International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 3, Pages 419-429

Local expansions and accretive mappings

W. A. Kirk

Department of Mathematics, The University of Iowa, lowa City 52242, lowa, USA

Received 14 February 1983; Revised 24 March 1983

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.


Let X and Y be complete metric spaces with Y metrically convex, let DX be open, fix u0X, and let d(u)=d(u0,u) for all uD. Let f:X2Y 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 xD has a neighborhood N for which dist(f(u),f(v))c(max{d(u),d(v)})d(u,v) for all u,vN. Then, given yY, it is shown that yf(D) iff there exists x0D such that for xX\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