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Common best proximity pairs via the concept of complete proximal normal structure

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Abstract

We introduce a concept of complete proximal normal structure and used to investigate the existence of a best proximity point for an arbitrary family of cyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces. We also prove that every bounded, closed and convex pair in uniformly convex Banach spaces as well as every compact and convex pairs in Banach spaces has complete proximal normal structure. Furthermore, we consider a class of cyclic relatively nonexpansive mappings in the sense of Suzuki and establish a new best proximity point theorem in the setting of Hilbert spaces.

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References

  1. Abkar, A., Gabeleh, M.: Proximal quasi-normal structure and a best proximity point theorem. J. Nonlinear Convex Anal. 14, 653–659 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Belluce, L.P., Kirk, W.A.: Fixed-point theorems for families of contraction mappings. Pac. J. Math. 18, 213–218 (1966)

    Article  MathSciNet  Google Scholar 

  3. Belluce, L.P., Kirk, W.A.: Nonexpansive mappings and fixed-points in Banach spaces. Illinois J. Math. 11, 474–479 (1967)

    Article  MathSciNet  Google Scholar 

  4. Brodskii, M.S., Milman, D.P.: On the center of a convex set. Dokl. Akad. Nauk. USSR 59, 837–840 (1948). (in Russian)

    MathSciNet  Google Scholar 

  5. Eldred, A.A., Sankar Raj, V.: On common best proximity pair theorems. Acta Sci. Math. (Szeged) 75, 707–721 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Espínola, R.: A new approach to relatively nonexpansive mappings. Proc. Am. Math. Soc. 136, 1987–1996 (2008)

    Article  MathSciNet  Google Scholar 

  9. Espínola, R., Gabeleh, M.: On the structure of minimal sets of relatively nonexpnsive mappings. Numer. Funct. Anal. Optim. 34, 845–860 (2013)

    Article  MathSciNet  Google Scholar 

  10. Espínola, R., Sankara Raju Kosuru, G., Veeramani, P.: Pythagorean property and best-proximity point theorems. J. Optim. Theory Appl. 164, 534–550 (2015)

    Article  MathSciNet  Google Scholar 

  11. Gabeleh, M.: Convergence of Picard’s iteration using projection algorithm for noncyclic contractions. Indag. Math. 30, 227–239 (2019)

    Article  MathSciNet  Google Scholar 

  12. Goebel, K., Kirk, W.A.: Iteration processes for nonexpansive mapping. Contemp. Math. 21, 115–123 (1983)

    Article  MathSciNet  Google Scholar 

  13. Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics. Wiley-Interscience, New York (2001)

    Book  Google Scholar 

  14. Kirk, W.A.: A fixed point theorem for mappings which do not increase distances. Am. Math. Monthly 72, 1004–1006 (1965)

    Article  MathSciNet  Google Scholar 

  15. Lim, T.C.: Characterization of normal structure. Proc. Am. Math. Soc. 43, 313–319 (1974)

    Article  MathSciNet  Google Scholar 

  16. Sankara Raju Kosuru, G., Veeramani, P.: On existence of best proximity pair theorems for relatively nonexpansive mappings. J. Nonlinear Convex Anal. 11, 71–77 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 340, 1088–1095 (2008)

    Article  MathSciNet  Google Scholar 

  18. Zizler, V.: On some rotundity and smoothness properties of Banach spaces. Diss. Math. 87, 1–33 (1971)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referee for the careful reading of the manuscript and many useful comments.

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Authors and Affiliations

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

    Moosa Gabeleh

  2. 528 Rover Blvd., Los Alamos, NM, 87544, USA

    Jack Markin

Authors
  1. Moosa Gabeleh
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  2. Jack Markin
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Correspondence to Moosa Gabeleh.

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Communicated by Patrick Dowling.

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Gabeleh, M., Markin, J. Common best proximity pairs via the concept of complete proximal normal structure. Ann. Funct. Anal. 11, 831–847 (2020). https://doi.org/10.1007/s43034-020-00057-x

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  • DOI: https://doi.org/10.1007/s43034-020-00057-x

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