Endpoints of set-valued contractions in metric spaces

Abstract

Suppose (X, d) be a complete metric space, and suppose F : X → C B (X) be a set-valued map satisfies H (F x, F y) ≤ ψ (d (x, y)), for each x, y ∈ X, where ψ : [0, ∞) → [0, ∞) is upper semicontinuous, ψ (t) < t for each t > 0 and satisfies lim inft → ∞ (t - ψ (t)) > 0. Then F has a unique endpoint if and only if F has the approximate endpoint property. © 2009.