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Best approximations in metric spaces with property strongly UC

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Abstract

In this article, we introduce a geometrical notion, called property strongly UC which is stronger than property UC and prove the existence of best approximations for a new class of almost cyclic \(\psi\)-contraction maps defined on a pair of subsets of a metric space. As a particular case of this existence theorem, we obtain the main results of [Sadiq Basha, S., Best approximation theorems for almost cyclic contractions. J. Fixed Point Theory Appl. 23 (2021)] and [Eldred, A. Anthony; Veeramani, P., Existence and convergence of best proximity points. J. Math. Anal. Appl. 323 (2006)]. Moreover, we study the existence of a best approximation and continuity properties of almost cyclic contractions in the context of a reflexive Banach space and a metric space.

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References

  1. Al-Thagafi, M.A., Naseer, S.: Convergence and existence results for best proximity points. Nonlinear Anal. 70(10), 3665–3671 (2009)

    Article  MathSciNet  Google Scholar 

  2. Basha, S.: Best approximation theorems for almost cyclic contractions. J. Fixed Point Theory Appl. 23(2), 32 (2021)

    Article  MathSciNet  Google Scholar 

  3. Basha, S., Veeramani, P.: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103(1), 119–129 (2000)

    Article  MathSciNet  Google Scholar 

  4. Beer, Gerald, Pai, Devidas: Proximal maps, prox maps and coincidence points. Numer. Funct. Anal. Optim. 11(5–6), 429–448 (1990)

    Article  MathSciNet  Google Scholar 

  5. Digar, A., Kosuru, G., Sankara, R.: Existence of best proximity pairs and a generalization of Carathéodory theorem. Numer. Funct. Anal. Optim. 41(15), 1901–1911 (2020)

    Article  MathSciNet  Google Scholar 

  6. Eldred, A., Anthony, K.W.A., Veeramani, P.: Proximal normal structure and relatively nonexpansive mappings. Stud. Math. 171(3), 283–293 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Espínola, R.: A new approach to relatively nonexpansive mappings. Proc. Amer. Math. Soc. 136(6), 1987–1995 (2008)

    Article  MathSciNet  Google Scholar 

  9. Espínola, R., Fernández-León, A.: On best proximity points in metric and Banach spaces. Can. J. Math. 63(3), 533–550 (2011)

    Article  MathSciNet  Google Scholar 

  10. Fan, K.: Extensions of two fixed point theorems of F. E Browde. Mathematische Zeitschrift 112, 234–240 (1969)

    Article  MathSciNet  Google Scholar 

  11. Fernández-León, Aurora: Existence and uniqueness of best proximity points in geodesic metric spaces. Nonlinear Anal. 73(4), 915–921 (2010)

    Article  MathSciNet  Google Scholar 

  12. Goebel, K., Kirk, W. A.: Topics in metric fixed point theory, Cambridge University Press, Cambridge, (1990)

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

    Article  MathSciNet  Google Scholar 

  14. Kosuru, G., Sankara, R., Veeramani, P.: A note on existence and convergence of best proximity points for pointwise cyclic contractions. Numer. Funct. Anal. Optim. 32(7), 821–830 (2011)

    Article  MathSciNet  Google Scholar 

  15. Kosuru, G., Sankara, R.: Extensions of Edelstein’s theorem on contractive mappings. Numer. Funct. Anal. Optim. 36(7), 887–900 (2015)

    Article  MathSciNet  Google Scholar 

  16. Megginson, RE.: An introduction to Banach space theory, Springer-Verlag, New York, vol 183), (1998)

  17. Rajesh, S., Veeramani, P.: Best proximity point theorems for asymptotically relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 37(1), 80–91 (2016)

    Article  MathSciNet  Google Scholar 

  18. Rockafellar, R.: Tyrrell. Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, Convex analysis (1997)

    Google Scholar 

  19. Shunmugaraj, P., Thota, V.: Uniform convexity, strong convexity and property UC. J. Math. Anal. Appl. 446(2), 1769–1775 (2017)

    Article  MathSciNet  Google Scholar 

  20. Suzuki, T., Kikkawa, M., Vetro, C.: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 71(7–8), 2918–2926 (2009)

    Article  MathSciNet  Google Scholar 

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Funding

The research of the author is supported by the Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, India. The author  would like to thank Prof. P. Veeramani for his suggestions in improving Theorem 2.10.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, Indian Institute of Technology Kanpur, Kalyanpur, Uttar Pradesh, 208 016, India

    Abhik Digar

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  1. Abhik Digar
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Correspondence to Abhik Digar.

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Communicated by Timur Oikhberg.

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Digar, A. Best approximations in metric spaces with property strongly UC. Adv. Oper. Theory 9, 24 (2024). https://doi.org/10.1007/s43036-024-00323-y

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