On nondensely defined semilinear stochastic functional differential equations with nonlocal conditions

M Benchohra,S K Ntouyas,A Ouahab

doi:10.1155/jamsa/2006/69584

Abstract

The nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions for first-order semilinear stochastic functional differential equations in Hilbert spaces.

Highlights

This paper is concerned with the existence of integral solutions for initial value problems for first-order stochastic semilinear functional differential equations with nonlocal conditions in Hilbert spaces
Where f : J × M2([−r, 0], H) → H is a given function, A : D(A) ⊂ H → H is a nondensely defined closed linear operator on H, the function w(t) is a Hilbert space Q-valued Wiener process, φ ∈ M2([−r, 0], D(A)), 0 < r < ∞, is a suitable initial random function independent of w(t), h : M2([−r, 0], D(A)) → D(A), H a real separable Hilbert space with inner product ·, · and norm | · |, and M2 is a class of H-valued stochastic processes that will be specified later
The nonlocal condition can be applied in physics

Summary

Introduction

This paper is concerned with the existence of integral solutions for initial value problems for first-order stochastic semilinear functional differential equations with nonlocal conditions in Hilbert spaces. Random differential and integral equations play an important role in characterizing many social, physical, biological, and engineering problems; see, for instance, the monographs of Da Prato and Zabczyk [6] and Sobczyk [14]. Balasubramaniam and Ntouyas [2] studied the semilinear stochastic evolution delay equations with nonlocal conditions, where A is a densely defined linear operator. Our goal here is to extend the results of Balasubramaniam and Ntouyas [2], where A is nondensely defined. These results can be seen as a contribution to the literature

Preliminaries

Stochastic functional differential equations

Main result