Kaczor, Wiesława; Kuczumow, Tadeusz; Michalska, Małgorzata Convergence of ergodic means of orbits of semigroups of nonexpansive mappings in sets with the \(\varGamma\)-Opial property. (English) Zbl 1133.47048 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 7, 2122-2130 (2007). The authors generalize a T.Suzuki’s result [Nonlinear Anal., Theory Methods Appl.58, No.3–4 (A), 441–458 (2004; Zbl 1068.47078)] on the set of common fixed points of one-parameter semigroups of nonexpansive mappings in Banach spaces with the Opial property. As an application, they construct an ergodic nonexpansive retraction onto the common fixed point set of a nonexpansive semigroup. Reviewer: Ioan A. Rus (Cluj-Napoca) Cited in 3 Documents MSC: 47H20 Semigroups of nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. Keywords:\(\varGamma\)-Opial property; means of orbits of semigroups; nonexpansive mappings; nonexpansive semigroups; nonexpansive retracts Citations:Zbl 1068.47078 PDFBibTeX XMLCite \textit{W. Kaczor} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 7, 2122--2130 (2007; Zbl 1133.47048) Full Text: DOI References: [1] Bessaga, C.; Pełczyński, A., On bases and unconditional convergence of series in Banach spaces, Studia Math., 17, 151-164 (1958) · Zbl 0084.09805 [2] Budzyńska, M.; Kuczumow, T.; Michalska, M., The \(\Gamma \)-Opial property, Bull. Austral. Math. Soc., 73, 473-476 (2006) · Zbl 1108.46013 [3] Dalby, T.; Sims, B., Duality map characterisations for Opial conditions, Bull. Austral. Math. Soc., 53, 413-417 (1996) · Zbl 0884.46014 [4] Van Dulst, D., Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc., 25, 139-144 (1982) · Zbl 0453.46017 [5] Edelstein, M.; O’Brien, R. C., Nonexpansive mappings, asymptotic regularity and successive approximations, J. London Math. Soc., 17, 547-554 (1978) · Zbl 0421.47031 [6] Engelking, R., (General Topology. General Topology, Mathematical Monographs, vol. 60 (1977), PWN—Polish Scientific Publishers: PWN—Polish Scientific Publishers Warsaw) [7] Goebel, K.; Kirk, W. A., Topics in Metric Fixed Point Theory (1990), Cambridge University Press · Zbl 0708.47031 [8] Goebel, K.; Kuczumow, T., Irregular convex sets with the fixed point property for nonexpansive mappings, Colloq. Math., 40, 259-264 (1978) · Zbl 0418.47031 [9] Gossez, J.-P.; Lami Dozo, E., Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math., 40, 565-573 (1972) · Zbl 0223.47025 [10] Kaczor, W.; Prus, S., Asymptotical smoothness and its applications, Bull. Austral. Math. Soc., 66, 405-418 (2002) · Zbl 1031.47037 [11] Kelley, J. L., General Topology (1975), Springer: Springer Berlin, NY · Zbl 0306.54002 [12] Lim, T.-C., Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math., 90, 135-143 (1980) · Zbl 0454.47046 [13] Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces. I. Sequence Spaces (1977), Springer-Verlag: Springer-Verlag Berlin, NY · Zbl 0362.46013 [14] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902 [15] Prus, S., Geometrical background of metric fixed point theory, (Kirk, W. A.; Sims, B., Handbook of Metric Fixed Point Theory (2001), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 93-132 · Zbl 1018.46010 [16] B. Sims, A support map characterization of the Opial conditions, Miniconference on linear analysis and function spaces (Canberra, 1984), in: Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 9, Austral. Nat. Univ., Canberra, 1985, pp. 259-264; B. Sims, A support map characterization of the Opial conditions, Miniconference on linear analysis and function spaces (Canberra, 1984), in: Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 9, Austral. Nat. Univ., Canberra, 1985, pp. 259-264 [17] Suzuki, T., Some remarks on the set of common fixed points of one-parameter semigroups of nonexpansive mappings in Banach spaces with the Opial property, Nonlinear Anal., 58, 441-458 (2004) · Zbl 1068.47078 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.