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)≤ψ(d(x,y)), , where ψ:0,∞)→0,∞) is upper semicontinuous, ψ(t)<t for each t>0 and satisfies lim inf_t→∞(t−ψ(t))>0. Then F has a unique endpoint if and only if F has the approximate endpoint property.