Abstract
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.
Highlights
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
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
Summary
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. 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 These derivatives will provide us new chances to get generalized solutions of fractional differential equations [1,2,3,4,5,6]. 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. 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.
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