International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 3, Pages 419-429
doi:10.1155/S016117128300037X

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