Common fixed points for semigroups of mappings
Authors:
Anthony T. Lau and Chi Song Wong
Journal:
Proc. Amer. Math. Soc. 41 (1973), 223228
MSC:
Primary 54H15; Secondary 47H10
MathSciNet review:
0322837
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a compact convex subset of a strictly convex Banach space. Let be a Hausdorff topological semigroup which is either left amenable or left reversible. Then for any generalised nonexpansive (jointly) continuous action of on contains a common fixed point of .
 [1]
Mahlon
M. Day, Amenable semigroups, Illinois J. Math.
1 (1957), 509–544. MR 0092128
(19,1067c)
 [2]
K.
de Leeuw and I.
Glicksberg, Applications of almost periodic compactifications,
Acta Math. 105 (1961), 63–97. MR 0131784
(24 #A1632)
 [3]
Ralph
DeMarr, Common fixed points for commuting contraction
mappings, Pacific J. Math. 13 (1963),
1139–1141. MR 0159229
(28 #2446)
 [4]
M.
Edelstein, On fixed and periodic points under contractive
mappings, J. London Math. Soc. 37 (1962),
74–79. MR
0133102 (24 #A2936)
 [5]
E.
Granirer, Extremely amenable semigroups, Math. Scand.
17 (1965), 177–197. MR 0197595
(33 #5760)
 [6]
G.
E. Hardy and T.
D. Rogers, A generalization of a fixed point theorem of Reich,
Canad. Math. Bull. 16 (1973), 201–206. MR 0324495
(48 #2847)
 [7]
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)
 [8]
R.
Kannan, Some results on fixed points. II, Amer. Math. Monthly
76 (1969), 405–408. MR 0257838
(41 #2487)
 [9]
Anthony
ToMing Lau, Invariant means on subsemigroups of locally compact
groups, Rocky Mountain J. Math. 3 (1973),
77–81. MR
0326296 (48 #4640)
 [10]
Anthony
ToMing Lau, Invariant means on almost periodic functions and fixed
point properties, Rocky Mountain J. Math. 3 (1973),
69–76. MR
0324313 (48 #2665)
 [11]
Theodore
Mitchell, Fixed points of reversible semigroups of nonexpansive
mappings, Kōdai Math. Sem. Rep. 22 (1970),
322–323. MR 0267414
(42 #2316)
 [12]
Theodore
Mitchell, Topological semigroups and fixed points, Illinois J.
Math. 14 (1970), 630–641. MR 0270356
(42 #5245)
 [13]
I.
Namioka, On certain actions of semigroups on
𝐿spaces, Studia Math. 29 (1967),
63–77. MR
0223863 (36 #6910)
 [14]
Simeon
Reich, Some remarks concerning contraction mappings, Canad.
Math. Bull. 14 (1971), 121–124. MR 0292057
(45 #1145)
 [15]
Wataru
Takahashi, Fixed point theorem for amenable semigroup of
nonexpansive mappings., Kōdai Math. Sem. Rep.
21 (1969), 383–386. MR 0262896
(41 #7501)
 [16]
Chi Song Wong, Fixed point theorems on compact Hausdorff spaces (to appear).
 [17]
, Fixed points of generalised nonexpansive mappings on an into Proc. Amer. Math. Soc. (to appear).
 [1]
 M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509544. MR 19, 1067. MR 0092128 (19:1067c)
 [2]
 K. deLeeuw and I. Glickberg, Application of almost periodic compactification, Acta. Math. 105 (1961), 6397. MR 24 #A1632. MR 0131784 (24:A1632)
 [3]
 R. DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math. 13 (1963), 11391141. MR 28 #2446. MR 0159229 (28:2446)
 [4]
 M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 7479. MR 24 #A2936. MR 0133102 (24:A2936)
 [5]
 E. E. Granirer, Extremely amenable semigroups, Math. Scand. 17 (1965), 177197. MR 33 #5760. MR 0197595 (33:5760)
 [6]
 G. Hardy and T. Roger, A generalisation of a fixed point theorem of S. Reich, Canad. Math. Bull. (to appear). MR 0324495 (48:2847)
 [7]
 R. D. Holmes and A. Lau, Nonexpansive actions of topological semigroups and fixed points, J. London Math. Soc. 5 (1972), 330336. MR 0313895 (47:2447)
 [8]
 R. Kannan, Some results on fixed points. II, Amer. Math. Monthly 76 (1969), 405408. MR 41 #2487. MR 0257838 (41:2487)
 [9]
 A. T. Lau, Invariant means on subsemigroups of locally compact groups, Rocky Mountain J. Math. 3 (1973), 7781. MR 0326296 (48:4640)
 [10]
 , Invariant means on almost periodic functions and fixed point properties, Rocky Mountain J. Math. 3 (1973), 6976. MR 0324313 (48:2665)
 [11]
 T. Mitchell, Fixed points of reversible semigroups of nonexpansive maps, Kodai Math. Sem. Rep. 22 (1970), 322323. MR 42 #2316. MR 0267414 (42:2316)
 [12]
 T. Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630641. MR 42 #5245. MR 0270356 (42:5245)
 [13]
 I. Namioka, On certain actions of semigroups on spaces, Studia Math. 29 (1967), 6377. MR 36 #6910. MR 0223863 (36:6910)
 [14]
 S. Reich, Some remarks concerning contractive mappings, Canad. Math. Bull. 14 (1971), 121124. MR 45 #1145. MR 0292057 (45:1145)
 [15]
 W. Takahashi, Fixed point theorem for amenable semigroup of nonexpansive mappings, Kodai Math. Sem. Rep. 21 (1969), 383386. MR 41 #7501. MR 0262896 (41:7501)
 [16]
 Chi Song Wong, Fixed point theorems on compact Hausdorff spaces (to appear).
 [17]
 , Fixed points of generalised nonexpansive mappings on an into Proc. Amer. Math. Soc. (to appear).
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
54H15,
47H10
Retrieve articles in all journals
with MSC:
54H15,
47H10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197303228373
PII:
S 00029939(1973)03228373
Keywords:
Common fixed points,
equicontinuous mapping,
generalized nonexpansive mapping,
left amenable topological semigroup,
left reversible topological semigroup,
strictly convex Banach space,
strongly almost periodic function
Article copyright:
© Copyright 1973 American Mathematical Society
