zbMATH — the first resource for mathematics

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.

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.
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.