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 inf t → ∞ ( t − ψ ( t ) ) > 0 . Then F has a unique endpoint if and only if F has the approximate endpoint property.