On the Stability of Fractional Differential Equations Involving Generalized Caputo Fractional Derivative

Tran, Minh Duc; Ho, Vu; Van, Hoa Ngo

doi:https://doi.org/10.1155/2020/1680761

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2020 | Article ID 1680761 | https://doi.org/10.1155/2020/1680761

On the Stability of Fractional Differential Equations Involving Generalized Caputo Fractional Derivative

Minh Duc Tran,^1,2Vu Ho,³and Hoa Ngo Van^2,4

Academic Editor: Gisele Mophou

Received28 Sept 2019

Accepted16 Jan 2020

Published15 Feb 2020

Abstract

This work presents the results of the global existence for fractional differential equations involving generalized Caputo derivative with the case of the fractional order derivative . In addition, the Ulam–Hyers–Mittag-Leffler stability of the given problems is also established.

1. Introduction

In recent years, there are a vast number of various concepts for fractional integrals and derivatives, such as Riemann–Liouville, Riesz, Grünwald–Letnikov, Hadamard, and Caputo derivatives and/or integrals. One can notice that most of the research results on the topics of fractional differential equations involving Riemann–Liouville and Hadamard fractional derivatives have been paid more and more attention by a large number of mathematicians because of the interesting and their applications. For more details on fractional calculus theory and interesting applications, one can see the monographs and the interesting papers in [1–6] and the references cited therein. However, both of the definitions of Hadamard and Riemann–Liouville fractional derivatives have their own disadvantages as well; one of which is that the derivative of a constant is not equal to zero. Then, to overcome the disadvantage of two types of these fractional derivatives, the Caputo and Caputo–Hadamard fractional derivatives were proposed. In the past decade, in [7, 8], Katagampola has proposed a new generalized concept of the fractional derivative, the so-called Caputo–Katugampola, that unifies the definitions of Caputo and Caputo-Hadamard fractional derivatives into a single form. The parameter family ρ of Caputo–Katugampola fractional derivative, , of the noninteger order α allows one to interpolate two types of the Caputo and Caputo–Hadamard fractional derivatives. Other approaches of fractional operators based on using very general kernel functions have been also proposed in [1, 9]. These approaches relate to the various real data corresponding to different complex systems requiring different kernel functions. For more details, on Caputo–Katugampola fractional derivative and interesting applications, one can see the papers [1, 2, 10–15] and the references cited therein. Very recently, the motivation behind the approach of Caputo–Katugampola fractional operator relates to the chaos problems in fractional dynamical systems suggested in the security of image encryption [16, 17] and in quantum mechanics [12].

During the past two decades, a large number of mathematicians have paid great attention to the studies of the concepts of Ulam’s stability because of its usefulness in many applications such as numerical analysis and optimization, where finding the exact solutions is quite difficult. In fact, it is not easy to get exact solutions to most of the problems of fractional differential equations. Therefore, it is vital to develop the concepts of Ulam’s stability for these problems because we need not obtain the exact solutions of the given problems when we study the properties of Ulam’s stability. This theory helps us getting an efficient and reliable technique for approximately solving fractional differential equations because there exists a close exact solution when the given problem is Ulam stable. More details from historical point of view and recent developments of such stabilities are reported in [9, 17–30] and the references cited therein. So, the motivation for the elaboration of this paper is the investigation of some kinds of the Ulam–Hyers stability for the following problem involving the concept of Caputo–Katugampola fractional derivative with the case of the :where is a real parameter, is a nonlinear continuous function, and is the Caputo–Katugampola fractional derivative. Based on (1), the parameter ρ allows one to get the initial value problem involving the Caputo fractional derivative if ρ tends to 1, and the initial value problem with the concept of Caputo–Hadamard fractional derivative if ρ tends to . Our aim in this paper is to discuss the global existence of solutions of problem (1) by using Schauder’s and Weissinger’s fixed point theorem. In addition, some kinds of the Ulam–Hyers stability of problem (1) are also established. The rest of this paper is arranged as follows: some fundamental theories of Caputo–Katugampola fractional calculus are introduced in Section 2. Section 3 is devoted to discuss the global existence of solutions of problem (1), and the stability of problem (1) is presented in Section 4.

2. Fundamental Theorems of Fractional Analysis

In this section, some definitions and basic results will be briefly presented which will be used throughout the paper. Let be the space of vector-valued continuous functions ψ from endowed with the norm where is the vector norm in the -dimensional Euclidean space. Denote by the weighted space of continuous functions given bywhere .

Let , then the Riemann–Liouville generalized fractional integral of ψ is defined by (see [7])

Let , then the Riemann–Liouville generalized derivative of ψ is defined by (see [7])where

Let , then the Caputo-generalized fractional derivative of ψ denoted by is defined by

By putting we have thatwhere . If , then we have that (see [7])

We observe that

Remark 1 (see [7]). Let , then the following properties are satisfied:(i)(ii)(iii)For , we have

Remark 2. Let and then we have, for

Remark 3. Let and , then we have that, for ,where

Theorem 1 (see Theorem 8 in [10]). Let be two integrable functions and be a continuous function on Assume that are nonnegative, and r is nonnegative and nondecreasing. Ifthen

Furthermore, if the function q is nondecreasing, then

The existence and uniqueness results are proved according to the following Schauder’s and Weissinger’s fixed point theorem [31].

Theorem 2. Assume that is a complete metric space, and let be a closed convex subset of . Furthermore, let be the map such that the set is relatively compact in . Then, the operator has at least one fixed point such that

Theorem 3. Assume that is a nonempty complete metric space and let for every such that the series converges. Furthermore, let the mapping satisfy the inequalityfor every and for any Then, the operator has a unique fixed point Moreover, for any the sequence converges to the fixed point

3. The Existence and Uniqueness of the Solution

In this section, we reconsider the following fractional differential equations:

A function is said to be a solution of problem (16); if ψ is continuous, , , and .

Theorem 4. Let the function belong to , where . Then, problem (16) is equivalent to the fractional integral equation:where

Proof. Let be a solution of (16), then from (16) and Remark 3 we have that, for ,for Because of the continuous hypotheses of the function f and from (16), it yields thatConsequently, by (18) and (3) we get the necessity condition. Conversely, let satisfy the integral equation (17). Using the continuity of f yields that is continuous on and . Indeed, it follows from the hypothesis that and, for all there exists a positive constant C such thatThis infers thatTaking the limit when , we observe that the right-hand side of (21) tends to 0. Furthermore, and . Next, by taking the Caputo–Katugampola fractional derivative, , on the two sides of (17) and by Remark 1 one has that

In the below theorems, we will present the existence and uniqueness of the local solution to problem (16) by using the Schauder fixed point theorem. We set . Let be a given constant, and define.

Theorem 5. Let be a continuous function and . Then, problem (16) has at least one solution on . Furthermore, if the following Lipschitz condition is held,where L is a positive constant, then problem (16) has a unique solution on .

Proof. LetwhereWe observe that is nonempty, bounded, closed, and convex subset. Define the operator byThe proof of this theorem is divided into two steps. In the first step, we shall prove that the operator has at least one fixed point by using Schauder’s fixed point principle and in the second one we also verify that the operator has a unique solution by using Weissinger’s fixed point theorem.
Step 1. We shall show that the conditions of Theorem 2 are satisfied.
We show that . For any we obtainOn the other hand, let such that in as . By the continuity of the function f one has in as . Sincewe have . Then, tends to in as . This yields is continuous. Thus, we infer that , for , i.e.,
Next, we show that is a relatively compact set. For and , sincewe conclude that the set is uniformly boundedness. Furthermore, for Applying the mean value theorem, one obtainsfor some Therefore, if we havewhere , is fixed, and This shows that is equicontinuous. Hence, by Arzela–Ascoli theorem (see Theorem 1.8 in [5]), this yields that is relatively compact. So, according to the conditions of Theorem 2, we can conclude that the operator has a fixed point.
Step 2. For the uniqueness of solution, we suppose that is another solution for problem (16) on and . Then, we obtainThus, it follows thatBy the induction method, we will verify thatwherefor Indeed, we assume that (35) is satisfied for the case of , and for , one has thatSetting we observe that the series converges to the Mittag-Leffler function . So, we can conclude that the series is convergent. Based on Theorem 3, the operator has a unique fixed point according to the conditions of Weissinger’s fixed point theorem.

Theorem 6. Assume that the function is a continuous function and . Let be a solution of problem (16) on . If is bounded on for some , then

Proof. This assertion will be divided into two steps. First of all, we verify exists. It means there exists such that, for , , for any given i.e.,Since is bounded on , there exists a sequence and a positive constant such that , , and . In addition, as is a bounded convergent subsequence, it follows that This implies that there exists such that , and for , we haveIf (38) is not true, then for , there exists such that and Then, from (39) and triangle inequality one hasThen, for sufficiently large , we haveThis implies the contradiction that exists. For the next step, we show that is continuable. Define Then, is continuous on Consider the operator as follows:where As a result of Theorem 5, since on the interval equation (42) possesses a solution, and we can rewrite equation (42) as follows:whereis the known function. Let Because of the continuation of f on , denote Again letwhereBy the same argument as in Theorem 5, we show that has a fixed point. Let , such that in as By the continuity hypothesis of the function f we have in as Sincewe have as , which yields that operator is continuous. Next, it follows that is relatively compact. For and , we haveThus, . Furthermore, for one hasWith the same argument as in the proof of Theorem 5, since is continuous on , it follows that is equicontinuous. Hence, by Arzela–Ascoli theorem, we deduce that is relatively compact. Therefore, by the Schauder’s fixed point theorem, operator has a fixed point , i.e., whereAccording to Theorem 4, we conclude that is a solution of problem (16) on . By the assumption of this theorem again, it follows that can be extended beyond β. So, the solution of problem (16) exists on and so .

Theorem 7. Let be a continuous function and there exists a continuous function such that , where is a constant. Then, problem (16) has at least one solution on . Furthermore, if the following Lipschitz condition is satisfied,then problem (16) has a unique solution on .

Proof. It follows from the assertion of Theorem 5 that there exists a solution of problem (16). By Theorem 4, satisfies the following integral equation

Step 3 (the existence of solution). Suppose that the solution ψ admits a maximal existence interval, denoted by . By the condition of the function f, one has the estimateBy putting , then (54) leads toWe now prove that is bounded on . Define the functionalThen, one has that , and by iterating consecutively, we also obtain for In addition, by mathematical induction and for , then (56) leads to the following estimation:Indeed, for , (58) is obvious. If (58) is valid for , then by using Dirichlet’s formula, we obtainWith the change of variables we obtainThus, (58) is valid for . Besides, we observe that as Hence, from (57), we haveFurthermore, if the function is nondecreasing, then for all , we have and soTherefore, we deduce that is bounded on . It follows from Theorem 6 that the solution can be extended to the right side of β. This obviously contradicts the assumption that is the maximal existence interval. This consequently implies that

Step 4 (the uniqueness of solution). By the Lipschitz condition and inequality (62) the uniqueness of solution can be proved. Indeed, let us assume that is another solution for problem (16) and Then, one has thatwhere . Define , and note that since we have . Applying Theorem 1, we get This implies that which completes the proof.

4. The Stability of Problem (16)

In the sequel, some kinds of the Ulam–Hyers–Mittag-Leffler stability for problem (16) will be investigated. Denote Let and . We consider the inequalities as follows:

Remark 4. From inequalities (64) and (65), we observe that(i)If a function ξ is a solution of (64), then there exists a function such that , for all , and (ii)If a function ξ is a solution of (65), then there exists a function such that , for all , and Definition 1. Let . roblem (16) is said to be(i)Ulam–Hyers–Mittag-Leffler stable if there is a constant such that, for each and for each solution ξ of (64), there exists a solution ψ of problem (16) satisfying the estimate(ii)Ulam–Hyers–Rassias–Mittag-Leffler stable with respect to if there is a constant such that, for each and for each solution ξ of (65), there exists a solution ψ of problem (16) satisfying the estimateIn the following theorem, the Ulam–Hyers–Mittag-Leffler stability for problem (16) is presented.

Theorem 8. Let be a continuous function which satisfies the assumption as follows. There exists a positive constant L such that, for all , one has

Then, we have the following assertions: (A1) For every , if a function ξ satisfies (64) for all , then there exists a unique solution ψ of problem (16) with initial conditions , which satisfies

Furthermore, this yields that problem (16) is Ulam–Hyers–Mittag-Leffler stable. (A2) Assume that the following hypothesis holds: Let in the inequality (65) be a nondecreasing function for all . We assume that there exists a positive constant which satisfies , and for ,

Then, for every , if a function ξ satisfies (65), for all , there exists a unique solution ψ of problem (16) with initial conditions , and problem (16) is Ulam–Hyers–Rassias–Mittag-Leffler stable.

Proof. In the view of Remark 4 and Theorem 4, we notice that if functions satisfy inequalities (64) and (65), respectively, then there exists respectively, such that , and for respectively, where and To show the results of this theorem, the method of successive approximations will be used.
Prove the assertion (A1): we define , and we consider the sequence which is defined as follows:By the abovementioned definition of successive approximations, for and from (71), one hasBy the Lipschitz condition of the function f, for any and , one hasSo, from (74) and for , one obtainsand for , we also obtainBy using the mathematical induction method, for , we haveNow, if we assume that (78) holds for , then by (4) one obtainswhich is the inequality (78) for . This yields that the inequality (78) is satisfied for all Then, one hasSince the series of the right-hand side of the above inequality is convergent tothat is,This yields that the series is uniformly convergent on J with respect to the norm . Now, assume thatThen, we take which is the partial of series (83) of the formBy (83) and (84), we noticeSet for . We prove that the limit function ψ is a solution of the following integral equation on J:By using definition of successive approximation for any and by the Lipschitz condition of f, we haveOn the other hand, (83) and (84) yieldTherefore, we obtain for So, it follows from (87) that, for Then, by taking limit , the right-hand side of (90) tends to 0. This yields (86) is a solution of problem (16) with initial conditions . In addition, from (82) and (83), we also get the following estimation between the solution of problem (16) and the solution of inequality (64):Then, by Definition 1-(i) we can deduce problem (16) is Ulam–Hyers–Mittag-Leffler stable, where and .
Prove the assertion (A2): similar to the proof of the assertion (I), we also consider the sequence given by , and for ,By similar processing as the case of (A1), we get for and for ,In order to check the validity of (93), for , from (72) and by using the definition of successive approximations (92), one obtainsSuppose that (93) is true for , that is,Then, by the Lipschitz condition of f and the hypothesis of the assertion (A2), we obtainThis proves inequality (93) is valid for all . On the other hand, by the hypothesis and from (93) we have that, for Since the function is continuous on J, it is bounded. So, inequality (97) yields that the series is absolutely and uniformly convergent on J with respect to the distance . Therefore, we set, for Then, similar to the proof of (A1) we also get the estimation between the solution of problem (16) and the solution of inequality (65) as follows:Taking the integral on of the order on both sides of (99) and using hypothesis (H3), one obtainsThis yields thatTherefore, we haveSet Then, equation (102) becomesBy using the Gronwall inequality in 1, (103) yields thatFurthermore, since the function is nondecreasing on , we obtainConsequently, we obtainThen, by Definition 1-(iii) we can deduce problem (16) is Ulam–Hyers–Rassias–Mittag-Leffler stable, where and .

5. Conclusion

In this work, the global existence and uniqueness of the solution to the Caputo-generalized fractional differential equation are investigated. We discussed the stability in various Ulam–Hyers–Mittag-Leffler’s types via the successive approximation method.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the Vietnam National Foundation for Science and Technology Development (NAFOSTED), under grant number 107.02-2017.319.

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Copyright © 2020 Minh Duc Tran et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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