Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving <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>Caputo Derivative in Banach and Fréchet Spaces

Derbazi, Choukri; Baitiche, Zidane; Benchohra, Mouffak; N’Guérékata, G.

doi:https://doi.org/10.1155/2020/6383916

International Journal of Differential Equations

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Research Article | Open Access

Volume 2020 | Article ID 6383916 | https://doi.org/10.1155/2020/6383916

Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving -Caputo Derivative in Banach and Fréchet Spaces

Choukri Derbazi,¹Zidane Baitiche,¹Mouffak Benchohra,²and G. N’Guérékata³

Academic Editor: Mayer Humi

Received06 Jul 2020

Revised26 Aug 2020

Accepted16 Sept 2020

Published14 Oct 2020

Abstract

Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leffler–Ulam stability results for a Cauchy problem involving -Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger’s fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir–Keeler condensing operators combined with the technique of Hausdorff measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Grönwall’s inequality, the Mittag–Leffler–Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo’s fixed point theorem in Fréchet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems.

1. Introduction

Fractional differential equations gained much attention due to their applications in various fields of science and engineering (see, for instance, [1–7] and the references therein). For more information about the basic theory of fractional differential equations, we can refer to the monographs [8–11] and references cited therein. Besides the classical and fractional-order differential and integral operators, there is another kind of fractional derivatives that appears in the literature called -Caputo fractional derivative, which was introduced by Almeida in [12], where the kernel operator contains a special function of an arbitrary exponent. According to this idea, a wide class of well-known fractional derivatives are obtained like Caputo and Caputo–Hadamard for particular choices of . Additionally, some interesting details about the -fractional derivatives and integrals can be found in [13–23]. Moreover, fixed point theory is a very useful tool in the theory of the existence of solutions to functional and differential equations; the reader is advised to see references [24–29] in which many scholars turned to the existence and uniqueness of solutions for differential equations involving different kinds of fractional derivatives under various boundary conditions. On the contrary, the notion of measure of noncompactness was first introduced by Kuratowski [30] in 1930 which was further extended to general Banach spaces by Banás and Goebel [31]. Later Darbo formulated his celebrated fixed point theorem in 1955 for the case of the Kuratowski measure of noncompactness (cf. [32]) which generalizes both the classical Schauder fixed point principle and (a special variant of) Banach’s contraction mapping principle. After that, the Darbo fixed point theorem has been generalized in many different directions; we suggest some works for reference [33–36]. The reader may also consult [37–43] and references therein where several applications of the measure of noncompactness can be found.

Very recently Dudek [44] proved a new fixed point theorem using the concept of measures of noncompactness in Fréchet spaces which generalize the famous Darbo’s fixed point theorem. To see more applications about the usefulness of this new fixed point theorem to prove the existence of solutions for certain classes of functional integral equations in Fréchet spaces, the reader can refer to [45–50].

On the contrary, one of the important parts of the qualitative theory of linear and nonlinear differential equations is the Ulam–Hyers stability, first formulated by Hyers and Ulam in 1940 [51–53]. Furthermore, the fractional Ulam stability was introduced by Wang et al. [54]. For some recent results of stability analysis by different types of fractional derivative operator, we refer the reader to articles [55–64], as well as to the recent book by Abbas et al. [65] and the references cited therein. More recently, some authors explored another form of stability known as Mittag–Leffler–Ulam–Hyers for the solutions of fractional differential equations [66–73].

Inspired by the above works, our goal is to extend the studies in [29, 37, 45, 72]. More precisely, we consider first the problem of the existence, uniqueness, and Mittag–Leffler–Ulam–Hyers stability for the following initial value problem of the fractional differential equation with constant coefficient in Banach spaces of the form:where is the -Caputo fractional derivatives such that is a given function satisfying some assumptions that will be specified later, is a Banach space with norm , and . Moreover, we also extend the above problem to give a uniqueness results on unbounded domains in a Banach space via Banach contraction principle coupled with Bielecki-type norm.

Next, we turn our attention to the existence of solutions for the same problem (1) in the Fréchet spaces. In precise terms, we investigate the existence of solutions for the following problem:

The structure of the present work is organized as follows: in Section 2, we collect some basic concepts on the fractional integrals and derivatives, auxiliary results, lemmas and notions of measures of noncompactness, and fixed point theorems that are used throughout this paper. In Section 3, based on Weissinger’s fixed point theorem combined with the Chebyshev norm, we give a uniqueness result for problem (1) on a compact interval in a Banach space. In Section 4, using the ideas of Hausdorff measure of noncompactness and Meir–Keeler condensing operator, we present the existence of solutions of IVP (1) in Banach spaces. In Section 5, we discuss the Mittag–Leffler–Ulam stability results for the problem at hand. In Section 6, we apply the Banach fixed point theorem coupled with a Bielecki-type norm to derive the uniqueness of solution on unbounded domains in a Banach space. In Section 7, we look into the existence of solutions for the IVP (2) in the Fréchet spaces via Darbo’s fixed point theorem. The last section provides a couple of examples to illustrate the applicability of the results developed.

2. Preliminaries and Background Materials

In this section, we present some basic notations, definitions, and preliminary results, which will be used throughout this paper.

Let be a finite interval and be an increasing function with , for all , and let be the Banach space of all continuous functions from into with the supremum (uniform) norm:

A measurable function is Bochner integrable if and only if is Lebesgue integrable.

By , we denote the space of the Bochner integrable functions , with the norm

Now, we define the Hausdorff measure of noncompactness and give some of its important properties.

Definition 1. (see [31]). Let be a Banach space and a bounded subset of . Then the Hausdorff measure of noncompactness of is defined byTo discuss the problem in this paper, we need the following lemmas.

Lemma 1. Let be bounded. Then the Hausdorff measure of noncompactness has the following properties. For more details and the proof of these properties see [31]:(1) is relatively compact(2)(3)(4), where and represent the closure and the convex hull of , respectively(5), where (6), for any Now, we recall some fixed point theorems that will be used later

Theorem 1 (Weissinger’s fixed point theorem [74]). Assume to be a nonempty complete metric space and let for every such that converges. Furthermore, let the mapping satisfy the following inequality:for every and every . Then, has a unique fixed point . Moreover, for any , the sequence converges to this fixed point .
On the contrary, in 1969, the concepts of the Meir–Keeler contraction mapping were introduced by Meir and Keeler.

Definition 2 (see [75]). Let be a metric space. Then a mapping on is said to be a Meir–Keeler contraction (MKC, for short); if for any , there exists such thatIn [34], the authors defined the notion of the Meir–Keeler condensing operator on a Banach space and gave some fixed point results.

Definition 3 (see [34]). Let be a nonempty subset of a Banach space and arbitrary measure of noncompactness on . We say that an operator is a Meir–Keeler condensing operator if for any , there exists such thatfor any bounded subset of .
The following fixed point theorem with respect to the Meir–Keeler condensing operator which is introduced by Aghajani et al. [34] plays a key role in the proof of our main results.

Theorem 2 (see [34]). Let be a nonempty, bounded, closed, and convex subset of a Banach space . Also, let be an arbitrary measure of noncompactness on . If is a continuous and Meir–Keeler condensing operator, then has at least one fixed point and the set of all fixed points of in is compact.
The following lemmas are needed in our argument.

Lemma 2 (see [76]). Let be a Banach space. If is bounded, then for any , where , and is the Hausdorff measure of noncompactness defined on the bounded sets of . Furthermore if is equicontinuous, then is continuous on , and

Lemma 3 (see [77]). Let be a Banach space and let be bounded. Then for each , there is a sequence , such thatWe call uniformly integrable if there exists such that

Lemma 4 (see [78]). If is uniformly integrable, then is measurable, andBefore introducing the basic facts on fractional operators, we recall three types of functions that are important in fractional calculus: the gamma, beta, and Mittag–Leffler functions

Definition 4 (see [79]). The gamma function, or the second-order Euler integral, denoted is defined as

Definition 5 (see [79]). The beta function, or the first-order Euler function, can be defined asWe use the following formula which expresses the beta function in terms of the gamma function:The next function is a direct generalization of the exponential series.

Definition 6 (see [79]). The one-parameter Mittag–Leffler function is defined asFor , this function coincides with the series expansion of , i.e.,

Definition 7 (see [79]). The two-parameter Mittag–Leffler function is defined asNow, we give some results and properties from the theory of fractional calculus. We begin by defining -Riemann–Liouville fractional integrals and derivatives, in what follows.

Definition 8 (see [2, 12]). For , the left-sided -Riemann–Liouville fractional integral of order for an integrable function with respect to another function that is an increasing differentiable function such that , for all , is defined as follows:where is the gamma function.
Note that equation (19) is reduced to the Riemann–Liouville and Hadamard fractional integrals when and , respectively.
The integer order of the differential operator with respect to another function that is an increasing differentiable function such that , for all is defined byFurthermore, for , we use the symbol to indicate the n-th composition of with itself; that is, we put

Definition 9 (see [12]). Let and let , be two functions such that is increasing and , for all . The left-sided –Riemann–Liouville fractional derivative of a function of order is defined bywhere .

Definition 10 (see [12]). Let and let , be two functions such that is increasing and , for all . The left-sided -Caputo fractional derivative of of order is defined bywhere for and for .
From the definition, it is clear thatThis generalization (24) yields the Caputo fractional derivative operator when . Moreover, for , it gives the Caputo–Hadamard fractional derivative.
Some basic properties are listed in the following lemma.

Lemma 5 (see [2, 12]). Let , and . Then for each , we have(1)(2)(3)(4)(5)

Remark 1. Note that for an abstract function , the integrals which appear in the previous definitions are taken in Bochner’s sense (see, for instance, [80]).
In the sequel, we will make use of the following generalizations of Grönwall’s lemmas

Theorem 3 (see [23]). Let be two integrable functions and continuous, with domain . Let be an increasing function such that . Assume that(1) and are nonnegative(2) is nonnegative and nondecreasing.Ifthen

Corollary 1 (see [23]). Under the hypotheses of Theorem 3, let be a nondecreasing function on . Then, we havewhere is a Mittag–Leffler function with one parameter.

Lemma 6 (see [14]). Let . Then for all , we have

Remark 2. Observe that from Lemma 6 if , we can get the following inequality:For the existence of solutions for the problem (1), we need the following lemma.

Lemma 7. Let . Then is the solution ofif and only if it is the solution of the integral equation:

Proof. Let be a solution of the problem (30). Define . Then,Sincetaking the -Riemann–Liouville fractional integral of order to the above equation, we getSincewe getUsing the definition of , we obtain equation (31). Conversely, suppose that is the solution of the equation (31). Then, it can be written aswhere . Since is continuous and is constant, operating the -Caputo fractional differential operator on both sides of equation (37), we obtainUsing Lemma 5, the following is obtained:From equation (37), we get . This proves that is the solution of the Cauchy problem (30) which completes the proof.
Now, we are ready to present our main results.

3. Uniqueness Result with respect to the Chebyshev Norm and Weissinger’s Fixed Point Theorem

First of all, we define what we mean by a solution of equation (1).

Definition 11. A function is said to be a solution of equation (1) if satisfies the equation on and the condition .

Theorem 4. Let the following assumptions hold: (H1) The function is continuous. (H2) There exists a constant such thatfor any and .
Then there exists a unique solution of equation (1) on .

Proof. In view of Lemma 7, we introduce an operator associated with equation (1) as follows:Clearly, the fixed points of the operator are solutions of equation (1). Weissinger’s fixed point theorem will be used to prove that has a fixed point. For this reason, we shall show that is a contraction. Let . Then, for every and , using (H2), we haveTherefore, we concludefor each and all .
Now letBy Definition (6), we haveTherefore, the existence of the unique fixed point of follows from Weissinger’s fixed point theorem. That is, (1)fd1 has a unique solution. This completes the proof.

4. Existence Result via Meir–Keeler Condensing Operators

In this section, we can weaken the condition (H2) to a linear growth condition. But now Theorem 2 that we apply will only guarantee the existence not also the uniqueness of the solution.

Theorem 5. Assume that the hypothesis (H1) holds. Furthermore, we impose the following: (H3) There exist continuous functions such that for any and . (H4) For each bounded set , and each , the following inequality holds:Then the problem (1) has at least one solution defined on provided thatwhere

Proof. Consider the operator defined by equation (41) and define a bounded closed convex setwithWe shall show that the operator satisfies all the assumptions of Theorem 2. We split the proof into four steps: Step 1. The operator maps the set into itself. By the assumption (H3), we have Thus This proves that transforms the ball into itself. Step 2. The operator is continuous. Suppose that is a sequence such that in as . It is easy to see that due to the continuity of . On the contrary, taking (H3) into consideration, we get the following inequality: We notice that since the function is the Lebesgue integrable over . This fact together with the Lebesgue dominated convergence theorem implies that It follows that , which implies the continuity of the operator . Step 3. is equicontinuous. For any and , we get where we have used the fact that . Therefore, As , the right-hand side of the above inequality tends to zero independently of . Hence, we conclude that is bounded and equicontinuous. Step 4. Now, we prove that is a Meir–Keeler condensing operator. To do this, suppose is given. We will prove that there exists such that For every bounded and equicontinuous subset and using Lemma 3 and the properties of , there exist sequences such that Next, by Lemma 4 and (H4), we have As the last inequality is true, for every , we infer Since is bounded and equicontinuous, we know from Lemma 3 that Therefore, we have Observe that from the last estimates Let us now take so we get which means that is a Meir–Keeler condensing operator. It follows from Theorem 2 that the operator defined by (41) has at least one fixed point , which is just the solution of the initial value problem (1). This completes the proof of Theorem 5.

5. Mittag–Leffler–Ulam–Hyers Stability Analysis

In this section, we discuss the Mittag–Leffler–Ulam–Hyers stability analysis of the solutions to the proposed problem (1).

Now, we consider the Mittag–Leffler–Ulam–Hyers stability for problem (1).

Let and be a continuous function. We consider the following inequalities:

Definition 12 (see [71, 73]). Equation (1) is Mittag–Leffler–Ulam–Hyers stable, with respect to if there exists a real number such that for each and for each solution of the inequality (67), there exists a solution of equation (1) with

Definition 13 (see [71, 73]). Equation (1) is generalized Mittag–Leffler–Ulam–Hyers stable, with respect to if there exists with such that for each and for each solution of the inequality (67), there exists a solution of equation (1) with

Remark 3 (see [71, 73]). It is clear that Definition 12 Definition 13,

Remark 4 (see [71, 73]). A function is a solution of the inequality (67) if and only if there exists a function (which depends on solution ) such that(i)(ii)Now we are ready to state our Mittag–Leffler–Ulam–Hyers stability of solution to the problem (1). The arguments are based on the Grönwall inequality equation (27).

Theorem 6 Assume that (H1) and (H2) hold. Then problem (1) is Mittag–Leffler–Ulam–Hyers stable on and consequently generalized Mittag–Leffler–Ulam–Hyers stable.

Proof. Let and let be a function which satisfies the inequality (67) and let be the unique solution of the following problem:By Lemma 7, we haveSince we have assumed that is a solution of the inequality (67), we have the following by Remark 4:Again by Lemma 7, we haveOn the contrary, we have, for each ,Hence, using Remark 2 and part (i) of Remark 4 and (H2), we can getApplying Corollary 1 (the Grönwall inequality equation (27)) to the above inequality with , and . Since is a nondecreasing function on , we conclude thatwhich yieldsTaking for simplicitythen (78) becomesIn consequence, it follows thatThus, the problem (1) is Mittag–Leffler–Ulam–Hyers stable. Furthermore, if we set , then the problem (1) is generalized Mittag–Leffler–Ulam–Hyers stable. This completes the proof.

6. Uniqueness Result with respect to the Bielecki Norms and Banach’s Fixed Point Theorem on an Unbounded Domain

In this section, we present an uniqueness result concerning the problem (1) on unbounded domain, i.e., in the case .

Theorem 7. If be a continuous function that satisfies the Lipschitz condition with respect to the second variable, i.e., there exists a positive constant such thatfor all and each .
Then, the problem (1) possesses a unique solution defined on .

Proof. Consider the Banach space equipped with a Bielecki norm type defined as below:where will be chosen later and is the Mittag–Leffler function which is given in Definition 6. (for more properties on the Bielecki-type norm, see [25, 28]).
Consider the operator defined bywhereNow, we prove that the operator is a contraction mapping on with respect to the Bielecki norm. To this end, we apply the Banach fixed point theorem to prove that has a fixed point. Given and , using the inequality (82), and Lemma 6, we can getHence, we haveNote that is a monotone increasing function on , then we getSince we can choose sufficiently large such thatit follows that the mapping is a contraction with respect to the Bielecki norm. Hence, by the Banach fixed point theorem, has a unique fixed point which is a unique solution of the initial value problem (1) in the space . This completes the proof.

7. An Existence Result in Fréchet Spaces via Darbo’s Fixed Point Theorem

By using Darbo’s fixed point theorem, we give in this section our last existence theorem concerning the IVP (2) in the Fréchet spaces.

Firstly, we need to fix the notation. Let and let .

In this section, we let to be the Fréchet space of all continuous functions from into , equipped with the family of seminorms:

Definition 14. A nonempty subset is said to be bounded ifNext we present some facts concerning the notion of a sequence of measures of noncompactness in the Fréchet spaces [44, 50].
In what follows, will be a real Fréchet space. If is a nonempty subset of , then and denote the closure and the closed convex closure of , respectively. Also, we denote by the family of all nonempty and bounded subsets of and the family of all relatively compact subsets of .

Definition 15. A family of functions where is said to be a family of measures of noncompactness in the real Fréchet space if it satisfies the following conditions:(1)The family is nonempty and (2), for all and (3), for all and (4)If is a sequence of closed subsets of such that for , and for each , then The following lemmas are needed in our argument.

Lemma 8 (see [77]). If is a bounded subset of a Fréchet space , then for each , there is a sequence , such that

Lemma 9 (see [78]). If is uniformly integrable, then is measurable, for andfor each .

Definition 16. Let be a nonempty subset of a Fréchet space , and let be a continuous operator which transforms bounded subsets of onto bounded ones. One says that satisfies the Darbo condition with constants with respect to a family of measures of noncompactness , iffor each bounded set and .
If , then is called a contraction with respect to .
The following generalization of the classical Darbo fixed point theorem for the Fréchet spaces plays a key role in the proof of our main results.

Theorem 8. (see [44]). Let be a nonempty, bounded, closed, and convex subset of a Fréchet space and let be a continuous mapping. Suppose that is a contraction with respect to a family of measures of noncompactness . Then has at least one fixed point in the set .

Definition 17. A function is said to be a solution of equation (2) if satisfies the equation on , and the condition .
Now, we shall prove the following theorem concerning the existence of solutions of problem (2).

Theorem 9. Let be a continuous function such that the following assumptions hold: (H5) There exists a continuous functions such that (H6) For each bounded set and for each , we havewhere is a measure of noncompactness on the Banach space .
For , letDefine on the family of measures of noncompactness bywhere is a bounded and equicontinuous set of , and .
Iffor each , then the problem (2) has at least one solution.

Proof. Consider the operator defined by (84), but in the Fréchet space .
Note that, the fixed points of the operator are solutions of the problems (2). For any , let be a positive real number withand we consider the ballNotice that is closed, convex, and bounded subset of the Fréchet space . We shall show that the operator satisfies all the assumptions of Theorem 8. We split the proof into three steps. Step 1. The operator maps the set into itself. For any , and each and , by (H5), we have Thus This proves that transforms the ball into itself. Step 2. The operator is continuous. Suppose that is a sequence such that in as . Then for each , we have Since as , the Lebesgue dominated convergence theorem implies that which implies the continuity of the operator . Step 3. Our aim in this step is to show that is contraction on . For every bounded equicontinuous subset and using Lemma 8, there exist sequences such that for all , we have Next, by Lemma 9 and (H6), we have As the last inequality is true, for every , we infer Thus Using the condition (99), we claim that is a contraction on . It follows from Theorem 8 that the operator defined by (84) has at least one fixed point , which is just the solution of initial value problem (2). This completes the proof of Theorem 9.

8. Examples

In this section, we give a couple of examples to illustrate the usefulness of our main result.

Letbe the Banach space of real sequences converging to zero, endowed its usual norm

Example 1. Consider the following initial value problem of a fractional differential posed in :Note that, this problem is a particular case of IVP (1), whereand given byIt is clear that condition (H1) holds. Moreover, for any and , we havewhere we have used the trigonometric identity and . Hence condition (H2) holds with . Therefore, by Theorem 4, the IVP (112) has a unique solution . Moreover, the inequality appearing in (67) has the following expression:By Theorem 6, the IVP (112) is Mittag–Leffler–Ulam–Hyers stable withwhereLet nowbe the Banach space with the norm

Example 2. Consider the following initial value problem of a fractional differential posed in :In this case, we takeand given byIt is clear that condition (H1) holds and asHence condition (H3) holds with ; we get easily . On the contrary, for any bounded set , we haveHence, (H4) is satisfied. Now, we shall check that condition (48) is satisfied. IndeedThen can be chosen as . Consequently, all the hypothesis of Theorem 5 are satisfied, and we conclude that the IVP (121) has at least one solution .

Example 3. Consider the following scalar initial value problem:whereClearly, is continuous. On the contrary, for any , we haveHence the inequality (82) holds with . Moreover, if we choose, , it follows that the mapping is a contraction. Hence by Theorem 7, the IVP (127) has a unique solution on .

Example 4. Let us consider problem (2) with specific data:and be the Fréchet space of all continuous functions from into , equipped with the family of seminorms:In order to illustrate Theorem 9, we take where is the sigmoid function [19] which can be expressed as in the following form:and a convenience of the sigmoid function is its derivative:Taking also given byThe hypothesis (H5) is satisfied with . So, for any , we haveIt follows from Theorem 9 that the problem (2) with the data (130), (132), and (134) has at least one solution defined on .

Data Availability

There are no data used in this work.

Conflicts of Interest

The authors declare they have no conflict of interest.

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