On implicit impulsive Langevin equation involving mixed order derivatives

Akbar Zada,Rizwan Rizwan,Jiafa Xu,Zhengqing Fu

doi:10.1186/s13662-019-2408-6

Abstract

In this paper, we consider a nonlocal boundary value problem of nonlinear implicit impulsive Langevin equation involving mixed order derivatives. Sufficient conditions are constructed to discuss the qualitative properties like existence and Ulam’s stability of the proposed problem. The main result is verified by an example.

Highlights

An equation of the form m d2z dw2 = λ dz dw + η(w) is calledLangevin equation, introduced byPaul Langevin in 1908
It is very important to learn the idea of fractional Langevin equations; for more details, see [1–4]
It has been observed that Fractional differential equations (FDEs) are more accurate than the integer-order derivatives

Summary

Introduction

Many mathematicians have 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 [73–75]. Zada et al [77] studied the existence and uniqueness of solutions by using Diaz Margolis’s fixed point theorem and presented different types of Ulam–Hyers stability for a class of nonlinear implicit fractional differential equations with non-instantaneous integral impulses and nonlinear integral boundary conditions:. M, Zada et al [78] studied the existence and uniqueness of solutions by using Diaz Margolis’s fixed point theorem and presented different types of Ulam–Hyers stability for a class of nonlinear implicit sequential fractional differential equations with non-instantaneous integral impulses and multi-point boundary conditions:. We give an example which supports our main result

Preliminaries

Conclusion

Methods