Skip to main content
SpringerLink
Log in

A global optimality result using nonself mappings

  • Theoretical Article
  • Published:
OPSEARCH Aims and scope Submit manuscript

Abstract

In this paper we consider a classical global optimization problem of obtaining minimum distance between pairs of closed sets by using an extension of the weak contraction mapping. The extension is accomplished with the help of three control functions. P-property of pairs of closed sets has been used. The result has several corollaries and an illustrative example. One corollary extends an existing work on best proximity point problem. The example shows that the extension is actual.

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. Schutz, B.F.: Geometrical Methods of Mathematical Physics. Cambridge University Press (1980)

  2. Hobson, M.P., Efstathiou, G., Lasenby, A.N.: General Relativity. Cambridge University Press, New York (2006)

    Book  Google Scholar 

  3. Kirk, W.A., Srinivasan, P.S., Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4(1), 79–89 (2003)

    Google Scholar 

  4. Eldred, A.A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323(2), 1001–1006 (2006)

    Article  Google Scholar 

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

    Article  Google Scholar 

  6. Abkar, A., Gabeleh, M.: Best proximity points for cyclic mappings in ordered metric spaces. J. Optim. Theory Appl. 151, 418–424 (2011)

    Article  Google Scholar 

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

    Article  Google Scholar 

  8. Karpagam, S., Agrawal, S.: Best proximity points for cyclic orbital Meir-Keeler contractions. Nonlinear Anal. 74, 1040–1046 (2011)

    Article  Google Scholar 

  9. Vetro, C.: Best proximity points: Convergence and existence theorems for p-cyclic mappings. Nonlinear Anal. 73, 2283–2291 (2010)

    Article  Google Scholar 

  10. Mongkolkeha, C., Kumam, P.: Best proximity point Theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. 215–226 (2012)

  11. Mongkolkeha, C., Kumam, P.: Some common best proximity points for proximity commuting mappings. Optim. Lett. (2012). doi:10.1007/s11590-012-0525-1

    Google Scholar 

  12. Amini-Harandi, A.: Best proximity points for proximal generalized contractions in metric spaces. Optim. Lett. doi:10.1007/s11590-012-0470-z

  13. Sankar Raj, V.: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. TMA 74(14), 4804–4808 (2011)

    Article  Google Scholar 

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

    Article  Google Scholar 

  15. Alber, Ya.I., Guerre-Delabriere, S.: Principle of weakly contractive maps in Hilbert spaces. In: New Results in Operator Theory and its Applications. Birkhuser, Basel, 98, pp. 7–22 (1997)

  16. Rhoades, B.E.: Some theorems on weakly contractive maps. Nonlinear Anal. TMA 47, 2683–2693 (2001)

    Article  Google Scholar 

  17. Choudhury, B.S., Metiya, N.: Fixed points of weak contractions in cone metric spaces. Nonlinear Anal. 72(34), 1589–1593 (2010)

    Article  Google Scholar 

  18. Choudhury, B.S., Konar, P., Rhoades, B.E., Metiya, N.: Fixed point theorems for generalized weakly contractive mappings. Nonlinear Anal. 74(6), 2116–2126 (2011)

    Article  Google Scholar 

  19. Choudhury, B.S., Kundu, A.: \((\psi ,\alpha ,\beta )\)-weak contractions in partially ordered metric spaces. 25(1), 6–10 (2012)

  20. Doric, D.: Common fixed point for generalized (ψ, ϕ)-weak contractions. Appl. Math. Lett. 22, 1896–1900 (2009)

    Article  Google Scholar 

  21. Popescu, O.: Fixed points for ψ, ϕ-weak contractions. Appl. Math. Lett. 24, 1–4 (2011)

    Article  Google Scholar 

  22. Zhang, Q., Song, Y.: Fixed point theory for generalized ψ-weak contractions. Appl. Math. Lett. 22, 75–78 (2009)

    Article  Google Scholar 

  23. Choudhury, B.S., Kundu, S.: A viscosity type iteration by weak contraction for approximating solutions of generalized equilibrium problem. J. Nonlinear Sci. Appl. 5, 243–251 (2012)

    Google Scholar 

  24. Khan, M.S., Swaleh, M., Sessa, S.: Fixed point theorems by altering distances between the points. Bull. Australian Math. Soci. 30(1), 1–9 (1984)

    Article  Google Scholar 

  25. Choudhury, B.S.: A common unique fixed point result in metric spaces involving generalised altering distances. Math. Commun. 10, 105–110 (2005)

    Google Scholar 

  26. Dutta, P.N., Choudhury, B.S.: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl. 2008, 8 (2008). Article ID 406368

    Article  Google Scholar 

  27. Sastry, K.P.R., Babu, G.V.R.: Some fixed point theorems by altering distances between the points. Indian J. Pure Appl. Math. 30(6), 641–647 (1999)

    Google Scholar 

  28. Shatanawi, W., Samet, B.: On (ψ, ϕ)-weakly contractive condition in partially ordered metric spaces. Comput. Math. Appl. 62(8), 3204–3214 (2011)

    Article  Google Scholar 

  29. Kirk, W.A., Reich, S., Veeramani, P.: Proximinal retracts and best proximity pair theorems. Numer. Func. Anal. Optim. 24(7–8), 851–862 (2003)

    Article  Google Scholar 

  30. Eslamian, M., Abkar, A.: A fixed point theorems for generalised weakly contractive mappings in complete metric space. Ital. J. Pure Appl. Math. (in press)

Download references

Acknowledgments

The work is supported by the Council of Scientific and Industrial Research, Government of India, under Research Project No - 25(0168)/09/EMR-II . The support is gratefully acknowledged. The valuable suggestions of the learned referee are acknowledged.

Author information

Authors and Affiliations

  1. Department of Mathematics, Bengal Engineering and Science University, Shibpur, P.O.-B. Garden, Howrah, 711103, West Bengal, India

    Binayak S. Choudhury, Pranati Maity & P. Konar

Authors
  1. Binayak S. Choudhury
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pranati Maity
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Konar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Binayak S. Choudhury.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Choudhury, B.S., Maity, P. & Konar, P. A global optimality result using nonself mappings. OPSEARCH 51, 312–320 (2014). https://doi.org/10.1007/s12597-013-0147-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12597-013-0147-0

Keywords

AMS Subject Classifications

Search

Navigation