Abstract
In this paper, we introduce the \({\mathcal {F}}\)-metric space concept, which generalizes the metric space notion. We define a natural topology \(\tau _{{\mathcal {F}}}\) in such spaces and we study their topological properties. Moreover, we establish a new version of the Banach contraction principle in the setting of \({\mathcal {F}}\)-metric spaces. Several examples are presented to illustrate our study.
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References
An, T.V., Tuyen, L.Q., Dung, N.V.: Stone-type theorem on \(b\)-metric spaces and applications. Topol. Appl. 185–186, 50–64 (2015)
Branciari, A.: A fixed point theorem of Banach–Caccioppoli type on a class of generalized metric spaces. Publ. Math. Debr. 57, 31–37 (2000)
Czerwik, S.: Contraction mappings in \(b\)-metric spaces. Acta Math. Univ. Ostrav. 1(1), 5–11 (1993)
Fagin, R., Kumar, R., Sivakumar, D.: Comparing top \(k\) lists. SIAM J. Discret. Math. 17(1), 134–160 (2003)
Gähler, V.S.: 2-metrische Räume und ihre topologische struktur. Math. Nachr. 26, 115–118 (1963/1964)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)
Ha, K.S., Cho, Y.J., White, A.: Strictly convex and 2-convex 2-normed spaces. Math. Jpn. 33, 375–384 (1988)
Jleli, M., Samet, B.: A generalized metric space and related fixed point theorems. Fixed Point Theory Appl. 2015, 14 (2015)
Khamsi, M.A., Hussain, N.: KKM mappings in metric type spaces. Nonlinear Anal. 7(9), 3123–3129 (2010)
Kirk, W., Shahzad, N.: Fixed Point Theory in Distance Spaces. Springer, Cham (2014)
Matthews, S.G.: Partial metric topology. In: Proceedings of the 8th Summer Conference on General Topology and Applications. Annals of the New York Academy of Sciences, vol. 728, pp. 183–197 (1994)
Mustafa, Z., Sims, B.: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 7(2), 289–297 (2006)
Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project No RGP-1436-034.
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Jleli, M., Samet, B. On a new generalization of metric spaces. J. Fixed Point Theory Appl. 20, 128 (2018). https://doi.org/10.1007/s11784-018-0606-6
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DOI: https://doi.org/10.1007/s11784-018-0606-6