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Best proximity point theorems for supreme proximal contractions

  • S. Sadiq BashaEmail author
Original Paper

Abstract

This article presents some best proximity point theorems for new classes of non-self mappings, called supreme proximal contractions and supreme proximal cyclic contractions, which are brought forth to generalize the notion of self-contraction. Eventually, these results explore the existence of best proximity points that serve as optimal approximate solutions to the fixed point equations of the form \(Tx=x\), where T is a supreme proximal contraction or a supreme proximal cyclic contraction. Further, it is interesting to observe that such results generalize the most celebrated and elegant Banach’s contraction to the case of non-self mappings.

Keywords

Best proximity point Fixed point Proximal contraction Optimal approximate solution 

Mathematics Subject Classification

47H10 47H09 

References

  1. 1.
    Espinola, R., Kosuru, G.S.R., Veeramani, P.: Pythagorean property and best-proximity point theorems. J. Optim. Theory Appl. 164, 534–550 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gabeleh, M., Otafudu, O.O.: Nonconvex proximal normal structure in convex metric spaces. Banach J. Math. Anal. 10, 400–414 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Karpagam, S., Agrawal, S.: Best proximity points for cyclic contractions (preprint) Google Scholar
  4. 4.
    Sadiq Basha, S.: Best proximity point theorems generalizing the contraction principle. Nonlinear Anal. 74, 5844–5850 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Sadiq Basha, S.: Best proximity points: optimal solutions. J. Optim. Theory Appl. 151, 210–216 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Sadiq Basha, S.: Best proximity point theorems. J. Approx. Theory 163, 1772–1781 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sadiq Basha, S., Shahzad, N.: Best proximity point theorems for generalized proximal contractions. Fixed Point Theory Appl. 2012, 42 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Witthayarat, U., Cho, Y.J., Cholamjiak, P.: On solving proximal split feasibility problems and applications. Ann. Funct. Anal. 9, 111–122 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Wlodarczyk, K., Plebaniak, R., Banach, A.: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 70, 3332–3341 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wlodarczyk, K., Plebaniak, R., Banach, A.: Erratum to: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 71, 3585–3586 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wlodarczyk, K., Plebaniak, R., Obczynski, C.: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Anal. 72, 794–805 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Department of MathematicsAnna UniversityChennaiIndia

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