Under different criteria, we prove the existence of solutions for sequential fractional differential inclusions containing Riemann–Liouville and Caputo type derivatives and supplemented with generalized fractional integral boundary conditions. Our existence results rely on the endpoint theory, the Krasnosel’skiĭ’s fixed point theorem for multivalued maps and Wegrzyk’s fixed point theorem for generalized contractions. We demonstrate the application of the obtained results with the help of examples.
Boundary value problems involving fractional differential equations and inclusions with a variety of boundary conditions have been investigated in recent years. For application details, we refer the reader to [1,2,3,4,5,6,7] and the references cited therein, while some recent results on boundary value problems of fractional differential inclusions can be found in [8,9,10,11,12,13,14,15,16,17,18,19,20,21].
In a recent work , the authors studied the existence of solutions for a nonlinear sequential Riemann–Liouville and Caputo fractional differential equation subject to generalized fractional integral conditions. The objective of the present paper is to investigate the multivalued analogue of the problem considered in . Precisely, we consider the following inclusions problem:
where and denote the left Riemann–Liouville and left Caputo fractional derivatives of order q and r, respectively; is a multivalued map; is the family of all nonempty subsets of ; is the generalized fractional integral of order with parameters , , , , , and for all , . For definitions of fractional derivatives and integrals involved in the problem (1) and (2), see . Here we emphasize that the boundary conditions (2) correspond to different kinds of integral boundary conditions for appropriate choice of the parameters; for details, see Remark 2 in .
In the sequel, we need the following known lemma.
 Let , , , , q, r, , , , , , , for , and . The unique solution of the following linear sequential fractional differential equation
subject to the generalized fractional integral conditions
is given by the formula
where it is assumed that ,
In view of Lemma 1, we can define a solution of problem (1) and (2) as follows.
A function is called a solution of problem (1) and (2) if there exists a function with a.e. in such that and
For the sake of computational convenience, we set
2. Main Results
In this section, we present our main results in different subsections by applying a variety of fixed point theorems for multivalued maps.
2.1. Existence Result via Endpoint Theory
Denote by the Banach space of all continuous functions from into with the norm The space of functions such that is denoted by
Let be a metric space induced from the normed space . Consider given by where and . Then is a metric space (see ).
Now we state the endpoint fixed point theorem that will be applied to prove our first result.
() Let be a complete metric space and be a multifunction such that where is the collection of all nonempty closed and bounded subsets of X and is an upper semi-continuous function such that Then has a unique endpoint if and only if has an approximate endpoint property.
For more details about endpoint theory, we refer the reader to the article .
By Lemma 1, we define an operator as follows:
for where denotes the set of selections of F defined by
Suppose that is a nondecreasing upper semi-continuous mapping such that and for all . Moreover, assume that is an integrable bounded multifunction such that is measurable for all here denotes the collection of all nonempty compact subsets of . In addition, we assume that there exists a function such that
where Φ is defined by (6). If the multifunction (defined by (7)) has the approximate endpoint property, then the inclusion problem (1) and (2) has a solution.
Our proof will be complete once it is shown that the multifunction defined by (7) has an endpoint. To do this, we show that the operator is a closed subset of for all . Note that the multivalued map is measurable and has closed values for all and therefore has a measurable selection. So is nonempty for all . Let be a sequence in with for For every , choose such that
In view of the compactness of F, we deduce that the sequence has a subsequence which converges to some . Let this subsequence be denoted by again. Clearly and for all
Thus and consequently is closed-valued. On the other hand, is a bounded set for all as F is a compact multivalued map.
Now we establish that . Let and . Choose such that
for almost all . Since
there exists provided that
Let us consider the multivalued map given by
The multifunction is measurable, since and are measurable. Choose such that
We define the element as follows:
Let . Then, one can get
which implies that Therefore for all . Therefore there exists such that , since the multifunction (by the hypothesis) has an approximate endpoint property. Therefore we deduce that the problem (1) and (2) has a solution The proof is complete. □
Consider the following inclusions problem containing Riemann–Liouville and Caputo derivative operators with generalized fractional integral boundary conditions
Here , , , , , , , , , , , , , , , , , , , , . From the given data, we find that , , , , and
Let be a multi-valued map defined by
Let us take and note that it is nondecreasing upper semi-continuous on satisfying and for all . Set , such that . Therefore, we have
for x, . Letting define an operator by
Observe that in view of Therefore, the operator has the approximate endpoint property. Clearly the hypothesis of Theorem 1 is satisfied. So the conclusion of Theorem 1 applies to the problem (8) with given by (9).
2.2. Existence Result via Krasnosel’skiĭ’s Multi-Valued Fixed Point Theorem
This subsection is concerned with the second existence result for the problem (1) and (2) when the map F in (1) is Carathéodory and convex valued. The proof of this result relies on Krasnosel’skiĭ’s fixed point theorem for multivalued maps .
In the following we need the following assumptions:
(H1). is Carathéodory multivalued map, where is the collection of all nonempty compact and convex subsets of ;
(H2). There exists a function such that
(H3). There exists a function satisfying for a.e and all and that where
If the assumptions H1–H3 are satisfied, then the problem (1) and (2) have at least one solution on
Define the multivalued operators by
Notice that where is given by (7). In several steps, it will be shown that and satisfy the hypothesis of Krasnosel’skiĭ’s multivalued fixed point theorem.
Let be a bounded set in . The operators and define the multivalued operators Observe that the operator is equivalent to the composition , where is the continuous linear operator on into , defined by
For an arbitrary element , let be a sequence in . Then for almost all . As is compact for all , we can find a convergent subsequence of (also labeled as ) converging in measure to some for almost all . On the other hand, continuity of implies that pointwise on .
To ensure the uniform convergence, we have to establish that is an equi-continuous sequence. Take with and Then
Obviously the right hand of the above inequality tends to zero as independent of . So is an equi-continuous sequence. In consequence, the Arzelá-Ascoli theorem applies and hence there exists a uniformly convergent subsequence (labeled as again) such that . So and hence is compact for all . Therefore, is compact.
In order to show that is convex for all let . We select such that
for almost all . Then, for we have
So is convex as . Thus Therefore, is convex-valued. The proof for is similar.
We split the remaining proof in several steps and claims.
is a multi-valued contraction on . Take and Then, for each , there exists such that
Since therefore, we can find satisfying
Define Then the multivalued operator U defined by is measurable and nonempty. Let be a measurable selection for which exists by Kuratowski-Ryll-Nardzewski’s selection theorem . Then and for each , we have a.e. on
For each , let us define
It follows that and
Interchanging the roles of x and we obtain an analogous inequality:
which, together with the condition implies that is a multivalued contraction.
We show that is compact and upper semicontinuous through certain claims.
Claim I: maps bounded sets into bounded sets in .
Let be a bounded ball in with Then, for each , we can find satisfying
Then we have
Claim II: maps bounded sets into equi-continuous sets.
Let with and For each we obtain
independently of Then, by the Arzelá-Ascoli theorem, we deduce that is completely continuous.
Thus it follows by Claims I and II that is completely continuous. Hence, by Proposition 1.2 in , it will be upper semicontinuous once it is shown to be closed graph. This will be shown in the next claim.
Claim III: has a closed graph.
Letting and we show that For we can find such that, for each
We will show that there exists such that for each
If we consider the linear operator given by
then we note that
Thus, it follows by a closed graph result  that is a closed graph operator. Further, let Since we have that
for some . Hence has a closed graph. Thus, the operator is compact and upper semicontinuous.
Now, we establish that for all Take arbitrary elements with ( defined by (6)) and Then, selecting we have
Then we have
which leads to
This shows that for all
Thus, the operators and verify the hypothesis of Krasnosel’skiĭ’s multivalued fixed point theorem and hence there exists a solution in Therefore there exists a solution of the problem (1) and (2) in which completes the proof.
Consider problem (8) with the multi-valued map defined by
and for By using the data in problem (8), we find that . Setting , we find that . Clearly all the assumptions of Theorem 2 hold true, and consequently, problem (8) with given by (12) has a solution by Theorem 2 on .
2.3. Existence Result via Wegrzyk’s Fixed Point Theorem
In this subsection we apply Wegrzyk’s fixed point theorem  to prove an existence result for problem (1) and (2) when the right hand side of the inclusions (1) is not necessarily nonconvex valued.
Let us first recall that a multivalued operator is a generalized contraction if and only if there is a strict comparison function (continuous, strictly increasing and , for each ) satisfying Here
(Wegrzyk’s fixed point theorem ). Let be a complete metric space. If is a generalized contraction, then
is such that is measurable for each ();
There exists with for almost all and a strictly increasing function such that
If is a strict comparison function, where (Φ is defined by (6)), then the boundary value problem (1) and (2) has at least one solution on
Suppose that is a strict comparison function. Notice that is measurable and has a measurable selection by the assumptions and (see Theorem III.6 ). As we have
Thus the set is nonempty for each .
Now we verify that the operator satisfies the hypothesis of Lemma 3. Let us first show that for each Let be such that in as Then and we can find such that, for each ,
Since F is compact valued, we pass onto a subsequence (if necessary) to get that converges to v in So and for each , we have
Next we establish that
Let and . Then there exists such that, for each ,
By , we have
In consequence, we can find satisfying
By Proposition III.4 , the multivalued operator is measurable and there exists a function which is a measurable selection for . Hence, and we have , for each .
For each , let us define
Hence Analogously, one can obtain by interchanging the roles of x and that for each Thus is a generalized contraction. So, by Lemma 3, we deduce that has a fixed point which corresponds to a solution of problem (1) and (2). This finishes the proof. □
Fix such that for almost all and From Example 1, we have In consequence Letting all the conditions of Theorem 3 are satisfied. Therefore the problem (8) with given by (13) has has at least one solution on by the conclusion of Theorem 3.
In this paper, we presented three existence results for sequential fractional differential inclusions involving Riemmann–Liouville and Caputo type derivatives, subject to generalized fractional integral boundary conditions. These results provide different criteria for the existence of solutions for the problem at hand. The first result (Theorem 1) is obtained with the aid of the endpoint theory, while Krasnosel’skiĭ’s fixed point theorem for multivalued maps is applied to derive the second result (Theorem 2). In the third result (Theorem 3), we used Wegrzyk’s fixed point theorem for generalized contractions to establish the existence of solutions for the given problem. It is imperative to note that Wegrzyk’s fixed point theorem is a generalization of Covit and Nadler’s fixed point theorem  in the sense that it deals with the generalized contractions. Thus Theorem 3 holds for several values of the function (defined in ); for example, , (contraction case), etc. Some interesting special cases of our results follow by fixing the parameters involved in the boundary conditions (2). For instance, our results correspond to Dirichlet boundary conditions if we take for all and for all . Fixing for all and for some , we obtain the existence results for the boundary conditions of the form: . On the other hand, we obtain the existence results for (1) associated with the boundary conditions: by letting for some , and for all . We emphasize that the existence results indicated for special forms of the boundary conditions are new.
Conceptualization, J.T., S.K.N. and B.A.; formal analysis, J.T., S.K.N., B.A. and A.A.; funding acquisition, J.T.; methodology, J.T., S.K.N., B.A. and A.A. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-61-KNOW-021.
Conflicts of Interest
The authors declare no conflict of interest.
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