On two fractional differential inclusions.

We investigate in this manuscript the existence of solution for two fractional differential inclusions. At first we discuss the existence of solution of a class of fractional hybrid differential inclusions. To illustrate our results we present an illustrative example. We study the existence and dimension of the solution set for some fractional differential inclusions.

Background

As you know, fractional dynamical systems be used in modeling of some real processes and there are many published works about the existence of solutions for many fractional differential equations (see for example Baleanu et al. 2013a, b, c; Chai 2013 and the references therein) and inclusions (see for example, Benchohra and Hamidi 2010; Agarwal et al. 2013; Ahmad et al. 2013; Nieto et al. 2013; Ouahab 2008; Phung and Truong 2013; Bragdi et al. 2013 and the references therein). For finding more details about elementary notions and definitions of fractional differential equations and inclusions one can study well-known books (see for example Aubin and Ceuina 1984; Deimling 1992; Kilbas et al. 2006; Kisielewicz 1991; Podlubny 1999). Recently, it has been published many useful works about modeling of fractional differential equations via providing different applications in some fields (see for example Atangana 2016; Atangana and Alkahtani 2016; Atangana and Koca 2016a, b). In this article, we first review the existence solution for the fractional hybrid derivative inclusion with boundary conditions and , where , , , , denotes Caputo fractional derivative of order , is continuous and is a multifunction via some properties. Also, we review existence and dimension of the solution set of fractional derivative inclusionwith boundary condition , where and for , is a multifunction via some properties, are two mappings with the properties and and the functions and are defined by and .

Preliminaries

Suppose that be a metric space. Denote by and the class of all subsets and the class of all nonempty subsets of respectively. Here, , , and denote the class of all closed, bounded, convex and compact subsets of respectively. A mapping is called a multifunction on and is called a fixed point of whenever (Deimling 1992). A multifunction is called lower semi-continuous whenever the set is open for each open subset A of (Kisielewicz 1991). If the set is open for each open set A of , then we say that is upper semi-continuous (Kisielewicz 1991). A multifunction is called compact whenever is a compact for each bounded subsets S of (Aubin and Ceuina 1984). A multifunction is said to be measurable whenever the function is measurable for all and (Deimling 1992). The Pompeiu–Hausdorff metric on into is defined by , where (Berinde and Pacurar 2013). Then is a metric space and is a generalized metric space (Berinde and Pacurar 2013). A multifunction is called a contraction whenever there exists such that for all (Covitz and Nadler 1970). Covitz and Nadler (1970) proved that each closed valued contractive multifunction on a complete metric space has a fixed point. We say that is a Caratheodory multifunction whenever is measurable for all and is an upper semi-continuous map for almost all (see Aubin and Ceuina 1984; Deimling 1992; Kisielewicz 1991). Also, a Caratheodory multifunction is called -Caratheodory whenever for each there exists such thatfor all and for almost all (see Aubin and Ceuina 1984; Deimling 1992; Kisielewicz 1991).

Lemma 1

(Deimling 1992) Ifis upper semi-continuous, thenGr(G) is a closed subset of. IfGis completely continuous and has a closed graph, then it is upper semi-continuous.

Lemma 2

(Lasota and Opial 1965) Suppose thatis a Banach space,an-Caratheodory multivalued anda linear continuous mapping fromto. Then the mappingdefined byis a closed graph mapping in.

Theorem 3

(Dhage 2006) Suppose thatis a Banach algebra space,andandtwo multifunctions satisfying the following conditions

is Lipschitz with a Lipschitz constantk,

is upper semi-continuous and compact,

is a convex subsetSfor all,

, where

Then, there existssuch that.

Lemma 4

(Agarwal et al. 2013) Suppose thatis a measurable map such that the Lebesgue measureof the setis zero. Then there are arbitrarily many linearly independent measurable selectionsofG.

Theorem 5

(Agarwal et al. 2013) Suppose thatCis a nonempty closed convex subset of Banach space. Letb a-contraction. Iffor all, then.

Main results

First, we review the fractional hybrid differential inclusionwith the boundary conditions and , where , , , , denotes Caputo fractional derivative of order , is continuous and is a multifunction via some properties.

Lemma 6

Suppose that, and. The unique solution of the fractional differential problemwith the boundary value conditionsandis given by

Proof

The general solution of the equation is where are arbitrary constants (see Kilbas et al. 2006; Podlubny 1999). By using the boundary conditions, we get and . Hence, and This completes the proof.

is solution for the problem (1) whenever it satisfies the boundary conditions and there exists a function such thatwhere .

Theorem 7

Letbe a Caratheodory multifunction, is a continuous and bounded function with boundKand there exist continuous functionssuch thatandfor all. Ifthen the problem (1) has a solution.

Define , whereclearly S is a closed, bounded and convex subset of the Banach algebra space . Now, consider the multivalued operators byandThus, the problem (1) is tantamount to the problem . We prove that the multifunctions and well-defined the conditions of Theorem 3. Note that, the operator , where is the continuous linear operator on into defined byLet be arbitrary and a sequence in . Then, for almost . Because is compact for all , there exists a convergent subsequence of (we show it again ) to some . Since is continuous, pointwise on J. Because we will show that the convergence is uniform, we have to prove that is an equi-continuous sequence. Suppose that . So, we haveHence the right hand of above inequalities tends to 0 as and so the sequence is equi-continuous. By using the Arzela–Ascoli theorem, it has a uniformly convergent subsequence. Thus, there is a subsequence of (we show it again by ) such that . Hence, . Thus, is compact for all . Now, we show that is convex for all . Let and . Choose such thatfor almost all . Let . Then, we haveSince G is convex valued, . Cleary, is bounded, closed and convex valued. We prove that is a convex subset of S for all . Suppose that and . Choose such thatfor almost all . Hence, we getSince G is convex valued, . So, is convex subset of for all . But, we havefor all . So, and is a convex subset of S for all . Here, We show that operator is compact. For showing this, it is enough to prove that is uniformly bounded and equi-continuous. Let . Choose such that for some . Hence,and so In this part, prove that maps S to equi-continuous subsets of . Suppose that with , and . Choose such that . Then, we haveSo the right side of inequality towards to 0 as . Hence by using the Arzela–Ascoli theorem, is compact. Here, we show that has a closed graph. Suppose that and for all n such that and . We show that . For each natural number n, choose such thatfor all . Again, consider the continuous linear operator such thatBy using Lemma 2, is a closed graph operator. Since and for all n, there is such thatHence, . This implies that, has a closed graph and thus the operator is upper semi-continuous. Now, we show that is a contractive multifunction. Note that,for all . So, and satisfy the conditions of Theorem 3 and thus the operator inclusions has a solution in S. Therefore, the problem (1) has a solution.

To illustrate our main results, we present the following example:

Example 1

Here, we investigation the problemwith the boundary conditions and . Put , , , , , , , , and for . Note that, ,and . By using the Theorem 7, the problem has a solution.

Now, we review existence and dimension of the solution set of the fractional differential inclusion problemwith boundary condition , where and for , is a multifunction via some properties, are two mappings with the properties and and the functions and are defined by and .

Lemma 8

Suppose that, andwithfor. Then solution of the problemwith the boundary condition is

The general solution of the problem is formed bywhere is arbitrary constant and (Podlubny 1999). Thus, we obtainandfor all . By using the boundary condition, we getHence,This completes the proof.

An element is a solution for the problem (2) whenever it satisfies the boundary condition and there is a function such thatfor almost all andPut with the norm is a Banach space (Su 2009). Define selection set of G at by

Theorem 9

Suppose that, , wherefor, is a multifunction such that the mapis measurable,andfor almost alland. Then the inclusion problem (2) has a solution.

Note that, the multivalued mapis measurable and closed valued for all . Hence, it has a measurable selection and so the set is nonempty. Now, consider the operator defined bywherefor all . Here, we prove that is a closed subset of for each . Let and be a sequence in with . For each n, choose such thatfor almost all . because G has compact values, has a subsequence which converges to a . We write it again by . Clearly andfor all . This implies that . Thus, the multifunction has closed values. Now, we show that is a contractive multifunction with constant . Suppose that and . Consider such thatfor almost all . Sincefor almost all , there issuch thatfor almost all . Define the multifunction byThe multifunctionis measurable. Thus, we can choose such thatfor almost all . Now, define bySoand and so we get . This implies that the multifunction N is a contraction via closed values. By using the well-known theorem of Covitz and Nadler, N has a fixed point which is a solution for the inclusion problem (2).

Lemma 10

Suppose that, is a multifunction such that the mapis measurable,for almost allandandis defined bywhereThen, for all.

Note that the operator , where is the continuous linear operator on into which is defined bySuppose that and is a sequence in . so,for almost . Sinceis compact for all , there exists a convergent subsequence of (we show it again by ) which converges to some . Since is continuous, pointwise on J. Because we show that the convergence is uniform, we must prove that is an equi-continuous sequence. Let . ThenNote that, the right side of the inequality towards to zero when . So, the sequence is equi-continuous and so by using the Arzela–Ascoli theorem there is a uniformly convergent subsequence. Thus, there exists a subsequence of (we show it again by ) such that . This implies that . Hence, is compact for all . Now, we prove that is convex for each . Let . Choose such thatandfor almost all . Let . Then, we haveSince is convex, . This completes the proof.

One can check that the fixed point set of is equal to the set of all solutions of the problem (2).

Theorem 11

Suppose that, is a multifunction such that the mapis measurable,andfor almost alland. If Lebesgue measure of the setis zero and , then the set of all solutions of the problem (2) is infinite dimensional, wherelis defined in Theorem9.

Define the operator bywhereBy using Lemma 10, for each . Like to the Theorem 9, is contraction. we show that for each and . Let andfor all . By using Lemma 4, there exist linearly independent measurable selections for F. Putfor . Assume that for almost . By using the Caputo derivatives, we get for almost . Hence, . This implies that are linearly independent. Thus, . Now by using Theorem 5, the set of fixed points of is infinite dimensional.

Conclusions

The existence of solution for fractional differential inclusions is an important task which can be used successfully in solving real world problems from many fields of science and engineering. Thus, in our paper we analyze firstly the existence of solution of a given class of fractional hybrid differential inclusions. An example was give in order to show the reported results Secondly we concentrate our attention on proving the existence and dimension of the solution set for some fractional differential inclusions. These results are useful for the numerical studies involving the investigated equations.