Existence of Solutions for Fractional Differential Inclusions with Antiperiodic Boundary Conditions
© B. Ahmad and V. Otero-Espinar. 2009
Received: 21 January 2009
Accepted: 18 March 2009
Published: 4 May 2009
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.
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 [1–14] and the references therein.
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 [23–27] and the references therein. For some recent development on differential inclusions, we refer the reader to the references [28–32].
Chang and Nieto  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
Let be a Banach space. Then a multivalued map is convex (closed) valued if is convex (closed) for all The map is bounded on bounded sets if is bounded in for any bounded set of (i.e., . is called upper semicontinuous (u.s.c.) on if for each the set is a nonempty closed subset of , and if for each open set of containing there exists an open neighborhood of such that . is said to be completely continuous if is relatively compact for every bounded subset of If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph, that is, imply In the following study, denotes the set of all nonempty bounded, closed, and convex subset of . has a fixed point if there is such that
In passing, we remark that the Caputo derivative becomes the conventional derivative of the function as 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 .
For the forthcoming analysis, we need the following assumptions:
Now we state the following lemmas which are necessary to establish the main result of the paper.
Lemma 2.5 (Bohnenblust-Karlin ).
Lemma 2.6 ().
3. Main Result
Next we show that there exists a positive number such that where Clearly is a bounded closed convex set in for each positive constant If it is not true, then for each positive number , there exists a function with and
Hence, we conclude that 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, has a fixed point which is a solution of the problem (1.2).
As an application of Theorem 3.1, we discuss two cases in relation to the nonlinearity in (1.2), namely, 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 such that for each In this case, For the linear growth, the nonlinearity satisfies the relation for each In this case and the condition (3.1) modifies to In both the cases, the antiperiodic problem (1.2) has at least one solution on
We consider and in (1.2). Here, Clearly satisfies the assumptions of Theorem 3.1 with (condition (3.1). Thus, by the conclusion of Theorem 3.1, the antiperiodic problem (1.2) has at least one solution on
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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