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

In this paper we show that some of the recent results on fixed 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 Δ-convergence and applied to study some aspects on the geometry of CAT(0) spaces. At this point, two recently posed questions in [W.A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (12) (2008) 3689–3696] are answered in the negative. The work finishes with the study of the Lifšic characteristic and property (P) of Lim–Xu to derive fixed point results for uniformly lipschitzian mappings in CAT(k) spaces. A conjecture raised in [S. Dhompongsa, W.A. Kirk, B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear Anal. 65 (2006) 762–772] on the Lifšic characteristic function of CAT(k) spaces is solved in the positive.