Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps

Abstract

In this paper, we study a class of second-order neutral stochastic evolution equations with infinite delay and Poisson jumps (SNSEEIPs), in which the initial value belongs to the abstract space \documentclass[12pt]{minimal}\begin{document}$\mathcal {B}$\end{document}B. We establish the existence and uniqueness of mild solutions for SNSEEIPs under non-Lipschitz condition with Lipschitz condition being considered as a special case by means of the successive approximation. Furthermore, we give the continuous dependence of solutions on the initial data by means of a corollary of the Bihari inequality. An application to the stochastic nonlinear wave equation with infinite delay and Poisson jumps is given to illustrate the theory.