Existence and Dependence Results for Semilinear Functional Stochastic Differential Equations with Infinite Delay in a Hilbert Space

Abstract

Using techniques of measures of noncompactness, we prove existence, uniqueness, and dependence results for semilinear stochastic differential equations with infinite delay on an abstract phase space of Hilbert space valued functions defined axiomatically, where the unbounded linear part generates a noncompact semigroup and the nonlinear parts satisfies some growth condition and, with respect to the second variable, a condition weaker than the Lipschitz one. These results are applied to a stochastic parabolic partial differential equation with infinite delay.Mathematics Subject Classification. Primary 34K50; Secondary 47H08, 35K58, 60H15.