In this paper, we study the existence of solutions for a fractional boundary value problem involving Hadamard-type fractional differential inclusions and integral boundary conditions. Our results include the cases for convex as well as non-convex valued maps and are based on standard fixed point theorems for multivalued maps. Some illustrative examples are also presented.
MSC: 34A60, 34A08.
Keywords:Hadamard fractional derivative; integral boundary conditions; fixed point theorems
The theory of fractional differential equations and inclusions has received much attention over the past years and has become an important field of investigation due to its extensive applications in numerous branches of physics, economics and engineering sciences [1-4]. Fractional differential equations and inclusions provide appropriate models for describing real world problems, which cannot be described using classical integer order differential equations. Some recent contributions to the subject can be seen in [5-21] and references cited therein.
It has been noticed that most of the work on the topic is based on Riemann-Liouville and Caputo-type fractional differential equations. Another kind of fractional derivatives that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in 1892 , which differs from the preceding ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains a logarithmic function of arbitrary exponent. Details and properties of the Hadamard fractional derivative and integral can be found in [1,23-27].
In this paper, we study the following boundary value problem of Hadamard-type fractional differential inclusions:
We aim to establish a variety of results for inclusion problem (1.1) by considering the given multivalued map to be convex as well as non-convex valued. The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, while the third result is obtained by using the fixed point theorem for contractive multivalued maps due to Covitz and Nadler.
We emphasize that the main idea of the present research is to introduce Hadamard-type fractional differential inclusions supplemented with Hadamard-type integral boundary conditions and develop some existence results for the problem at hand. It is imperative to note that our results are absolutely new in the context of Hadamard-type integral boundary value problems and provide a new avenue to the researchers working on fractional boundary value problems.
The paper is organized as follows. In Section 2, we solve a linear Hadamard-type integro-differential boundary value problem and recall some preliminary concepts of multivalued analysis that we need in the sequel. Section 3 contains the main results for problem (1.1). In Section 4, some illustrative examples are discussed.
This section is devoted to the basic concepts of Hadamard-type fractional calculus and multivalued analysis. We also establish an auxiliary lemma to define the solutions for the given problem.
2.1 Fractional calculus
The Hadamard fractional integral of order q for a function g is defined as
provided the integral exists.
Lemma 2.3 (Auxiliary lemma)
is given by
Proof As argued in , the solution of the Hadamard differential equation in (2.1) can be written as
2.2 Basic concepts of multivalued analysis
Lemma 2.4 [, Proposition 1.2]
3 Existence results
3.1 The upper semicontinuous case
Our first main result for Carathéodory case is established via the nonlinear alternative of Leray-Schauder for multivalued maps.
Lemma 3.2 (Nonlinear alternative for Kakutani maps )
Theorem 3.3Assume that:
where Ω is given by (2.3).
for (defined by (2.6)). Observe that the fixed points of the operator ℱ correspond to the solutions of problem (1.1). We will show that ℱ satisfies the assumptions of the Leray-Schauder nonlinear alternative (Lemma 3.2). The proof consists of several steps.
Then we have
By Lemma 2.4, ℱ will be upper semicontinuous (u.s.c.) if we prove that it has a closed graph since ℱ is already shown to be completely continuous.
Step 4. ℱ has a closed graph.
Using the computations of the second step above, we have
Consequently, we have
Note that the operator is upper semicontinuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.2), we deduce that ℱ has a fixed point which is a solution of problem (1.1). This completes the proof. □
3.2 The lower semicontinuous case
In what follows, we consider the case when F is not necessarily convex valued and obtain the existence result by combining the nonlinear alternative of Leray-Schauder type with the selection theorem due to Bressan and Colombo  for lower semicontinuous maps with decomposable values.
LetYbe a separable metric space, and letbe a lower semicontinuous (l.s.c.) multivalued operator with nonempty closed and decomposable values. ThenNhas a continuous selection, that is, there exists a continuous function (single-valued) such thatfor every.
Theorem 3.7Assume that (H2), (H3) and the following condition holds:
Proof It follows from (H2) and (H4) that F is of l.s.c. type . Then, by Lemma 3.6, there exists a continuous function such that for all , where is the Nemytskii operator associated with F, defined as
Consider the problem
3.3 The Lipschitz case
where and . Then is a metric space (see ).
To show the existence of solutions for problem (1.1) with a nonconvex valued right-hand side, we need a fixed point theorem for multivalued maps due to Covitz and Nadler .
Theorem 3.10Assume that the following conditions hold:
Proof We transform problem (1.1) into a fixed point problem by means of the operator defined by (3.1) and show that the operator ℱ satisfies the assumptions of Lemma 3.9. The proof will be given in two steps.
Since the set-valued map is measurable with the measurable selection theorem (e.g., [, Theorem III.6]), it admits a measurable selection . Moreover, by assumption (H6), we have
By (H6), we have
Since the multivalued operator is measurable (Proposition III.4 ), there exists a function which is a measurable selection for U. So and for each , we have .
Since ℱ is a contraction, it follows by Lemma 3.9 that ℱ has a fixed point x which is a solution of (1.1). This completes the proof. □
In this section we present some concrete examples to illustrate our results.
Let us consider the boundary value problem
Then we have
The authors declare that they have no competing interests.
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.
All authors are members of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
This research was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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