Abstract
In this work, we study existence and convergence of best proximity points of a cyclic contraction mapping in a complete \(\mathrm{{CAT}}_\mathrm{{p}}(0)\) metric space, with \(p \ge 2\). The case of coupled best proximity points of a pair of cyclic contraction mappings is also discussed. As an application, we provide sufficient conditions to obtain an extension of the Banach Contraction Principle for coupled fixed points.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bridson, M., Haefliger, A.: Metric spaces of non-positive curvature. Springer-Verlag, Berlin (1999)
Eldred, A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001–1006 (2006)
Eldred, A., Kirk, W.A., Veeramani, P.: Proximal normal structure and relatively nonexpansive mappings. Stud. Math. 3, 283–293 (2005)
Espinola, R., Fernández-León, A.: On best proximity points in metric and Banach spaces. Canad. J. Math. 63, 533–550 (2011)
Göhde, G.: Zumprinzip der kontraktiven Abbildung. Math. Nachr. 30, 251–258 (1965)
Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Series of Monographs and Textbooks in Pure and Applied Mathematics, 83. Dekker, New York (1984)
Khamsi, M.A., Shukri, S.: Generalized CAT(0) spaces. Bull. Belg. Math. Soc. Simon Stevin 24, 417–426 (2017)
Khan, A.R., Shukri, S.: Best proximity points in the Hilbert ball. J. Nonlinear Convex Anal. 17, 1083–1094 (2016)
Kirk, W.A., Reich, S., Veeramani, P.: Proximinal retracts and best proximity pair theorems. Numer. Func. Anal. Optim. 24, 851–862 (2003)
Kumam, W., Pakkaranang, N., Kumam, P., Cholamjiak, P.: Convergence analysis of modified Picard-S hybrid iterative algorithms for total asymptotically nonexpansive mappings in Hadamard spaces. Int. J. Comput. Math. 97, 175–188 (2018)
Pakkarananga, N., Kumama, P., Chod, Y.J., Saiparaa, P., Padcharoena, A., Khaofonga, C.: Strong convergence of modified viscosity implicit approximation methods for asymptotically nonexpansive mappings in complete CAT(0) spaces. J. Math. Computer Sci. 17, 345–354 (2017)
Raj, V.S.: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 74, 4804–4808 (2011)
Raj, V.S., Eldred, A.A.: A Characterization of Strictly Convex Spaces and Applications. J. Optim. Theory Appl. 160, 703–710 (2014)
Shukri, S., Khan, A.R.: Best proximity points in partially ordered metric spaces. Adv. Fixed Point Theory 8, 118–130 (2017)
Sintunavarat, W., Kumam, P.: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012, 93 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shukri, S. Existence and convergence of best proximity points in \(\mathrm{{CAT}}_\mathrm{{p}}(0)\) spaces. J. Fixed Point Theory Appl. 22, 48 (2020). https://doi.org/10.1007/s11784-020-00785-6
Published:
DOI: https://doi.org/10.1007/s11784-020-00785-6
Keywords
- Best proximity points
- coupled best proximity points
- fixed points
- coupled fixed points
- \(\mathrm{{CAT}}_\mathrm{{p}}(0)\) spaces
- contraction mappings
- cyclic contraction mappings