Abstract. In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We take two different approaches, each one leading to different results that complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT(0) spaces where we study the particular behavior of these spaces regarding the problems we are concerned with.

We study the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. We also study the existence of fixed points for setvalued nonexpansive mappings in the same class of spaces. Our results do not assume convexity of the metric which makes a big difference when studying the existence of fixed points for set-valued mappings.

Kadec-Klee property Fixed points Uniformly lipschitzian mappingsIn 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 Lifsic 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 Lifsic characteristic function of CAT(k) spaces is solved in the positive.

We study the nonexpansivity of reflection mappings in geodesic spaces and apply our findings to the averaged alternating reflection algorithm employed in solving the convex feasibility problem for two sets in a nonlinear context. We show that weak convergence results from Hilbert spaces find natural counterparts in spaces of constant curvature. Moreover, in this particular setting, one obtains strong convergence.

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