A Langevin-Type q-Variant System of Nonlinear Fractional Integro-Difference Equations with Nonlocal Boundary Conditions

Hana Al-Hutami,Ravi P. Agarwal,Bashir Ahmad

doi:10.3390/fractalfract6010045

Abstract

We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the given problem. Illustrative examples for the obtained results are also presented.

Highlights

The Langevin equation provides a decent approach to describe the evolution of fluctuating physical phenomena
The failure of the ordinary Langevin equation for correct description of the dynamical systems in complex media led to its several generalizations
Keeping in mind the importance of the fractional Langevin equation, we introduce and study a new problem consisting of a coupled system of Langevin-type nonlinear fractional q-integro-difference equations complemented with nonlocal multipoint boundary conditions

Summary

Introduction

The Langevin equation provides a decent approach to describe the evolution of fluctuating physical phenomena. The failure of the ordinary Langevin equation for correct description of the dynamical systems in complex media led to its several generalizations One such example is that of the Langevin equation, involving fractional-order derivative operators, which provides a more flexible model for fractal processes. One can find interesting results on nonlinear boundary value problems involving fractional q-derivative and q-integral operators, and different kinds of boundary conditions in the articles [24,25,26,27,28,29,30,31,32,33,34,35,36,37]. Keeping in mind the importance of the fractional Langevin equation, we introduce and study a new problem consisting of a coupled system of Langevin-type nonlinear fractional q-integro-difference equations complemented with nonlocal multipoint boundary conditions.

Preliminaries on Fractional q-Calculus

Existence and Uniqueness Results

Illustration of Theorem 1

Illustration of Theorem 2

Conclusions