Fixed Point and Endpoint Theories for Two Hybrid Fractional Differential Inclusions with Operators Depending on an Increasing Function

Rezapour, Sh.; Iqbal, M. Q.; Hussain, A.; Zada, A.; Etemad, S.

doi:https://doi.org/10.1155/2021/4512223

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Advances in Fractional Functional Analysis

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Research Article | Open Access

Volume 2021 | Article ID 4512223 | https://doi.org/10.1155/2021/4512223

Fixed Point and Endpoint Theories for Two Hybrid Fractional Differential Inclusions with Operators Depending on an Increasing Function

Sh. Rezapour,^1,2M. Q. Iqbal,³A. Hussain,³A. Zada,⁴and S. Etemad²

Academic Editor: Emanuel Guariglia

Received26 Jun 2021

Accepted17 Aug 2021

Published02 Sept 2021

Abstract

The main concentration of the present research is to explore several theoretical criteria for proving the existence results for the suggested boundary problem. In fact, for the first time, we formulate a new hybrid fractional differential inclusion in the -Caputo settings depending on an increasing function subject to separated mixed -hybrid-integro-derivative boundary conditions. In addition to this, we discuss a special case of the proposed -inclusion problem in the non--hybrid structure with the help of the endpoint notion. To confirm the consistency of our findings, two specific numerical examples are provided which simulate both -hybrid and non--hybrid cases.

1. Introduction

Arbitrary order calculus theory is considered as an important topic of research for all mathematicians, researchers, engineers, and scientists due to the applicability of mentioned theory in several contexts in engineering and applied science and its flexibility to model different systems and phenomena having memory effects (see, e.g., [1–3] and reference therein). Several arbitrary order derivatives have been introduced in the past decade, and the most common of them are Riemann-Liouville, Caputo, and Hadamard derivatives. Hence, arbitrary order boundary value problems (BVPs) can be formulated in the framework of different operators. In the meantime, some recent research investigations have been conducted with the aid of these operators to establish the relevant analytical results for proposed BVPs. For instance, Alzabut et al. [4] investigated the oscillatory behavior of a kind of fractional differential equations (FDEs) supplemented with damping and forcing terms by terms of generalized proportional operators. In [5], Baleanu et al. modeled an applied instrument in engineering in the context of a hybrid Caputo FBVP and studied its existence theory. Also, the same authors [6] established similar results by means of Caputo and Riemann-Liouville conformable derivation and integration operators. In 2019, Matar et al. [7] devoted their focus on solvability of nonlinear systems of FDEs via nonlocal initial value problems by terms of fixed point methods and after that, Mohammadi et al. [8] utilized another fractional operator entitled Caputo–Hadamard for modeling a hybrid FBVP with Hadamard integral boundary conditions. Zhou et al. [9] presented a fractional antiperiodic model of Langevin equation and investigated qualitative aspects of its solutions with the aid of techniques appeared in functional analysis. Similarly, one can find some papers on applications of fractional operators [10–13].

In 2017, a generalization of the Caputo fractional operator known as -Caputo derivative (-CF) was presented by Almeida [14] in which its kernel is with respect to a given increasing function . One of the most important advantages of the -CF derivative operator is its ability to produce all previous fractional derivatives, and also, it involves the semigroup property. As a result, -CF derivative is known as an extended structure of arbitrary order derivative operators.

To get acquainted with some previous research works done based on -CF operators so far, we refer to a paper published by Wahash et al. [15]. In that paper, Wahash et al. designed a generalized -fractional differential equation with a simple integral condition as where , , , and , and also, stands for a continuous function along with . The lower-upper solution is a technique implemented in that article by Wahash et al. in which they utilized a fixed point method on cones. Further, lower-upper control maps are provided with respect to the nonlinear term without a certain monotonicity criterion [15]. Similarly, by using the newly introduced -CF operator and its generalizations, several articles have been published like as [16] in which Almeida et al. considered a FDE via a Caputo derivative with respect to a kernel function and reviewed some applications of them. Derbazi et al. [17] used such a generalized operator to investigate a nonlinear initial value problem via monotone iterative method. Samet et al. derived some Lyapunov-type inequalities in relation to an antiperiodic FBVP involving -Caputo operator [18]. The analysis of the stability to an -Hilfer impulsive FDE is another instance of applications of such generalized operators which was studied by Sousa et al. in [19]. In 2020, Tariboon et al. [20] turned to establishment of existence theorems to sequential generalized inclusion FBVP, and then, Thabet et al. [21] achieved to similar findings on a new structure of the pantograph inclusion FBVP. In a higher level, Vivek et al. [22] defined generalized -operators in the context of partial operators and analyzed a PDE in the -Caputo settings.

With regard to ideas of aforesaid research works, we consider the following -hybrid fractional differential inclusion in the sense of Caputo represented as

supplemented with separated mixed -hybrid-integro-derivative boundary conditions

where with , , , , , and . Two notations and stand for the -CF derivative and the -Riemann-Liouville integral (-RLF), respectively. Also, notice that . Besides, is assumed to be a nonzero continuous single-valued operator, and is assumed to be a set-valued operator equipped with some required properties. Notice that by putting , the given -hybrid Caputo fractional differential inclusion BVP (2) and (3) is transformed into a non--hybrid separated inclusion BVP presented by

Note that by taking into account the authors’ knowledge, there are no research manuscripts on -CF operators involving mixed -hybrid-integro-derivative boundary conditions simultaneously. In addition, this given structure is formulated in a unique and general form in which we can consider some standard special cases studied before. Here, we derive some analytical criteria to prove the existence results for the proposed novel -hybrid fractional differential inclusion in the -Caputo settings (2) equipped with separated mixed -hybrid-integro-derivative boundary conditions (3). The applied approach to achieve desired purposes is based on Dhage’s fixed point result. In addition, we discuss the special case of the proposed -inclusion problem in the non--hybrid version with the aid of the endpoint notion. We organize the present manuscript as the following construction. In Section 2, we briefly collect auxiliary preliminaries on the -fractional operators and some required notions on the multifunctions and related properties. In Section 3, the existence criteria of solutions for both proposed -hybrid and non--hybrid BVPs (2)–(4) are derived by two different analytical methods. To confirm the applicability of our analytical findings, two simulative numerical examples are formulated in Section 4 which cover both -hybrid and non--hybrid cases.

2. Auxiliary Preliminaries

By continuing the path ahead, we assemble and recall several auxiliary and fundamental notions in the direction of our theoretical methods implemented in this paper. The concept of RLF integral for of order is defined as

provided that the integral has finite value [23, 24]. In this position, let us take in which . Regarding a continuous function , the RLF derivative of order is defined as

provided that the integral has finite value [23, 24]. In the next step, for an absolutely continuous -times differentiable real-valued function on , the derivative in the Caputo settings of order is defined as

such that it is finite-valued [23, 24]. Now, let be increasing with . Then, an integral in the sense of -Riemann-Liouville for an integrable of order depending on increasing function is defined as

provided that the RHS of above equality involves the finite value [25, 26]. It is clear that if we take , then , and thus by inserting them into (8), we see that the -RLF integral is converted to the standard RLF integral given by (5). For a continuous function , a derivative in the sense of -RL of order is given by

provided that the RHS of above equality exists [25, 26]. If , then the -RLF derivative (9) is converted to the standard RLF derivative (6). Motivated by such operators, Almeida gave a -version of the CF derivative as follows:

provided that the RHS of above equality exists [14]. If , then the -CF derivative (10) is converted to the standard CF derivative (7). Some useful properties of the -CF and -RLF operators can be seen in the following.

Lemma 1 [14, 24]. Let and be increasing with . Then,
(i1)
(i2)
(i3)
(i4)
For instance, we plot the graph of -RLF integral and -CF derivative of for in Figure 1.

Lemma 2 [14]. Let . Then, for each ,

In accordance with above lemma, the authors proved that the series solution for given homogeneous differential equation has such a form where and [14].

We consider the normed space by notation . Also, we introduce the notations , , , , and for the category of all nonempty subsets, all bounded subsets, all closed subsets, all compact subsets, and all convex subsets of , respectively. In the subsequent path, a metric function attributed to Pompeiu-Hausdorff is defined by so that and [27]. We say that is Lipschitzian with constant if . Also, is a contraction if [27].

We represent the collection of all existing selections of at point by for almost all [27, 28]. We note that is an endpoint for given set-valued operator whenever we have [29]. Also, the mapping possesses an approximate endpoint property (APXEndP-property) whenever

[29]. We need next results.

Theorem 3 (Closed graph theorem [30]). Let be a separable Banach space, be -Carathéodory and be a linear continuous map. Then, is another operator in with action having closed graph property.

Theorem 4 (Dhage’s theorem [31]). Consider the Banach algebra , and the operators and satisfying the following: (i) is Lipschitzian (with )(ii) is compact upper semicontinuous(iii) with Then, either is unbounded, or a solution, belonging to , exists for which .

Theorem 5 (Endpoint theorem [29]). Suppose that be complete and admits the upper semicontinuity via and . Besides, we assume that is such that for each . Then, an endpoint (uniquely) exists for iff involves the APXEndP-property.

3. New Existence Criteria

In two previous sections, we assembled some auxiliary and useful notions to achieve our main goals. Now in the following, we first establish a required lemma to derive the main existence results. To do this, we need to consider a sup–norm given by on the space . In this case, the Banach space along with the multiplication action defined as is a Banach algebra for all .

Lemma 6. Let , , , , , , and . An element is a solution for given -hybrid fractional equation supplemented with separated mixed -integro-derivative boundary conditions which is given by the following: so that is a positive real constant given as

Proof. At first, the element is assumed to be a solution for the hybrid -Caputo differential Equation (16). Then, there exist such that or more precisely, we have In view of the notion of fractional derivative in the -Caputo framework, we get the following relations for : In the following, by taking integral of order in the -Riemann-Liouville settings on both sides of (22) and (23), we obtain In this step, by considering the first boundary condition in (17), we find that and so In addition, the second integro-derivative boundary condition given in (17) yields In the last step, if we insert the values and obtained in (25) and (26) into (21), then we get The resultant integral equation confirms that satisfies the mentioned -integral Equation (18), and the proof is completed.

Now, by considering Lemma 6, we can present the following definition.

Definition 7. An absolutely continuous function is called a solution function for the -hybrid inclusion BVP in the sense of -Caputo (2) and (3) if there is with for almost all which satisfies separated mixed -integro-derivative boundary conditions and also for each .

Now, we are in a position that we can prove the first existence result about the hybrid -Caputo inclusion BVP (2) and (3).

Theorem 8. Assume that is a set-valued operator and a function is continuous. In addition, let
(₁) a bounded function exists such that for each and (₂) is -Caratheodory
(₃) a function exists such that for all and for almost all
(₄) a real number exists so that where , , , and If , then the -hybrid inclusion BVP (2) and (3) has at least a solution.

Proof. For each , the collection of all existing selections of is defined as for every and for almost all . Define a set-valued map by where for some and for almost all . It is obvious that the function is a solution to the -hybrid BVP (2) and (3) if is a fixed point of . Now, define by and by where for some and for almost all . This implies . We show that both operators and satisfy Theorem 4. We at first prove that is Lipschitzian. Let . We have for all . Therefore, for all , we get Hence, is Lipschitz with constant . In the current moment, we check the convexity of . For this, let . Choose such that for almost all . Let . Then, for almost all . With due attention to the convexity of , has convex values. This implies that for almost all . Therefore, is convex for each . Next, we claim that is completely continuous. To confirm this claim, we verify that the set is equicontinuous and uniformly bounded. Firstly, we prove that corresponds bounded sets to bounded sets contained in . For , define the bounded ball . For every and , there exists a function such that for almost all . Then, where is given in (33). Thus, , and so the set is uniformly bounded. Now, we want to prove that corresponds bounded sets to equicontinuous sets. Take , and choose so that for almost all . Let with , Then, The right-hand side of the latter inequalities tends to zero (independent of ) as tends to . Application of Arzela–Ascoli theorem gives the complete continuity of . We here discuss that has a closed graph, and this finding implies that is upper semicontinuous. To achieve this aim, let and with and . We claim that . For every and , choose such that for almost all . It is suffices to find that there is a member so that for almost all . Define a linear continuous operator as for almost all . Hence, Application of Theorem 3 shows that has a closed graph. Besides, since and , so there exists such that for almost all . Hence, , and so possesses closed graph which implies that is upper semicontinuous. On the other hand, because of the compactness of values of , it is immediately deduced that is compact and upper semicontinuous. Utilizing (), we get Put . We have . Utilizing Theorem 4, we prove that one of the items i or ii is possible. First, we check that the item i is not the case. From Theorem 4 and the assumption (), consider an arbitrary member of with . Then, for all . Choosing a function , for each , we have for almost all . Thus, one can write Hence, we get which is a contradiction. Hence, the item ii indicated in Theorem 4 is valid. Thus, exists so that . In consequence, the operator has a fixed point. So the -hybrid inclusion BVP (2) and (3) has a solution, and this completes the proof.

Definition 9. An absolutely continuous function is called a solution for the non--hybrid inclusion BVP (4) in the sense of Caputo if there is with for almost all which satisfies separated mixed -integro-derivative boundary conditions for almost all .

For each , the collection of all existing selections of is defined as for almost all . Define as where

By making use of endpoints for the multifunction , we prove the following theorem.

Theorem 10. Consider as a set-valued operator. Let
(₅) be increasing and upper semicontinuous with and
(₆) the multifunction be integrable and bounded so that be measurable for each
(₇) a function exists such that for almost all and all , where and (₈) the operator given by (60) possesses APXEndP-property
Then, a solution exists to the non--hybrid inclusion FBVP (4).

Proof. In such an argument, we try to prove the existence of endpoint to the set-valued operator defined by (60). To proceed this, we first investigate that is closed for each . By taking into account the hypothesis (), is a closed-valued measurable multifunction for each . In consequence, has a measurable selection . Now, we show that is closed for all . Consider the sequence contained in with . For each , there exists such that for almost all . Since is compact multifunction, we acquire a subsequence tending to . Hence, we have and for almost all . Hence, which indicates that is closed-valued. In addition, is bounded for each since is compact. Finally, we investigate if holds. Let , and . Choose such that for almost all . Since for almost all , so there exists such that for almost all . Define the multifunction given by Since and are measurable, thus we choose such that for almost all . Select such that for almost all . Hence, we get This gives and shows that for all . Also from (), we realize that has approximate endpoint property. Application of Theorem 5 gives that has a unique endpoint, i.e., there exists such that . In conclusion, is a solution to the non--hybrid inclusion BVP (4).

4. Some Examples

This section involves two different numerical simulation examples corresponding to the relevant -hybrid and non--hybrid fractional inclusion boundary problems to guarantee the applicability of proved theorems.

Example 1. With due attention to (2) and (3), we design the Caputo -hybrid differential inclusion BVP as supplemented with separated mixed -integro-derivative boundary conditions where , , , , , , , , and . We define by which is nonzero and continuous. Notice that . Moreover, the function is Lipschitz, that is, for each , we have If , then . Now, define by Since therefore, we have Then, . By using above values, we have . Also, we can find with . Finally, we have . Thus, all assertions of Theorem 8 are verified. Hence, the -hybrid Caputo differential inclusion BVP (73) supplemented with separated mixed -integro-derivative boundary conditions (74) has a solution.

Example 2. With due attention to (4), we design the Caputo non--hybrid differential inclusion BVP as supplemented with separated mixed -integro-derivative boundary where denotes the -CF derivative of order , , , , , , , , and . Using these values, we have . We consider the Banach space equipped with . Now, we define a multivalued map by for almost all . We define by . Obviously, and for all . Now, for each , we have where and is defined as for all . Then, . Lastly, we introduce by in which Thus, all assertions of Theorem 10 are verified. Hence, the non--hybrid Caputo differential inclusion BVP (79) with separated mixed -integro-derivative boundary (80) has a solution.

5. Conclusion

In the current research study, we derived some theoretical criteria to prove the existence results to a new -hybrid fractional differential inclusion in the Caputo settings depending on the increasing function with separated mixed -hybrid-integro-derivative boundary conditions. The applied method to achieve desired purposes is based on Dhage’s fixed point result. In addition, we discussed a special case of the proposed -inclusion problem in the non--hybrid structure with the help of the endpoint notion. To confirm the applicability of our theoretical findings, two specific numerical examples are provided which simulate both -hybrid and non--hybrid cases. Hence, this research work can motivate other researchers in this field to concentrate on various investigations of different -hybrid structures formulated by other fractional operators.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest

All authors declare that they have no competing interests.

Authors’ Contributions

All authors declare that the present study was realized in collaboration with equal responsibility. All authors read and approved the final version of the current manuscript.

Acknowledgments

The first and fifth authors would like to thank Azarbaijan Shahid Madani University. The authors also acknowledge the reviewers for their constructive remarks on our work.

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