Skip to main content
SpringerLink
Log in

Best Proximity Point Theorems via Proximal Non-self Mappings

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

Abstract

We prove a best proximity point theorem for proximal generalized contractive type mappings in metric spaces, which is a generalization of recent best proximity point theorems and some famous fixed point theorems due to Berinde and Suzuki. We also introduce a new class of proximal non-self mappings and obtain sufficient conditions, which ensure the existence of a best proximity point. Moreover, we define algorithms and prove that they find a best proximity point for these classes of non-self mappings in the setting of metric and Banach spaces.

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. Fan, K.: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 122, 234–240 (1969)

    Article  Google Scholar 

  2. Prolla, J.B.: Fixed point theorems for set valued mappings and existence of best approximations. Numer. Funct. Anal. Optim. 5, 449–455 (2010)

    Article  MathSciNet  Google Scholar 

  3. Reich, S.: Approximate selections, best approximations, fixed points and invariant sets. J. Math. Anal. Appl. 62, 104–113 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Seghal, V.M., Singh, S.P.: A generalization of multifunctions of Fans best approximation theorem. Proc. Amer. Math. Soc. 102, 534–537 (1988)

    MathSciNet  Google Scholar 

  5. Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium problems: nonsmooth optimization and variational inequality models, Nonconvex Optim. Appl. 58, (2004). doi:10.1007/b101835

  6. Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70, 3665–3671 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. De la Sen, M.: Fixed and best proximity points of cyclic jointly accretive and contractive self-mappings. J. Appl. Math. 29 (2012)

  8. Di Bari, C., Suzuki, T., Vetro, C.: Best proximity points for cyclic Meir- Keeler contractions. Nonlinear Anal. 69, 3790–3794 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Eldred, A.A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001–1006 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Espinola, R.: A new approach to relatively nonexpansive mappings. Proc. Amer. Math. Soc. 136, 1987–1996 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Farajzadeh, A.P., Plubtieng, S., Ungchittrakool, K.: On best proximity point theorems without ordering. Abstract Appl. Anal. 5 (2014)

  12. Samet, B.: Some results on best proximity points. J. Optim. Theory Appl. 159, 281–291 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Suzuki, T., Kikkawa, M., Vetro, C.: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 71, 2918–2926 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Basha, S.S., Veeramani, P.: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103, 119–129 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Basha, S.S.: Best proximity points: optimal solutions. J. Optim. Theory Appl. 151, 210–216 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fernandes-Leon, A.: Best proximity points for proximal contractions. J. Nonlinear Convex Anal. 15, 313–324 (2014)

    MathSciNet  Google Scholar 

  17. Amini-Harandi, A.: Best proximity points for proximal generalized contractions in metric spaces. Optim. Lett. 7, 913–921 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gabeleh, M.: Best proximity points for weak proximal contractions. Bull. Malaysian Math. Sci. Soc., (2014, in press)

  19. Basha, S.S., Shahzad, N.: Best proximity point theorems for generalized proximal contractions. Fixed Point Theory Appl. 2012, 42 (2012)

    Article  Google Scholar 

  20. Gabeleh, M.: Proximal weakly contractive and proximal nonexpansive non-self mappings in metric and Banach spaces. J. Optim. Theory Appl. 158, 615–625 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Suzuki, T.: A generalized Banach contraction principle which characterizes metric completeness. Proc. Amer. Math. Soc. 136, 1861–1869 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Berinde, V.: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 9, 43–53 (2004)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Goebel, K., Kirk, W.A.: Iteration processes for nonexpansive mapping. Contemp. Math. 21, 115–123 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sankar Raj, V.: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 74, 4804–4808 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Abkar, A., Gabeleh, M.: Global optimal solutions of noncyclic mappings in metric spaces. J. Optim. Theory Appl. 153, 298–305 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Abkar, A., Gabeleh, M.: A note on some best proximity point theorems proved under P-property. Abstract Appl. Anal. 3 (2013)

Download references

Acknowledgments

The author thanks the referees for their valuable suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran

    Moosa Gabeleh

Authors
  1. Moosa Gabeleh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moosa Gabeleh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gabeleh, M. Best Proximity Point Theorems via Proximal Non-self Mappings. J Optim Theory Appl 164, 565–576 (2015). https://doi.org/10.1007/s10957-014-0585-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-014-0585-8

Keywords

Mathematics Subject Classification

Search

Navigation