Skip to main content
SpringerLink
Log in

Invariant means on weakly almost periodic functions and generalized fixed point properties

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

In this paper, we prove common fixed point theorems for Generalized Suzuki Contractions (abbreviated as GSC) involving two semi-topological semigroups of self-mappings \(S_{1}\) and \(S_{2}\), besides establishing the existence of a left invariant mean (abbreviated as LIM) on the space of all weakly almost periodic functions on \(S_{1}\cap S_{2}\) (abbreviated as \(WAP(S_{1}\cap S_{2})\)).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Benavides, T.D., and M.A.J. Pineda. 2005. Fixed points of nonexpansive mappings in spaces of continuous functions. Proceedings of the American Mathematical Society 133: 3037–3046.

    Article  MathSciNet  Google Scholar 

  2. Benavides, T.D., M.A.J. Pineda, and S. Prus. 2004. Weak compactness and fixed point property for affine mappings. Journal of Functional Analysis 209: 1–15.

    Article  MathSciNet  Google Scholar 

  3. Browder, F.E. 1965. Non-expansive nonlinear operators in Banach spaces. Proceedings of the National Academy of Sciences of the United States of America 54: 1041–1044.

    Article  MathSciNet  Google Scholar 

  4. Clifford, A.H., and G.B. Preston. 1961. The algebraic theory of semigroups. Mathematical surveys, no. 7, vol. 1. Providence: American Mathematical Society.

    Google Scholar 

  5. Day, M.M. 1957. Amenable semigroups. Illinois Journal of Mathematics 1: 509–544.

    Article  MathSciNet  Google Scholar 

  6. De Marr, R.E. 1963. Common fixed points for commuting contraction mappings. Pacific Journal of Mathematics 13: 1139–1141.

    Article  MathSciNet  Google Scholar 

  7. Dowling, P.N., C.J. Lennard, and B. Turett. 2004. Weak compactness is equivalent to the fixed point property in c0. Proceedings of the American Mathematical Society 132: 1659–1666.

    Article  MathSciNet  Google Scholar 

  8. Goebel, K., and W.A. Kirk. 1990. Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  9. Goebel, K., and W.A. Kirk. 2001. Classical theory of nonexpansive mappings. In Handbook of metric fixed point theory, ed. Kirk W.A., Sims B, 49-91. Dordrecht: Kluwer Academic Publishers

  10. Holmes, R.D., and A. Lau. 1972. Nonexpansive actions of topological semigroups and fixed points. Journal of the London Mathematical Society s2–5 (2): 330–336.

    Article  Google Scholar 

  11. Kang, J. 2008. Fixed point set of semigroups of non-expansive mappings and amenability. Journal of Mathematical Analysis and Applications 341: 1445–1456.

    Article  MathSciNet  Google Scholar 

  12. Kirk, W.A. 1965. A fixed point theorem for mappings which do not increase distances. American Mathematical Monthly 72: 1004–1006.

    Article  MathSciNet  Google Scholar 

  13. Lau, A.T.-M. 1973. Invariant means on almost periodic functions and fixed point properties. Rocky Mountain Journal Of Mathematics 3: 69–76.

    Article  MathSciNet  Google Scholar 

  14. Lau, A.T.-M. 1991. Amenability and fixed point property for semigroup of nonexpansive mappings. In Fixed point theory and applications. Pitman research notes in mathematics series, vol. 252, ed. M.A. Thera, and J.B. Baillon, 303–313. Longman Scientific and Technical: Harlow.

    Google Scholar 

  15. Lau, A.T.-M., and Y. Zhang. 2008. Fixed point properties of semigroups of non-expansive mappings. Journal of Functional Analysis 254: 2534–2554.

    Article  MathSciNet  Google Scholar 

  16. Mitchell, T. 1970. Fixed points of reversible semigroups of nonexpansive mappings. Ködai Mathematical Seminar Report 22: 322–323.

    Article  MathSciNet  Google Scholar 

  17. Sahu, D.R., Donal O’Regan, and Ravi P. Agarwal. 2009. Fixed point theory for Lipschitzian-type mappings with applications, 6. Berlin: Springer.

    Book  Google Scholar 

  18. Schroder, L. 2008. Linearizability of non-expansive semigroup actions on metric space. Topology and its Applications 155: 1576–1579.

    Article  MathSciNet  Google Scholar 

  19. Singh, S.L., R. Chugh, R. Kamal, and A. Kumar. 2014. A new common fixed point theorem for Suzuki–Meir–Keeler contractions. Filomat 28 (2): 257–626.

    Article  MathSciNet  Google Scholar 

  20. Soliman, A.H. 2014. A coupled fixed point theorem for nonexpansive one parameter semigroup. Journal of Advanced Mathematical Studies 7 (2): 28–37.

    MathSciNet  MATH  Google Scholar 

  21. Suzuki, T. 2008. Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. Journal of Mathematical Analysis and Applications 340: 1088–1095.

    Article  MathSciNet  Google Scholar 

  22. Takahashi, W. 1969. Fixed point theorem for amenable semigroups of nonexpansive mappings. Ködai Mathematical Seminar Report 21: 383–386.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

All the authors are thankful to the anonymous referee for their fruitful suggestions/comments.

Author information

Author notes

  1. Md Ahmadullah

    Present address: Department of Mathematics, Rammohan College, Calcutta University, Kolkata, WB, 700009, India

Authors and Affiliations

  1. Department of Mathematics, King Khaled University, Abha, 9004, Saudi Arabia

    Ahmed H. Soliman

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, UP, 202002, India

    Mohammad Imdad & Md Ahmadullah

Authors
  1. Ahmed H. Soliman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad Imdad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Md Ahmadullah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Imdad.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animal participations

All the authors read and approved the final manuscript.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Soliman, A.H., Imdad, M. & Ahmadullah, M. Invariant means on weakly almost periodic functions and generalized fixed point properties. J Anal 29, 177–189 (2021). https://doi.org/10.1007/s41478-020-00254-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-020-00254-w

Keywords

Mathematics Subject Classification

Search

Navigation