Existence and Location of a Unique Solution of Caputo-Liouville Type Langevin Equation with Finitely Many Nonlinearities and Nonlocal Boundary Conditions

Bashir Ahmad ,Hanan Al-Johany ,Sotiris K Ntouyas ,Ahmed Alsaedi

doi:10.28924/2291-8639-19-2021-465

Abstract

In this paper, we discuss the existence of a unique solution of Caputo-Liouville type Langevin equation involving two fractional orders and finitely many nonlinearities, equipped with nonlocal boundary conditions via Banach contraction mapping principle. The location of the unique solution of the given problem is also presented. In addition, we discuss the existence of solutions for the problem at hand by means of Krasnoselskii's fixed point theorem. Examples are constructed for the illustration of the obtained results. The paper concludes with some interesting remarks.

Highlights

Fractional order differential equations received overwhelming attention of many researchers as these equations extensively appear in the mathematical modeling of several scientific and technical phenomena
We discuss the existence of a unique solution of Caputo-Liouville type Langevin equation involving two fractional orders and finitely many nonlinearities, equipped with nonlocal boundary conditions via Banach contraction mapping principle
By using Banach contraction mapping principle we prove the existence of a unique solution of boundary value problem (1.1)-(1.2) and we study the location of the unique solution

Summary

Introduction

Fractional order differential equations received overwhelming attention of many researchers as these equations extensively appear in the mathematical modeling of several scientific and technical phenomena. The failure of classical Langevin equation to describe the complex systems led to its several generalizations, which successfully modeled the physical phenomena in disordered regions [7], anomalous diffusion processes in complex and viscoelastic environment [8, 9], etc Among these generalizations includes the one obtained by replacing the ordinary derivative by fractional order derivative in it; the resulting form is known as fractional Langevin equation and can take care of the fractal and memory properties of the phenomena under investigation. Modern tools of functional analysis have played a key role in developing the theory (existence and uniqueness of solutions) for fractional order initial and boundary value problems, for example, see [25]- [30]. − ω ξ2 (ξ2 − s)β+α−1 0 Γ(β + α) i=1 ds + μω ξ2 (ξ2 − s)β−1 y(s)ds

Uniqueness and location results

An existence result