CAT(k)-spaces, weak convergence and fixed points

In this paper we show that some of the recent results on ¯xed point for CAT(0) spaces still hold true for CAT(1) spaces, and so for any CAT(k) space, under natural boundedness conditions. We also introduce a new notion of convergence in geodesic spaces which is related to the ¢-convergence and applied to study some aspects on the geometry of CAT(0) spaces. At this point, two recently posed questions in [12] (W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (12) (2008), 3689-3696) are answered in the negative. The work ¯nishes with the study of the Lif¸sic characteristic and property (P) of Lim-Xu to derive ¯xed point results for uniformly lipschitzian mappings in CAT(k) spaces. A conjecture raised in [4] (S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear Anal., 65 (2006), 762{772) on the Lif¸sic characteristic function of CAT(k) spaces is solved in the positive.