Nonlinear functional differential equations of monotone-type in Hilbert spaces

In this paper, we study a class of semilinear functional evolution equations with finite delay in which the nonlinearity satisfies the weak condition of demicontinuity with respect to functional variable and also a semimonotone condition. We will prove the existence, uniqueness and measurability of the mild solutions based on an extension of the analogous results for non-delay initial value problems on Banach spaces and a version of recently developed random Schauder’s fixed point theorem. This will be done first for the generalized solutions and then we follow an approximating procedure to obtain the same results for the mild solutions.