Abstract
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.
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1 Introduction
In this paper, we apply the Bohnenblust-Karlin fixed point theorem to prove the existence of solutions for a fractional differential inclusion with integral boundary conditions given by
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.
2 Preliminaries
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 .
Now we recall some basic definitions on multi-valued maps [11–14].
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 .
Let us record some definitions on fractional calculus [15–18].
Definition 2.1 For an at least -times continuously differentiable function , the Caputo derivative of fractional order q is defined as
where Γ denotes the gamma function.
Definition 2.2 The Riemann-Liouville fractional integral of order q for a function g is defined as
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.
Lemma 2.3 For a given , the solution of the boundary value problem
is given by the integral equation
In view of Lemma 2.3, a function is a solution of the problem (1.1) if there exists a function such that a.e. on and
Now we state the following lemmas which are necessary to establish the main result.
Lemma 2.4 (Bohnenblust-Karlin [19])
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 [20]
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.
(A2) For each , there exists a function such that , , for each with , and
where .
Furthermore, we set
3 Main result
Theorem 3.1 Suppose that the assumptions (A1) and (A2) are satisfied, and
where γ is given by (2.4) and
Then the boundary value problem (1.1) has at least one solution on .
Proof In order to transform the problem (1.1) into a fixed point problem, we define a multi-valued map as
Now we prove that the multi-valued map N satisfies all the assumptions of Lemma 2.4, and thus N has a fixed point which is a solution of the problem (1.1). In the first step, we show that is convex for each . For that, let . Then there exist such that for each , we have
Let . Then, for each , we have
Since is convex (F has convex values), therefore it follows that .
Next it will be shown that there exists a positive number r such that , where . Clearly is a bounded closed convex set in for each positive constant r. If it is not true, then for each positive number r, there exists a function , with , and
On the other hand, using (A2), we have
Dividing both sides by r and taking the lower limit as , we find that
which contradicts (3.1). Hence there exists a positive number such that .
Now we show that is equi-continuous. Let with . Let and , then there exists such that for each , we have
from which we obtain
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.
In our next step, we show that N has a closed graph. Let , and . Then we need to show that . Associated with , there exists such that for each ,
Thus we have to show that there exists such that for each ,
Let us consider the continuous linear operator given by
Observe that
Thus, it follows by Lemma 2.5 that is a closed graph operator. Further, we have . Since , therefore, Lemma 2.5 yields
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. □
Special cases
By fixing the parameters in the boundary conditions of (1.1), we obtain some new results. As the first case, by taking , , , our main result with corresponds to the problem
In case we fix , , , we obtain a new result for the problem
with .
Discussion
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)
Consider the following problem:
where , , , , , and
With the given data, , , ,
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.
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Acknowledgements
This research was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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Each of the authors, BA, SKN and AA contributed to each part of this work equally and read and approved the final version of the manuscript.
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Ahmad, B., Ntouyas, S.K. & Alsaedi, A. An existence result for fractional differential inclusions with nonlinear integral boundary conditions. J Inequal Appl 2013, 296 (2013). https://doi.org/10.1186/1029-242X-2013-296
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DOI: https://doi.org/10.1186/1029-242X-2013-296