Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

T E Govindan

doi:10.1155/2011/186206

Abstract

This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.

Highlights

In this paper, a neutral stochastic partial functional differential equation is considered in a real separable Hilbert space of the form dxtft, xtAx t a t, xt dt b t, xt dw t, t > 0, 1.1 x t φ t, t ∈ −r, 0 0 ≤ r < ∞, 1.2 where xt s x t s, −r ≤ s ≤ 0
Adimy and Ezzinbi 5 studied the following neutral partial functional differential equation: dD xt dt AD xt F xt, t > 0, 1.3 x t φ t, t ∈ −r, 0, 1.4 where D is a bounded linear operator from C : C −r, 0, X into X a Hilbert space defined by D φ φ 0 − D0 φ, for φ ∈ C, where the operator D0 is given by
We introduce the concept of a mild solution of the problem 1.1 - 1.2

Summary

Introduction

A neutral stochastic partial functional differential equation is considered in a real separable Hilbert space of the form dxtft, xt. Adimy and Ezzinbi 5 studied the following neutral partial functional differential equation: dD xt dt AD xt F xt , t > 0, 1.3 x t φ t , t ∈ −r, 0 , 1.4 where D is a bounded linear operator from C : C −r, 0 , X into X a Hilbert space defined by D φ φ 0 − D0 φ , for φ ∈ C, where the operator D0 is given by. The authors from 5 developed a basic theory on such equations and studied an existence result, among others, using a global Lipschitz condition on F u. We refer to Ezzinbi et al 7 for yet another class of deterministic neutral equations which is again a particular case of 1.1 wherein the authors study existence and regularity problems using global Lipschitz conditions.

Mathematical Formulation

Existence and Uniqueness of a Solution

An Example