Taiwanese Journal of Mathematics

INVARIANT MEANS AND FIXED POINT PROPERTIES OF SEMIGROUP OF NONEXPANSIVE MAPPINGS

Anthony To-Ming Lau

Full-text: Open access

Abstract

This paper outlines some of my recent joint works with Q. Takahashi on fixed point properties or ergodic properties for semigroup of nonexpansive mappings on closed convex subsets of a Banach space and their relationship with existence of left invariant mean on certain subspaces of bounded realvalued functions on the semigroup.

Article information

Source
Taiwanese J. Math., Volume 12, Number 6 (2008), 1525-1542.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405038

Digital Object Identifier
doi:10.11650/twjm/1500405038

Mathematical Reviews number (MathSciNet)
MR2444870

Zentralblatt MATH identifier
1181.47062

Subjects
Primary: 47A10: Spectrum, resolvent
Secondary: 43A07: Means on groups, semigroups, etc.; amenable groups 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions

Keywords
fixed property ergodic property nonexpansive mappings invariant mean amenability weakly compact convex set left reversible semigroups topological semigroups

Citation

Lau, Anthony To-Ming. INVARIANT MEANS AND FIXED POINT PROPERTIES OF SEMIGROUP OF NONEXPANSIVE MAPPINGS. Taiwanese J. Math. 12 (2008), no. 6, 1525--1542. doi:10.11650/twjm/1500405038. https://projecteuclid.org/euclid.twjm/1500405038


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References

  • D. Alspach,\! A fixed point free nonexpansive\! map, Proc. Amer. Math. Soc., 82\! (1981), 423-424.
  • S. Atsushiba, A. T.-M. Lau and W. Takahashi, Nonlinear strong ergodic theorems for commutative nonexpansive semigroups on strictly convex Banach spaces, Journal of Nonlinear and Convex Analysis, 1 (2000), 213-231.
  • J.B. Baillon, Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C.R. Acad. Sci. Paris, Sér A-B, 280 (1975), 1511-1514.
  • L.P. Belluce and W.A. Kirk, Nonexpansive mappings and fixed points in Banach spaces, Illinois J. Math., 11 (1967), 474-479.
  • J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on Semigroups, John Wiley & Sons, New York, 1989.
  • W. Bartoszek, Nonexpansive actions of topological semigroups on strictly convex Banach space and fixed points, Proc. Amer. Math. Soc., 104 (1988), 809-811.
  • M. M. Day, Amenable semigroups, Illinois J. Math., 1 (1957), 509-544.
  • M. M. Day, Mean for bounded functions and ergodicity of bounded representations of semigroups, Trans. Amer. Math. Soc., 69 (1950), 276-291.
  • R. DeMarr, Common fixed-points for commuting contraction mappings, Pacific J. Math., 13 (1963), 1139-1141.
  • M. Despic and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canad. Math. Bulletin, 37 (1994), 165-167.
  • J. Diestel, Geometry of Banach spaces, Selected Topics, Lecture Notes in Mathematics, 485, Springer, Berlin, 1985.
  • K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, 28, Cambridge, 1990.
  • E. E. Graniner and A. T.-M. Lau, Invariant means on locally compact groups, Illinois J. Math., 15 (1971), 249-257.
  • B. Halpern, Fixed points of nonexpansive maps, Bull. of Amer. Math. Soc., 73 (1967), 957-961.
  • E. Hewitt, On two problems of Urysohn, Annals. of Math., 47 (1946), 503-509.
  • N. Hirano, K. Kido and W. Takahashi, Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces, Nonlinear Anal., 12 (1988), 1269-1281.
  • R. H. Holmes and A. T.-M. Lau, Nonexpansive actions of topological semigroups and fixed points, J. London Math. Soc., 5 (1972), 330-336.
  • R. H. Holmes and A. T.-M. Lau, Fixed points of semigroups of non-expansive mappings, Studia Math., 43 (1972), 217-218.
  • R. H. Holmes and A. T.-M. Lau, Asymptotically non-expansive actions of topological semigroups and fixed points, Bull. London Math. Soc., 3 (1971), 343-347.
  • R. Hsu, Topics on weakly almost periodic functions, Ph.D. Thesis, SUNY at Buffalo, 1985.
  • O. Kada, A. T.-M. Lau and W. Takahashi, Asymptotically invariant net and fixed point set for semigroup of nonexpansive mappings, Nonlinear Anal., 29 (1997), 537-550.
  • J. I. Kang, Fixed point set of semigroups of non-expansive mappings and amenability, Journal of Mathematical Analysis and Applications, 341 (2008), 1445-1456.
  • J. I. Kang, Fixed point of non-expansive mappings associated to invariant means in a Banach space, Nonlinear Analysis (to appear).
  • E. Kaniuth, A. T.-M. Lau and J. Pym, On character amenability of Banach algebras, Journal of Mathematical Analysis and Applications, (to appear).
  • A. T.-M. Lau, nvariant means on almost periodic functions and fixed point properties, Rocky Mountain J. Math., 3 (1973), 69-76.
  • A. T.-M. Lau, Some fixed point theorems and $W^*$-algebras in “Fixed Point Theory and Its Applications” (S. Swaminathan, Ed.), Academic Press, Orlando, FL., 1976, pp. 121-129.
  • A. T.-M. Lau, Semigroup of operators on dual Banach spaces, Proc. Amer. Math. Soc., 54 (1976), 393-396.
  • A. T.-M. Lau, Semigroup of nonexpansive mappings on a Hilbert space, J. Math. Anal. Appl., 105 (1985), 514-522.
  • A. T.-M. Lau, Amenability of semigroups, in “The Analytic and Topological Properties of Semigroups” (K. H. Hofmann, J. D. Lawson and J. S. Pym, Eds.) pp. 313-334, De Gruyter Expositions in Mathematics 1, Berlin, New York, 1990.
  • A. T.-M. Lau, Amenability and fixed point property for semigroup of nonexpansive mappings, in: Fixed Point Theory and Applications (M. A. Théra and J. B. Baillon, Eds.), Pitman Research Notes in Mathematical Series, vol. 252, 1991, pp. 303-313.
  • A. T.-M. Lau and M. Leinert, Fixed point property and the Fourier algebra of locally compact group, Trans. Amer. Math. Soc. (to appear).
  • A. T.-M. Lau and P. F. Mah, Quasi-normal structures for certain spaces of operators on a Hilbert space, Pacific Journal of Mathematics, 121 (1986), 109-118.
  • A. T.-M. Lau and P. F. Mah, Normal structure in dual Banach spaces associated with a locally compact group, Trans. Amer. Math. Soc., 310 (1988), 341-353.
  • A. T.-M. Lau, P. F. Mah and A. Ülger, Fixed point property and normal structure for Banach spaces associated to locally compact groups, Trans. Amer. Math. Soc., 125 (1997), 2021-2027.
  • A. T.-M. Lau, H. Miyake and W. Takahashi, Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach space, Nonlinear Analysis, 67 (2007), 1211-1225.
  • A. T.-M. Lau, K. Nishiura and W. Takahashi, Nonlinear ergodic theorems for semigroups of nonexpansive mappings and left ideals, Nonlinear Anal., 26 (1996), 1411-1427.
  • A. T.-M. Lau, K. Nishiura and W. Takahashi, Convergence of almost orbits of nonexpansive semigroups in Banach spaces, Proc. Amer. Math. Soc., 135 (2007), 3143-3150.
  • A. T.-M. Lau and A. L. T. Paterson, Group amenability properties for von Neuman algebras, Indiana Univ. Math. J., 55 (2006), 1363-1388.
  • A. T.-M. Lau, N. Shioji and W. Takahashi, Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces, J. Func. Anal., 161 (1999), 62-75.
  • A. T.-M. Lau and W. Takahashi, Weak convergence and non-linear ergodic theorems for reversible semigroups of nonexpansive mappings, Pacific J. Math., 126 (1987), 277-294.
  • A. T.-M. Lau and W. Takahashi, Invariant means and semigroups of nonexpansive mappings on uniformly convex Banach spaces, J. Math. Anal. Appl., 153 (1990), 497-505.
  • A. T.-M. Lau and W. Takahashi, Invariant means and fixed point properties for non-expansive representations of topological semigroups, Topol. Methods Nonlinear Anal., 5 (1995), 39-57.
  • A. T.-M. Lau and W. Takahashi, Invariant submeans and semigroups of nonexpansive mappings on Banach spaces with normal structure, J. Func. Anal., 25 (1996), 79-88.
  • A. T.-M. Lau and W. Takahashi, Fixed point and non-linear ergodic theorems for semigroups of nonlinear mappings, Handbook of Metric Fixed Point Theory (W.A. Kirk and B. Sims (Eds.), Kluwer Academic Publishers, 2001, pp. 517-555.
  • A. T.-M. Lau and W. Takahashi, Nonlinear submeans on semigroups, Topological Methods Nonlinear Analysis, 22 (2003), 345-353.
  • A. T.-M. Lau and W. Takahashi, Fixed point properties for semigroup of nonexpansive mappings on Fréchet spaces, (preprint).
  • A. T. Lau and A. Ülger, Some geometric properties on the Fourier and Fourier-Stieltjes algebras of locally compact groups, Arens regularity and related problems, Trans. Amer. Math. Soc., 337 (1993), 321-359.
  • A. T.-M. Lau and C. S. Wong, Common fixed points for semigroups of nonexpansive mappings, Proc. Amer. Math. Soc., 41 (1973), 223-228.
  • A. T.-M. Lau and Y. Zhang. Fixed point properties of semigroup of non-expansive mappings, Journal of Functional Analysis, 254 (2008), 2534-2554.
  • C. Lennard, $C_1$ is uniformly Kadec-Klee, Proc. Amer. Math. Soc., 109 (1990), 71-77.
  • G. Li, Weak convergence and non-linear ergodic theorems for reversible semigroups of non-Lipschitzian mappings, J. Math. Anal. Appl., 206 (1997), 451-464.
  • T. C. Lim, Characterizationsof normal structure, Proc. Amer. Math. Soc., 43 (1973), 313-319.
  • T. C. Lim, A fixed point theorem for families of nonexpansive mappings, Pacific J. Math., 53 (1974), 484-493.
  • T. C. Lim, Asymptotic centres and nonexpansive mappings in some conjugate Banach spaces, Pacific J. Math., 90 (1980), 135-143.
  • W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
  • T. Mitchell, Fixed points of reversible semigroups of non-expansive mappings, K\u odai Math. Sem. Rep., 22 (1970), 322-323.
  • T. Mitchell, Topological semigroups and fixed points, Illinois J. Math., 14 (1970), 630-641.
  • T. Mitchell, Talk in Richmond, Virginia Conference on “Analysis on Semigroups”, 1984.
  • N. Mizogudi and W. Takahashi, On the existence of fixed points and ergodic retractions for Lipschitzian semigroups of Hilbert spaces, Nonlinear Analysis, 14 (1990), 69-80.
  • I. Namioka, Følner's conditions for amenable semigroups, Math. Scand., 15 (1964), 18-28.
  • A. L. T. Paterson, Amenability, Amer. Math. Soc. Survey and Monograph, 29 (1988).
  • J. P. Pier, Amenable Locally Compact Group, John Wiley & Sons, New York, (1984).
  • S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 272-276.
  • S. Reich, Asymptotic behavior of contractions in Banach spaces, J. Math. Analy. Appl., 44 (1973), 57-70.
  • G. Rodé, An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl., 85 (1982), 172-178.
  • G. Schectman, On commuting families of nonexpansive operations, Proc. Amer. Math. Soc., 84 (1982), 373-376.
  • W. Takahashi, Fixed point theorem for amenable semigroups of non-expansive mappings, K\u odai Math. Sem. Rep., 21 (1969), 383-386.
  • 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.
  • W. Takahashi, Nonlinear Functional Analysis, Yokohama, 2000.
  • W. Takahashi, Fixed point theorem and nonlinear ergodic theorem for nonexpansive semigroups without convexity, Canadian J. Math., 44 (1992), 880-887.
  • W. Takahashi, Fixed point theorems and nonlinear ergodic theorems for nonlinear semigroups and their applications, Nonlinear Anal., 30 (1997), 1283-1293.
  • J. C. S. Wong, Topologically stationary locally compact groups and amenability, Trans. Amer. Math. Soc., 144 (1969), 351-363.