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:
where is the Hadamard fractional derivative of order α, is the Hadamard fractional integral of order β and is a multivalued map, is the family of all nonempty subsets of ℝ.
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 derivative of fractional order q for a function is defined as
where denotes the integer part of the real number q and .
The Hadamard fractional integral of order q for a function g is defined as
provided the integral exists.
Lemma 2.3 (Auxiliary lemma)
For and , the unique solution of the problem
is given by
Proof As argued in , the solution of the Hadamard differential equation in (2.1) can be written as
Using the given boundary conditions, we find that and
Substituting the values of and in (2.4), we obtain (2.2). This completes the proof. □
2.2 Basic concepts of multivalued analysis
Let denote a Banach space of continuous functions from into ℝ with the norm . Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .
For a normed space , let , , , and . A multi-valued map :
(i) is convex (closed) valued if is convex (closed) for all ;
(ii) is bounded on bounded sets if is bounded in X for all (i.e., );
(iii) is called upper semicontinuous (u.s.c.) on X if for each , the set is a nonempty closed subset of X and if for each open set N of X containing , there exists an open neighborhood of such that ;
(iv) G is lower semicontinuous (l.s.c.) if the set is open for any open set B in E;
(v) is said to be completely continuous if is relatively compact for every ;
(vi) is said to be measurable if for every , the function
(vii) has a fixed point if there is such that . The fixed point set of the multivalued operator G will be denoted by FixG.
For each , define the set of selections of F by
We define the graph of G to be the set and recall two results for closed graphs and upper-semicontinuity.
Lemma 2.4 [, Proposition 1.2]
If is u.s.c., then is a closed subset of ; i.e., for every sequence and , if when , , and , then . Conversely, ifGis completely continuous and has a closed graph, then it is upper semicontinuous.
LetXbe a separable Banach space. Let be measurable with respect totfor each and u.s.c. with respect toxfor almost all and for any , and let Θ be a linear continuous mapping from to . Then the operator
is a closed graph operator in .
3 Existence results
Definition 3.1 A function is called a solution of problem (1.1) if there exists a function with , a.e. on such that , a.e. on and , .
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 )
LetEbe a Banach space, Cbe a closed convex subset ofE, Ube an open subset ofCand . Suppose that is an upper semicontinuous compact map. Then either
(i) Fhas a fixed point in , or
(ii) there are and with .
Theorem 3.3Assume that:
(H1) is Carathéodory, i.e.,
(i) is measurable for each ;
(ii) is u.s.c. for almost all ;
(H2) there exist a continuous nondecreasing function and a function such that
(H3) there exists a constant such that
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 by
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.
Step 1. is convex for each .
This step is obvious since is convex (F has convex values), and therefore we omit the proof.
Step 2. ℱ maps bounded sets (balls) into bounded sets in .
For a positive number ρ, let be a bounded ball in . Then, for each , , there exists such that
Then we have
Step 3. ℱ maps bounded sets into equicontinuous sets of .
Let with and , where is a bounded set of as in Step 2. For each , we obtain
Obviously the right-hand side of the above inequality tends to zero independently of as . In view of Steps 1-3, the Arzelá-Ascoli theorem applies and hence is completely continuous.
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 , 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 linear operator given by
Thus, it follows by Lemma 2.5 that is a closed graph operator. Further, we have . Since , therefore, we have
for some .
Step 5. We show that there exists an open set with for any and all .
Let for some . Then there exists with such that, for , we have
Using the computations of the second step above, we have
Consequently, we have
In view of (H3), there exists M such that . Let us set
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.
Definition 3.4 Let A be a subset of . A is measurable if A belongs to the σ-algebra generated by all sets of the form , where is Lebesgue measurable in I and is Borel measurable in ℝ.
Definition 3.5 A subset of is decomposable if for all and measurable , the function , where stands for the characteristic function of .
LetYbe a separable metric space, and let be 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 that for every .
Theorem 3.7Assume that (H2), (H3) and the following condition holds:
(H4) is a nonempty compact-valued multivalued map such that
(a) is measurable,
(b) is lower semicontinuous for each .
Then problem (1.1) has at least one solution on .
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
Observe that if is a solution of problem (3.2), then x is a solution to problem (1.1). In order to transform problem (3.2) into a fixed point problem, we define an operator as
It can easily be shown that is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 3.3. So we omit it. This completes the proof. □
3.3 The Lipschitz case
Let be a metric space induced from the normed space . Consider given by
where and . Then is a metric space (see ).
Definition 3.8 A multivalued operator is called
(a) γ-Lipschitz if and only if there exists such that
(b) a contraction if and only if it is γ-Lipschitz with .
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 .
Let be a complete metric space. If is a contraction, then .
Theorem 3.10Assume that the following conditions hold:
(H5) is such that is measurable for each .
(H6) for almost all and with and for almost all .
Then problem (1.1) has at least one solution on if
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.
Step 1. is nonempty and closed for every .
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
that is, and hence F is integrably bounded. Therefore, . Moreover, for each . Indeed, let be such that ( ) in . Then and there exists such that, for each ,
As F has compact values, we pass onto a subsequence (if necessary) to obtain that converges to g in . Thus, and for each , we have
Step 2. Next we show that there exists such that
Let and . Then there exists such that, for each ,
By (H6), we have
So, there exists such that
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 .
For each , let us define
Analogously, interchanging the roles of x and , we obtain
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
Here , , ,
Example 4.1 Let be a multivalued map given by
For , we have
Here , , with , . It is easy to verify that . Then, by Theorem 3.3, problem (4.1) with given by (4.2) has at least one solution on .
Example 4.2 Consider the multivalued map given by
Then we have
Let . Then , and . By Theorem 3.10, problem (4.1) with given by (4.3) has at least one solution on .
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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