Existence Results for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 11.3962 13.4804" width="11.3962pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-245" d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z"/></g></svg>-</span>Hilfer Fractional Integro-Differential Hybrid Boundary Value Problems for Differential Equations and Inclusions

Kiataramkul, Chanakarn; Ntouyas, Sotiris K.; Tariboon, Jessada

doi:https://doi.org/10.1155/2021/9044313

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9044313 | https://doi.org/10.1155/2021/9044313

Existence Results for -Hilfer Fractional Integro-Differential Hybrid Boundary Value Problems for Differential Equations and Inclusions

Chanakarn Kiataramkul,¹Sotiris K. Ntouyas,^2,3and Jessada Tariboon¹

Academic Editor: Maria L. Gandarias

Received27 Jun 2021

Accepted25 Aug 2021

Published13 Sept 2021

Abstract

In this research work, we study a new class of -Hilfer hybrid fractional integro-differential boundary value problems with nonlocal boundary conditions. Existence results are established for single and multivalued cases, by using suitable fixed-point theorems for the product of two single or multivalued operators. Examples illustrating the main results are also constructed.

1. Introduction

Some real-world problems in physics, mechanics, and other fields can be described better with the help of fractional differential equations. So, differential equations of fractional order has recently received a lot of attention and now constitutes a significant branch of nonlinear analysis. Numerous monographs have appeared devoted to fractional differential equations, for example, see [1–8]. Recently, differential equations and inclusions equipped with various boundary conditions have been widely investigated by many researchers, see [9–14] and the references cited therein.

Hybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers [15–18].

We will give a brief history on the subject of hybrid differential and fractional differential equations. In 2010, Dhage and Lakshmikantham [19] initiated the study of the first-order hybrid differential equation where and They established the existence, uniqueness results, and some fundamental differential inequalities.

In 2011, Zhao et al. [15] discussed the following hybrid fractional initial value problem where is the Riemann-Liouville fractional derivative of order , and

Sun et al. [16] studied the following hybrid fractional boundary value problem where is the Riemann-Liouville fractional derivative of order , and

In [20], the authors studied the existence of solutions for a nonlocal boundary value problem of hybrid fractional integro-differential equations given by where is the Caputo fractional derivative of order with , is the Riemann-Liouville fractional integral of order , for , , , a functional , and The main result was obtained by means of a hybrid fixed-point theorem for three operators in a Banach algebra due to Dhage [21].

The existence of solutions for an initial value problem of hybrid fractional integro-differential equations, given by was studied in [22]. Here, is the Caputo fractional derivative of order with ; is the Riemann-Liouville fractional integral of order , for , , . A generalization of Krasnoselskii fixed-point theorem due to Dhage [21] was used in the proof of the existence result.

The problem (5) was extended in [23] to boundary value problems of the form where is the Caputo fractional derivative of order with , ; is the Riemann-Liouville fractional integral of order , for , , . An existence result is proved via Dhage’s [21] fixed-point theorem.

For recent results on hybrid boundary value problems of fractional differential equations and inclusions, we refer to [24–26] and references cited therein. In the literature, there do exist several definitions of fractional integrals and derivatives. One of them is the Hilfer fractional derivative, which composites both Riemann-Liouville and Caputo fractional derivatives [27]. Fractional differential equations involving Hilfer derivative have many applications, and we refer to [28] and the references cited therein. There are actual world occurrences with uncharacteristic dynamics such as atmospheric diffusion of pollution, signal transmissions through strong magnetic fields, the effect of the theory of the profitability of stocks in economic markets, the theoretical simulation of dielectric relaxation in glass forming materials, and network traffic. See [29, 30] and references cited therein.

In [31], an initial value problem was discussed for hybrid fractional differential equations involving -Hilfer fractional derivative of the form where is the -Hilfer fractional derivative with , and For some recent results on -Hilfer fractional initial value problems, see [32–37] and references cited therein.

In the present work, we study a -Hilfer hybrid fractional integro-differential nonlocal boundary value problem of the form where is the -Hilfer fractional derivative operator of order with , ; is -Riemann-Liouville fractional integral of order , for , , , with for . An existence result is established via a fixed-point theorem for the product of two operators due to Dhage [21].

As a second problem, we investigate the existence of solutions for the following inclusion -Hilfer fractional hybrid integro-differential equations with nonlocal boundary conditions of the form where is a multivalued map, is the family of all subsets of , and the other quantities are the same as in boundary value problem (8). Here, the existence result is based on a multivalued fixed-point theorem for the product of two operators due to Dhage [38].

The rest of the paper is arranged as follows: in Section 2, we recall some notations, definitions, and lemmas from fractional calculus needed in our study. Also, we prove an auxiliary lemma helping us to transform the hybrid boundary value problem (8) into an equivalent integral equation. The main existence result for the -Hilfer hybrid boundary value problem (8) is contained in Section 3. The obtained result is illustrated by a numerical example. Section 4 is devoted in the study of the inclusion case of the hybrid boundary value problem (8) by considering the multivalued hybrid boundary value problem (9). Some special cases are discussed in Section 5.

2. Preliminaries

This section is assigned to recall some notation in relation to fractional calculus. We denote by the -times absolutely continuous functions given by

Definition 1 (see [2]). Let , , be a finite or infinite interval of the half-axis and . Also, let be an increasing and positive monotone function on , having a continuous derivative on . The -Riemann-Liouville fractional integral of a function with respect to another function on is defined by where is the Euler Gamma function.

Definition 2 (see [2]). Let and , . The Riemann-Liouville derivatives of a function with respect to another function of order is defined by where , is represent the integer part of the real number .

Definition 3 (see [32]). Let with , is the interval such that and two functions such that is increasing and , for all . The -Hilfer fractional derivative of a function of order and type is defined by where ; represents the integer part of the real number with .

Lemma 4 (see [2]). Let , then we have the following semigroup property given by Next, we present the -fractional integral and derivatives of a power function.

Proposition 5 (see [2, 32]). Let , , and , then, -fractional integral and derivative of a power function are given by

Lemma 6 (see [33]). Let , , , , , and . If , then

Lemma 7 (see [32]). If , , , and , then for all , where .

Lemma 8. Let , , , and . Then, is a solution of the -Hilfer hybrid fractional integro-differential nonlocal boundary value problem of the form if and only if satisfies the equation

Proof. Let be a solution of the problem (18). Applying the operator to both sides of (18) and using Lemma 7, we obtain where , . By using the first boundary condition, , we get the constant . From the second boundary condition, , we find that Substituting the value of and in (20), we obtain (19).
Conversely, by a direct computation, it is easy to show that the solution given by (19) satisfies the problem (18). The proof of Lemma 8 is completed.☐

Let be the Banach space of continuous real-valued functions defined on , equipped with the norm and a multiplication Then, clearly, is a Banach algebra with above-defined supremum norm and multiplication in it.

Lemma 9 (see [21]). Let be a nonempty, closed convex, and bounded subset of the Banach algebra and two operators such that (a) is Lipschitzian with a Lipschitz constant (b) is completely continuous(c) for all (d) where Then, the operator equation has a solution.

3. Existence Result for the Problem (8)

In view of Lemma 8, we define an operator by

Notice that the problem (8) has solutions if and only if the operator has fixed points.

Theorem 10. Assume that:
(A₁) The function is continuous and there exists a positive function , with bound , such that for and ,
(A₂)
(A₃) There exists a positive constant such that (A₄) There exists a positive real number such that Then, the -Hilfer hybrid fractional integro-differential nonlocal boundary value problem (8) has at least one solution on , provided that

Proof. We consider a subset of defined by , where satisfies the inequality (25). Observe that is a closed, convex, and bounded subset of the Banach space . We set , and
Next, we define two more operators and as follows: Clearly, . In the next steps, we show that the operators and fulfil all the assumptions of Lemma 9. The proof is divided into three steps.☐

Step 1. We show that the operator is Lipschitzian with Lipschitz constant , i.e., condition of Lemma 9 is fulfilled. Let Then we have Consequently, Hence, the operator is Lipschitzian with Lipschitz constant .

Step 2. We show that the condition of Lemma 9 is satisfied, i.e., the operator is completely continuous on First, we will prove its continuity. Let be a sequence of functions in converging to a function Then, by the Lebesgue dominant theorem, for each , we have Therefore, is a continuous operator on Next, we show that the operator is uniformly bounded on For any , we have Hence, , which shows that the operator is uniformly bounded on Now, we show that the operator is equicontinuous. Let and Then, we have As , the right-hand side tends to zero, independently of . Thus, is equicontinuous. Therefore, it follows by Aezelá-Ascoli theorem that is a completely continuous operator on

Step 3. We show that the third condition of Lemma 9 is fulfilled. For any , we have which implies , and so,

Moreover, by (26), it holds which is fulfilled condition of Lemma 9. Hence, all the conditions of Lemma 9 are satisfied, and consequently, the operator equation has at least one solution in Therefore, there exists a solution of the -Hilfer hybrid fractional integro-differential nonlocal boundary value problem (8) in The proof is finished.

Now, we present an example of -Hilfer hybrid fractional integro-differential boundary value problem to illustrate our main result.

Example 11. Consider the boundary value problem of the form where Here, , , , , , , , and . Then, we can find that and we choose . In addition, , , , . Then, we have , , , , and which lead to Therefore, by Theorem 10, the -Hilfer hybrid fractional integro-differential nonlocal boundary value problem (34)-(35) has at least one solution on , such that , where is satisfies

4. Existence Result for the Problem (9)

First of all, we recall some basic concepts for multivalued maps [39–41]. For a normed space , let For each , define the set of selections of by

Lemma 12 (see [42]). Let be an Carathéodory multivalued map and let be a linear continuous mapping from to . Then, the operator is a closed graph operator in

Remark 13. We recall that a multivalued map is said to be Carathéodory if is measurable for each ; (ii) is upper semicontinuous for almost all ; (iii) for each , there exists such that for all with and for a.e.

The following multivalued fixed-point theorem for the product of two operators in a Banach algebra, due to Dhage [38, Theorem 4.13], plays a key role in proving the existence result for the nonlocal boundary value problem (9).

Lemma 14 (see [38]). Let be a Banach algebra and let be a single valued and be a multivalued operator satisfying the conditions: (a) is single-valued Lipschitz with a Lipschitz constant (b) is compact and upper semicontinuous(c), where Then, either (i) the operator inclusion has a solution or (ii) the set is unbounded.

Definition 15. A function is a solution of the problem (9) if , and there exists function such that a.e. on and

Theorem 16. Assume that (A₁) and (A₃) hold. In addition, we suppose that
(B₁) is -Carathéodory multivalued map;
(B₂) The functions and satisfy condition ;
(B₃) There exists a continuous function such that Then, the nonlocal -Hilfer hybrid inclusion boundary value problem (9) has at least one solution on , provided that

Proof. To transform the boundary value problem (9) into a fixed-point problem, by using Lemma 8, we define a multivalued operator as Next, we introduce the operator as in (27) and the multivalued operator by We will show that the operators and satisfy the hypotheses of Lemma 14. The proof is given in a series of steps.
Step 1. is convex valued. Let Then, there exist such that For any , we have Since is convex, for all , and so, Thus, , which means that is convex valued on
Step 2. is single-valued Lipschitz operator on It is proved in Step 1 of Theorem 10.
Step 3. The operator is completely continuous and upper semicontinuous on Let be a bounded set of Then, there exists a constant such that for all We prove first that the operator is completely continuous. Let Then, there exists such that for any Then, we have for all , which implies that . Therefore, the operator is uniformly bounded on
Next, we show that is an equicontinuous set in Let , with and . Then, we have As , the right-hand side of the above inequality tends to zero independently of , and thus, the operator is equicontinuous. In consequence, the operator is completely continuous by the Arzelá-Ascoli theorem.
Next, we show that is an upper semicontinuous multivalued mapping on It is known, by [[39], Proposition 1.2], that will be upper semicontinuous if we establish that it has a closed graph, since already shown to be completely continuous. Thus, we will prove that has a closed graph.
Let be a sequence in such that Let be a sequence such that and We shall show that . Since , there exists a such that We must prove that there is a such that Consider the continuous linear operator defined by Observe that as . From Lemma 12, it follows that is a closed graph operator. Further, we have Since , therefore, we have As a result, we have that is a compact and upper semicontinuous operator on
Step 4. We show that the condition (c) of Lemma 14 holds, that is, . This is obvious by (44).
Step 5. Finally, we show that the conclusion (ii) of Lemma 14 does not hold.
Let be any solution of the boundary value problem (9) such that for some Then, there is a such that Then, we have Taking the supremum for of the above inequality, we obtain a constant such that which means that the set is bounded.
As a result, the conclusion (ii) of Lemma 14 does not hold. Hence, the conclusion (i) holds, and consequently, the boundary value problem (9) has at least one solution on This completes the proof.☐

Example 17. Consider the boundary value problem of the form where Here, , , , , , , , , . Now, we find that , , and which yield , , . In addition, we have which implies . Hence, Therefore, by applying Theorem 16, the -Hilfer hybrid fractional integro-differential nonlocal boundary value problem (60)-(61) has at least one solution on .

5. Special Cases

The problem (8) considered in the present work is general in the sense that it includes the following classes of new boundary value problems of -Hilfer fractional differential equations. (I)Let and for all and Then, the problem (8) reduces to the following -Hilfer fractional boundary value problem

In the case when where , the corresponding -Hilfer fractional boundary value problem was studied in [34] for . (II)Let for all and Then, the problem (8) reduces to the following -Hilfer fractional boundary value problem (III)Let for all and Then, the problem (8) reduces to the following -Hilfer fractional boundary value problem

Therefore, the main result of this paper also includes the existence results for the solutions of the above-mentioned -Hilfer boundary value problems of fractional differential equations as special cases.

6. Conclusions

In this paper, we studied a new class of -Hilfer hybrid fractional integro-differential boundary value problems with nonlocal boundary conditions. After proving a basic lemma, helping us to transform the considered system into a fixed-point problem, an existence result is proved via a fixed-point theorem for the product of two operators due to Dhage [21]. The multivalued analogue is also studied, and an existence theorem was established with the help of a multivalued fixed-point theorem for the product of two operators due to Dhage [38]. Numerical examples illustrating the obtained results are also presented. Some special cases are also discussed. The obtained results are new and enrich the existing literature on hybrid fractional differential equations and inclusions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was funded by the King Mongkut’s University of Technology North Bangkok (contract no. KMUTNB-61-KNOW-034).

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