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Best proximity points: global minimization of multivalued non-self mappings

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Abstract

In this article, we give a best proximity point theorem for generalized contractions in metric spaces with appropriate geometric property. We also, give an example which ensures that our result cannot be obtained from a similar result due to Amini-Harandi (Best proximity points for proximal generalized contractions in metric spaces. Optim Lett, 2012). Moreover, we prove a best proximity point theorem for multivalued non-self mappings which generalizes the Mizoguchi and Takahashi’s fixed point theorem for multivalued mappings.

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References

  1. Abkar, A., Gabeleh, M.: Best proximity points for asymptotic cyclic contraction mappings. Nonlinear Anal. 74, 7261–7268 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. 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 

  3. Abkar, A., Gabeleh, M.: The existence of best proximity points for multivalued non-self-mappings. RACSAM (2012). doi:10.1007/s13398-012-0074-6

  4. Abkar, A., Gabeleh, M. : A note on some best proximity point theorems proved under P-property. arXiv:1301.7147v1 [math.FA] (2013)

  5. Amini-Harandi, A.: Best proximity points for proximal generalized contractions in metric spaces. Optim. Lett. (2012). doi:10.1007/s11590-012-0470-z

  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. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Eldred, A., Anuradha, J., Veeramani, P.: On equivalence of generalized multivalued contractions and Nadler’s fixed point theorem. J. Math. Anal. Appl. 336, 751–757 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

  11. Mizoguchi, N., Takahashi, W.: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141, 177–188 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Nadler, S.B.Jr.: Multivalued contraction mapping. Pac. J. Math. 30, 475–488 (1969)

    Google Scholar 

  13. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Sadiq Basha, S.: Best proximity point theorems. J. Approx. Theory 163, 1772–1781 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

  17. Suzuki, T.: Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s. J. Math. Anal. Appl. 340, 752–755 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. 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 

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Acknowledgments

The author is grateful to the referee for his-her helpful comments leading to improvement of the presentation of the work.

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

    Moosa Gabeleh

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  1. Moosa Gabeleh
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Correspondence to Moosa Gabeleh.

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Gabeleh, M. Best proximity points: global minimization of multivalued non-self mappings. Optim Lett 8, 1101–1112 (2014). https://doi.org/10.1007/s11590-013-0628-3

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  • DOI: https://doi.org/10.1007/s11590-013-0628-3

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