A nonlinear ergodic theorem for a reversible semigroup of Lipschitzian mappings in a Hilbert space
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- by Hajime Ishihara and Wataru Takahashi PDF
- Proc. Amer. Math. Soc. 104 (1988), 431-436 Request permission
Abstract:
Let $C$ be a nonempty closed convex subset of a Hilbert space, $S$ a right reversible semitopological semigroup, $\mathcal {S} = \{ {T_t}:t \in S\}$ a continuous representation of $S$ as Lipschitzian mappings on a closed convex subset $C$ into $C$, and $F(\mathcal {S})$ the set of common fixed points of mappings ${T_t},t \in S$. Then we deal with the existence of a nonexpansive retraction $P$ of $C$ onto $F(\mathcal {S})$ such that $P{T_t} = {T_t}P = P$ for each $t \in S$ and ${P_x}$ is contained in the closure of the convex hull of $\left \{ {{T_t}x:t \in S} \right \}$ for each $x \in C$.References
- 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
- Kazimierz Goebel and Simeon Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. MR 744194
- K. Goebel, W. A. Kirk, and R. L. Thele, Uniformly Lipschitzian families of transformations in Banach spaces, Canadian J. Math. 26 (1974), 1245–1256. MR 358453, DOI 10.4153/CJM-1974-119-9
- Norimichi Hirano and Wataru Takahashi, Nonlinear ergodic theorems for nonexpansive mappings in Hilbert spaces, Kodai Math. J. 2 (1979), no. 1, 11–25. MR 531784
- Hajime Ishihara and Wataru Takahashi, Fixed point theorems for uniformly Lipschitzian semigroups in Hilbert spaces, J. Math. Anal. Appl. 127 (1987), no. 1, 206–210. MR 904222, DOI 10.1016/0022-247X(87)90152-1
- Hajime Ishihara, Fixed point theorems for Lipschitzian semigroups, Canad. Math. Bull. 32 (1989), no. 1, 90–97. MR 996128, DOI 10.4153/CMB-1989-013-3
- Anthony To Ming Lau, Semigroup of nonexpansive mappings on a Hilbert space, J. Math. Anal. Appl. 105 (1985), no. 2, 514–522. MR 778484, DOI 10.1016/0022-247X(85)90066-6
- Teck Cheong Lim, On asymptotic centers and fixed points of nonexpansive mappings, Canadian J. Math. 32 (1980), no. 2, 421–430. MR 571935, DOI 10.4153/CJM-1980-033-5
- Gregory B. Passty, Construction of fixed points for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 84 (1982), no. 2, 212–216. MR 637171, DOI 10.1090/S0002-9939-1982-0637171-7
- R. R. Phelps, Convex sets and nearest points, Proc. Amer. Math. Soc. 8 (1957), 790–797. MR 87897, DOI 10.1090/S0002-9939-1957-0087897-7
- 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, A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 97 (1986), no. 1, 55–58. MR 831386, DOI 10.1090/S0002-9939-1986-0831386-4
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 431-436
- MSC: Primary 47H20; Secondary 47A35
- DOI: https://doi.org/10.1090/S0002-9939-1988-0962809-X
- MathSciNet review: 962809