ORBITAL FIXED POINT CONDITIONS IN GEODESIC SPACES

Many metric fixed point results can be formulated in an abstract 'convexity structure' setting. This discussion contains a review of some of these, as well as a discussion of other results which seem to require a bit more structure on the space. A metric space (X, d) is said to be Gamma-uniquely geodesic if Gamma is a family of geodesic segments in X and for each x, y is an element of X there is a unique geodesic [x, y] is an element of Gamma with endpoints x and y. Let X be Gamma-uniquely geodesic and let rho(X) denote the family of all bounded closed convex (relative to Gamma) subsets of X. Assume that the family rho(X) is compact in the sense that every descending chain of nonempty subsets of rho(X) has a nonempty intersection. This is a brief discussion of what additional conditions on a mapping T : K -> K for K is an element of rho(X) always assure that has at least one fixed point. In particular it is shown that if the balls in X are Gamma-convex and if the closure of a Gamma-convex set in X is again Gamma-convex then a mapping T : K -> K always has a fixed point if it is nonexpansive with respect to orbits in the sense of Amini-Harandi, et al., and if for each x is an element of K with x not equal T (x). inf (m is an element of N) {lim sup (n ->infinity) d (T-m (x), T-n (x))} < diam (O (x)). Mappings of the above type include those which are pointwise contractions in the sense that for each x is an element of K there exists alpha (x) is an element of (0, 1) such that d(T (x), T (y)) <= alpha (x) d (x, y) for all y is an element of K. The results discussed here extend known results if K is a weakly compact (convex) subset of a Banach space. A number of open questions are raised in connection with characterizations of normal structure in certain geodesic spaces.