On the Existence and Uniqueness of Solution to Fractional-Order Langevin Equation

Abstract

In this work, we give sufficient conditions to investigate the existence and uniqueness of solution to fractional-order Langevin equation involving two distinct fractional orders with unprecedented conditions (three-point boundary conditions including two nonlocal integrals). The problem is introduced to keep track of the progress made on exploring the existence and uniqueness of solution to the fractional-order Langevin equation. As a result of employing the so-called Krasnoselskii and Leray-Schauder alternative fixed point theorems and Banach contraction mapping principle, some novel results are presented in regarding to our main concern. These results are illustrated through providing three examples for completeness.

Compact and Noncompact Solutions to Generalized Sturm–Liouville and Langevin Equation with Caputo–Hadamard Fractional Derivative

“…Besides, this aspect together with the solution's uniqueness aspect of the same equation have been well explored in [24]. However, for further recent outcomes, we refer the reader to papers [25,26] and references therein.…”

Measure of noncompactness for an infinite system of fractional Langevin equation in a sequence space

“…It worth pointing out that the Langevin equations (formulated by Langevin in 1908) present an accurate way to describe the evolution of physical phenomena in a fluctuating environment [14]. There is a clear progress on fractional Langevin equations in physics [15][16][17][18][19]. New results on the existence of solutions for fractional Langevin equations under variety of boundary value conditions have been published; see [20][21][22][23][24][25][26][27][28][29] and the references mentioned therein.…”

Positive Solvability for Conjugate Fractional Differential Inclusion of (k, n − k) Type without Continuity and Compactness

“…It is worth mentioning that the literature on the existence and uniqueness of solutions to fractional differential equations is expanding at present, and this problem has drawn the attention of many contributors [19][20][21][22][23][24][25][26][27].…”