Abstract
We introduce a concept of pointwise cyclic relatively nonexpansive mapping involving orbits to investigate the existence of best proximity points using a geometric property defined on a nonempty and convex pair of subsets of a Banach space X, called weak proximal normal structure. Examples are given to support our main conclusions. We also introduce a notion of proximal diametral sequence and establish a characterization of proximal normal structure and show that every nonempty and convex pair in uniformly convex in every direction Banach spaces has weak proximal normal structure. As an application, we give a new existence theorem for cyclic contractions in reflexive Banach spaces without strictly convexity condition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abkar, A., Gabeleh, M.: Best proximity points for asymptotic cyclic contraction mappings. Nonlinear Anal. 74, 7261–7268 (2011)
Abkar, A., Gabeleh, M.: Proximal quasi-normal structure and a best proximity point theorem. J. Nonlinear Convex Anal. 14, 653–659 (2013)
Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70, 3665–3671 (2009)
Brodskii, M.S., Milman, D.P.: On the center of a convex set. Dokl. Akad. Nauk. USSR 59, 837–840 (1948). (in Russian)
Day, M.M., James, R.C., Swaminathan, S.: Normed linear spaces that are uniformly convex in every directions. Can. J. Math. XXIII, 1051–1059 (1971)
Eldred, A.A., Kirk, W.A., Veeramani, P.: Proximal normal structure and relatively nonexpansive mappings. Stud. Math. 171, 283–293 (2005)
Eldred, A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001–1006 (2006)
Espínola, R.: A new approach to relatively nonexpansive mappings. Proc. Am. Math. Soc. 136, 1987–1996 (2008)
Emmanuele, G.: Asymptotic behavior of iterates of nonexpansive mappings in Banach spaces with Opial’s condition. Proc. Am. Math. Soc. 94, 103–109 (1985)
Fernández-León, A.: Existence and uniqueness of best proximity points in geodesic metric spaces. Nonlinear Anal. 73, 915–921 (2010)
Gabeleh, M.: Best proximity points and fixed point results for certain maps in Banach spaces. Numer. Funct. Anal. Optim. 36, 1013–1028 (2015)
Gabeleh, M., Shahzad, N.: Best proximity points, cyclic Kannan maps and geodesic metric spaces. J. Fixed Point Theory Appl. (2016). doi:10.1007/s11784-015-0272-x
Garkavi, A.L.: On the Cebysev center of a set in a normed space. Investigations of Contemporary Problems in the Constructive Theory of Functions, Moscow, pp. 328–331 (1961)
Garkavi, A.L.: The best possible net and the best possible cross-section of a set in a normed space. Izv. Akad. Nauk SSSR Ser. Mat. 26, 87–106. Am. Math. Soc. Transi. Ser. 2 39, 111–132 (1964)
Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics. Wiley, New York (2001)
Kirk, W.A.: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004–1006 (1965)
Markin, J., Shahzad, N.: Best proximity points for relatively U-continuous mappings in Banach and hyperconvex spaces. Abstr. Appl. Anal. 2013, 5 (2013)
Zizler, V.: On some rotundity and smoothness properties of Banach spaces. Diss. Mat. 87, 1–33 (1971)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was in part supported by a Grant from IPM (No. 94470047).
Rights and permissions
About this article
Cite this article
Gabeleh, M. A characterization of proximal normal structure via proximal diametral sequences. J. Fixed Point Theory Appl. 19, 2909–2925 (2017). https://doi.org/10.1007/s11784-017-0460-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-017-0460-y