Fixed points of asymptotically contractive mappings

Abstract Let X and Y be metric spaces. We say that a continuous mapping T : X → Y is asymptotically- k -contractive if lim n , m ; n ≠ m d ( Tx n , Tx m ) ⩽ k lim n , m ; n ≠ m d ( x n , x m ) for every sequence { x n } such that both limits exist. It is proved that every k -set-contractive mapping T : X → Y is asymptotically k -contractive and that every asymptotically k -contractive mapping T : C → C has a fixed point when C is a bounded closed convex set of a Banach space.