## Journal of Applied Mathematics

### On Fuzzy Modular Spaces

#### Abstract

The concept of fuzzy modular space is first proposed in this paper. Afterwards, a Hausdorff topology induced by a $\beta$-homogeneous fuzzy modular is defined and some related topological properties are also examined. And then, several theorems on $\mu$-completeness of the fuzzy modular space are given. Finally, the well-known Baire’s theorem and uniform limit theorem are extended to fuzzy modular spaces.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 576237, 8 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394807903

Digital Object Identifier
doi:10.1155/2013/576237

Mathematical Reviews number (MathSciNet)
MR3035184

Zentralblatt MATH identifier
1266.46058

#### Citation

Shen, Yonghong; Chen, Wei. On Fuzzy Modular Spaces. J. Appl. Math. 2013 (2013), Article ID 576237, 8 pages. doi:10.1155/2013/576237. https://projecteuclid.org/euclid.jam/1394807903

#### References

• H. Nakano, “Modular Semi-Ordered Spaces,” Tokoyo, Japan, 1959.
• J. Musielak and W. Orlicz, “On modular spaces,” Studia Mathematica, vol. 18, pp. 49–65, 1959.
• W. M. Kozłowski, “Notes on modular function spaces–-I,” Commentationes Mathematicae, vol. 28, no. 1, pp. 87–100, 1988.
• W. M. Kozłowski, “Notes on modular function spaces–-II,” Commentationes Mathematicae, vol. 28, no. 1, pp. 101–116, 1988.
• W. M. Kozłowski and G. Lewicki, “Analyticity and polynomial approximation in modular function spaces,” Journal of Approximation Theory, vol. 58, no. 1, pp. 15–35, 1989.
• S. J. Kilmer, W. M. Kozłowski, and G. Lewicki, “Best approximants in modular function spaces,” Journal of Approximation Theory, vol. 63, no. 3, pp. 338–367, 1990.
• M. A. Khamsi, W. M. Kozłowski, and S. Reich, “Fixed point theory in modular function spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 14, no. 11, pp. 935–953, 1990.
• T. Dominguez Benavides, M. A. Khamsi, and S. Samadi, “Uniformly Lipschitzian mappings in modular function spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 46, pp. 267–278, 2001.
• N. Hussain, M. A. Khamsi, and A. Latif, “Banach operator pairs and common fixed points in modular function spaces,” Fixed Point Theory and Applications, vol. 2011, article 75, 2011.
• M. A. Japón, “Some geometric properties in modular spaces and application to fixed point theory,” Journal of Mathematical Analysis and Applications, vol. 295, no. 2, pp. 576–594, 2004.
• M. A. Khamsi and W. M. Kozlowski, “On asymptotic pointwise contractions in modular function spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 9, pp. 2957–2967, 2010.
• M. A. Khamsi and W. M. Kozlowski, “On asymptotic pointwise nonexpansive mappings in modular function spaces,” Journal of Mathematical Analysis and Applications, vol. 380, no. 2, pp. 697–708, 2011.
• C. Mongkolkeha and P. Kumam, “Fixed point theorems for generalized asymptotic pointwise $\rho$-contraction mappings involving orbits in modular function spaces,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1285–1290, 2012.
• K. Nourouzi, “Probabilistic modular spaces,” in Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, 2007.
• K. Nourouzi, “Baire's theorem in probabilistic modular spaces,” in Proceedings of the World Congress on Engineering (WCE '08), vol. 2, pp. 916–917, 2008.
• K. Fallahi and K. Nourouzi, “Probabilistic modular spaces and linear operators,” Acta Applicandae Mathematicae, vol. 105, no. 2, pp. 123–140, 2009.
• A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395–399, 1994.
• B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–334, 1960.
• P. Klement and R. Mesiar, “Triangular norms,” Tatra Mountains Mathematical Publications, vol. 13, pp. 169–193, 1997.
• H. Dutta, I. H. Jebril, B. S. Reddy, and S. Ravikumar, “A generalization of modular sequence spaces by Cesaro mean of order one,” Revista Notas De Matematica, vol. 7, no. 1, pp. 1–13, 2011.
• H. Dutta and F. Başar, “A generalization of Orlicz sequence spaces by Cesàro mean of order one,” Acta Mathematica Universitatis Comenianae, vol. 80, no. 2, pp. 185–200, 2011.
• V. Karakaya and H. Dutta, “On some vector valued generalized difference modular sequence spaces,” Filomat, vol. 25, no. 3, pp. 15–27, 2011.