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A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space


Author: Wataru Takahashi
Journal: Proc. Amer. Math. Soc. 81 (1981), 253-256
MSC: Primary 47H20; Secondary 47H09, 47H10
DOI: https://doi.org/10.1090/S0002-9939-1981-0593468-X
MathSciNet review: 593468
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Abstract: We prove a nonlinear ergodic theorem for noncommutative semigroups of nonexpansive mappings in a Hilbert space. Furthermore, we give a necessary and sufficient condition for a noncommutative semigroup to have a fixed point.


References [Enhancements On Off] (What's this?)

  • [1] J. B. 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), 1511 - 1514. MR 0375009 (51:11205)
  • [2] -, Quelques propriétés de convergence asymptotique pour les semigroupes de contractions impaires, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), 75-78. MR 0425702 (54:13655)
  • [3] J. B. Baillon and H. Brézis, Une remarque sur le comportement asymptotique des semigroupes non linéaires, Houston J. Math. 2 (1976), 5-7. MR 0394328 (52:15131)
  • [4] H. Brézis and F. E. Browder, Remarks on nonlinear ergodic theory, Adv. in Math. 25 (1977), 165-177. MR 0461233 (57:1218)
  • [5] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544. MR 0092128 (19:1067c)
  • [6] N. Hirano and W. Takahashi, Nonlinear ergodic theorems for nonexpansive mappings in Hilbert spaces, Kōdai Math. J. 2 (1979), 11-25. MR 531784 (80j:47064)
  • [7] A. Pazy, Asymptotic behavior of contractions in Hilbert space, Israel J. Math. 9 (1971), 235-240. MR 0282276 (43:7988)
  • [8] -, On the asymptotic behavior of iterates of nonexpansive mappings, Israel J. Math. 26 (1977), 197-204. MR 0440428 (55:13303)
  • [9] -, On the asymptotic behavior of semigroups of nonlinear contractions in Hilbert space, J. Funct. Anal. 3 (1978), 292-307. MR 0477932 (57:17432)
  • [10] W. Takahashi, Fixed point theorem for amenable semigroup of nonexpansive mappings, Kōdai Math. Sem. Rep. 21 (1969), 383-386. MR 0262896 (41:7501)
  • [11] -, Invariant functions for amenable semigroups of positive contractions on $ {L^1}$, Kōdai Math. Sem. Rep. 23 (1971), 131-143. MR 0296256 (45:5317)
  • [12] -, Ergodic theorems for amenable semigroups of positive contractions on $ {L^1}$, Sci. Rep. Yokohama Nat. Univ. Sect. I 19 (1972), 5-11.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0593468-X
Keywords: Ergodic theorem, amenable semigroup, nonexpansive mapping, fixed point
Article copyright: © Copyright 1981 American Mathematical Society

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