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Existence and Convergence Theorems for Global Minimization of Best Proximity Points in Hilbert Spaces

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Abstract

In order to solve global minimization problems involving best proximity points, we introduce general Mann algorithm for nonself nonexpansive mappings and then prove weak and strong convergence of the proposed algorithm under some suitable conditions in real Hilbert spaces. Furthermore, we also provide numerical experiment to illustrate the convergence behavior of our proposed algorithm.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Basha, S.S.: Best proximity points: global optimal approximate solutions. J. Glob. Optim. 49, 15–21 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Basha, S.S., Shahzad, N., Jeyaraj, R.: Best proximity points: approximation and optimization. Optim. Lett. 7, 145–155 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Gabeleh, M.: Best proximity point theorems via proximal non-self mappings. J. Optim. Theory Appl. 164, 565–576 (2015)

    Article  MathSciNet  Google Scholar 

  8. Gabeleh, M., Shahzad, N.: Best proximity points, cyclic Kannan maps and geodesic metric spaces. J. Fixed Point Theory Appl. 18, 167–188 (2016)

    Article  MathSciNet  Google Scholar 

  9. Haddadi, M.R.: Best proximity point iteration for nonexpansive mapping in Banach spaces. J. Nonlinear Sci. Appl. 7, 126–130 (2014)

    Article  MathSciNet  Google Scholar 

  10. Jacob, G.K., Postolache, M., Marudai, M., Raja, V.: Norm convergence iterations for best proximity points of non-self non-expansive mappings. Politeh. Univ. Buchar. Sci. Bull. Ser. A Appl. Math. Phys. 79, 49–56 (2017)

    MathSciNet  Google Scholar 

  11. Kim, W.K., Lee, K.H.: Existence of best proximity pairs and equilibrium pairs. J. Math. Anal. Appl. 316, 433–446 (2006)

    Article  MathSciNet  Google Scholar 

  12. Kim, W.K., Kum, S., Lee, K.H.: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 68, 2216–2227 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Pirbavafa, S., Vaezpour, S.M.: Equilibria of free abstract economies via best proximity point theorems. Bol. Soc. Mat. Mexicana 24, 471–481 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Raj, V.S.: Best proximity point theorems for non-self mappings. Fixed Point Theory 14, 447–454 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43, 53–159 (1991)

    Article  MathSciNet  Google Scholar 

  18. Srinivasan, P.S., Veeramani, P.: On existence of equilibrium pair for constrained generalized games. Fixed Point Theory Appl. 2004, 704376 (2004)

    Article  MathSciNet  Google Scholar 

  19. Suantai, S.: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 311, 506–517 (2005)

    Article  MathSciNet  Google Scholar 

  20. Suparatulatorn, R., Suantai, S.: A new hybrid algorithm for global minimization of best proximity points in Hilbert spaces. Carpath. J. Math. 35(1), 95–102 (2019)

    MathSciNet  Google Scholar 

  21. 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  MathSciNet  Google Scholar 

  22. Wang, L., Cho, Y.J., Huang, N.J.: The robustness of generalized abstract fuzzy economies in generalized convex spaces. Fuzzy Sets Syst. 176, 56–63 (2011)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

R. Suparatulatorn and S. Suantai would like to thank the Royal Golden Jubilee (RGJ) Ph.D. Programme (PHD/0021/2559) and Chiang Mai University for the financial support. W. Cholamjiak would like to thank the Thailand Research Fund under the project MRG6080105 and University of Phayao.

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Authors and Affiliations

  1. Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Raweerote Suparatulatorn & Suthep Suantai

  2. School of Science, University of Phayao, Phayao, 56000, Thailand

    Watcharaporn Cholamjiak

Authors
  1. Raweerote Suparatulatorn
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  2. Watcharaporn Cholamjiak
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  3. Suthep Suantai
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Correspondence to Suthep Suantai.

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Suparatulatorn, R., Cholamjiak, W. & Suantai, S. Existence and Convergence Theorems for Global Minimization of Best Proximity Points in Hilbert Spaces. Acta Appl Math 165, 81–90 (2020). https://doi.org/10.1007/s10440-019-00242-8

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  • DOI: https://doi.org/10.1007/s10440-019-00242-8

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