Skip to main content
SpringerLink
Log in

Weak fixed point property for nonexpansive mappings with respect to orbits in Banach spaces

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

Abstract

In this paper, we first show that a Banach space X has weak normal structure if and only if X has the weak fixed point property for nonexpansive mappings with respect to (wrt) orbits. Then, we give a counterexample to show that the Goebel–Karlovitz lemma does not hold for minimal invariant sets of nonexpansive mappings wrt orbits, and we present a modified version of the Goebel–Karlovitz lemma.

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. Aksoy A. G., Khamsi M. A.: Nonstandard Methods in Fixed Point Theory. Springer, New York (1990)

    Book  MATH  Google Scholar 

  2. Ayerbe Toledano J. M., Domínguez Benavides T., López Acedo G.: Measures of Noncompactness in Metric Fixed Point Theory. Birkhäuser, Basel (1997)

    Book  MATH  Google Scholar 

  3. Brodskiĭ M. S., Mil’man D. P.: On the center of a convex set. Dokl. Akad. Nauk SSSR (N.S.) 59, 837–840 (1948)

    MathSciNet  Google Scholar 

  4. Domínguez Benavides T.: The failure of the fixed point property for unbounded sets in c 0. Proc. Amer. Math. Soc. 140, 645–650 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. García-Falset J., Llorens-Fuster E., Mazcuñan-Navarro E. M.: Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. J. Funct. Anal. 233, 494–514 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Goebel K., Kirk W. A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  7. Karlovitz L. A.: Existence of fixed points of nonexpansive mappings in a space without normal structure. Pacific J. Math. 66, 153–158 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kirk W. A.: A fixed point theorem for mappings which do not increase distances. Amer. Math. Monthly 72, 1004–1006 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  9. W. A. Kirk, Some questions in metric fixed point theory. In: Recent Advances on Metric Fixed Point Theory (Seville, 1995), Ciencias 48, Univ. Sevilla, Seville, 1996, 73–97.

  10. Kirk W. A., Sims B.: Handbook of Metric Fixed Point Theory. Kluwer Academic Publishers, Dordrecht (2001)

    Book  MATH  Google Scholar 

  11. Lennard C., Nezir V.: Reflexivity is equivalent to the perturbed fixed point property for cascading nonexpansive maps in Banach lattices. Nonlinear Anal. 95, 414–420 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Nicolae, Generalized asymptotic pointwise contractions and nonexpansive mappings involving orbits. Fixed Point Theory Appl. 2010 (2010), Article ID 458265, 19 pages.

  13. Ray W. O.: The fixed point property and unbounded sets in Hilbert space. Trans. Amer. Math. Soc. 258, 531–537 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sine R.: On the converse of the nonexpansive map fixed point theorem for Hilbert space. Proc. Amer. Math. Soc. 100, 489–490 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, Isfahan, 81745-163, Iran

    A. Amini-Harandi & M. Fakhar

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

    A. Amini-Harandi & M. Fakhar

  3. Department of Mathematics, Khansar Faculty of Mathematics and Computer Sciences, P.O. Box 87916854163, Khansar, Iran

    H. R. Hajisharifi

Authors
  1. A. Amini-Harandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Fakhar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. R. Hajisharifi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Amini-Harandi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amini-Harandi, A., Fakhar, M. & Hajisharifi, H.R. Weak fixed point property for nonexpansive mappings with respect to orbits in Banach spaces. J. Fixed Point Theory Appl. 18, 601–607 (2016). https://doi.org/10.1007/s11784-016-0310-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-016-0310-3

Mathematics Subject Classification

Keywords

Search

Navigation