Existence of solutions of a semilinear functional-differential evolution nonlocal problem

Abstract

In this paper we study the existence of mild and classical solutions of a nonlocal Cauchy problem for a semilinear functional-differential evolution equation. Methods of the functional analysis concerning to a compact C0 semigroup of operators and Schauder's fixed point theorem are applied. The nonlocal Cauchy problem considered here is of the following form: u (t)+ Au (t)= f (t; u (t); u (b1 (t));:::; u (bm (t))); t∈(t0; t0+ a]; m∈ N;(1.1) u (t0)+ g (u)= u0;(1.2) where t0≥ 0, a¿ 0,− A is the infinitesimal generator of a compact C0 semigroup of operators