Abstract
In this paper, we study the existence of solutions for a fractional boundary value problem involving Hadamardtype fractional differential inclusions and integral boundary conditions. Our results include the cases for convex as well as nonconvex 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 theorems1 Introduction
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 [14]. 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 [521] and references cited therein.
It has been noticed that most of the work on the topic is based on RiemannLiouville and Caputotype fractional differential equations. Another kind of fractional derivatives that appears side by side to RiemannLiouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in 1892 [22], 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 [1,2327].
In this paper, we study the following boundary value problem of Hadamardtype fractional differential inclusions:
where
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 nonconvex valued. The first result relies on the nonlinear alternative of LeraySchauder type. In the second result, we combine the nonlinear alternative of LeraySchauder type for singlevalued 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 Hadamardtype fractional differential inclusions supplemented with Hadamardtype 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 Hadamardtype 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 Hadamardtype integrodifferential 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.
2 Preliminaries
This section is devoted to the basic concepts of Hadamardtype fractional calculus and multivalued analysis. We also establish an auxiliary lemma to define the solutions for the given problem.
2.1 Fractional calculus
Definition 2.1[1]
The Hadamard derivative of fractional order q for a function
where
Definition 2.2[1]
The Hadamard fractional integral of order q for a function g is defined as
provided the integral exists.
Lemma 2.3 (Auxiliary lemma)
For
is given by
where
Proof As argued in [1], the solution of the Hadamard differential equation in (2.1) can be written as
Using the given boundary conditions, we find that
which gives
Substituting the values of
2.2 Basic concepts of multivalued analysis
Here we outline some basic definitions and results for multivalued maps [28,29].
Let
For a normed space
(i) is convex (closed) valued if
(ii) is bounded on bounded sets if
(iii) is called upper semicontinuous (u.s.c.) on X if for each
(iv) G is lower semicontinuous (l.s.c.) if the set
(v) is said to be completely continuous if
(vi) is said to be measurable if for every
is measurable;
(vii) has a fixed point if there is
For each
We define the graph of G to be the set
Lemma 2.4 [[28], Proposition 1.2]
If
Lemma 2.5[30]
LetXbe a separable Banach space. Let
is a closed graph operator in
3 Existence results
Definition 3.1 A function
3.1 The upper semicontinuous case
Our first main result for Carathéodory case is established via the nonlinear alternative of LeraySchauder for multivalued maps.
Lemma 3.2 (Nonlinear alternative for Kakutani maps [31])
LetEbe a Banach space, Cbe a closed convex subset ofE, Ube an open subset ofCand
(i) Fhas a fixed point in
(ii) there are
Theorem 3.3Assume that:
(H_{1})
(i)
(ii)
(H_{2}) there exist a continuous nondecreasing function
(H_{3}) there exists a constant
where Ω is given by (2.3).
Then problem (1.1) has at least one solution on
Proof In view of Lemma 2.3, we define an operator
for
Step 1.
This step is obvious since
Step 2. ℱ maps bounded sets (balls) into bounded sets in
For a positive number ρ, let
Then we have
Thus
Step 3. ℱ maps bounded sets into equicontinuous sets of
Let
Obviously the righthand side of the above inequality tends to zero independently
of
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.
Let
Thus we have to show that there exists
Let us consider the linear operator
Observe that
as
Thus, it follows by Lemma 2.5 that
for some
Step 5. We show that there exists an open set
Let
Using the computations of the second step above, we have
Consequently, we have
In view of (H_{3}), there exists M such that
Note that the operator
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 LeraySchauder type with the selection theorem due to Bressan and Colombo [32] for lower semicontinuous maps with decomposable values.
Definition 3.4 Let A be a subset of
Definition 3.5 A subset of
Lemma 3.6[32]
LetYbe a separable metric space, and let
Theorem 3.7Assume that (H_{2}), (H_{3}) and the following condition holds:
(H_{4})
(a)
(b)
Then problem (1.1) has at least one solution on
Proof It follows from (H_{2}) and (H_{4}) that F is of l.s.c. type [33]. Then, by Lemma 3.6, there exists a continuous function
Consider the problem
Observe that if
It can easily be shown that
3.3 The Lipschitz case
Let
where
Definition 3.8 A multivalued operator
(a) γLipschitz if and only if there exists
(b) a contraction if and only if it is γLipschitz with
To show the existence of solutions for problem (1.1) with a nonconvex valued righthand side, we need a fixed point theorem for multivalued maps due to Covitz and Nadler [35].
Lemma 3.9[35]
Let
Theorem 3.10Assume that the following conditions hold:
(H_{5})
(H_{6})
Then problem (1.1) has at least one solution on
Proof We transform problem (1.1) into a fixed point problem by means of the operator
Step 1.
Since the setvalued map
that is,
As F has compact values, we pass onto a subsequence (if necessary) to obtain that
Hence,
Step 2. Next we show that there exists
Let
By (H_{6}), we have
So, there exists
Define
Since the multivalued operator
For each
Thus,
Hence,
Analogously, interchanging the roles of x and
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. □
4 Examples
In this section we present some concrete examples to illustrate our results.
Let us consider the boundary value problem
Here
and
Example 4.1 Let
For
Here
Example 4.2 Consider the multivalued map
Then we have
and
Let
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Authors’ information
All authors are members of Nonlinear Analysis and Applied Mathematics (NAAM)  Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
Acknowledgements
This research was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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