Local expansions and accretive mappings

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