Existence of fixed points for asymptotically nonexpansive type actions of semigroups

  • S. Saeidi
  • F. Golkar
  • A. M. Forouzanfar


We prove the existence of common fixed points for reversible semitopological semigroups acting asymptotically nonexpansive type on nonempty compact convex sets in the context of locally convex spaces. This answers a problem raised by Lau (Fixed Point Theory Appl, 2010), for an asymptotically nonexpansive type action.


Semigroup action asymptotically nonexpansive reversible semitopological semigroup 

Mathematics Subject Classification

Primary 47H10 Secondary 20M30 47H09 47H20 



The authors would like to thank the anonymous reviewer for his or her thoughtful review of the manuscript, constructive comments and suggesting the idea of Theorem 2.7.


  1. 1.
    Paterson, A.L.T.: Amenability. American Mathematical Society, Providence, RI (1988)CrossRefzbMATHGoogle Scholar
  2. 2.
    Berglund, J.F., Junghenn, H.D., Milnes, P.: Analysis on Semigroups. Wiley, New York (1988)zbMATHGoogle Scholar
  3. 3.
    Holmes, R.D., Lau, A.T.-M.: Nonexpansive actions of topological semigroups and fixed points. J. Lond. Math. Soc. 5, 330–336 (1972)CrossRefzbMATHGoogle Scholar
  4. 4.
    Mitchell, T.: Fixed points of reversible semigroups of nonexpansive mappings. Kodai Math. Semin. Rep. 22, 322–323 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    DeMarr, R.: Common fixed points for commuting contraction mappings. Pac. J. Math. 13, 1139–1141 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Takahashi, W.: Fixed point theorem for amenable semigroup of nonexpansive mappings. Kodai Math. Semin. Rep. 21, 383–386 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lau, A.T.-M.: Invariant means on almost periodic functions and fixed point properties. Rocky Mt. J. Math. 3, 69–76 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fendler, G., Lau, A.T.-M., Leinert, M.: Weak* fixed point property and asymptotic centre for the Fourier–Stieltjes algebra of a locally compact group. J. Funct. Anal. 264, 288–302 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lau, A.T.-M., Mah, P.F.: Fixed point property for Banach algebras associated to locally compact groups. J. Funct. Anal. 258, 357–372 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lau, A.T.-M., Zhang, Y.: Fixed point properties of semigroups of non-expansive mappings. J. Funct. Anal. 254, 2534–2554 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lau, A.T.-M., Zhang, Y.: Fixed point properties for semigroups of nonlinear mappings and amenability. J. Funct. Anal. 263, 2949–2977 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Randrianantoanina, N.: Fixed point properties of semigroups of nonexpansive mappings. J. Funct. Anal. 258, 3801–3817 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Borzdynski, S., Wisnicki, A.: A common fixed point theorem for a commuting family of weak* continuous nonexpansive mappings. Stud. Math. 225, 173–181 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Saeidi, S.: Common fixed point property through analysis of retractions. J. Fixed Point Theory Appl. 17, 483–494 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Saeidi, S.: Nonexpansive retractions and fixed point properties. J. Math. Anal. Appl. 391, 99–106 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mohamadi, I., Saeidi, S.: Existence and structure of the common fixed points based on tvs. Filomat 31, 1773–1779 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Amini, M., Medghalchi, A., Naderi, F.: Semi-asymptotic non-expansive actions of semi-topological semigroups. Bull. Korean Math. Soc. 53, 39–48 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Holmes, R.D., Narayanaswami, P.P.: On asymptotically nonexpansive semigroups of mappings. Can. Math. Bull. 13, 209–214 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Holmes, R.D., Lau, A.T.-M.: Asymptotically nonexpansive actions of topological semigroups and fixed points. Bull. Lond. Math. Soc. 3, 343–347 (1971)CrossRefzbMATHGoogle Scholar
  20. 20.
    Lau, A.T.-M.: Normal structure and common fixed point properties for semigroups of nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. (2010) Art. ID 580956Google Scholar
  21. 21.
    Kirk, W.A.: A fixed point theorem for mappings which do not increase distance. Am. Math. Monthly 72, 1002–1004 (1965)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lim, T.C.: Characterizations of normal structure. Proc. Am. Math. Soc. 43, 313–319 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lau, A.T.-M., Takahashi, W.: Invariant means and fixed point properties for nonexpansive representations of topological semigroups. Topol. Methods Nonlinear Anal. 5, 39–57 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Xu, H.K.: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Anal. 16, 1139–1146 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Benavides, T.D., Ramirez, P.L.: Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 129, 3549–3557 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tingley, D.: An asymptotically nonexpansive commutative semigroup with no fixed points. Proc. Am. Math. Soc. 97, 107–113 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KurdistanSanandajIran
  2. 2.Department of Mathematics, Faculty of Mathematical Sciences and ComputerShahid Chamran University of AhvazAhvazIran

Personalised recommendations