Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations

Abstract

This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results.

Existence and Uniqueness of Mild Solutions of Stochastic Partial Integro-Differential Impulsive Equations with Infinite Delay via Resolvent Operator

“…Since many control systems arising for realistic models depends heavily on histories (that is, effect of infinite delay on the state equations), there is a real need to discuss the existence results for impulsive partial stochastic neutral integro-differential equations with state-dependent delay. Recently, the problem of the existence of solutions for partial impulsive functional differential equations with infinite delay has been investigated in many publications such as [2,3,5,[7][8][9][10][11]19] and the references therein. Motivated by the previously mentioned works, in this paper, we will extend some such results of mild solutions for (1.1).…”