a solution of the problem (1.2) if there exists a function Open image in new window such that Open image in new window a.e. on Open image in new window and
which, in terms of Green's function Open image in new window , can be expressed as
Here, we remark that the Green's function Open image in new window for Open image in new window takes the form (see [22])

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

Lemma 2.5 (Bohnenblust-Karlin [36]).

Let Open image in new window be a nonempty subset of a Banach space Open image in new window , which is bounded, closed, and convex. Suppose that Open image in new window is u.s.c. with closed, convex values such that Open image in new window and Open image in new window is compact. Then G has a fixed point.

Lemma 2.6 ([37]).

Let Open image in new window be a compact real interval. Let Open image in new window be a multivalued map satisfying Open image in new window and let Open image in new window be linear continuous from Open image in new window then the operator Open image in new window is a closed graph operator in Open image in new window

3. Main Result

Theorem 3.1.

Suppose that the assumptions Open image in new window and Open image in new window are satisfied, and

Then the antiperiodic problem (1.2) has at least one solution on Open image in new window

Proof.

To transform the problem (1.2) into a fixed point problem, we define a multivalued map Open image in new window as
Now we prove that Open image in new window satisfies all the assumptions of Lemma 2.6, and thus Open image in new window has a fixed point which is a solution of the problem (1.2). As a first step, we show that Open image in new window is convex for each Open image in new window For that, let Open image in new window Then there exist Open image in new window such that for each Open image in new window we have

Since Open image in new window is convex ( Open image in new window has convex values), therefore it follows that Open image in new window

In order to show that Open image in new window is closed for each Open image in new window let Open image in new window be such that Open image in new window in Open image in new window Then Open image in new window and there exists a Open image in new window such that

As Open image in new window has compact values, we pass onto a subsequence to obtain that Open image in new window converges to Open image in new window in Open image in new window Thus, Open image in new window and

Hence Open image in new window

Next we show that there exists a positive number Open image in new window such that Open image in new window where Open image in new window Clearly Open image in new window is a bounded closed convex set in Open image in new window for each positive constant Open image in new window If it is not true, then for each positive number Open image in new window , there exists a function Open image in new window with Open image in new window and

On the other hand, in view of Open image in new window , we have
where we have used the fact that

Dividing both sides of (3.8) by Open image in new window and taking the lower limit as Open image in new window , we find that Open image in new window which contradicts (3.1). Hence there exists a positive number Open image in new window such that Open image in new window

Now we show that Open image in new window is equicontinuous. Let Open image in new window with Open image in new window Let Open image in new window and Open image in new window then there exists Open image in new window such that for each Open image in new window we have

Using (3.8), we obtain

Obviously the right-hand side of the above inequality tends to zero independently of Open image in new window as Open image in new window Thus, Open image in new window is equicontinuous.

As Open image in new window satisfies the above assumptions, therefore it follows by Ascoli-Arzela theorem that Open image in new window is a compact multivalued map.

Finally, we show that Open image in new window has a closed graph. Let Open image in new window and Open image in new window We will show that Open image in new window By the relation Open image in new window we mean that there exists Open image in new window such that for each Open image in new window

Thus we need to show that there exists Open image in new window such that for each Open image in new window
Let us consider the continuous linear operator Open image in new window so that
Observe that
Thus, it follows by Lemma 2.6 that Open image in new window is a closed graph operator. Further, we have Open image in new window Since Open image in new window therefore, Lemma 2.6 yields

Hence, we conclude that Open image in new window 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, Open image in new window has a fixed point Open image in new window which is a solution of the problem (1.2).

Remark 3.2.

If we take Open image in new window where Open image in new window 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 Open image in new window in (1.2), namely, Open image in new window 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 Open image in new window such that Open image in new window for each Open image in new window In this case, Open image in new window For the linear growth, the nonlinearity Open image in new window satisfies the relation Open image in new window for each Open image in new window In this case Open image in new window and the condition (3.1) modifies to Open image in new window In both the cases, the antiperiodic problem (1.2) has at least one solution on Open image in new window

Examples
  1. (a)

    We consider Open image in new window and Open image in new window in (1.2). Here, Open image in new window Clearly Open image in new window satisfies the assumptions of Theorem 3.1 with Open image in new window (condition (3.1). Thus, by the conclusion of Theorem 3.1, the antiperiodic problem (1.2) has at least one solution on Open image in new window

     
  2. (b)

    As a second example, let Open image in new window be such that Open image in new window and Open image in new window in (1.2). In this case, (3.1) takes the form Open image in new window As all the assumptions of Theorem 3.1 are satisfied, the antiperiodic problem (1.2) has at least one solution on Open image in new window

     

Notes

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.

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Copyright information

© B. Ahmad and V. Otero-Espinar. 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.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad de Santiago de CompostelaSantiago de CompostelaSpain

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