Abstract
In the setting of convex metric spaces, we introduce the two geometric notions of uniform convexity in every direction as well as sequential convexity. They are used to study a concept of proximal normal structure. We also consider the class of noncyclic relatively nonexpansive mappings and analyze the min-max property for such mappings. As an application of our main results we conclude with some best proximity pair theorems for noncyclic mappings.
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)
Brodskii, M.S., Milman, D.P.: On the center of a convex set. Dokl. Akad. Nauk USSR 59, 837–840 (1948). (in Russian)
Busemann, H.: Spaces with non-positive curvature. Acta Math. 80, 259–310 (1948)
Day, M.M., James, R.C., Swaminathan, S.: Normed linear spaces that are uniformly convex in every direction. Canad. J. Math. 23, 1051–1059 (1971)
Eldred, A.A., Kirk, W.A., Veeramani, P.: Proximal normal structure and relatively nonexpansive mappings. Studia Math. 171, 283–293 (2005)
Fernández-León, A., Nicolae, A.: Best proximity pair results for relatively nonexpansive mappings in geodesic spaces. Numer. Funct. Anal. Optim. 35, 1399–1418 (2014)
Gabeleh, M.: Minimal sets of noncyclic relatively nonexpansive mappings in convex metric spaces. Fixed Point Theory 16, 313–322 (2015)
Gabeleh, M.: Remarks on minimal sets for cyclic mappings in uniformly convex Banach spaces. Numer. Funct. Anal. Optim. 38, 360–375 (2017)
Gabeleh, M.: A characterization of proximal normal structure via proximal diametral sequences. J. Fixed Point Theory Appl. 19, 2909–2925 (2017)
Gabeleh, M., Olela, O.: Otafudu, Markov-Kakutani's theorem for best proximity pairs in Hadamard spaces. Indag. Math. 28, 680–693 (2017)
A. L. Garkavi, On the Chebyshev center of a set in a normed space, in: Investigations of Contemporary Problems in the Constructive Theory of Functions, Fizmatgiz. (Moscow, 1961), pp. 328–331 (in Russian).
W. A. Kirk, Geodesic geometry and fixed point theory, II, in: Proceedings of the International Conference on Fixed Point Theory and Applications, (Valencia, Spain, July 2003), Yokohama Publ. (Yokohama, 2004), pp. 113–142.
Kirk, W.A.: A fixed point theorem for mappings which do not increase distances. Amer. Math. Monthly 72, 1004–1006 (1965)
Kohlenbach, U.: Some logical metatheorems with applications in functional analysis. Trans. Amer. Math. Soc. 357, 89–128 (2005)
Künzi, H.-P.A., Yıldız, F.: Convexity structures in \(T_0\)-quasi-metric spaces. Topology Appl. 200, 2–18 (2016)
V. Sankar Raj, A best proximity point theorem for weakly contractive non-self-mappings, Nonlin. Anal., 74 (2011), 4804–4808
Shimizu, T., Takahashi, W.: Fixed points of multivalued mappings in certain convex metric spaces. Topol. Methods Nonlinear Anal. 8, 197–203 (1996)
Takahashi, W.: A convexity in metric space and nonexpansive mappings. Kodai Math. Sem. Rep. 22, 142–149 (1970)
Zizler, V.: On some rotundity and smoothness properties of Banach spaces. Dissertationes Mat. 87, 1–33 (1971)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was partially supported by a grant from IPM (No. 96470046).
The second author acknowledges partial support from the National Research foundation of South Africa under grant 114773.
Rights and permissions
About this article
Cite this article
Gabeleh, M., Künzi, HP.A. Min-max property in metric spaces with convex structure. Acta Math. Hungar. 157, 173–190 (2019). https://doi.org/10.1007/s10474-018-0857-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-018-0857-0
Key words and phrases
- proximal normal structure
- noncyclic relatively nonexpansive mapping
- uniformly in every direction convex metric space