Some Existence Results for Systems of Impulsive Stochastic Differential Equations

Sliman Mekki,Abdelghani Ouahab,Juan J Nieto,Tayeb Blouhi

doi:10.2478/amsil-2020-0028

Abstract

Abstract In this paper we study a class of impulsive systems of stochastic differential equations with infinite Brownian motions. Sufficient conditions for the existence and uniqueness of solutions are established by mean of some fixed point theorems in vector Banach spaces. An example is provided to illustrate the theory.

Highlights

Differential equations with impulses were considered for the first time by Milman and Myshkis ([29]) and followed by a period of active research which culminated with the monograph by Halanay and Wexler ([19])
In [6], the authors used the idea of fixed point theory in generalized Banach spaces to prove the existence of mild solutions of impulsive coupled systems of stochastic differential equations with fractional Brownian motion
An application of Schaefer’s and Perov fixed point theorems in generalized Banach spaces are used to prove the existence of solutions to problem (1.1)

Summary

Introduction

Differential equations with impulses were considered for the first time by Milman and Myshkis ([29]) and followed by a period of active research which culminated with the monograph by Halanay and Wexler ([19]). In [6], the authors used the idea of fixed point theory in generalized Banach spaces to prove the existence of mild solutions of impulsive coupled systems of stochastic differential equations with fractional Brownian motion. In the absence of random effect and stochastic analysis many authors studied the existence of solutions for systems of differential and difference equations with and without impulses by using the vector version of the fixed point theorem (see [5, 3, 20, 17, 22, 31, 32, 35, 30], the monograph of Graef et al [15], and the references therein). An application of Schaefer’s and Perov fixed point theorems in generalized Banach spaces are used to prove the existence of solutions to problem (1.1)

Preliminaries

Generalized metric and Banach spaces

Existence and uniqueness results

An example