An existence result for fractional differential inclusions with nonlinear integral boundary conditions
© Ahmad et al.; licensee Springer 2013
Received: 25 November 2012
Accepted: 26 April 2013
Published: 19 June 2013
This paper studies the existence of solutions for a fractional differential inclusion of order with nonlinear integral boundary conditions by applying Bohnenblust-Karlin’s fixed point theorem. Some examples are presented for the illustration of the main result.
MSC: 34A40, 34A12, 26A33.
where denotes the Caputo fractional derivative of order q, , are given continuous functions and with .
Differential inclusions of integer order (classical case) play an important role in the mathematical modeling of various situations in economics, optimal control, etc. and are widely studied in literature. Motivated by an extensive study of classical differential inclusions, a significant work has also been established for fractional differential inclusions. For examples and details, see [1–10] and references therein.
Let denote a Banach space of continuous functions from into ℝ with the norm . Let be the Banach space of functions which are Lebesgue integrable and normed by .
Let be a Banach space. Then a multi-valued map is convex (closed) valued if is convex (closed) for all . The map G is bounded on bounded sets if is bounded in X for any bounded set of X (i.e., ). G is called upper semi-continuous (u.s.c.) on X if for each , the set is a nonempty closed subset of X, and if for each open set of X containing , there exists an open neighborhood of such that . G is said to be completely continuous if is relatively compact for every bounded subset of X. If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., , , imply . In the following study, denotes the set of all nonempty bounded, closed and convex subsets of X. G has a fixed point if there is such that .
where Γ denotes the gamma function.
provided the right-hand side is pointwise defined on .
To define the solution for (1.1), we consider the following lemma. We do not provide the proof of this lemma as it employs the standard arguments.
Now we state the following lemmas which are necessary to establish the main result.
Lemma 2.4 (Bohnenblust-Karlin )
Let D be a nonempty subset of a Banach space X, which is bounded, closed and convex. Suppose that is u.s.c. with closed, convex values such that and is compact. Then G has a fixed point.
Lemma 2.5 
Let I be a compact real interval. Let F be a multi-valued map satisfying (A1) and let Θ be linear continuous from , then the operator , is a closed graph operator in .
For the forthcoming analysis, we need the following assumptions:
(A1) Let ; be measurable with respect to t for each , u.s.c. with respect to x for a.e. , and for each fixed , the set is nonempty.
3 Main result
Then the boundary value problem (1.1) has at least one solution on .
Since is convex (F has convex values), therefore it follows that .
which contradicts (3.1). Hence there exists a positive number such that .
Obviously, the right-hand side of the above inequality tends to zero independently of as . Thus, N is equi-continuous.
As N satisfies the above three assumptions, therefore it follows by the Ascoli-Arzelá theorem that N is a compact multi-valued map.
for some .
Hence, we conclude that N is a compact multi-valued map, u.s.c. with convex closed values. Thus, all the assumptions of Lemma 2.4 are satisfied. Hence the conclusion of Lemma 2.4 applies and, in consequence, N has a fixed point x which is a solution of the problem (1.1). This completes the proof. □
As an application of Theorem 3.1, we discuss two cases for nonlinearities , , : (a) sub-linear growth in the second variable of the nonlinearities; (b) linear growth in the second variable (state variable). In case of sub-linear growth, there exist functions , with such that , , for each . In this case, , , , and the condition (3.1) is . For the linear growth, the nonlinearities F, g, h satisfy the relation , , for each . In this case , , , and the condition (3.1) becomes . In both cases, the boundary value problem (1.1) has at least one solution on .
Example 3.2 (linear growth case)
Clearly, . Thus, by Theorem 3.1, the problem (3.2) has at least one solution on .
Example 3.3 (sub-linear growth case)
Letting , , in Example 3.2, we find that . Hence there exits a solution for the sub-linear case of the problem (3.2) by Theorem 3.1.
This research was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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