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Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces. (English) Zbl 1191.47077
The authors prove a convergence theorem for the sequence of Mann iterations, for a strongly continuous semigroup of nonexpansive mappings acting on a closed convex subset of a complete CAT(0) space, to a common fixed point of all mappings in the semigroup. They also prove a result concerning the limits of subsequences of Mann iterations of multivalued nonexpansive mappings on metric spaces of hyperbolic type.

##### MSC:
 47H20 Semigroups of nonlinear operators 54H25 Fixed-point and coincidence theorems (topological aspects) 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
##### Keywords:
geodesic metric space; nonexpansive semigroup; fixed point
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##### References:
 [1] Browder, F.E., Convergence of approximates to fixed points of nonexpansive mappings in Banach spaces, Arch. ration. mech. anal., 24, 82-90, (1967) · Zbl 0148.13601 [2] Goebel, K.; Kirk, W.A., Iteration processes for nonexpansive mappings, Contemp. math., 21, 115-123, (1983) · Zbl 0525.47040 [3] Goebel, K.; Kirk, W.A., Topics in metric fixed point theory, (1990), Cambridge Univ. Press Cambridge · Zbl 0708.47031 [4] Halpern, B., Fixed points of nonexpansive maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101 [5] Kirk, W.A., Krasnoselskii’s iteration process in hyperbolic space, Numer. funct. anal. optim., 4, 371-381, (1981-1982) [6] W.A. Kirk, Geodesic geometry and fixed point theory, in: Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colecc. Abierta, 64, Univ. Seville Secr. Publ. Seville, 2003, pp. 195-225 [7] Kirk, W.A.; Panyanak, B., A concept of convergence in geodesic spaces, Nonlinear anal., (2007) · Zbl 1132.54025 [8] Sims, B., A class of spaces with weak normal structure, Bull. austral. math. soc., 49, 523-528, (1994) · Zbl 0807.47047 [9] Suzuki, T., On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. amer. math. soc., 131, 2133-2136, (2002) · Zbl 1031.47038 [10] Takahashi, W., A convexity in metric space and nonexpansive mappings I, Kodai math. sem. rep., 22, 142-149, (1970) · Zbl 0268.54048
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