Journal of Nonlinear Analysis and Application

Volume 2018, No. 1 (2018), Pages 57-63

Article ID jnaa-00376, 7 Pages

doi: 10.5899/2018/jnaa-00376

Research Article

Best Proximity Points of Contractive-type and Nonexpansive-type Mappings

R. Kavitha1 *, S. Somasundaram1

1Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, India.

* Corresponding author. Email address:

Received: 23 February 2017; Accepted: 15 March 2017

Copyright © 2018 R. Kavitha and S. Somasundaram. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The purpose of this paper is to obtain best proximity point theorems for multivalued nonexpansive-type and contractive-type mappings on complete metric spaces and on certain closed convex subsets of Banach spaces. We obtain a convergence result under some assumptions and we prove the existence of common best proximity points for a sequence of multivalued contractive-type mappings.

Keywords: Best proximity point; Contractive-type and nonexpansive-type multivalued maps; Common best proximity point.


  1. A. Abkar, M. Gabeleh, Best Proximity Points for Asymptotic Cyclic Contraction Mappings, Nonlinear Analysis, 74 (2011) 7261-7268.

  2. A. Abkar, M. Gabeleh, Proximal Quasi-normal Structure and a Best Proximity Point Theorem, Journal of Nonlinear and Convex Analysis, 14 (2013) 653-659.

  3. Akbar Azam, Common Fixed Point Theorems in Complex Metric spaces, Numerical Functional Analysis and Optimization, 32 (3) (2011) 243-253.

  4. A. Anthony Eldred, V. Sankar Raj, On Common Best Proximity Pair Theorems, Acta Sci. Math. (Szeged) 75 (2009) 707-721.

  5. F. E. Browder, Fixed Point Theorems for Non-compact Mappings in Hilbert Space, Proc. Nat. Acad. Sci. U. S. A, 53 (1965) 1272-1276.

  6. F. E. Browder, Non-Expansive Non-Linear Operators in a Banach Space, Proc. Nat. Acad. Sci. U. S. A, 54 (1965) 1041-1044.

  7. M. Edelstein, An Extension of Banach's Contraction Principle, Proc. Amer. Math. Soc, 12 (1961) 7-10.

  8. M. Gabeleh, Best Proximity Points and Fixed Point Results for Certain Maps in Banach Spaces, Numerical Functional Analysis and Optimization, 36 (8) (2015) 1013-1028.

  9. M. Gabeleh, Best Proximity Point Theorems via Proximal Non-self mappings, Journal of Optimization Theory and Applications: Newyork, 164 (2015) 565-576.

  10. M. Gabeleh, Proximal Weakly Contractive and Proximal Nonexpansive non-self- mappings in Metric and Banach Spaces, J. Optim. Theory. App., 158 (2) (2013) 615-625.

  11. T. Husain, Abdul Latif, Fixed Points of Multivalued Nonexpansive Maps, Internat. J. Math. and Math. sci, 14 (3) (1991) 421-430.

  12. T. Husain, E. Tarafdar, Fixed Point Theorems for Multivalued Mappings of Nonexpansive Type, Yokohama Math. J, 28 (1980) 1-6.

  13. L. A. Karlovitz, On Non-Expansive Mapping, Proc. Amer. Math. Soc, 55 (1976) 321-325.

  14. S. Kasahara, On Some Generalizations of the Banach Contraction Theorem, Publ. RISM Kyoto Univ, 12 (1976) 427-437.

  15. Dozo E. Lami, Multivalued Non-Expansive Mappings and Opial's Condition, Proc. Amer. Math. Soc, 38 (1973) 286-292.

  16. Basha S. Sadiq, Extensions of Banach's Contraction Principle, Numerical Functional Analysis and Optimization, 31 (5) (2010) 569-576.

  17. Raj V. Sankar, A Best Proximity Point Theorem for Weakly Contractive Non-self-mappings, Nonlinear Anal, 74 (2011) 4804-4808.

  18. N. Shahzad, S. Sadiq Basha, R. Jeyaraj, Common Best Proximity Points: Global Optimal Solutions, Journal of Optimization Theory and Applications, 148 (1) (2011) 69-78.