Research Article | Open Access
Existence Results for Fractional Differential Inclusions with Multivalued Term Depending on Lower-Order Derivative
This paper is concerned with a class of fractional differential inclusions whose multivalued term depends on lower-order fractional derivative with fractional (non)separated boundary conditions. The cases of convex-valued and non-convex-valued right-hand sides are considered. Some existence results are obtained by using standard fixed point theorems. A possible generalization for the inclusion problem with integral boundary conditions is also discussed. Examples are given to illustrate the results.
Recently, the subject of fractional differential equations has emerged as an important area of investigation. Indeed, we can find numerous applications of fractional-order derivatives in the mathematical modeling of physical and biological phenomena in various fields of science and engineering [1–3]. A variety of results on initial and boundary value problems of fractional differential equations and inclusions can easily be found in the literature on this topic. For some recent results, we can refer to, for instance, [4–20] (equations) [21–27] (inclusions) and the references therein.
Ahmad and Ntouyas  considered a boundary value problem of fractional differential inclusions with fractional separated boundary conditions given by where denotes the Caputo fractional derivative of order , is a multivalued map, and , , () are real constants, with .
In Cernea , the following multipoint boundary value problem for a fractional-order differential inclusion was studied where is the standard Riemann-Liouville fractional derivative, , , , , , , and is a multivalued map.
In Khan et al. , the authors studied the existence and uniqueness results of nonlinear fractional differential equation of the type where , , , (or , ) and , are the Caputo fractional derivatives. The results in [11, 22, 24] are obtained by using appropriate standard fixed point theorems.
Motivated by the papers cited above, in this paper, we consider the existence results for a new class of fractional differential inclusions of the form where denotes the Caputo fractional derivative of order , is a multivalued map, , , and . We study (1.4) subject to two families of boundary conditions: separated boundary conditions Nonseparated boundary conditions where , , , are real constants and .
The results of this paper can easily to be generalized to the boundary value problems of fractional differential inclusions (1.4) with the following integral boundary conditions: where are given functions.
We remark that when the third variable of the multifunction in (1.4) vanishes, the problem (1.4), (1.5) reduces to the case considered in . When , , and , the problem (1.4), (1.6) reduces to an antiperiodic fractional boundary value problem (the case of a given continuous function was studied in [4, 15]). Our results generalize some results from the literature cited above and constitute a contribution to this emerging field of research.
The rest of the paper is organized as follows: in Section 2 we present the notations and definitions and give some preliminary results that we need in the sequel, Section 3 is dedicated to the existence results of the fractional differential inclusion (1.4) with boundary conditions (1.5) and (1.6), in Section 4 we indicate a possible generalization for the inclusion problem (1.4) with integral boundary conditions (1.7) and (1.8), and two illustrative examples are given in Section 5.
In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper.
Let be a normed space. We use the notations , , , , , , and so on.
Let ; the Pompeiu-Hausdorff distance of , is defined as
A multivalued map is convex (closed) valued if is convex (closed) for all . is said to be completely continuous if is relatively compact for every . is called upper semicontinuous on , if for every , the set is a nonempty closed subset of , and for every open set of containing , there exists an open neighborhood of such that . Equivalently, is upper semicontinuous if the set is open for any open set of . is called lower semicontinuous if the set is open for each open set in . If a multivalued map is completely continuous with nonempty compact values, then is upper semicontinuous if and only if has a closed graph, that is, if and , then implies .
A multivalued map is said to be measurable if, for every , the function is a measurable function.
Definition 2.1. A multivalued map is called-Lipschitz if there exists such that a contraction if it is -Lipschitz with .
Definition 2.2. A multivalued map is said to be Carathéodory if is measurable for each ; is upper semicontinuous for a.e. . Further, a Carathéodory function is said to be - Carathéodory if for each , there exists such that for all , and a.e. .
Lemma 2.3 (see ). Let be a Banach space. Let be an - Carathéodory multivalued map and a linear continuous map from to , then the operator is a closed graph operator in .
Here for a.e. .
Definition 2.4 (see ). The Riemann-Liouville fractional integral of order for a function is defined as provided the integral exists.
Definition 2.5 (see ). For at least -times differentiable function , the Caputo derivative of order is defined as where denotes the integer part of the real number .
Lemma 2.6 (see ). Let ; then the differential equation has solutions and here , , .
The following lemma obtained in  is useful in the rest of the paper.
Lemma 2.7 (see ). For a given , the unique solution of the fractional separated boundary value problem is given by where
We notice that the solution (2.10) of the problem (2.9) does not depend on the parameter , that is to say, the parameter is of arbitrary nature for this problem. And by (2.10), we should assume that and .
Lemma 2.8. For any , the unique solution of the fractional nonseparated boundary value problem is given by
Proof. For , by Lemma 2.6, we know that the general solution of the equation can be written as where , are arbitrary constants. Since ( is a constant), , (see ), from (2.14), we have Using the boundary conditions, we obtain Therefore, we have Substituting the values of , in (2.14), we obtain (2.13). This completes the proof.
From the proof of the above lemma, we notice that the solution (2.13) of the problem (2.12) does not depend on the parameter , that is to say, the parameter is of arbitrary nature for this problem. In this situation, we need to assume that and .
Let be the space of all continuous functions defined on . Define the space and () endowed with the norm . We know that is a Banach space (see ).
We end this section with two fixed point theorems, which will be used in the sequel.
Theorem 2.11 (nonlinear alternative of Leray-Schauder type ). Let be a Banach space, a closed convex subset of , an open subset of with . Suppose that is an upper semicontinuous compact map. Then either has a fixed point in , or there is a and such that .
Theorem 2.12 (Covitz and Nadler Jr. ). Let be a complete metric space. If is a contraction, then has a fixed point.
3. Existence Results
For each , define the set of selections of by In view of Lemmas 2.7 and 2.8, we define operators as with It is clear that if is a fixed point of the operator (the operator ), then is a solution of the problem (1.4), (1.5) (the problem (1.4), (1.6)).
Now we are in a position to present our main results. The methods used to prove the existence results are standard; however, their exposition in the framework of problems (1.4), (1.5) and (1.4), (1.6) is new.
3.1. Convex Case
We consider first the case when is convex valued.: is a Carathéodory multivalued map; there exist and continuous, nondecreasing such that for and a.e. .
Proof. Consider the operator defined by (3.2). From , we have for each , the set is nonempty . For , let and , that is, , we have
where is a constant given by
Hence we know that the operator is well defined.
We put where Here means that the constant defined by (3.9) is related to .
We will show that satisfies the requirements of the nonlinear alternative of Leray-Schauder type. The proof will be given in five steps.
Step 1 ( is convex valued). Since is convex valued, we know that is convex and therefore it is obvious that is convex for each .
Step 2 ( maps bounded sets into bounded sets in ). Let be a bounded subset of such that for any , , . We prove that there exists a constant such that for each , one has for each . Let and , then there exists such that By simple calculations, we have Hence we obtain
Step 3 ( maps bounded sets into equicontinuous sets in ). Let be a bounded set of as in Step 2. Let and . For each , then there is such that . Since we obtain that (since , and ) and the limits are independent of and .
Step 4 ( has a closed graph). Let , , and ; we need to show . Now implies that there exists such that for . Let us consider the continuous linear operator given by and denote . Then and We apply Lemma 2.3 to find that has closed graph and from the definition of we get . Since , , it follows the existence of such that . This means that .
Step 5 (a priori bounds on solutions). Let for some . Then there exists such that for . With the same arguments as in Step 2 of our proof, for each , we obtain Thus Now we set Clearly, is an open subset of and . As a consequence of Steps 1–4, together with the Arzela-Ascoli theorem, we can conclude that is upper semicontinuous and completely continuous. From the choice of the , there is no such that for some . Therefore, by the nonlinear alternative of Leary-Schauder type (Theorem 2.11), we deduce that has a fixed point , which is a solution of the problem (1.4), (1.5). This completes the proof.
Proof. To obtain the result, the main aim is to study the properties of the operator defined in (3.3). The proof of them is similar to those of Theorem 3.1, so we omit the details. Here we just give some estimations, which are needed in the following theorems. Let and ; then there exists such that We put and here and are constants given by By simple calculations, we have Hence we obtain This is the end of the proof.
3.2. Nonconvex Case
Now we study the case when is not necessarily convex valued.
A subset of is decomposable if for all and Lebesgue measurable, then , where stands for the characteristic function.: is a multivalued map such that is measurable; is lower semicontinuous for a.e. .
Proof. From (H1)(1.5), (H2), and [33, Lemma 4.1], the map
is lower semicontinuous and has nonempty closed and decomposable values. Then from a selection theorem due to Bressan and Colombo , there exists a continuous function such that for all . That is to say, we have for a.e. . Now consider the problem
with the boundary conditions (1.5). Note that if is a solution of the problem (3.29), then is a solution to the problem (1.4), (1.5).
Problem (3.29) is then reformulated as a fixed point problem for the operator defined by It can easily be shown that is continuous and completely continuous and satisfies all conditions of the Leray-Schauder nonlinear alternative for single-valued maps . The remaining part of the proof is similar to that of Theorem 3.1, so we omit it. This completes the proof.
The proof of this theorem is similar to that of Theorem 3.3.: is a multivalued map such that is integrably bounded and the map is measurable for all ; there exists such that for a.e. and all , , , ,
Proof. From , we have that the multivalued map is measurable [28, Proposition ] and closed valued for each . Hence it has measurable selection [28, Theorem ] and the set is nonempty. Let be defined in (3.2). We will show that, under this situation, satisfies the requirements of Theorem 2.12.
Step 1. For each , . Let , such that in . Then and there exists , such that By , the sequence is integrable bounded. Since has compact values, we may pass to a subsequence if necessary to get that converges to in . Thus and for each This means that and is closed.
Step 2. There exists such that Let , and ; then there exists such that From , we deduce Hence, for a.e. , there exists such that Consider the multivalued map given by Since , are measurable, [35, Theorem ] implies that is measurable. It follows from that the map is measurable. Hence by (3.38) and [28, Proposition ], the multivalued map is measurable and nonempty closed valued. Therefore, we can find such that for a.e. ,