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A note on fixed point sets in CAT(0) spaces. (English) Zbl 1101.54040
Summary: We show that the fixed point set of a quasi-nonexpansive selfmap of a nonempty convex subset of a CAT(0) space is always closed, convex and contractible. Moreover, we give a construction of a continuous selfmap of a CAT(0) space whose fixed point set is prescribed.

MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text:
References:
 [1] Ahmed, M. A.; Zeyada, F. M.: On convergence of a sequence in complete metric spaces and its applications to some iterates of quasi-nonexpansive mappings. J. math. Anal. appl. 274, 458-465 (2002) · Zbl 1024.47036 [2] Bridson, M.; Haefliger, A.: Metric spaces of non-positive curvature. (1999) · Zbl 0988.53001 [3] Khamsi, M. A.; Kirk, W. A.: An introduction to metric spaces and fixed point theory. (2001) · Zbl 1318.47001 [4] Kirk, W. A.: Geodesic geometry and fixed point theory. Seminar of mathematical analysis, 195-225 (2003) · Zbl 1058.53061 [5] W.A. Kirk, Geodesic geometry and fixed point theory II, in: Proceedings of the International Conference in Fixed Point Theory and Applications, Valencia, Spain, 2003, pp. 113 -- 142 · Zbl 1083.53061