Advertisement

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

  • Raweerote Suparatulatorn
  • Watcharaporn Cholamjiak
  • Suthep SuantaiEmail author
Article
  • 28 Downloads

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.

Keywords

General Mann algorithm Global minimization problem Best proximity point problem Nonexpansive mapping 

Mathematics Subject Classification (2010)

41A29 90C26 47H09 

Notes

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.

References

  1. 1.
    Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70, 3665–3671 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Basha, S.S.: Best proximity points: global optimal approximate solutions. J. Glob. Optim. 49, 15–21 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Basha, S.S.: Best proximity points: optimal solutions. J. Optim. Theory Appl. 151, 210–216 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Basha, S.S., Veeramani, P.: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103, 119–129 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Basha, S.S., Shahzad, N., Jeyaraj, R.: Best proximity points: approximation and optimization. Optim. Lett. 7, 145–155 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Di Bari, C., Suzuki, T., Vetro, C.: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 69, 3790–3794 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gabeleh, M.: Best proximity point theorems via proximal non-self mappings. J. Optim. Theory Appl. 164, 565–576 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gabeleh, M., Shahzad, N.: Best proximity points, cyclic Kannan maps and geodesic metric spaces. J. Fixed Point Theory Appl. 18, 167–188 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Haddadi, M.R.: Best proximity point iteration for nonexpansive mapping in Banach spaces. J. Nonlinear Sci. Appl. 7, 126–130 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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) MathSciNetGoogle Scholar
  11. 11.
    Kim, W.K., Lee, K.H.: Existence of best proximity pairs and equilibrium pairs. J. Math. Anal. Appl. 316, 433–446 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kirk, W.A., Reich, S., Veeramani, P.: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851–862 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pirbavafa, S., Vaezpour, S.M.: Equilibria of free abstract economies via best proximity point theorems. Bol. Soc. Mat. Mexicana 24, 471–481 (2018) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Raj, V.S.: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 74, 4804–4808 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Raj, V.S.: Best proximity point theorems for non-self mappings. Fixed Point Theory 14, 447–454 (2013) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43, 53–159 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Srinivasan, P.S., Veeramani, P.: On existence of equilibrium pair for constrained generalized games. Fixed Point Theory Appl. 2004, 704376 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Suantai, S.: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 311, 506–517 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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) Google Scholar
  21. 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) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand
  2. 2.School of ScienceUniversity of PhayaoPhayaoThailand

Personalised recommendations