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On Higher-Order Sequential Fractional Differential Inclusions with Nonlocal Three-Point Boundary Conditions
We study a nonlinear three-point boundary value problem of sequential fractional differential inclusions of order with , . Some new existence results for convex as well as nonconvex multivalued maps are obtained by using standard fixed point theorems. The paper concludes with an example.
The topic of fractional differential equations has attracted a great attention in the recent years. It is mainly due to the intensive development of the theory and applications of fractional calculus. In fact, the tools of fractional calculus have considerably improved the modeling of several real world phenomena in physics, chemistry, bioengineering, etc. The systematic development of theory, methods, and applications of fractional differential equations can be found in [1–6]. For some recent results on fractional differential equations and inclusions, see [7–23] and the references cited therein.
In this paper, we study the following boundary value problem: where is the Caputo fractional derivative, is the ordinary derivative, is a multivalued map, is the family of all subsets of , , is a positive real number, and is a real number.
The present work is motivated by a recent paper of the authors , where the problem (1) was considered for a single-valued case. The existence of solutions for the given multivalued problem is discussed for three cases: (a) convex-valued maps; (b) not necessarily convex-valued maps; (c) nonconvex-valued maps. To establish the existence results, we make use of nonlinear alternative for Kakutani maps, nonlinear alternative of Leray-Schauder type for single-valued maps, selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, and a fixed point theorem for contractive multivalued maps due to Covitz and Nadler. The tools employed in this paper are standard; however, their exposition in the framework of the problem at hand is new.
The paper is organized as follows: in Section 2 we recall some preliminary facts that we used in the sequel. Section 3 contains the main results and an example. In Section 4, we summarize the work obtained in this paper and discuss some special cases.
Definition 1. For -times absolutely continuous function , the Caputo derivative of fractional order is defined as where denotes the integer part of the real number .
Definition 2. The Riemann-Liouville fractional integral of order is defined as provided the integral exists.
Definition 3. A function is called a solution of problem (1) if there exists a function with , a.e. , such that , a.e. , and , , , and .
For the forthcoming analysis, we define Furthermore, we assume the nonresonance condition, that is, for and , we choose such that
3. Existence Results
Let denote a Banach space of continuous functions from into with the norm . Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .
Let denote a normed space. Then we define
Definition 5. A multivalued map is convex- (closed-) valued if is convex (closed) for all .
Definition 6. The map is bounded on bounded sets if is bounded in for all (i.e., ).
Definition 7. is called upper semicontinuous (u.s.c.) on if for each , the set is a nonempty closed subset of , and if, for each open set of containing , there exists an open neighborhood of such that .
Definition 8. is said to be completely continuous if is relatively compact for every .
If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph; that is, , , and imply that . has a fixed point if there is such that . The fixed point set of the multivalued operator will be denoted by .
Definition 9. A multivalued map is said to be measurable if for every , the function is measurable.
3.1. The Carathéodory Case
Definition 10. A multivalued map is said to be Carathéodory if (i) is measurable for each ;(ii) is upper semicontinuous for almost all .Further a Carathéodory function is called -Carathéodory if (iii)for each , there exists such that for all and for .
For each , define the set of selections of by
For the forthcoming analysis, we need the following lemmas.
Lemma 11 (nonlinear alternative for Kakutanimaps ). Let be a Banach space, a closed convex subset of , an open subset of , and . Suppose that is an upper semicontinuous compact map. Then either(i) has a fixed point in , or(ii)there is a and with .
Lemma 12 (see ). Let be a Banach space. Let be an -Carathéodory multivalued map and let be a linear continuous mapping from to . Then the operator is a closed graph operator in .
Now we are in a position to prove the existence of the solutions for the boundary value problem (1) when the right-hand side is convex-valued.
Theorem 13. Assume that the nonresonance condition (6) holds. In addition, we suppose that(H1) is Carathéodory and has nonempty compact and convex values;(H2)there exist a continuous nondecreasing function and a function such that (H3)there exists a constant such that where ( is defined in (4) and (5)). Then the boundary value problem (1) has at least one solution on .
Proof. Define the operator by
for . We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that is convex for each . This step is obvious since is convex ( has convex values), and therefore we omit the proof.
In the second step, we show that maps bounded sets (balls) into bounded sets in . For a positive number , let be a bounded ball in . Then, for each , there exists such that Then for , we have Consequently,
Now we show that maps bounded sets into equicontinuous sets of . Let and . For each , we obtain
Obviously the right-hand side of the above inequality tends to zero independently of as . As satisfies the above three assumptions, therefore it follows from the Ascoli-Arzelá theorem that is completely continuous.
In our next step, we show that has a closed graph. Let , , and . Then we need to show that . Associated with , there exists such that, for each , Thus, it suffices to show that there exists such that, for each , Let us consider the linear operator given by Observe that as .
Thus, it follows from Lemma 12 that is a closed graph operator. Further, we have . Since , therefore, we have for some .
Finally, we show that there exists an open set with for any and all . Let and . 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 , there exists such that . Let us set Note that the operator is upper semicontinuous and completely continuous. From the choice of , there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 11), we deduce that has a fixed point which is a solution of the problem (1). This completes the proof.
Remark 14. The condition in the statement of Theorem 13 may be replaced with the following one.
There exists a constant such that where is the same as defined in .
3.2. The Lower Semicontinuous Case
As a next result, we study the case when is not necessarily convex-valued. Our strategy to deal with this problem is based on the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo  for lower semicontinuous maps with decomposable values.
Let be a nonempty closed subset of a Banach space and let be a multivalued operator with nonempty closed values. is lower semicontinuous (l.s.c.) if the set is open for any open set in . Let be a subset of . is measurable if belongs to the -algebra generated by all sets of the form , where is Lebesgue measurable in and is Borel measurable in . A subset of is decomposable if, for all and measurable , the function , where stands for the characteristic function of .
Definition 15. Let be a separable metric space and let be a multivalued operator. We say has a property (BC) if is lower semicontinuous (l.s.c.) and has nonempty closed and decomposable values.
Let be a multivalued map with nonempty compact values. Define a multivalued operator associated with as which is called the Nemytskii operator associated with .
Definition 16. Let be a multivalued function with nonempty compact values. We say is of lower semicontinuous type (l.s.c. type) if its associated Nemytskii operator is lower semicontinuous and has nonempty closed and decomposable values.
Lemma 17 (see ). Let be a separable metric space and let be a multivalued operator satisfying the property (BC). Then has a continuous selection; that is, there exists a continuous function (single-valued) such that for every .
Theorem 18. Assume that , , and the following condition hold:(H4) is a nonempty compact-valued multivalued map such that(a) is measurable,(b) is lower semicontinuous for each .Further the nonresonance condition (6) holds. Then the boundary value problem (1) has at least one solution on .
Proof. It follows from and that is of l.s.c. type. Then from Lemma 17, there exists a continuous function such that for all .
Consider the problem
Observe that if is a solution of (32), then is a solution to the problem (1). In order to transform the problem (32) into a fixed point problem, we define the 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 13. 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 and is a generalized metric space (see ).
Definition 19. A multivalued operator is called(a)-Lipschitz if and only if there exists such that (b)a contraction if and only if it is -Lipschitz with .
Lemma 20 (see ). Let be a complete metric space. If is a contraction, then .
Theorem 21. Assume that the nonresonance condition (6) holds. In addition, suppose that the following conditions hold: (H5) is such that is measurable for each ;(H6) for almost all and with and for almost all .Then the boundary value problem (1) has at least one solution on if
Proof. Observe that the set is nonempty for each by the assumption , so has a measurable selection (see Theorem ). Now we show that the operator , defined in the beginning of proof of Theorem 13, satisfies the assumptions of Lemma 20. To show that for each , let be such that in . Then and there exists such that, for each ,
As has compact values, we pass onto a subsequence (if necessary) to obtain that converges to in . Thus, and, for each , we have
Next we show that there exists such that Let and . Then there exists such that, for each ,
By , we have So, there exists such that
Define by Since the multivalued operator is measurable (Proposition ), there exists a function which is a measurable selection for . So and for each , we have .
For each , let us define
Analogously, interchanging the roles of and , we obtain
Since is a contraction, it follows from Lemma 20 that has a fixed point which is a solution of (1). This completes the proof.
Example 23. Consider the problem
Here, , , , , , and is a multivalued map given by For , we have Thus, with . In this case By the condition , that is, we find that with . Therefore, it follows from Theorem 13 that problem (49) has at least one solution.
In this paper, we have solved a three-point boundary value problem of Caputo-type sequential fractional differential inclusions of an arbitrary order . The existence of solutions for the given problem with the convex-valued map is obtained by means of nonlinear alternative for Kakutani maps, while the existence result for not necessarily convex-valued map is established by combining 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 decomposable values. The nonconvex-valued case relies on a fixed point theorem for contractive multivalued maps due to Covitz and Nadler. Some new existence results follow by fixing the parameters involved in the given problem. For instance, by taking , our results correspond to a two-point Caputo-type multivalued problem of an arbitrary order , while the results for sequential differential inclusions of order can be obtained by fixing in the results of this paper.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU. Sotiris K. Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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