A new approach to generalize metric functions

Document Type : Research Paper

Authors

Department of Mathematics, Siksha-Bhavana, Visva-Bharati, Santiniketan-731235, Birbhum, West Bengal, India

Abstract

This article consists of a new concept of generalized metric space, called $\phi$-metric space which is developed by making a suitable modification in the `triangle inequality. The notion of $\phi$-metric generalizes the concept of some existing metrizable generalized spaces like S-metric, b-metric, etc.  It is shown that one can easily construct a $\phi$-metric from those generalized metric functions and the notion of convergence of a sequence on those generalized metric spaces are identical with the respective induced $\phi$-metric spaces. Moreover, $\phi$-metric space is metrizable and its properties are pretty similar to the metric functions. So $\phi$-metric functions substantially may play the role of metric functions. Also, the structure of $\phi$-metric space is studied and some well-known fixed point theorems are established.

Keywords

[1] M. Fr´echet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 1—72.
[2] F. Hausdorff, Grundzuge der Mengenlehre (Fundamentals of Set Theory), Leipzig, Von Veit, 1914.
[3] S Gahler, 2-metrische raume und ihre topologische struktur, Math. Nachr. 26 (1963), 115–148.
[4] B. C. Dhage, Generalized metric spaces mappings with fixed point, Bull. Calcutta Math. Soc. 84 (1992), 329—336.
[5] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006), 289—297.
[6] S. Sedghi, N. Shobe and A. Aliouche, A generalization of fixed point theorems in S-metric space, Math. Vesn. 64 (2012), 258–266.
[7] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Univ. Ostrav. 1 (1993), 5–11.
[8] S. Czrerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Semin Mat. Fis. Univ. Modena Reggio Emilia 46 (1998), 263–276.
[9] P. Chaipunya and P. kumam, On the distance between three arbitrary points, J. Funct. Spaces Appl. 2013 (2013) Article ID 194631, 7 pages.
[10] M.A. Khamsi and N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal. Theory Methods Appl. 73 (2010), 3123–3129.
[11] W. Kirk and N. Shahzad, Fixed point theory in distance spaces, Springer Cham, 2014.
[12] T.V. An, L.Q. Tuyen and N.V. Dung, Stone-type theorem on b-metric spaces and applications, Topol. Appl. 185-186 (2015), 50–64.
[13] R. Engelking, General Topology, Sigma Series in Pure Mathematics, Heldermann Verlag, Berlin 6 (1989).
[14] F. Siwiec, On defining a space by a weak-base, Pac. J. Math. 52 (1974), 233–245.
[15] S.P. Franklin, Spaces in which sequences suffice, Fundam. Math. 57 (1965), 107–115.
[16] R.H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175–186.
[17] M. Kir and H. Kiziltunc, On some well known fixed point theorems in b-metric spaces, Turk. J. Anal. Number Theory 1 (2013), 13–16.
[18] M.E. Gordji, M. Rameni, M. De La Sen and Y. Je Cho, On orthogonal sets and Banach fixed point theorem, Fixed Point Theory 18 (2017), 569–578.
[19] W.S. Dua and T.M. Rassias, Simultaneous generalizations of known fixed point theorems for a Meir-Keeler type condition with applications, Int. J. Nonlinear Anal. Appl. 11 (2020), 55–56.
[20] M. Ramezani and H. Baghani, Contractive gauge functions in strongly orthogonal metric spaces, Int. J. Nonlinear Anal. Appl. 8 (2017), 23—28.
[21] S. Khalehoghli, H. Rahimi and M.E. Gordji, Fixed point theorems in R-metric spaces with applications, AIMS Math. 5 (2020), 3125—3137.
[22] M. Paknazar, M. E. Gordji, M. De La Sen and S. M. Vaezpour, N-fixed point theorems for nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl. 2013 (2013), 111.
[23] R.P. Agarwal, M. Meehan and D. O’Regan, Fixed point theory and application, Cambridge University Press 2004.
[24] Z.E.D.D. Olia, M.E. Gordji and D.E. Bagha, Banach fixed point theorem on orthogonal cone metric spaces, FACTA Univ. (NIS) Ser. Math. Inf. 35 (2020), 1239–1250.
[25] A. Das and T. Bag, Some fixed point theorems in extended cone b-metric spaces, Commun. Math. Appl. 13 (2022), 647–659.
[26] A. Das and T. Bag, A generalization to parametric metric spaces, Int. J. Nonlinear Anal. Appl. In Press, 1–16, 2022. DOI: http://dx.doi.org/10.22075/ijnaa.2022.26832.3420.
[27] M.E. Gordji, M.R. Delavar and M. De La Sen, On ϕ-convex functions, J. Math. Inequal. 10 (2016), 173–183.
[28] L.N. Mishra, V. Dewangan, V.N. Mishra and S. Karateke, Best proximity points of admissible almost generalized weakly contractive mappings with rational expressions on b-metric spaces, J. Math. Comput. Sci. 22 (2021), 97–109.
[29] G. Abd-Elhamed, Fixed point results for (β, α)-implicit contractions in two generalized b-metric spaces, J. Nonlinear Sci. Appl. 14 (2021), 39–47.
[30] S. Rawat, RC. Dimri and A. Bartwal, F-Bipolar metric spaces and fixed point theorems with applications, J. Math. Computer Sci. 26 (2022), 184—195.
[31] Z. Mustaf and M.M.M. Jaradat, Some remarks concerning D∗ -metric spaces, J. Math. Comput. Sci. 22 (2021), 128—130.
[32] Y. Kowsar, M. Moshtaghi, E. Velloso, J.C. Bezdek, L. Kulik and C. Leckie, Shape-Sphere: A metric space for analysing time series by their shape, Inf. Sci. 582 (2022), 198–214.
[33] Z. Badreddine and H. Frankowska, Hamilton-Jacobi inequalities on a metric space, J. Differ. Equ. 271 (2021), 1058–1091.
Volume 14, Issue 3
March 2023
Pages 279-298
  • Receive Date: 14 June 2022
  • Revise Date: 02 September 2022
  • Accept Date: 18 September 2022