Langevin Equations with Generalized Proportional Hadamard–Caputo Fractional Derivative

We look at fractional Langevin equations (FLEs) with generalized proportional Hadamard–Caputo derivative of different orders. Moreover, nonlocal integrals and nonperiodic boundary conditions are considered in this paper. For the proposed equations, the Hyres–Ulam (HU) stability, existence, and uniqueness (EU) of the solution are defined and investigated. In implementing our results, we rely on two important theories that are Krasnoselskii fixed point theorem and Banach contraction principle. Also, an application example is given to bolster the accuracy of the acquired results.


Introduction
In recent years, fractional calculus has gained great importance by numerous renowned mathematicians. e most essential feature of this topic is that it allows us to execute integrations and differentiations in any order, not necessarily integer ones Such a benefit has been encouraged by the applications in various areas, conceivably including fractal phenomena which appear in many sciences such as physics and engineering. Also, discovering the fractional derivatives and their generalizations was done by well-known mathematicians such as Riemann, Caputo, Hadamard, Euler, Liouville, Laplace, Laurent, and Fourier and was constantly the major path of research in the fractional calculus area. Moreover, these derivatives will provide us new chances to get generalized solutions of fractional differential equations [1][2][3][4][5][6]. It is worth to mention that the most important comprehensive treatments of differential equations with fractional order were initiated by Caputo [7] in 1969, and it was later called Caputo's derivative. ere are many generalizations and modifications of this derivative [8][9][10][11][12], for example, the authors in [9] used Caputo's derivative to modify the Hadamard derivatives to a more beneficial concept that called Hadamard-Caputo derivatives (HCD). In 2019, Rahman et al. [12] introduced an integral form of Hadamard fractional derivatives which give generalized forms than the HCD and have shorted as GHCD. Many authors used HCD and their generalizations to study the existence, uniqueness, and stability of fractional differential equations [1,6,13,14]. e researchers in [14] utilized the HCDs to present some results on the existence and stability for solutions of fractional Langevin equations with some conditions related to nonperiodic type boundary and nonlocal integral. Recently, Devi et al. [1] used two fixed point theorems due to Krasnoselskii and Banach as well as the HCDs of distinctive orders connected with nonlocal integral to establish the existence, uniqueness, and HU stability of solutions for fractional Langevin equations with nonperiodic boundary conditions.
According to varied literature, like [15][16][17], it has verified that the FLEs are perfectly mathematical models for singlefile prevalence and the conduct of unshackled particles driven by internal noises. is motivates us to examine the solutions of the FLEs and their features. So, in the present work, we use the generalized proportional Hadamard-Caputo derivative to investigate the FLEs with nonlocal integrals and nonperiodic boundary conditions. For the proposed equations, the HU stability and EU of the solution are defined and analyzed. In implementing our results, we relied on two main fixed point theorems, namely, Krasnoselskii's theorem and Banach contraction operator. Furthermore, to reinforce the accuracy of the gained results, an application example is presented with adequate values for the parameters.
Our paper arranged as follows. In Section 2, we present all basic concepts related to the HCDs and their generalizations with some previous results which serve our results. Section 3 devoted to present our main results which in turn was divided into two subsections, namely, 3.1, introduced to the existence and uniqueness of the solutions of the FLEs, and 3.2, prepared to study the stability results for the solutions of the FLEs.

Preliminaries
We use the set AC r [1, e] to refer to all absolutely continuous functions ξ such that its derivative of order (r − 1) is absolutely continuous on [1, e], where r ∈ N and e is the renowned Euler's number.
Definition 1 (see [18]). If ξ: [1, ∞) ⟶ R is both integrable and continuous, the one side fractional integral of Hadamard of q th -order is given by Definition 2 (see [18]). e q th -order Hadamard fractional derivative is given by where m � [q] + 1 and q > 0.
Definition 3 (see [18]). e q th -order fractional HCD is given by Rahman et al. [12] recently defined a generalized proportional Hadamard fractional integral of order q.
Finally, Jarad et al. [10] presented a broader range of fractional proportional integrals and derivatives.
Theorem 1 (see [8]). Let K be a nonempty convex, bounded, and closed subset of a Banach space W. Consider J 1 and J 2 are two operators from K to W such that

Main Results
is section comprises our new results for the fractional Langevin equations with generalized proportional Hadamard-Caputo derivative. By using Krasnoselskii and Banach fixed point theorems, we investigate the existence, uniqueness, and HU stability for these FLEs. Before we can show our results, we have to prove the following helpful lemmas. en, Also, the fractional differential equation where Proof.
Also, Computational Intelligence and Neuroscience By applying equation (11) in (12), we have where the replacing of the integer m − r by l has been modified. Hence, the solution of the FDE (9) is given by □ Lemma 3. e solution of the following fractional Langevin equation in the integral form is given by Proof. Applying the generalized proportional Hadamard fractional integral ℷ q,ϱ to equation (15), we get Repeating integration by using generalized proportional Hadamard fractional integral operator of order p, we obtain en, by substituting in (18) from (19) and (20), we get where M 0 , M 1 , M 2 , M 3 , and M 4 are real constants. By using the boundary condition, we get the following.
From the conditions, By putting t � 1 in (17), we get Also, by putting t � e in (17) and (22) and from equation (24), we have Computational Intelligence and Neuroscience 5 Substituting in (22) from equations (24) and (25), we obtain 6 Computational Intelligence and Neuroscience It should be noted that if the operator J has a fixed point, the solution of equation (15) exists.

Existence Result.
We now introduce the following conditions to confirm our existence result with the support of fixed point procedure.

Theorem 2.
Let us imagine that the continuous function ξ: [1, e] × R ⟶ R satisfying the following inferences:
Finally, we shall demonstrate that J 1 is continuous and compact.
To begin with, since ξ is a continuous function on t ∈ [1, e], operator J 1 is also continuous. J 1 must be uniformly bounded and equicontinuous on B r in order to be seen to be compact.
is proves that J 1 is uniformly bounded. e compactness of operator J 1 is then demonstrated. For each 1 < t 1 < t 2 < e, we get Taking advantage of the e ((ϱ− 1)/ϱ)ln(t 2 /u) < e ((ϱ− 1)/ϱ)(ln(t 1 /u)) ≤ 1, e ((ϱ− 1)/ϱ)(ln(t 2 /t 1 )) ≤ 1, for each 1 ≤ u < t 1 < t 2 ≤ e, and from the first mean value theorem, we get Taking the limit as J 1 is an equicontinuous as a result of this. We may conclude that J 1 compacts on B r based on the Arzela Ascoli theorem. As a result, in B r , there is a point z so that z � J 1 z. As a consequence, equation (1) [1,e] then the FLEs (15) have a unique solution on [1, e].