Existence Results of Langevin Equations with Caputo–Hadamard Fractional Operator

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Introduction
In recent years, it has been noted that several real-world phenomena cannot be accurately modeled by ordinary diferential equations, partial diferential equations, or classical diferential equations defned using standard derivatives and integrals. Fractional calculus generalizes the concept of integer order to arbitrary order, making it a useful tool for modeling these complex phenomena. Over the past decades, fractional calculus has gained signifcant attention due to its wide range of applications in various scientifc felds. It has been used in felds, such as biology, blood fow phenomena, nonlinear oscillation of earthquakes, image processing, capacitor theory, groundwater problems, viscoelasticity, aerodynamics, geophysics, biophysics, polymer rheology, and electrical circuits. For more applications of FDEs, we refer to sufcient works [1][2][3][4][5].
Te Langevin equation (LE) describes the evolution of physical phenomena in a fuctuating environment. Te concept of the classical form of the Langevin equation was frst introduced by Paul Langevin in 1908. Te Brownian motion of particles was frst expressed by the classical Langevin equation [5]. Te classical Langevin equation failed to give sufcient results for some fractal disorder domains. So to explain the dynamic dealing of a few fractal and inherited properties of the fractional diferential equations was discussed by Kubo et al. [6,7]. Lim et al. [8] frst used FLEs with two separate fractional orders in 2008. Te solutions of this type of equation provide more fexible models as compared with the solutions of one-order FDEs. Recently, fractional diferential equation with boundary conditions has been discussed spaciously [9]. Existence results for the FDEs with diferent types of boundary conditions via fxed point technique are discussed by Yu et al. [10]. Te existing result of FDEs for coupled systems is connected with different types of boundary conditions by using upper and lower solution method analyses by Baghani et al. [11]. Fazli and Nieto discuss the FDEs with nonlocal boundary conditions [12]. In the HU stability, an exact solution exists very near to the approximate solution for FDEs. Te stability and existence result of FLEs is discussed by Rizwan and Zada [13]. Wang and Li discussed the stability and EU results of the fractional diferential equation [14]. Te existence results of FDEs with strip conditions including the p-Laplace operator are discussed by Wang and Wang [15]. Te stability and existence results of FDEs for the coupled system are discussed by Matar et al. [16]. Te stability results for positive general FDEs via the fxed point technique are discussed by Chai [17]. For more results about stability, see [18][19][20][21][22][23][24][25][26][27][28][29][30].
Inspired by the above work, we discuss the EU result for a nonlinear Langevin equation with Hadamarad-Caputo and Caputo fractional derivatives of a distinct order: where λ ≠ β, CH D ] , C D ϖ denoted the Hadamard-Caputo and Caputo derivative of fractional order ] and ϖ, respectively. 1 < μ < e, 1 < ] + ϖ ≤ 3, β, ξ > 0, λ ∈ R. Te rest section of the manuscript is as follows: the 2nd section contains the basic defnition and desired lemmas. Te main result contains in the 3rd section. HU stability is discussed in the 4th section.
In 5th section, we introduce an application to show the result that is more clear.

Preliminaries Results
Here, we introduce certain defnitions, theorems, and desired lemmas, which are necessary to fnd the important result.
Putting the value of C 1 in equation (11), we get the solution of equation (7) in form integral equation (8). Tis accomplishes the proof. □ □ Theorem (see [18]). Let W be a nonempty bounded, closed, and convex subset of a Banach space E. Let A 1 , A 2 be the operator from W to E such that

is continuous and compact
Ten, there exists z * ∈ W such that z * � A 1 z * + A 2 z * .

Main Results
Suppose S � C[1, e] be the Banach space containing all continuous functions from [1, e] to R with the norm given by ‖S‖ � sup t∈ [1,e] |κ ⋆ (t)|. Now, we defne an operator H: S ⟶ S, It is recognized that the result of equation (1) exists if operator H has a fxed point.

Existence Results.
In this section, we establish the existing results by using the fxed point technique and the following assumption: [1,e] |Φ ⋆ (t, 0)| < ∞ Also, we use some positive constants mentioned as follows: where ∧ 3 defnes in equation (16).
Let (H 1 ) and (H 2 ) be the operator on B r are defned as follows: We observe that (Hκ ⋆ )(t) � (H 1 κ ⋆ )(t) + (H 2 κ ⋆ 1 )(t), κ ⋆ ∈ B r on [1, e]. □ We will use three conditions to prove the required results related to Teorem 7: Journal of Mathematics (2) Now, we prove that H 2 is contraction map. For any value κ ⋆ , κ ⋆ 1 ∈ S, From equation (17) Tis proves that H 1 is uniformly bounded on B r . Now, we prove that operator H 1 is compact.
We know that Lagrange mean value theorem: where d 1 and d 2 are independent of t. By using Lagrange mean value theorem, we get Tus, H 1 is equicontinuous. Terefore, by using Arzela-Ascoli theorem, we conclude that H 1 is compact on B r . All the conditions of Teorem 7 are satisfed. Terefore, by using Teorem 7, there exists a point κ ⋆ ∈ B r such that κ ⋆ � H 1 κ ⋆ + H 2 κ ⋆ . Hence, give equation (1) where ∧ 1 and ∧ 2 are defned in equations (14) and (15), respectively.
Proof. Let B R � κ ⋆ ∈ S: ‖S‖ ≤ R { } are closed bounded and convex subsets of S, where and we prove that HB R is bounded, i.e., HB R ⊆ B R and H is contractive. with