Stability analysis of boundary value problems for Caputo proportional fractional derivative of a function with respect to another function viaimpulsive Langevin equation

Abstract: In this paper, we discuss existence and stability results for a new class of impulsive fractional boundary value problems with non-separated boundary conditions containing the Caputo proportional fractional derivative of a function with respect to another function. The uniqueness result is discussed via Banach’s contraction mapping principle, and the existence of solutions is proved by using Schaefer’s fixed point theorem. Furthermore, we utilize the theory of stability for presenting different kinds of Ulam’s stability results of the proposed problem. Finally, an example is also constructed to demonstrate the application of the main results.


Introduction
Fractional calculus is the generalization of the ordinary differentiation and integration to non-integer order. It has been applied in various fields such as visco-elastic materials, aerodynamics, finance, chaotic dynamics, nonlinear control, signal processing, bioengineering, chemical engineering, and applied sciences. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of many materials and processes. However, for the last few years, the fractional calculus was developed by many researchers. There are different definitions of fractional operators (derivative and integral) that have been presented such as Riemann-Liouville, Caputo, Hadamard, Hilfer, Katugampola, and the generalized fractional operators, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein.
The impulsive differential equations have impulsive conditions at points of discontinuity. They have played an important role in discussing the dynamics process of various physical and evolutionary phenomena which have discontinuous jumps and abrupt changes in their state of systems. Such processes and phenomena appear in various applications. For some works on impulsive problems, we refer readers to [15][16][17][18][19] and references cited therein.
The Langevin differential equation (first introduced by Paul Langevin in 1908 to provide a complex illustration of Brownian motion [20]) is found an effective piece of equipment to explain the evolution of physical phenomena in fluctuating environments of mathematical physics. After that, the ordinary Langevin equation was replaced by the fractional Langevin equation in 1996 [21]. For some works on the fractional Langevin equation, see, for example, [22][23][24][25][26].
In recent years, many researchers attention studied the exclusive examination of the qualitative theory for fractional differential equations. It is existence and uniqueness theory and stability analysis. One of the most method used to examine the stability analysis of functional differential equations is the Ulam's stability such as Ulam-Hyers (UH) stability, generalized Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias (UHR) stability and generalized Ulam-Hyers-Rassias (UHR) stability [27][28][29][30][31][32][33][34]. It has helpfulness in the field of numerical analysis and optimization because solving the exact solutions of the problems of fractional differential equations is very difficult. Consequently, it is imperative to develop the concepts of Ulam's stability for these problems because we need not get the exact solutions of the purpose problems when we study the properties of Ulam's stability. The qualitative theory encourages us obtain an efficient and reliable technique for approximately finding fractional differential equations because there exists a close exact solution when the purpose problem is Ulam's stable. Recently, many researchers attentively initiated and examined the existence, uniqueness, and different types of Ulam's stability of the solutions for nonlinear fractional differential equations with/without impulsive conditions; see [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] and references cited therein. To the best of our knowledge, there is no paper on impulsive fractional Langevin differential equations containing the Caputo proportional fractional derivative of a function concerning function.

Preliminaries
In this section, we recall some notations, definitions, lemmas, and properties of proportional fractional derivative and fractional integral operators of a function with respect to another function that will be used throughout the remaining part of this paper. For more details, see [13,14,50].
Next, the lemma presents the impact of the proportional fractional integral operator on the Caputo proportional fractional derivative operator of the same order.
In the following, for the convenience for the reader, we set the functional equation F x (t) = f (t, x(t), x(µt)), and we express the proportional fractional integral operator defined in (2.3) of a nonlinear function F x by a subscript notation by In the sequel, for nonnegative integers a < b, we use the following notations: where i = 0, 1, 2, . . . , m. In Lemma 2.11, we prepare an important lemma, which is used as the main results of the problem (1.1).
By a similar way repeating the same process, for t ∈ J k = (t k , t k+1 ], k = 0, 1, 2, . . . , m, we have the integral equation From the given boundary conditions, we get the following system , Solving the above system for the constants c 1 and c 2 , we have where Ω 1 Ω 4 Ω 2 Ω 3 are defined by (2.11), (2.12), (2.13) and (2.14), respectively. Substituting these values of c 1 and c 2 in (2.22), yields the solution in (2.10). Conversely, it is easily to shown by direct calculuation that the solution x(t) is given by (2.10) satisfies the problem (2.9) under the given boundary conditions. This completes the proof.
The fixed point theorems play an important role in studying the existence theory for the problem (1.1). We collect here some well-known fixed point theorems for the sake of essential in the proofs of our existence and uniqueness results.
Theorem 2.12. (Banach's fixed point theorem [50]) Let D be a non-empty closed subset of a Banach space E. Then any contraction mapping T from D into itself has a unique fixed point.
Theorem 2.13. (Schaefer's fixed point theorem [50]) Let E be a Banach space and T : E → E be a completely continuous operator, and let the set D = {x ∈ E : x = σT x, 0 < σ ≤ 1} be bounded. Then T has a fixed point in E.

Existence and uniqueness results
In this section, we discuss the existence and uniqueness results for the problem (1.1) via Banach's and Schaefer's fixed point theorems.
In view of Lemma 2.11 to establish existence theorems, we consider the operator equation It is clear that the problem (1.1) has a solution if and only if the operator Q has fixed points.
To simplify the computations, we use the following constants: By applying classical fixed point theorems, we prove in the next subsections, for the problems (1.1), our main existence and uniqueness results.

Uniqueness result via Banach's fixed point theorem
The first result is an existence and uniqueness result for the problem (1.1) by applying Banach's fixed point theorem.
. . , m satisfy the following assumptions: (H 1 ) There exist a constant L 1 > 0 such that, for every t ∈ J and x 1 , Then, the problem (1.1) has a unique solution on J provided that Proof. Observe that the problem (1.1) is equivalent to a fixed point problem x = Qx, where the operator Q is defined by (3.1). Thus, we need to establish that the operator Q has a fixed point. This will be achieved by means of the Banach's fixed point theorem. Let K 1 , K 2 and K 3 be nonnegative constants such that . (3.11) Clearly, B r 1 is a bounded, closed, and convex subset of E. We complete the proof in two steps.
Step I. We show that QB r 1 ⊂ B r 1 .

Existence result via Schaefer's fixed point theorem
The second existence result is based on Schaefer's fixed point theorem.
Proof. We apply Schaefer's fixed point theorem. The proof is given in the following four steps.
Step I. We prove that the operator Q is continuous. Let x n be a sequence such that x n → x in E. Then, for any t ∈ J, we get By using the fact of 0 < e ρ−1 ρ (ψ a (u)−ψ a (s)) ≤ 1 for 0 ≤ s ≤ u ≤ T with the notations (2.6), (2.11)-(2.15) and (3.2)-(3.5), we obtain Since f , λ, ϕ k and ϕ * k are continuous, this implies that Q is also continuous. Then, F x n − F x → 0, and x n − x → 0, as n → ∞, and ϕ k ( Step II. We prove that the operator Q maps a bounded set into a bounded set in E. For r 2 > 0, there exists a constant N > 0 such that, for each x ∈ B r 2 = {x ∈ E : x ≤ r 2 }, then Qx ≤ N. Then, for any t ∈ J and x ∈ B r 2 , we have
Step III. We prove that Q maps a bounded set into an equicontinuous set of E. Let τ 1 , τ 2 ∈ J k for some k ∈ {0, 1, 2, . . . , m} with τ 1 < τ 2 . Then, for any x ∈ B r 2 , where B r 2 is as defined in Step II, by using the property of f is bounded on the compact set J × B r 2 , we have By using the notations (2.6), (2.11)-(2.15) and (3.2)-(3.5), we obtain that From the above inequality, we get that e ρ−1 This inequality is independent of unknown variable x ∈ B r 2 and tends to zero as τ 2 → τ 1 , which implies that (Qx)(τ 2 ) − (Qx)(τ 1 ) → 0 as τ 2 → τ 1 . Therefore by the Arzelá-Ascoli theorem, we can conclude that the operator Q : E → E is completely continuous.
Step IV. The set D = {x ∈ E : x = σQx, } is bounded (a priori bounds). Let x ∈ D, then x = σQx for some 0 < σ < 1. From (H 3 ) and (H 4 ), for each t ∈ J, we get the result by using the same process in Step II, Then, x ≤ Θ 1 h * 1 + 2h * 2 r 2 + (|λ|Λ 2 r 2 + mk 1 )Θ 2 + Θ 3 k * 1 + Θ 4 := N < ∞. This implies that the set D is bounded. By all the assumptions of Theorem 3.2, we conclude that there exists a positive constant N such that x ≤ N < ∞. By applying Schaefer's fixed point theorem (Theorem 2.13), the operator Q has at least one fixed point which is a solution of problem (1.1). The proof is completed.

Ulam stability results
This section is discussed the different type of Ulam's stability such as UH stable, generalized UH stable, UHR stable and generalized UHR stable of the problem (1.1). Now, we introduce Ulam's stability concepts for the problem (1.1).
Let φ ∈ C(J, R + ) be a nondecreasing function, > 0, υ ≥ 0, z ∈ E such that, for t ∈ J k , k = 1, 2, . . . , m, the following sets of inequalities are satisfied: Definition 4.1. If for > 0 there exists a constant C f > 0 such that, for any solution z ∈ E of inequality (4.1), there is a unique solution x ∈ E of system (1.1) that satisfies Definition 4.2. If for > 0 and set of positive real numbers R + there exists φ ∈ C(R + , R + ), with φ(0) = 0 such that, for any solution z ∈ E of inequality (4.2), there exist > 0 and a unique solution x ∈ E of system (1.1) that satisfies then system (1.1) is UHR stable with respect to (υ, φ).

Definition 4.4.
If there exists a real number C f > 0 such that, for any solution z ∈ E of inequality (4.2), there is a unique solution x ∈ E of system (1.1) that satisfies then system (1.1) is generalized UHR stable with respect to (υ, φ). Remark 4.6. The function z ∈ E is called a solution for inequality (4.1) if there exists a function w ∈ E together with a sequence w k , k = 1, 2, . . . , m (which depends on z) such that The function z ∈ E is called a solution for inequality (4.2) if there exists a function w ∈ E together with a sequence w k , k = 1, 2, . . . , m (which depends on z) such that ) + w k , t ∈ J. Remark 4.8. The function z ∈ E is called a solution for inequality (4.3) if there exists a function w ∈ E together with a sequence w k , k = 1, 2, . . . , m (which depends on z) such that

Ulam-Hyers stability
In this subsection, we establish the results related to UH stability of system (1.1).
Proof. Let z be any solution of inequality (4.1). Then, by Remark 4.6 (A 2 )-(A 4 ), we have By Lemma 2.11, the solution of (4.5) is given by From Remark 4.6 (A 1 ) with (H 1 ), (H 2 ) and the fact of 0 < e ρ−1 ρ (ψ a (u)−ψ a (s)) ≤ 1 for 0 ≤ s ≤ u ≤ T , it follows that This implies that By setting we end up with |z(t) − x(t)| ≤ C f . Hence, the system (1.1) is UH stable. The proof is completed.
Corollary 4.12. In Theorem 4.11, if we set = 1 then the system (1.1) is generalized UHR stable.

An example
This section give an example which illustrate the validity and applicability of main results.

Conclusions
In this paper, we have studied the existence, uniqueness, and stability of solutions for a new class of impulsive fractional differential equation augmented by non-separated boundary conditions involving Caputo proportional derivative of a function with respect to another function. The uniqueness of solutions is obtained by using Banach's contraction mapping principle, whereas the existence result is established via Schaefer's fixed point theorem. Moreover, by the application of qualitative theory and nonlinear functional analysis, we investigated results concerning to different kinds of Ulam-Hyers stability such as, Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability. The concerned results have been examined by a suitable example to illustrate the main results.