# SEMI-ASYMPTOTIC NON-EXPANSIVE ACTIONS OF SEMI-TOPOLOGICAL SEMIGROUPS

• Amini, Massoud (Department of Mathematics Faculty of Mathematical Sciences Tarbiat Modares University) ;
• Medghalchi, Alireza (Faculty of Mathematical Sciences and Computer Kharazmi University) ;
• Naderi, Fouad (Department of Mathematics Faculty of Mathematical Sciences Tarbiat Modares University)
• Published : 2016.01.31
• 62 7

#### Abstract

In this paper we extend Takahashi's fixed point theorem on discrete semigroups to general semi-topological semigroups. Next we define the semi-asymptotic non-expansive action of semi-topological semi-groups to give a partial affirmative answer to an open problem raised by A.T-M. Lau.

#### Keywords

non-expansive mappings;normal structure;semi-topological semigroups;amenable;left reversible

#### References

1. L. P. Belluce and W. A. Kirk, Nonexpansive mappings and fixed points in Banach spaces, Illinois J. Math. 11 (1967), 474-479.
2. D. E. Alspach, A fixed point free non-expansive map, Proc. Amer. Math. Soc. 82 (1981), no. 3, 423-424. https://doi.org/10.1090/S0002-9939-1981-0612733-0
3. L. P. Belluce andW. A. Kirk, Fixed point theorems for families of contraction mappings, Pacific J. Math. 18 (1966), 213-217. https://doi.org/10.2140/pjm.1966.18.213
4. J. F. Berglund, H. D. Junghen, and P. Milnes, Analysis on Semigroups, John Wiley & Sons Inc., New York, 1989.
5. M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544.
6. R. DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math. 13 (1963), 1139-1141. https://doi.org/10.2140/pjm.1963.13.1139
7. R. D. Holmes and A. T. M. Lau, asymptotically non-expansive actions of topological semigroups and fixed points, Bull. London. Math. Soc. 3 (1971), 343-347. https://doi.org/10.1112/blms/3.3.343
8. R. D. Holmes and A. T. M. Lau, Nonexpansive action of topological semigroups and fixed points, J. London Math. Soc. 5 (1972), 330-336.
9. R. D. Holmes and P. P. Narayanaswami, On asymptotically nonexpansive semigroups of mappings, Canad. Math. Bull. 13 (1970), 209-214. https://doi.org/10.4153/CMB-1970-042-1
10. A. T. M. Lau, Invariant means on almost periodic functions and fixed point properties, Rocky Mountain J. Math. 3 (1973), 69-76. https://doi.org/10.1216/RMJ-1973-3-1-69
11. A. T. M. Lau, Normal structure and common fixed point properties for semigroups of nonex-pansive mappings in Banach spaces, Fixed Point Theory Appl. 2010 (2010), Art. ID 580956, 14 pp.
12. A. T. M. Lau and Y. Zhang, Fixed point properties of semigroups of non-expansive mappings, J. Funct. Anal. 254 (2008), no. 10, 2534-2554. https://doi.org/10.1016/j.jfa.2008.02.006
13. A. T. M. Lau and Y. Zhang, Fixed point properties for semigroups of nonlinear mappings and amenability, J. Funct. Anal. 263 (2012), no. 10, 2949-2977. https://doi.org/10.1016/j.jfa.2012.07.013
14. T. Mitchell, Fixed points of reversible semigroups of nonexpansive mappings, Kodai Math. Sem. Rep. 21 (1970), 322-323.
15. A. L. Paterson, Amenability, American Mathematical Society, Providence, 1988.
16. W. Takahashi, Fixed point theorem for amenable semigroup of nonexpansive mappings, Kodai Math. Sem. Rep. 21 (1969), 383-386. https://doi.org/10.2996/kmj/1138845984

#### Cited by

1. Pointwise eventually non-expansive action of semi-topological semigroups and fixed points vol.437, pp.2, 2016, https://doi.org/10.1016/j.jmaa.2016.01.064
2. Existence of fixed points for asymptotically nonexpansive type actions of semigroups vol.20, pp.2, 2018, https://doi.org/10.1007/s11784-018-0548-z