Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses

Rizwan Rizwan,Akbar Zada,Xiaoming Wang

doi:10.1186/s13662-019-1955-1

Rizwan Rizwan, Akbar Zada + Show 1 more

Open Access

https://doi.org/10.1186/s13662-019-1955-1

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Abstract

In this paper, we consider a nonlocal boundary value problem of nonlinear implicit fractional Langevin equation with noninstantaneous impulses. We study the existence, uniqueness and generalized Ulam–Hyers–Rassias stability of the proposed model with the help of fixed point approach, over generalized complete metric space. We give an example which supports our main result.

Highlights

At Wisconsin university, Ulam raised a question about the stability of functional equations in 1940
In 1978, Rassias [23] provided a remarkable generalization of the Ulam–Hyers stability of mappings by considering variables
5 Conclusions In this article, we considered a nonlocal boundary value problem of nonlinear implicit fractional Langevin equation with noninstantaneous impulses

Summary

Introduction

At Wisconsin university, Ulam raised a question about the stability of functional equations in 1940. Impulsive fractional differential equations are used to describe both physical and social sciences. Many mathematicians devoted considerable attention to the existence, uniqueness and different types of Hyers–Ulam stability of the solutions of nonlinear implicit fractional differential equations with Caputo fractional derivative, see [4, 6, 7]. Zada et al [38] studied existence and uniqueness of solutions by using Diaz–Margolis’s fixed point theorem and presented Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability for a class of nonlinear implicit fractional differential equation with noninstantaneous integral impulses and nonlinear integral boundary condition:. M, Motivated by [34, 38], we consider the following nonlocal boundary value problem of nonlinear implicit fractional Langevin equation with noninstantaneous impulses:. We give an example to illustrate our main result

Solution framework of linear impulsive fractional Langevin equation

Generalized Ulam–Hyers–Rassias stability

Conclusions