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Convergence of ergodic means of orbits of semigroups of nonexpansive mappings in sets with the \(\varGamma\)-Opial property. (English) Zbl 1133.47048

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.

MSC:

47H20 Semigroups of nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 1068.47078
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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
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