The Kneser Property for Abstract Retarded Functional Differential Equations with Infinite Delay

We establish existence of mild solutions for a class of semilinear first-order abstract retarded functional differential equations (ARFDEs) with infinite delay and we prove that the set consisting of mild solutions for this problem is connected in the space of continuous functions. 1. Introduction. The purpose of this paper is to establish existence of mild solutions of a semilinear abstract retarded functional differential equation (ARFDE) with infinite delay of first order, and to show that under general conditions the set formed by the mild solutions is connected in the space of continuous functions. This property is known in the literature as the Kneser's property. We refer to [5] for the original result in the frame of differential equations and to [9] for a similar result for functional equations. We start with an abstract statement of this property. In this statement, we denote by V δ (B) the δ-neighborhood of a set B in a metric space.