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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
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.


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  • [1] 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 0375009
  • [2] Jean-Bernard Baillon, Quelques propriétés de convergence asymptotique pour les semi-groupes de contractions impaires, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 3, Ai, A75–A78 (French, with English summary). MR 0425702
  • [3] J. B. Baillon and H. Brezis, Une remarque sur le comportement asymptotique des semigroupes non linéaires, Houston J. Math. 2 (1976), no. 1, 5–7 (French). MR 0394328
  • [4] H. Brézis and F. E. Browder, Remarks on nonlinear ergodic theory, Advances in Math. 25 (1977), no. 2, 165–177. MR 0461233
  • [5] Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. MR 0092128
  • [6] 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
  • [7] A. Pazy, Asymptotic behavior of contractions in Hilbert space, Israel J. Math. 9 (1971), 235–240. MR 0282276
  • [8] A. Pazy, On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert space, Israel J. Math. 26 (1977), no. 2, 197–204. MR 0440428
  • [9] A. Pazy, On the asymptotic behavior of semigroups of nonlinear contractions in Hilbert space, J. Functional Analysis 27 (1978), no. 3, 292–307. MR 0477932
  • [10] Wataru Takahashi, Fixed point theorem for amenable semigroup of nonexpansive mappings., Kōdai Math. Sem. Rep. 21 (1969), 383–386. MR 0262896
  • [11] Wataru Takahashi, Invariant functions for amenable semigroups of positive contractions on 𝐿¹, Kōdai Math. Sem. Rep. 23 (1971), 131–143. MR 0296256
  • [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