Nonexpansive actions of topological semigroups on strictly convex Banach spaces and fixed points
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- by Wojciech Bartoszek PDF
- Proc. Amer. Math. Soc. 104 (1988), 809-811 Request permission
Abstract:
Let $C$ be a closed convex subset of a strictly convex Banach space $X$ and $\left \{ {{T_s}:s \in S} \right \}$ be a continuous representation of a semitopological semigroup $S$ as nonexpansive mappings of $C$ into itself. The main result establishes the fact that if for some $x \in C$ the trajectory $\left \{ {{T_s}x:s \in S} \right \}$ is relatively compact and $AP(S)$ has a left invariant mean then $K = \overline {\operatorname {conv} \{ {T_s}x:s \in S\} }$ contains a common fixed point for ${\left \{ {{T_s}} \right \}_{s \in S}}$.References
-
W. Bartoszek and T. Downarowicz, Compactness of trajectories of dynamical systems in uniform complete spaces, Suppl. Rend. Circ. Mat. Palermo (The 13th Winter School on Abstract Analysis, Srni, 1985).
- C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Functional Analysis 13 (1973), 97–106. MR 0346611, DOI 10.1016/0022-1236(73)90069-4
- Michael Edelstein, On non-expansive mappings of Banach spaces, Proc. Cambridge Philos. Soc. 60 (1964), 439–447. MR 164222
- Anthomy ToMing Lau, Invariant means on almost periodic functions and fixed point properties, Rocky Mountain J. Math. 3 (1973), 69–76. MR 324313, DOI 10.1216/RMJ-1973-3-1-69
- 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 W. Parry, Entropy and generators in ergodic theory, Yale Univ. Press, New Haven, Connecticut, 1966. W. A. Rochlin, Isbrannyje woprosy mietriczieskoj teorii dynamichieskich sistiem, Uspekhi Mat. Nauk 30 (1949), 57-128.
- Stephen F. Roehrig and Robert C. Sine, The structure of $\omega$-limit sets of nonexpansive maps, Proc. Amer. Math. Soc. 81 (1981), no. 3, 398–400. MR 597649, DOI 10.1090/S0002-9939-1981-0597649-0
- Chi Lin Yen, On the rest points of a nonlinear nonexpansive semigroup, Pacific J. Math. 45 (1973), 699–706. MR 331150
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 809-811
- MSC: Primary 47H20; Secondary 47H09, 47H10
- DOI: https://doi.org/10.1090/S0002-9939-1988-0964861-4
- MathSciNet review: 964861