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

Faculty of Electrical Electronics Engineering, Ton Duc ang University, Ho Chi Minh City, Vietnam Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc ang University, Ho Chi Minh City, Vietnam Faculty of Mathematical Economics, Banking University of Ho Chi Minh City, Vietnam Faculty of Mathematics and Statistics, Ton Duc ang University, Ho Chi Minh City, Vietnam


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][2][3][4][5][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. en, 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.
e parameter family ρ of Caputo-Katugampola fractional derivative, C D α,ρ a + , 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]. ese 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][11][12][13][14][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. erefore, 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.
is 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][18][19][20][21][22][23][24][25][26][27][28][29][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 α ∈ (1, 2): where 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 0 + . 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. e 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.

Fundamental Theorems of Fractional Analysis
In this section, some definitions and basic results will be briefly presented which will be used throughout the paper.
where c ∈ (0, 1]. Let α > 0, then the Riemann-Liouville generalized fractional integral of ψ is defined by (see [7]) Let α > 0, then the Riemann-Liouville generalized derivative of ψ is defined by (see [7]) where n � α. Let α > 0, then the Caputo-generalized fractional derivative of ψ denoted by By putting ψ (p,k) (t) ≔ (t 1− p (d/dt)) k ψ(t), we have that where , then we have that (see [7]) We observe that Remark 1 (see [7]). Let α, β, ρ > 0, then the following properties are satisfied: Remark 3. Let n − 1 < α ≤ n ∈ N and ψ ∈ C n ([a, b], R d ), then we have that, for t ∈ (a, b], where n � α. Theorem 1 (see Theorem 8 in [10]). Let p(t), q(t) be two integrable functions and r(t) be a continuous function on [a, b]. Assume that p and q are nonnegative, and r is nonnegative and nondecreasing. If then Furthermore, if the function q is nondecreasing, then e existence and uniqueness results are proved according to the following Schauder's and Weissinger's fixed point theorem [31].

Theorem 2.
Assume that (X, d) is a complete metric space, and let S be a closed convex subset of X. Furthermore, let P: S ⟶ S be the map such that the set Px: x ∈ S { } is relatively compact in X. en, the operator P has at least one fixed point x * ∈ S such that Px * � x * . Theorem 3. Assume that (X, d) is a nonempty complete metric space and let λ n ≥ 0 for every n ∈ N 0 such that the series ∞ n�0 λ n converges. Furthermore, let the mapping P: X ⟶ X satisfy the inequality d P n x, P n y ≤ λ n d(x, y), (15) for every n ∈ N and for any x, y ∈ X. en, the operator P has a unique fixed point x 0 . Moreover, for any x * ∈ X, the sequence (P n x * ) n ≥ 1 converges to the fixed point x 0 .

The Existence and Uniqueness of the Solution
In this section, we reconsider the following fractional differential equations:

en, problem (16) is equivalent to the fractional integral equation:
Mathematical Problems in Engineering (16), then from (16) and Remark 3 we have that, for α ∈ (1, 2), Because of the continuous hypotheses of the function f and from (16), it yields that Consequently, by (18) and (3) we get the necessity condition. Conversely, let ψ ∈ C 1 ([a, b], R d ) satisfy the integral equation (17). Using the continuity of f yields that Taking the limit when t ⟶ a + , we observe that the right-hand side of (21) tends to 0. Furthermore, ψ(a) � ψ 1 and ψ ′ (a) � ψ 2 . Next, by taking the Caputo-Katugampola fractional derivative, RL D α,ρ a + , 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 where L is a positive constant, then problem (16) has a unique solution on [a, b].

Proof.
Let where We observe that Ω T ⊆ C([a, T], R d ) is nonempty, bounded, closed, and convex subset. Define the operator T: e proof of this theorem is divided into two steps. In the first step, we shall prove that the operator T has at least one fixed point by using Schauder's fixed point principle and in the second one we also verify that the operator T has a unique solution by using Weissinger's fixed point theorem.
Step 1. We shall show that the conditions of eorem 2 are satisfied.
Next, we show that TΩ T � (Tψ)(t): ψ ∈ Ω T is a relatively compact set. For Tψ ∈ TΩ T and t ∈ [a, T], since we conclude that the set TΩ T is uniformly boundedness.
Applying the mean value theorem, one obtains where a > 0, ρ > 0 is fixed, and α ∈ (1, 2). is shows that TΩ T is equicontinuous. Hence, by Arzela-Ascoli theorem (see eorem 1.8 in [5]), this yields that TΩ T is relatively compact. So, according to the conditions of eorem 2, we can conclude that the operator T has a fixed point.
Step 2. For the uniqueness of solution, we suppose that en, we obtain By the induction method, we will verify that sup t∈[a,T] where for n ∈ N. Indeed, we assume that (35) is satisfied for the case of (n − 1) ∈ N, and for n ∈ N, one has that Setting K n ≔ (L n /Γ(nα + 1))((t ρ − a ρ )/ρ) nα , we observe that the series ∞ n�0 K n converges to the Mittag-Leffler function E α,1 (L((t ρ − a ρ )/ρ) α ). So, we can conclude that the series K n is convergent. Based on eorem 3, the operator T has a unique fixed point according to the conditions of Weissinger's fixed point theorem.
en, ψ(t) is continuous on [a, β]. Consider the operator T as follows: where is the known function. Let where h � min 1, By the same argument as in eorem 5, we show that T has a fixed point. Let With the same argument as in the proof of eorem 5, since ψ 1 (t) is continuous on [β, β + 1], it follows that TΩ h is equicontinuous. Hence, by Arzela-Ascoli theorem, we deduce that TΩ h is relatively compact. erefore, by the Schauder's fixed point theorem, operator T has a fixed point ψ(t) ∈ Ω h , i.e., According to eorem 4, we conclude that ψ(t) is a solution of problem (16) on (a, β + h]. By the assumption of this theorem again, it follows that ψ(t) can be extended beyond β. So, the solution of problem (16) exists on [a, ∞) and so β � +∞. such that ||f(t, ψ)|| ≤ M||ψ|| + q(t), where M > 0 is a constant. en, problem (16) has at least one solution on [a, +∞). Furthermore, if the following Lipschitz condition is satisfied, then problem (16) has a unique solution on [a, +∞).
Proof. It follows from the assertion of eorem 5 that there exists a solution ψ(t) of problem (16). By eorem 4, ψ(t) satisfies the following integral equation Step 3 (the existence of solution). Suppose that the solution ψ admits a maximal existence interval, denoted by [a, β)

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(β < + ∞). By the condition of the function f, one has the estimate (54) (55) We now prove that w(t) ≔ ||ψ(t)|| is bounded on [a, β). Define the functional en, one has that w(t) ≤ v(t) + Fw(t), and by iterating consecutively, we also obtain for n ∈ N, (57) In addition, by mathematical induction and v(t) > 0 for t ∈ [a, β), then (56) leads to the following estimation: (58) Indeed, for i � 1, (58) is obvious. If (58) is valid for i ∈ N, then by using Dirichlet's formula, we obtain With the change of variables z � (s ρ − a ρ )/(t ρ − a ρ ), we obtain (60) us, (58) is valid for i + 1. Besides, we observe that F n w(t) ⟶ 0 as n ⟶ ∞. Hence, from (57), we have Furthermore, if the function v is nondecreasing, then for all s ∈ [a, t], we have v(s) ≤ v(t) and so erefore, we deduce that ψ(t) is bounded on [a, β). It follows from eorem 6 that the solution ψ(t) can be extended to the right side of β. is obviously contradicts the assumption that [a, β) is the maximal existence interval. is consequently implies that β � +∞.
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 ξ(t) is another solution for problem (16) and ξ(a) � ψ(a), ξ ′ (a) � ψ ′ (a). en, one has that , which completes the proof.

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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 δ 1 ∈ C(J, R d ) such that ‖δ 1 ‖ ≤ ε, for all t ∈ J, and C D α,ρ a + ξ(t) � f(t, ξ(t)) + δ 1 (t) (ii) If a function ξ is a solution of (65), then there exists a function δ 2 ∈ C(J, R d ) such that ‖δ 2 ‖ ≤ εφ(t), for all t ∈ J, and C D α,ρ (16) is said to be (i) Ulam-Hyers-Mittag-Leffler stable if there is a constant C f > 0 such that, for each ε > 0 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 (φ, η) if there is a constant C φ > 0 such that, for each ε > 0 and for each solution ξ of (65), there exists a solution ψ of problem (16) satisfying the estimate In the following theorem, the Ulam-Hyers-Mittag-Leffler stability for problem (16) is presented.
Theorem 8. Let f: J × R d ⟶ R d be a continuous function which satisfies the assumption as follows.
(A2) Assume that the following hypothesis holds: Let φ: J ⟶ R + in the inequality (65) be a nondecreasing function for all t ∈ J. We assume that there exists a positive constant K φ which satisfies K φ L < 1, and for t ∈ J, respectively, where ‖δ 1 (t)‖ ≤ ε and ||δ 2 (t)|| ≤ εφ(t). To show the results of this theorem, the method of successive approximations will be used.

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By the abovementioned definition of successive approximations, for n � 1 and from (71), one has By the Lipschitz condition of the function f, for any t ∈ J and n � 2, 3, 4, . . ., one has So, from (74) and for n � 2, one obtains and for n � 3, we also obtain By using the mathematical induction method, for n ≥ 4, we have Now, if we assume that (78) holds for n � k, then by (4) one obtains which is the inequality (78) for n � k + 1. is yields that the inequality (78) is satisfied for all n ≥ 1. en, one has Since the series of the right-hand side of the above inequality is convergent to that is, is yields that the series ψ 0 (t) + ∞ n�1 [ψ n (t) − ψ n− 1 (t)] is uniformly convergent on J with respect to the norm ‖·‖. Now, assume that en, we take ψ k (t) which is the k th partial of series (83) of the form By (83) and (84), we notice Set ψ(t) � ψ(t) for t ∈ J. 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 t ∈ J and by the Lipschitz condition of f, we have On the other hand, (83) and (84) yield erefore, we obtain for t ∈ J So, it follows from (87) that, for t ∈ J, ] n+1 ρ α(n+1) Γ((n + 1)α + 1) .
Prove the assertion (A2): similar to the proof of the assertion (I), we also consider the sequence ψ n n ≥ 0 ∈ C([a, b], R d ) given by ψ 0 (t) � ξ(t), t ∈ [a, b], and for n ≥ 1, By similar processing as the case of (A1), we get for n ≥ 1 and for t ∈ J, In order to check the validity of (93), for n � 1, from (72) and by using the definition of successive approximations (92), one obtains Suppose that (93) is true for n � k ∈ N, that is, en, by the Lipschitz condition of f and the hypothesis of the assertion (A2), we obtain is proves inequality (93) is valid for all n ≥ 1. On the other hand, by the hypothesis K φ L ∈ (0, 1) and from (93) we have that, for t ∈ J, Since the function φ(t) is continuous on J, it is bounded. So, inequality (97) yields that the series ψ 0 (t) + ∞ n�1 [ψ n (t) − ψ n− 1 (t)] is absolutely and uniformly convergent on J with respect to the distance ‖ · ‖. erefore, we set, for t ∈ J, en, similar to the proof of (A1) we also get the estimation between the solution of problem (16) ψ(t) and the solution of inequality (65) ξ(t) as follows: Taking the integral I α,ρ a + on [a, b] of the order α ∈ (1, 2) on both sides of (99) and using hypothesis (H3), one obtains