Skip to main content
SpringerLink
Log in

Some fixed point results for generalized contractions of Suzuki type in Banach spaces

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we consider generalized contractions of Suzuki type. In particular, we give some geometrical properties of their fixed points. In addition, the convergence of some iterative processes and the problem of the existence of retractions associated with them are established.

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. Baillon, J.B.: Un théorème de type ergodique pour les contractions nonlinéaires dans un espace de Hilbert. C. R. Acad. Sci. Paris. Sér. A-B 280, 1511–1514 (1975)

    MathSciNet  MATH  Google Scholar 

  2. Baillon, J.B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4, 1–9 (1978)

    MathSciNet  MATH  Google Scholar 

  3. Belluce, L.P., Kirk, W.A., Steiner, E.F.: Normal structure in Banach spaces. Pac. J. Math. 26(3), 433–440 (1968)

    Article  MathSciNet  Google Scholar 

  4. Betiuk-Pilarska, A., Dominguez Benavides, T.: The fixed point property for some generalized nonexpansive mappings and renorming. J. Math. Anal. Appl. 429, 800–813 (2015)

    Article  MathSciNet  Google Scholar 

  5. Betiuk-Pilarska, A., Wisnicki, A.: On the Suzuki nonexpansive mappings. Ann. Funct. Anal. 4(2), 72–86 (2013)

    Article  MathSciNet  Google Scholar 

  6. Browder, F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Nat. Acad. Sci. USA 54, 1041–1044 (1965)

    Article  MathSciNet  Google Scholar 

  7. Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459–470 (1977)

    MathSciNet  MATH  Google Scholar 

  8. Dhompongsa, S., Kaewcharoen, A.: Fixed point theorems for nonexpansive mappings and Suzuki-generalized nonexpansive mappings on a Banach lattice. Nonlinear Anal. Theory Methods Appl. 71(11), 5344–5353 (2009)

    Article  MathSciNet  Google Scholar 

  9. Falset, J.G., Llorens-Fuster, E., Suzuki, T.: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 375, 185–195 (2011)

    Article  MathSciNet  Google Scholar 

  10. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  11. Hardy, G.E., Rogers, T.D.: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 16, 201–206 (1973)

    Article  MathSciNet  Google Scholar 

  12. Karlovitz, L.A.: Existence of fixed points of nonexpansive mappings in a space without normal structure. Pac. J. Math. 66(1), 153–159 (1976)

    Article  MathSciNet  Google Scholar 

  13. Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics. A Wiley-Interscience Series of Texts. Wiley, Oxford (2001)

    Google Scholar 

  14. Lau, A.T., Wong, C.S.: Common fixed points for semigroups of mappings. Proc. Am. Math. Soc. 41(1), 223–228 (1973)

    Article  MathSciNet  Google Scholar 

  15. Lau, A.T., Takahashi, W.: Invariant means and fixed point properties for nonexpansive representations of topological semigroups. Topol. Methods. Nonlinear. Anal. 5, 39–57 (1995)

    Article  MathSciNet  Google Scholar 

  16. Lau, A.T., Takahashi, W.: Invariant means and semigroups of nonexpansive mappings on uniformly convex Banach spaces. J. Math. Anal. Appl. 153(2), 497–505 (1990)

    Article  MathSciNet  Google Scholar 

  17. Lau, A.T., Takahashi, W.: Weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mappings. Pac. J. Math. 126(2), 277–294 (1987)

    Article  MathSciNet  Google Scholar 

  18. Lau, A.T., Zhang, Y.: Fixed point properties of semigroups of non-expansive mappings. J. Func. Anal. 254(10), 2534–2554 (2008)

    Article  MathSciNet  Google Scholar 

  19. Lau, A.T.: Invariant means on almost periodic functions and fixed point properties. Rocky. Mt. J. Math. 3(1), 69–76 (1973)

    Article  MathSciNet  Google Scholar 

  20. Lau, A.T.: Normal structure and common fixed point properties for semigroups of nonexpansive mappings in Banach spaces. Fixed Point. Theory Appl. 2010, 14 (2010). (Article ID580956)

    MathSciNet  MATH  Google Scholar 

  21. Lau, A.T.: Invariant means and fixed point properties of semigroup of nonexpansive mappings. Taiwan. J. Math. 12(6), 1525–1542 (2008)

    Article  MathSciNet  Google Scholar 

  22. Reich, S.: Some remarks concerning contraction mappings. Can. Math. Bull. 14(1), 121–124 (1971)

    Article  MathSciNet  Google Scholar 

  23. Reich, S.: Nonlinear evolution equations and nonlinear ergodic theorems. Nonlinear Anal. 1, 319–330 (1977)

    Article  MathSciNet  Google Scholar 

  24. Reich, S.: Almost convergence and nonlinear ergodic theorems. J. Approx. Theory 24(4), 269–272 (1978)

    Article  MathSciNet  Google Scholar 

  25. Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 340(2), 1088–1095 (2008)

    Article  MathSciNet  Google Scholar 

  26. Wong, C.S.: Approximation to fixed points of generalized nonexpansive mappings. Proc. Am. Math. Soc. 54(1), 93–97 (1976)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank anonymous referees for their valuable remarks and suggestions to improve the manuscript.

Author information

Authors and Affiliations

  1. Laboratory of Informatics and Mathematics, University of Souk-Ahras, P. O. Box 1553, 41000, Souk-Ahras, Algeria

    Sami Atailia, Najeh Redjel & Abdelkader Dehici

Authors
  1. Sami Atailia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Najeh Redjel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abdelkader Dehici
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the manuscript.

Corresponding author

Correspondence to Sami Atailia.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Funding

This work is supported by the research team RPC (Controllability and Perturbation Results) in the Laboratory of Informatics and Mathematics (LIM) at the University of Souk-Ahras (Algeria).

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

Atailia, S., Redjel, N. & Dehici, A. Some fixed point results for generalized contractions of Suzuki type in Banach spaces. J. Fixed Point Theory Appl. 21, 78 (2019). https://doi.org/10.1007/s11784-019-0717-8

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s11784-019-0717-8

Keywords

Mathematics Subject Classification

Search

Navigation