Endpoints of set-valued contractions in metric spaces

Suppose (X, d) be a complete metric space, and suppose F : X -> CB(X) be a set-valued map satisfies H(Fx, Fy) <= psi(d(x, y)), for each x, y is an element of X, where psi : [0, infinity) -> [0, infinity) is upper semicontinuous, psi(t) < t for each t > 0 and satisfies lim inf(t-infinity) (t - psi(t)) > 0. Then F has a unique endpoint if and only if F has the approximate endpoint property. (C) 2009 Elsevier Ltd. All rights reserved.