A Fixed-point Theorem for Asymptotically Contractive Mappings

We present fixed point theorems for a nonexpansive mapping from a closed convex subset of a uniformly convex Banach space into itself under some asymptotic contraction assumptions. They generalize results valid for bounded convex sets or asymptotically compact sets. In this note we generalize a famous result by Browder [3], Göhde [6] and Kirk [8], recently extended by Luc in [14], by using the notion of an asymptotically compact subset of a Banach space. Here no compactness assumption is involved. Instead we use uniform asymptotic concepts introduced in [19] and the following definition which differs from the one in [14]; a comparison will be made later on. Note that the meaning of the word “asymptotic” is not related with the iterations of the map as in [8], [9], [10], [21] but bears on the behavior of the map at infinity. Definition 1. Let C be a subset of a Banach space X. We say that a map f : C → X is asymptotically contractive on C if there exists x0 ∈ C such that lim sup x∈C, ‖x‖→∞ ‖f(x)− f(x0)‖ ‖x− x0‖ < 1. (1) It is easy to see that this condition is independent of the choice of x0 ∈ C; more precisely, when it is satisfied for some x0, then it is satisfied for any x1 ∈ C. The notion of an asymptotically contractive map enables us to extend to unbounded sets the results of [3], [6], [8] valid for nonexpansive self-mappings on closed convex bounded subsets of uniformly convex Banach spaces. Recall that a mapping f : C → X is nonexpansive if ‖f(x)− f(x′)‖ ≤ ‖x− x′‖ for any x, x′ in C. It is said to be demi-closed if its graph is sequentially closed in the product of the weak topology on C with the norm topology on X . Proposition 2. Let X be a reflexive Banach space and let C be a closed convex subset of X. Let f : C → X be a nonexpansive map which is asymptotically contractive on C and such that f(C) ⊂ C and I− f is demi-closed. Then f has a fixed point. Received by the editors February 19, 2002. 2000 Mathematics Subject Classification. Primary 47H10, 47H09, 54H25, 55M20.