Convergence of almost-orbits of nonexpansive semigroups in Banach spaces

Lau, Anthony; Nishiura, Koji; Takahashi, Wataru

doi:10.1090/S0002-9939-07-08999-X

Convergence of almost-orbits of nonexpansive semigroups in Banach spaces
HTML articles powered by AMS MathViewer

by Anthony To-Ming Lau, Koji Nishiura and Wataru Takahashi PDF

Proc. Amer. Math. Soc. 135 (2007), 3143-3150 Request permission

Abstract:

The purpose of this paper is to show that the study of mean ergodic theorems for almost-orbits of semigroups of nonexpansive mappings on closed convex subsets of a Banach space can be reduced to the study of orbits for semigroups of nonexpansive mappings. This provides a unified approach to various mean ergodic theorems for almost-orbits in the literature and new applications.

References

Sachiko Atsushiba, Anthony To-Ming Lau, and Wataru Takahashi, Nonlinear strong ergodic theorems for commutative nonexpansive semigroups on strictly convex Banach spaces, J. Nonlinear Convex Anal. 1 (2000), no. 2, 213–231. MR 1777140
Sachiko Atsushiba and Wataru Takahashi, A nonlinear strong ergodic theorem for nonexpansive mappings with compact domains, Math. Japon. 52 (2000), no. 2, 183–195. MR 1793266
Sachiko Atsushiba and Wataru Takahashi, Strong convergence theorems for one-parameter nonexpansive semigroups with compact domains, Fixed point theory and applications. Vol. 3, Nova Sci. Publ., Huntington, NY, 2002, pp. 15–31. MR 2081406
Jean-Bernard Baillon, Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), no. 22, Aii, A1511–A1514 (French, with English summary). MR 375009
Ronald E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math. 32 (1979), no. 2-3, 107–116. MR 531254, DOI 10.1007/BF02764907
Norimichi Hirano, Kazuo Kido, and Wataru Takahashi, Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces, Nonlinear Anal. 12 (1988), no. 11, 1269–1281. MR 969505, DOI 10.1016/0362-546X(88)90059-4
Hirobumi Kiuchi and Wataru Takahashi, Asymptotic behavior of almost-orbits of nonexpansive semigroups without convexity, Kodai Math. J. 15 (1992), no. 2, 185–192. MR 1185417, DOI 10.2996/kmj/1138039596
Anthony To-Ming Lau, Naoki Shioji, and Wataru Takahashi, Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces, J. Funct. Anal. 161 (1999), no. 1, 62–75. MR 1670206, DOI 10.1006/jfan.1998.3352
Isao Miyadera and Kazuo Kobayasi, On the asymptotic behaviour of almost-orbits of nonlinear contraction semigroups in Banach spaces, Nonlinear Anal. 6 (1982), no. 4, 349–365. MR 654811, DOI 10.1016/0362-546X(82)90021-9
Hirokazu Oka, Nonlinear ergodic theorems for commutative semigroups of asymptotically nonexpansive mappings, Nonlinear Anal. 18 (1992), no. 7, 619–635. MR 1157563, DOI 10.1016/0362-546X(92)90002-V
Gerd Rodé, An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl. 85 (1982), no. 1, 172–178. MR 647565, DOI 10.1016/0022-247X(82)90032-4
Wataru Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981), no. 2, 253–256. MR 593468, DOI 10.1090/S0002-9939-1981-0593468-X
Wataru Takahashi and Jong Yeoul Park, On the asymptotic behavior of almost-orbits of commutative semigroups in Banach spaces, Nonlinear and convex analysis (Santa Barbara, Calif., 1985) Lecture Notes in Pure and Appl. Math., vol. 107, Dekker, New York, 1987, pp. 271–293. MR 892798