Fixed point theorem for nonexpansive semigroup on Banach space

Authors:
Wataru Takahashi and Doo Hoan Jeong

Journal:
Proc. Amer. Math. Soc. **122** (1994), 1175-1179

MSC:
Primary 47H20; Secondary 47H09, 47H10

DOI:
https://doi.org/10.1090/S0002-9939-1994-1223268-8

MathSciNet review:
1223268

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let *C* be a nonempty closed convex subset of a uniformly convex Banach space, and let *S* be a semitopological semigroup such that has a left invariant submean. Then we prove a fixed point theorem for a continuous representation of *S* as nonexpansive mappings on *C*.

**[1]**V. Barbu and Th. Precupanu,*Convexity and optimization in Banach spaces*, Editura Academiei R. S. R., Bucuresti, 1978. MR**0461071 (57:1057)****[2]**W. Bartoszek,*Nonexpansive actions of topological semigroups on strictly convex Banach space and fixed points*, Proc. Amer. Math. Soc.**104**(1988), 809-811. MR**964861 (89i:47120)****[3]**A. T. Lau,*Semigroup of nonexpansive mappings on a Hilbert space*, J. Math. Anal. Appl.**105**(1985), 514-522. MR**778484 (86m:47085)****[4]**A. T. Lau and W. Takahashi,*Weak convergence and non-linear ergodic theorems for reversible semigroups of nonexpansive mappings*, Pacific J. Math.**126**(1987), 277-294. MR**869780 (88e:47113)****[5]**-,*Invariant means and semigroups of nonexpansive mappings on uniformly convex Banach spaces*, J. Math. Anal. Appl.**153**(1990), 497-505. MR**1080662 (91k:47134)****[6]**T. Mitchel,*Topological semigroups and fixed points*, Illinois J. Math.**14**(1970), 630-641. MR**0270356 (42:5245)****[7]**N. Mizoguchi and W. Takahashi,*On the existence of fixed points and ergodic retractions for Lipschitzian semigroups in Hilbert spaces*, Nonlinear Anal. TMA**14**(1990), 69-80. MR**1028248 (91h:47071)****[8]**W. Takahashi,*A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space*, Proc. Amer. Math. Soc.**81**(1981), 253-256. MR**593468 (82f:47079)****[9]**-,*Fixed point theorems for families of nonexpansive mappings on unbounded sets*, J. Math. Soc. Japan**36**(1984), 543-553. MR**759413 (86c:47080)****[10]**-,*A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space*, Proc. Amer. Math. Soc.**97**(1986), 55-58. MR**831386 (88f:47051)****[11]**K. K. Tan and H. K. Xu,*Continuous representation of semitopological semigroup as nonexpansive mappings on Banach space*(to appear). MR**1295494 (95h:47076)****[12]**H. K. Xu,*Inequalities in Banach spaces with applications*, Nonlinear Anal. TMA**16**(1991), 1127-1138. MR**1111623 (92e:47126)****[13]**C. Zalinescu,*On uniformly convex functions*, J. Math. Anal. Appl.**95**(1983), 344-374. MR**716088 (85a:26018)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47H20,
47H09,
47H10

Retrieve articles in all journals with MSC: 47H20, 47H09, 47H10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1223268-8

Keywords:
Fixed point,
nonexpansive mapping,
mean

Article copyright:
© Copyright 1994
American Mathematical Society