Abstract
In this paper we construct a counter-example to give a negative answer to Suzuki et al. (Open Math 13(1):510–517, 2015, Problem 5.1) on the metrization of \(\nu \)-generalized metric spaces. We also prove a sufficient condition for a \(\nu \)-generalized metric space with \(\nu \ge 4\) having a metric with the same convergence of sequences.
Similar content being viewed by others
References
Abtahi, M., Kadelburg, Z., Radenovic, S.: Fixed points of Ciric-Matkowski-type contractions in \(\nu \)-generalized metric spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111(1), 57–64 (2017)
An, T.V., Dung, N.V., Kadelburg, Z., Radenovi, S.: Various generalizations of metric spaces and fixed point theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109, 175–198 (2015)
Arshad, M., Ameer, E., Karapinar, E.: Generalized contractions with triangular \(\alpha \)-orbital admissible mapping on Branciari metric spaces. J. Inequal. Appl. 2016(63), 1–21 (2016)
Aydi, H. Karapinar, E., Zhang, D.: On common fixed points in the context of Brianciari metric spaces. Results Math. 71(1–2), 73–92 (2017)
Branciari, A.: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ. Math. Debr. 57(1–2), 31–37 (2000)
Das, P., Dey, L.K.: Fixed point of contractive mappings in generalized metric spaces. Math. Slovaca 59(4), 499–504 (2009)
Dung, N.V., An, T.V., Hang, V.T.L.: Remarks on Frink’s metrization technique and applications. Fixed Point Theory, 1–22 (2017) (Accepted)
Engelking, R.: General Topology. Sigma Series in Pure Mathematics, vol. 6. Heldermann Verlag, Berlin (1988)
Erhan, I.M., Karapinar, E., Sekulic, T.: Fixed points of \((\psi,\phi )\) contractions on rectangular metric spaces. Fixed Point Theory Appl. 2012(138), 1–12 (2012)
Kadelburg, Z., Radenovic, S.: On generalized metric spaces: a survey. TWMS J. Pure Appl. Math. 5(1), 3–13 (2014)
Karapinar, E., Du, W.S., Kumam, P., Petruşel, A., Romaguera, S. (eds.): Existence and Uniqueness of Fixed Point in Various Abstract Spaces and Related Applications. Abstract and Applied Analysis. Hindawi Publishing Corporation, Cairo (2015)
Kikina, L., Kikina, K.: On fixed point of a Ljubomir Ciric quasi-contraction mapping in generalized metric spaces. Publ. Math. Debr. 83(3), 1–6 (2013)
Kirk, W., Shahzad, N.: Fixed Point Theory in Distance Spaces. Springer, Cham (2014)
Kumam, P., Dung, N.V.: Remarks on generalized metric spaces in the Branciari’s sense. Sarajevo J. Math. 10(2), 209–219 (2014)
Sarma, I.R., Rao, J.M., Rao, S.S.: Contractions over generalized metric spaces. J. Nonlinear Sci. Appl. 2(3), 180–182 (2009)
Suzuki, T.: Generalized metric spaces do not have the compatible topology. Abstract Appl. Anal. 2014, 1–5 (2014). Article ID 458098
Suzuki, T., Alamri, B., Kikkawa, M.: Only 3-generalized metric spaces have a compatible symmetric topology. Open Math. 13(1), 510–517 (2015)
Suzuki, T.: Completeness of 3-generalized metric spaces. Filomat 30(13), 3575–3585 (2016)
Acknowledgements
The authors sincerely thank anonymous referees for several helpful comments. The authors also thank members of The Dong Thap Group of Mathematical Analysis and its Applications for their discussions on the manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Van Dung, N., Le Hang, V.T. On the metrization problem of \(\nu \)-generalized metric spaces. RACSAM 112, 1295–1303 (2018). https://doi.org/10.1007/s13398-017-0425-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-017-0425-4