CONSTRUCTION OF FLXED POINTS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS
Gregory B. Passty · 1982
In uniformly convex Banach spaces with Frechet differentiate norms (e.g. Lp, 1 <p < oo), fixed points for asymptotically nonexpansive mappings are constructed as weak limits of iterates of the mappings themselves or of related mappings. Let E be a uniformly convex Banach space, and let C be a closed convex subset of E. A mapping U of C into itself is said to be asymptotically nonexpansive (Goebel and Kirk (6)) if || Ux - Uy < k\\x - y for all x and/ in C, with lim k = 1. It was proved in (6) that if C is further assumed to be bounded, then an asymptotically nonexpansive self-map of C has a fixed point. We show here that if E has a Frechet differentiable norm, and if U is, for example, weakly continuous, then fixed points of U can be obtained by iterating U starting at a point of asymptotic regularity. Theorem 1 extends theorems of Feathers and Dotson (5) and of Bose (2) which were obtained in uniformly convex spaces with weakly continuous duality maps. The basic tool in both of these papers was Opial's Lemma (7). Because this lemma does not carry over to Lp, p ^ 2, new techniques are needed for this more general case. These were provided by Bâillon (1) and simplified by Bruck (4) when the norm is Frechet differentiable. We will present Theorem 1 in a slightly more general form, and then discuss applications to asymptotically nonexpansive mappings and to a conjecture of H. Schaefer (8). First we extend the definition of (6) to sequences of maps which are not necessarily powers of a given map. Definition 1. The sequence {T} ~_, of self-maps of C is asymptotically nonex- pansive if || Tnx — Tny < kn\\x — y for all x,y in C with lim,, kn = 1. Denote the set of fixed points of T by F(T), strong convergence by —», and weak convergence by -»•. We may now state Theorem 1. Let E be uniformly convex with a Frechet differentiable norm, and C a closed convex subset of E. Let F be a subset of C and S = { 7^ } *_, an asymptotically nonexpansive sequence of self-maps of C such that (a) F c D ^°_, F(Tn). Assume also that there exists x0 in C for which (b) T^Xq -* z implies z E F, and