Nonexpansive, continuous antirepresentations have common fixed points
Author:
Wojciech Bartoszek
Journal:
Proc. Amer. Math. Soc. 127 (1999), 10511055
MSC (1991):
Primary 47H10, 22A25; Secondary 28D05
MathSciNet review:
1469398
Fulltext PDF Free Access
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Abstract: Let be a closed convex subset of a Banach (dual Banach) space . By we denote an antirepresentation of a semitopological semigroup as nonexpansive mappings on . Suppose that the mapping is jointly continuous when has the weak (weak*) topology and the Banach space of bounded right uniformly continuous functions on has a right invariant mean. If is weakly compact (for some the set is weakly* compact) and norm separable, then has a common fixed point in .
 [1]
Dale
E. Alspach, A fixed point free nonexpansive
map, Proc. Amer. Math. Soc.
82 (1981), no. 3,
423–424. MR
612733 (82j:47070), http://dx.doi.org/10.1090/S00029939198106127330
 [2]
Wojciech
Bartoszek, Nonexpansive actions of topological
semigroups on strictly convex Banach spaces and fixed points, Proc. Amer. Math. Soc. 104 (1988), no. 3, 809–811. MR 964861
(89i:47120), http://dx.doi.org/10.1090/S00029939198809648614
 [3]
Wojciech
Bartoszek and Tomasz
Downarowicz, Compactness of trajectories of dynamical systems in
complete uniform spaces, Proceedings of the 13th winter school on
abstract analysis (Srní, 1985), 1985, pp. 13–16 (1986).
MR 894267
(88e:54022)
 [4]
C.M. Dafermos and M. Slemrod, Asymptotic behaviour of nonlinear contraction semigroup, J. Funct. Anal. 13 (1973), 97106.
 [5]
Mahlon
M. Day, Fixedpoint theorems for compact convex sets, Illinois
J. Math. 5 (1961), 585–590. MR 0138100
(25 #1547)
 [6]
M.
Hochster, Subsemigroups of amenable
groups, Proc. Amer. Math. Soc. 21 (1969), 363–364. MR 0240223
(39 #1575), http://dx.doi.org/10.1090/S00029939196902402230
 [7]
Wataru
Takahashi and Doo
Hoan Jeong, Fixed point theorem for nonexpansive
semigroup on Banach space, Proc. Amer. Math.
Soc. 122 (1994), no. 4, 1175–1179. MR 1223268
(95b:47087), http://dx.doi.org/10.1090/S00029939199412232688
 [8]
R.
D. Holmes and Anthony
T. Lau, Nonexpansive actions of topological semigroups and fixed
points, J. London Math. Soc. (2) 5 (1972),
330–336. MR 0313895
(47 #2447)
 [9]
Anthony
To Ming Lau and Wataru
Takahashi, Invariant means and semigroups of nonexpansive mappings
on uniformly convex Banach spaces, J. Math. Anal. Appl.
153 (1990), no. 2, 497–505. MR 1080662
(91k:47134), http://dx.doi.org/10.1016/0022247X(90)902288
 [10]
Theodore
Mitchell, Topological semigroups and fixed points, Illinois J.
Math. 14 (1970), 630–641. MR 0270356
(42 #5245)
 [1]
 D.E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 n 3 (1981), 423424. MR 82j:47070
 [2]
 W. Bartoszek, Nonexpansive actions of topological semigroups on strictly convex Banach space and fixed points, Proc. Amer. Math. Soc. 104 n 3 (1988), 809811. MR 89i:47120
 [3]
 W. Bartoszek and T. Downarowicz, Compactness of trajectories of dynamical systems in uniform complete spaces, Suppl. Rend. Circ. Mat. Palermo serie II, n 10 (1985), 1316. MR 88e:54022
 [4]
 C.M. Dafermos and M. Slemrod, Asymptotic behaviour of nonlinear contraction semigroup, J. Funct. Anal. 13 (1973), 97106.
 [5]
 M.M. Day, Fixedpoint theorems for compact sets, Illinois J. Math. 5 (1961), 585590. MR 25:1547
 [6]
 M. Hochster, Subsemigroups of amenable groups, Proc. Amer. Math. Soc. 21 (1969), 363364. MR 39:1575
 [7]
 D.H. Jeong and W. Takahashi, Fixed point theorem for nonexpansive semigroups on Banach space, Proc. Amer. Math. Soc. 122 n 4 (1994), 11751179. MR 95b:47087
 [8]
 R.D. Holmes and A.T. Lau, Nonexpansive actions of topological semigroups and fixed points, J. London Math. Soc. (2) 5 (1972), 330336. MR 47:2447
 [9]
 A.T. Lau and W. Takahashi, Invariant means and semigroups on nonexpansive mappings on uniformly convex Banach spaces, J.Math. Anal. Appl. 153 (1990), 497505. MR 91k:47134
 [10]
 T. Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630641. MR 42:5245
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Additional Information
Wojciech Bartoszek
Affiliation:
Department of Mathematics, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa
Email:
bartowk@alpha.unisa.ac.za
DOI:
http://dx.doi.org/10.1090/S0002993999045670
PII:
S 00029939(99)045670
Keywords:
Fixed point,
topological semigroup,
nonexpansive mapping
Received by editor(s):
July 14, 1997
Communicated by:
Dale Alspach
Article copyright:
© Copyright 1999
American Mathematical Society
