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
Highlights
D onto open sets in Y, and suppose f is locally expansive on D in the sense that there exists a
This result is applied to the study of existence of zeros of locally strongly accretive and -accretive mappings in Banach spaces
We first prove a general theorem for "local expansions" and we apply this result in special settings to the study of the existence of zeros of the locally strongly accretive and -accretive mappings
Summary
D onto open sets in Y, and suppose f is locally expansive on D in the sense that there exists a. This result is applied to the study of existence of zeros of (set-valued) locally strongly accretive and -accretive mappings in Banach spaces. We first prove a general theorem for "local expansions" and we apply this result in special settings to the study of the existence of zeros of the locally strongly accretive and -accretive mappings. Extensions of those of [i] even in the point-valued case Results similar to those obtained here may be found in Ray and Walker [2] and in. While it is likely that the methods of Ray-Walker and of Torrej6n could be modified to attain the generality we obtain, our approach, which is a refinement of the argument of Kirk-SchSneberg [1], seems more direct and more in the spirit of the original work of Browder [h, h]. AI for a subset A of X, we use to denote inf[llxll: x A]
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More From: International Journal of Mathematics and Mathematical Sciences
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