Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

Hossein Fazli,Hongguang Sun,Juan J Nieto

doi:10.3390/math8050743

Abstract

We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theorems.

Highlights

Dynamical behavior of physical processes are usually represented by differential equations
Materials [9]; the corresponding models can be described by the fractional differential equations
Fractional Langevin equation as a generalization of classical one gives a fractional Gaussian process parametrized by two indices, which is more flexible for modeling fractal processes [12,13,14,15,16]

Summary

Introduction

Dynamical behavior of physical processes are usually represented by differential equations. Authors in [23] studied nonlinear fractional Langevin equation involving two fractional orders in different intervals as a generalized form of three point third order nonlocal boundary value problem of nonlinear ordinary differential equations. As we have seen in the papers cited above about analysis of fractional Langevin equation, using various classical fixed point theorems is a common and useful technique for obtaining the existence and uniqueness results for fractional Langevin equation involving different initial or boundary conditions. The contractivity constant of the fixed point problem associated with the fractional Langevin equation depended on the friction coefficient λ. We obtain a new existence and uniqueness results under some weak conditions by using contractive mapping theorem and Weissinger’s fixed point theorem

Preliminaries and Auxiliary Results

Existence and Uniqueness

Illustrative Examples

Conclusions