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On asymptotic pointwise contractions in metric spaces. (English) Zbl 1176.54031
Let $M$ be a complete CAT(0) metric space and $C$ be a nonempty bounded closed convex subset of it. Then, for each pointwise asymptotically nonexpansive map $T:C\to C$, $\text{Fix}(T)$ is nonempty, closed and convex.

54H25Fixed-point and coincidence theorems in topological spaces
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
54C60Set-valued maps (general topology)
Full Text: DOI
[1] Kirk, W. A.: Fixed points of asymptotic contractions. J. math. Anal. appl. 277, 645-650 (2003) · Zbl 1022.47036
[2] W.A. Kirk, Asymptotic pointwise contractions, in: Plenary Lecture, the 8th International Conference on Fixed Point Theory and Its Applications, Chiang Mai University, Thailand, July 16--22, 2007
[3] Kirk, W. A.; Xu, Hong-Kun: Asymptotic pointwise contractions. Nonlinear anal. 69, 4706-4712 (2008) · Zbl 1172.47038
[4] Khamsi, M. A.; Kirk, W. A.: An introduction to metric spaces and fixed point theory. (2001) · Zbl 1318.47001
[5] Aronszajn, N.; Panitchpakdi, P.: Extension of uniformly continuous transformations and hyperconvex metric spaces. Pacific J. Math. 6, 405-439 (1956) · Zbl 0074.17802
[6] Baillon, J. B.: Nonexpansive mappings and hyperconvex spaces. Contemp. math. 72, 11-19 (1988) · Zbl 0653.54021
[7] Espinola, R.; Khamsi, M. A.: Introduction to hyperconvex spaces. Handbook of metric fixed point theory (2001)
[8] Khamsi, M. A.: On asymptotically nonexpansive mappings in hyperconvex metric spaces. Proc. amer. Math. soc. 132, 365-373 (2004) · Zbl 1043.47040
[9] Bridson, M.; Haefliger, A.: Metric spaces of non-positive curvature. (1999) · Zbl 0988.53001
[10] Bruhat, F.; Tits, J.: Groupes réductifs sur un corps local. I. données radicielles valuées. Inst. hautes études sci. Publ. math. 41, 5-251 (1972)
[11] Dhompongsa, S.; Kirk, W. A.; Sims, Brailey: Fixed points of uniformly Lipschitzian mappings. Nonlinear anal. 65, 762-772 (2006) · Zbl 1105.47050
[12] Garcia-Falset, J.; Fuster, E. L.; Prus, S.: The fixed point property for mappings admitting a center. Nonlinear anal. 66, 1257-1274 (2007) · Zbl 1118.47043
[13] Kaewcharoen, A.; Kirk, W. A.: Proximinality in geodesic spaces. Abstr. appl. Anal., 1-10 (2006) · Zbl 1141.51011
[14] Dhompongsa, S.; Kaewkhao, A.; Panayanak, B.: Lims theorem for multivalued mappings in $CAT(0)$ spaces. J. math. Anal. appl. 312, 478-487 (2005) · Zbl 1086.47019
[15] Gohde, D.: Zum prinzip der kontraktiven abbildung. Math. nachr. 30, 251-258 (1965) · Zbl 0127.08005