Partial Functional Evolution Equations with Finite Delay

Abstract

In this chapter, we study some first order classes of partial functional, neutral functional, integro-differential, and neutral integro-differential evolution equations on a positive line \(\mathbb{R}_{+}\) with local and nonlocal conditions when the historical interval H is bounded, i.e., when the delay is finite. In the literature devoted to equations with finite delay, the phase space is much of time the space of all continuous functions on H for r > 0, endowed with the uniform norm topology. Using a recent nonlinear alternative of Leray–Schauder type for contractions in Frechet spaces due to Frigon and Granas combined with the semigroup theory, the existence and uniqueness of the mild solution will be obtained. The method we are going to use is to reduce the existence of the unique mild solution to the search for the existence of the unique fixed point of an appropriate contraction operator in a Frechet space.