Some Results for Nonlinear ()-Term Fractional Integrodifferential Inclusions with Multipoint Boundary Conditions
We are concerned with the nonlinear -term fractional integrodifferential inclusions , , where , , are constant coefficients, and , , subject to the nonlocal conditions , The existence results are obtained by using two fixed-point theorems due to Bohnenblust-Karlin and Covitz-Nadler, respectively. Our results partly generalize and improve the known ones.
Fractional differential equations (FDEs) have received increasing interest for the last three decades. It is benefited by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science, engineering economics, and other fields, see for instance [1–12] and references therein.
Babakhani and Gejji  considered the existence of positive solutions for the nonlinear fractional differential equations where , , , , and is the standard Riemann-Liouville fractional derivative. Some existence results of positive solutions are obtained by using some fixed-point theorems on a cone.
Stojanović  considered the existenceuniqueness of solutions for a nonlinear -term fractional differential equation where , with initial data , .
On the other hand, realistic problems arising from economics, optimal control, and so on can be modeled as differential inclusions. Recently, El-Sayed and Ibrahim  initiated the study of fractional differential equations inclusions. Differential inclusions have been widely investigated by many authors, see [14–30] and references therein.
Very recently, in the survey paper , Agarwal et al. establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problems for fractional differential equations and inclusions involving the Caputo fractional derivative.
Ouahab  studied the following boundary value problem of fractional differential inclusions: where is the standard Riemann-Liouville fractional derivative, is a multivalued map with compact values.
Chang and Nieto  studied boundary value problem of fractional differential inclusions where is the Caputo's derivative, .
However, to the best of our knowledge, the existence of solutions for fractional integro-differential inclusions with multipoint boundary conditions has not been paid much attention. Our goal is to fill this gap in literature.
In the present work, we consider more general fractional integro-differential inclusions with multipoint boundary conditions where , , , , is the standard Riemann-Liouville fractional derivative, , , , , are two continuous functions on , and is a given multivalued function ( is the family of all nonempty subsets of ). We shall consider both the cases of convex and nonconvex valued right hand side and establish some sufficient conditions which admit that the integro-differential inclusions problem has at least one solution. These results obtained by applying two fixed point theorems due to Bohnenblust-Karlin and Covitz-Nadler are complement of previously known results.
The organization of this paper is as follows. In Section 2, we present some necessary definitions and preliminaries which are used throughout this paper. Main results and their proofs are given in Section 3.
2. Preliminaries and Several Lemmas
In this section, we recall some basic definitions and notations and give several lemmas which are useful in our discussion.
Definition 2.1. The Riemann-Liouville fractional integral of order of a function is given by provided the right side is pointwise defined on .
Definition 2.2. The Riemann-Liouville fractional derivative of function is given by where , , provided the right side is pointwise defined on .
Lemma 2.3. Let , then for some , , .
Remark 2.4. If the fractional derivative is integrable, then If is continuous on , then , and (2.4) reduces to
Lemma 2.5. Let , , and . Then is a solution of the BVP if and only if satisfies the integral equation
Proof. In view of Lemma 2.3 and Remark 2.4, (2.6) is equivalent to the integral equation
for some , that is,
The boundary condition implies . Thus,
In view of the boundary condition , we conclude that
Therefore, the solution of (2.6) and (2.7) satisfies (2.8).
Conversely, if is a solution of (2.8), it is easy to verify that satisfies (2.6) and (2.7). The proof is complete.
Now, we recall some facts from multivalued analysis.
Let be a metric space, and , , , , , , and so forth.
Consider , given by where , . Then is a metric space and is a generalized metric space (see ).
A multivalued map is said to be measurable if for each , the function , defined by is measurable.
A multivalued map is convex (closed) valued if is convex (closed) for all . is bounded on bounded sets if is bounded in for all . That is, . is called upper semicontinuous (u.s.c for short) on if for each , the set is nonempty closed subset of , and if for each open set of containing , there exists an open neighborhood of such that . is said to be completely continuous if is relatively compact for every . If the multivalued map is completely continuous with nonempty compact valued, then is u.s.c. if and only if has closed graph, that is, , , imply .
Definition 2.6. The multivalued map is -Carathéodory if (i) is measurable for each ; (ii) is upper semicontinuous for almost all ; (iii)for each , there exists such that for all and for almost all ,
For any , we define the set which is known as the set of selection functions.
Lemma 2.7 (see ). Let be a Banach space. Let be an -Carathéodory multivalued map with , and let be a linear continuous mapping from into . Then the operator is a closed graph operator in .
The following Bohnenblust-Karlin fixed-point lemma and the fixed-point theorem for contractive multivalued operators given by Covitz and Nadler are of great importance in the proofs of our main results. The proofs of these results can be found in Bohnenblust and Karlin  and in Covitz and Nadler .
Lemma 2.8 (see ). Let be a Banach space, be a nonempty subset of , which is bounded, closed, and convex. Suppose is u.s.c. with closed, convex values, and such that and is relatively compact. Then has a fixed point.
Lemma 2.9 (see ). Let be a complete metric space. If is a contraction operator, then .
For convenience, let us list some conditions. (H1);(H2) is Carethéodory multivalued map.
Lemma 2.10. Assume that hypothesis (H1) is satisfied. For any , the integral equation has a unique solution in .
Proof. We define the operator as follows: Obviously, is a map from into . Also, we have Therefore, , where . By (H1) and Banach contraction principle, the conclusion of lemma is true.
3. Main Results
In this section, we present our main results and prove them. Firstly, under convexity condition on the multivalued right-hand side, we are to establish the existence theorem of solutions for fractional differential inclusions (1.5) and (1.6), by employing the Bohnenblust-Karlin fixed-point theorem. Then, under nonconvexity condition on the multivalued right-hand side, the existence theorem of solutions are gotten, by employing the Covitz-Nadler fixed-point theorem.
Proof. To transform the problem into a fixed-point problem, we consider the multivalued operator, , where for any is defined by
For any , we have , , . In view of Lemma 2.10, is well defined. Moreover, it follows from the convexity of (because has convex values) that is convex for each . Clearly, the fixed points of are solutions of (1.5) and (1.6).
We shall show that has a fixed point in three steps.
Step 1. we claim that there exists a , such that , where .
In fact, if it is not true, then for any , there exists a function , but , that is and for some ,
On the other hand, from (H2), we obtain Thus,
Dividing both sides by and taking the lower limit as , we get
This is a contraction to (H3). Hence there exists a such that .Step 2. is equicontinuous.
Let , , , and . Then there exists such that for each , we have Therefore, where
The right-hand side of the above inequality tends to zeros independently of , as . This shows that is equicontinuous. Step 3. has a closed graph.
Let , and . We will prove that . Now implies that there exists such that for each ,
We need to show that there exists such that for each , In (2.18), taking In view of Lemma 2.10, we know that for any , the integral equation (2.18) has a unique solution, which we denote by . Because of this, we can define the operator It is easy to verify that the operator is linear. On the other hand, we can get where that is, , which implies is continuous. From Lemma 2.7, it follows that is a closed graph operator. Moreover, we know . Since , it follows from Lemma 2.7 that (3.11) holds for some .
Therefore, is a compact multivalued map, u.s.c. with convex closed values. As a consequence of Lemma 2.8, we immediately conclude that has a fixed-point which is a solution of the problem (1.5) and (1.6). The proof is complete.
As a direct corollary of Theorem 3.1, we can immediately obtain the following result when the nonlinearity has sublinear growth in the state variable.
Proof. We only need to verify that (H4) implies (H3) in Theorem 3.1. Taking , it is easy to prove because of . Then (H3) is satisfied. The proof is finished.
Theorem 3.3. Assume that the following hypotheses hold: (A1) ; is measurable for each ;(A2)There exist three functions such that for a.e. and , Then the BVP (1.5) and (1.6) has at least one solution in provided that where and defined as in Lemma 2.10, defined by (3.1) in Theorem 3.1.
Proof. Transform the problem into a fixed-point problem. Let be defined as in Theorem 3.1. We shall show that satisfies the assumptions of Lemma 2.9. The proof will be given in two steps.Step 1. for each .
Indeed, let such that in . Then there exists such that for each ,
For every , from (A2), we have Then, that is, , where
It is clear that is a multivalued integrable bounded map. Since , we may pass to a subsequence if necessary to get that converges weakly to in . From Mauz's lemma , there exists , then there exists a subsequence