Langevin equation with nonlocal boundary conditions involving a $ \psi $-Caputo fractional operators of different orders

This paper studies Langevin equation with nonlocal boundary conditions involving a $\psi$--Caputo fractional derivatives operator. By the aide of fixed point techniques of Krasnoselskii and Banach, we derive new results on existence and uniqueness of the problem at hand. Further, the $\psi $-fractional Gronwall inequality and $\psi $--fractional integration by parts are employed to prove Ulam--Hyers and Ulam--Hyers--Rassias stability for the solutions. Examples are gifted to demonstrate the advantage of our major results. The proposed results here are more general than the existing results in the literature and obtain them as particular cases.


Introduction
Lately, fractional calculus has played a very significant role in various scientific fields; see for instance [1,2] and the references cited therein. As a result of this, fractional differential equations have caught the attention of many investigators working in different desciplines [3,4]. However, most of researchers works have been conducted by using fractional derivatives that mainly rely on Riemann-Liouville, Hadamard, Katugampola, Grunwald Letnikov and Caputo approaches.
Fractional derivatives of a function with respect to another function have been considered in the classical monographs [1,5] as a generalization of Riemann-Liouville derivative. This fractional derivative is different from the other classical fractional derivative as the kernel appears in terms of another function ψ. We will call this derivative as ψ-fractional derivative. Recently, this derivative has been reconsidered by Almeida in [7] where the Caputo-type regularization of the existing definition and some interesting properties are provided. Several properties of this operator could be found in [1,5,6,8,9]. For some particular cases of ψ, one can realize that ψ-fractional derivative can be reduced to the Caputo fractional derivative [1], the Caputo-Hadamard fractional derivative [10] and the Caputo-Erdélyi-Kober fractional derivative [11].
On the other hand, the investigation of qualitative properties of solutions for different fractional differential (and integral ) equations is the key theme of applied mathematics research. Numerous interesting results concerning the existence, uniqueness, multiplicity, and stability of solutions or positive solutions by applying some fixed point techniques. However, most of the proposed problems have been handled concerning the classical fractional derivatives of the Riemann-Liouville and Caputo [17][18][19][20][21][22][23][24][25][26][27].
In parallel with the intense investigation of fractional derivative, a normal generalization of the Langevin differential equation appears to be replacing the classical derivative by a fractional derivative to produce fractional Langevin equation (FLE). FLE was first introduced in [31] and then different types of FLE were the object of many scholars [33][34][35][36][37][38][39]. In [40], the authors studied a nonlinear FLE involving two fractional orders at various intervals with three-point boundary conditions. FLE involving a Hadamard derivative type was considered in [34,35].
Alternatively, the stability problem of differential equations was discussed by Ulam in [44]). Thereafter, Hyers in [45] developed the concept of Ulam stability in the case of Banach spaces. Rassias provided a fabulous generalization of the Ulam-Hyers (U-H) stability of mappings by taking into account variables. His approach was refered to as Ulam-Hyers-Rassias (U-H-R) stability [47]. Recently, the Ulam stability problem of implicit differential equations was extended into fractional implicit differential equations by some authors [48][49][50][51]. A series of papers was devoted to the investigation of existence, uniqueness and U-H stability of solutions of the FLE within different kinds of fractional derivatives.
Motivated by the recent developments in ψ-fractional calculus, in the present work, we investigate the existence, uniqueness and stability in the sense U-H-R of solutions for the following FLE within ψ-Caputo fractional derivatives involving nonlocal boundary conditions where J δ,ψ a+,ξ and c D θ,ψ a+,t are ψ-fractional integral of order δ, ψ-Caputo fractional derivative of order θ ∈ {̺, ς, δ} respectively, 0 ≤ a < η < ξ < T < ∞, 1 < ̺ ≤ 2, 0 < δ < ς ≤ 1 and f : [a, T] × R × R → R + is a continuous function. It is worth mentioning here that the proposed results in this paper which rely on ψ-fractional integrals and ψ-Caputo fractional derivatives can generalize the existing results in the literature and obtain them as particular cases.
We propose the remarkable paper [9] in which some generalizations using ψ-fractional integrals and derivatives are described. In particular, we have there exists a solution u ∈ C [a, T] of the problem (1.1) with where Φ (ǫ) is only dependent on ǫ.
there exists a solution u ∈ C [a, T] of the problem (1.1) with Below, we generalize Gronwall's inequality for ψ-fractional derivative proved by Shi-you Lin in [43]. (iii) The constants ̺ i > 0 (i = 1, 2, ..., n) . If Remark 2.13. For n = 2 in the hypotheses of Lemma 2.12. Let v(t) be a nondecreasing function on a ≤ t < T. Then we have where E ̺ 1 ,ψ is the Mittag-Leffler function defined below.
The remaining portion of the paper, we make use of the next suppositions: (A 2 ) There exists an increasing function (A 4 ) There exists an increasing function Φ (t) ∈ (C [a, T] , R + ) and there exists l Φ > 0 such that for any t ∈ [a, T] , J ̺,ψ We adopt the following conventions: and Further, we assume where σ ij are constants.

Existence and uniqueness of solution
In order to study the nonlinear FLE (1.1), We first consider the linear associated FLE and conclude solving it.

Linear boundary problem
The following Lemma regards a linear variant of problem where F ∈ C([a, T], R).

Lemma 3.1. The unique solution of the ψ-Caputo linear problem (3.1) is given by the integral equation
Now applying J ς,ψ a+,t to both sides of (3.4) , we get where c 3 ∈ R.
Using the boundary conditions in (3.1-b), we obtain c 3 := c 3 (F) = 0 and J δ,ψ Then we also get a system of linear equations with respect to c 1 , c 2 as follows Because the determinant of coefficient for ∆ = 0. Thus, we have Substituting these values of c 1 and c 2 in (3.6), we finally obtain (3.2) as 12) i.e., the integral equation (3.12) can be written as (3.2) and (3.13) Differentiating the above relations one time we obtain (3.1-a), also it is easy to get that the condition (3.1-b) is satisfied. The proof is complete.
For convenience, we define the following functions and

Nonlinear problem
Thanks to Lemma 3.1, the following result is an immediate consequence.

Lemma 3.2.
Let λ > 0. Then the problem (1.1) is equivalent to the integral equation and d ij are defined in (3.14) and (3.15). In order to lighten the statement of our result, we adopt the following notation. and We are now in a position to establish the existence and uniqueness results.  (3.26) equipped with the norm Then, we may conclude that (E, . E ) is a Banach space.
To introduce a fixed point problem associated with (3.16) we consider an integral operator (3.28) holds. Then the problem (1.1) has a unique solution on E.
Proof. The proof will be given in two steps.
Step 1. The operator Ψ maps bounded sets into bounded sets in E.
For our purpose, consider a function u ∈ E It is clear that Ψu ∈ E. Also by (2.11), (3.17) and ( 3.28), we have Indeed, it is sufficient to prove that for any Denoting where φ u , d ij (t) and ρ ij defined by (3.17), (3.14-3.15) and (3.22-3.25) respectively.
Using (3.28) and (3.35), we obtain where ς is defined by (3.29) and choose The continuity of the functional f u would imply the continuity of (Ψu) and c D δ,ψ a+,t (Ψu) Hence, Ψ maps bounded sets into bounded sets in E.
Step 2. Now we show that Ψ is a contraction. By (A 1 ) and (3.28), for u, v ∈ E and t ∈ [a, T], we have and for all t ∈ [a, T], which implies Hence, we get Consequently, A similar argument shows that Combining (3.48) and (3.49), we obtain (3.50) Consequently, by (3.47) and (3.50), we have and choose ς = max {ς 11 , ς 12 , ς 21 , ς 22 } < 1. Hence, the operator Ψ is a contraction, therefore Ψ maps bounded sets into bounded sets in E. Thus, the conclusion of the theorem follows by the contraction mapping principle.
For simplicity of presentation, we let (3.55) We consider the space defined by (3.26) equipped with the norm (3.56) It is easy to know that (E, . E ) is a Banach space with norm (3.56). On this space, by virtue of Lemma 3.2, we may define the operator Ψ : where Ψ 1 and Ψ 2 the two operators defined on B r by where d ij (t) defined by (3.14) and (3.15).
Applying c D δ,ψ a+,t on both sides of (3.58) and (3.59), we have Proof. The proof will be specified in sundry steps: Step 1. Firstly, we prove that, for any u, v ∈ B r , Ψ 1 u + Ψ 2 v ∈ B r , it follows that which yields that Ψ 1 is bounded. On the opposite side Then, from (3.67) and (3.69), it follows that which concludes that Ψ 1 u + Ψ 2 v ∈ B r .for all u, v ∈ B r .
Step 3. The continuity of Ψ 1 follows from that of f u . Let {u n } be a sequence such that u n −→ u in E. Then for each t ∈ [a, T] By last equality with equation (3.58), we can write It follows from (3.65) that By (3.78), we have Consequently, by (3.79), we have Since f u is a continuous function, then by the Lebesgue dominated convergence theorem which implies Furthermore, Ψ 1 is uniformly bounded on B r as (Ψ 1 u) E ≤ (Λ 11 + Λ 12 ) × f u ∞ , due to (3.67).
Step 4. Finally, we establish the compactness of Ψ 1 . Let u, v ∈ B r , for t 1 , t 2 ∈ [a, T] , t 1 < t 2 , we have On the other hand Using (3.82) and ( 3.83), we get and Consequently, we have Thus, Ψ 1 is relatively compact on B r . Hence, by the Arzela-Ascoli Theorem, Ψ 1 is completely continuous on B r . Therefore, according to Theorem 2.7, the problem (1.1) has at least one solution on B r . This completes the proof.

Stability of solutions
Hereafter, we discuss the U-H and U-H-R stability of solutions of the FLE (1.1). In the proofs of Theorems 4.4 and 4.9, we use integration by parts in the settings of ψ-fractional operators.
where Φ (t) ≥ 0 if and only if there exists a function g ∈ C, (which depends onũ) such that
Proof. Letũ ∈ C be a solution of the inequality (4.2). Then by Remark 4.1-ii, we have that where with c j ( fũ + g) = c j ( fũ) + c j (g) , j = 1, 2, 3. (4.7) In view of (A 1 ) and (4.3), we obtain As an outcome of Lemma 4.2 we have the following result: thenũ is a solution of the following integral inequality where ς 13 is given by (3.19).

23) and
(4.24) It follows from (4.18) and (4.19), that Using Lemma 2.3 and (A 1 ), we have where where This means that Using Lemma 2.12, the above inequality implies the estimation for p (t) such as (4.35) Therefore, with (A 4 ), the inequality (4.34) can be rewritten as By Remark 2.13, one can obtain Thus, we complete the proof.

Applications
Will be provided in the revised submission.

Conclusion
The Langevin equation has been introduced to characterize dynamical processes in a fractal medium in which the fractal and memory features with a dissipative memory kernel are incorporated. Therefore, the consideration of Langevin equation in frame of fractional derivatives settings would be providing better interpretation for real phenomena. Consequently, scholars have considered different versions of Langevin equation and thus many interesting papers have been reported in this regard. However, one can notice that most of existing results have been carried out with respect to the classical fractional derivatives.
In this paper, we have tried to promote the current results and considered the FLE in a general platform. The boundary value problem of nonlinear FLE involving ψfractional operator was investigated. We employ the the newly accommodated ψfractional calculus to prove the following for the considered problem: We claim that the results of this paper are new and generalize some earlier results. For further investigation, one can propose to study the properties of the solution of the considered problem via some numerical computations and simulations. We leave this as promising future work. Results obtained in the present paper can be considered as a contribution to the developing field of fractional calculus via generalized fractional derivative operators.

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