Existence of mild solutions for the impulsive semilinear nonlocal problem with random effects

Shuorui Zhang,Jitao Sun

doi:10.1186/1687-1847-2014-194

Abstract

In this paper, we study the existence of impulsive semilinear nonlocal problems with random effects. Random perturbations are taken into consideration in abstract spaces. By applying the measure of noncompactness and a random fixed point theorem with stochastic domain, we get some existence results which improve and generalize many known results. Some weaker assumptions are established. Besides, we do not claim the semigroup to be compact.

Highlights

During the past few decades impulsive differential equations have been widely studied and significant progress has been made in [ – ] and references therein
Since many real world phenomena exhibit the presence of sudden state changes, the study of impulsive differential equations is becoming more important nowadays
We will give a very useful random fixed point theorem with stochastic domain

Summary

Introduction

During the past few decades impulsive differential equations have been widely studied and significant progress has been made in [ – ] and references therein. For impulsive semilinear nonlocal problems, in [ ], Liang et al obtained existence and uniqueness results by various assumptions of compactness or Lipschitz on T(t), f , g or Ik. Very recently, in [ , ] the authors put forward sufficient conditions on the existence of nonlocal impulsive differential equations by supposing the semigroup T(t) to be equicontinuous, but in their papers impulsive functions are always assumed to be compact or Lipschitzian. The nonlocal problem with impulses and differential equations with random effects were studied in, e.g., [ , ], respectively. We study the nonlocal problem with impulses combined with random effects. We prove existence results without compactness on T(t) or f and the Lipschitz condition on f , and we weaken assumptions on impulsive functions Ik. The rest of this paper is organized as follows.

Notations and preliminaries

Conclusion

Methods