Some results in metric modular spaces

Document Type : Research Paper


1 Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

2 Department of Mathematics, West Tehran Branch, Islamic Azad University, Tehran , Iran

3 Department of Mathematics, Faculty of Basic Science, Univercity of Bojnord, P. O. Box 1339, Bojnord, Iran


A metric modular on a set $X$ is a function $w : (0,\infty)\times X\times X\longrightarrow  [0,\infty]$ written as $(\lambda,x,y)\mapsto w_{\lambda}(x,y)$ satisfying, for all $x, y, z\in X$, the following three properties: $x = y$  if and only if  $w_{\lambda}(x, y) = 0$ for all $\lambda>0$; $w_{\lambda}(x, y) = w_{\lambda}(y, x)$ for all $\lambda>0$; $w_{\lambda+\mu}(x, y) \leq w_{\lambda}(x, z) + w_{\mu}(y, z)$ for all $\lambda, \mu>0$. In this paper we define a Hausdorff topology on  metric modular spaces and prove  some known results of metric spaces including  Baire's theorem and  Uniform limit theorem for metric modular spaces.


[1] H. Abobakr and R.A. Ryan, Modular Metric Spaces, Irish Math. Soc. Bull. 80 (2017), 35–44.
[2] V. Chistyakov, Metric modulars and their application, Dokl. Akad. Nauk. 406 (2006), no. 2, 165–168.
[3] V. Chistyakov,Modular metric spaces. I. Basic concepts, Nonlinear Anal. 72 (2010), no. 1, 1–14.
[4] H. Hosseinzadeh and V. Parvaneh, Meir-Keeler type contractive mappings in modular and partial modular metric
spaces, Asian-European J. Math. 13 (2020), no. 05, 2050087.
[5] M.A. Khamsi, A convexity property in modular function spaces, Math. Japon. 44 (1996), no. 2, 269–279.
[6] W.M. Kozlowski, Modular function spaces, Monographs and Text- books in Pure and Applied Mathematics, vol.
122, Marcel Dekker, Inc., New York, 1988.
[7] M. Krbec, Modular interpolation spaces, I. Z. Anal. Anwend. 1 (1982), 25–40.
[8] W.A.J. Luxemburg, Banach function spaces, Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands,
[9] L. Maligranda, Orlicz spaces and interpolation. Semin. Math., 5, Universidade Estadual de Campinas, Departamento de Matematica, Campinas, 1989.
[10] S. Mazur and W. Orlicz, On some classes of linear spaces, Studia Math. 17 (1958), 97–119.
[11] J. Musielak and W. Orlicz, On modular spaces, Studia Math. 18 (1959), 49–65.
[12] J. Musielak and W. Orlicz, Some remarks on modular spaces, Bull. Acad. Polon. Sci. Sr. Math. Astron. Phys. 7
(1959), 661–668.
[13] H. Nakano, Modulared Semi-Ordered Linear Spaces, Tokyo Math. Book Ser., 1, Maruzen Co., Tokyo, 1950.
[14] H. Nakano, Topology and Linear Topological Spaces, Tokyo Math. Book Ser., 3, Maruzen Co., Tokyo, 1951.
[15] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co., Ltd., Tokyo, 1950.
[16] W. Orlicz, Collected Papers, vols. I, II. PWN, Warszawa, 1988.
[17] C. Park, J.M. Rasias, A. Bodaghi and S.O. Kim, Approximate homomorphisms from ternary semigroups to
modular spaces, RACSAM 113 (2019), 2175–2188.
Volume 13, Issue 2
July 2022
Pages 983-988
  • Receive Date: 24 December 2021
  • Revise Date: 06 March 2022
  • Accept Date: 13 March 2022