W. A. Kirk

Copyright © 1983 W. A. Kirk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let X and Y be complete metric spaces with Y metrically convex, let D⊂X be open, fix u0∈X, and let d(u)=d(u0,u) for all u∈D. Let f:X→2Y be a closed mapping which maps open subsets of D onto open sets in Y, and suppose f is locally expansive on D in the sense that there exists a continuous nonincreasing function c:R+→R+ with ∫+∞c(s)ds=+∞ such that each point x∈D has a neighborhood N for which dist(f(u),f(v))≥c(max{d(u),d(v)})d(u,v) for all u,v∈N. Then, given y∈Y, it is shown that y∈f(D) iff there exists x0∈D such that for x∈X\D, dist(y,f(x0))≤dist(u,f(x)). This result is then applied to the study of existence of zeros of (set-valued) locally strongly accretive and ϕ-accretive mappings in Banach spaces