A leray-schauder type theorem and applications to boundary value problems for neutral equations

Abstract

Consider the following two theorems which motivate the main result of this paper. THEOREM 1.1 (Fitzpatrick and Petryshyn [l]). Let X be a Frechet space and let D be an open neighbourhood of 0. Suppose that F : o- X is a continuous x-condensing map such that (a) hx # F(x), x E W and h > 1. Then F has a fixed point. Here, for a subset D of X, i3D and D denote the boundary and closure of D, respectively. Further, x is the ball measure of noncompactness on X defined as in [2].