LOCAL EXPANSIONS AND ACCRETIVE MAPPINGS

W.A. KIRK

《International Journal of Mathematics and Mathematical Sciences》 1983卷 (1983/12) Pp. 419-429

https://doi.org/10.1155/S016117128300037X

摘要

Let X and Y be complete metric spaces with Y metrically convex, let D ⊂ X be open, fix u_0∈X, and let d(u)=d(u_0,u) for all u ∈ D. Let f : X→2^Y 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 x_0 ∈ D such that for x ∈ X\D, dist(y,f(x_0)) ≤ 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.

關鍵字

Local expansions ； accretive mappings ； nonexpansive mappings ； fixed points ； zeros