×

zbMATH — the first resource for mathematics

On fuzzy modular spaces. (English) Zbl 1266.46058
Summary: 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.
MSC:
46S40 Fuzzy functional analysis
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. Nakano, “Modular Semi-Ordered Spaces,” Tokoyo, Japan, 1959. · Zbl 0086.09104
[2] J. Musielak and W. Orlicz, “On modular spaces,” Studia Mathematica, vol. 18, pp. 49-65, 1959. · Zbl 0099.09202
[3] W. M. Kozłowski, “Notes on modular function spaces-I,” Commentationes Mathematicae, vol. 28, no. 1, pp. 87-100, 1988. · Zbl 0747.46023
[4] W. M. Kozłowski, “Notes on modular function spaces-II,” Commentationes Mathematicae, vol. 28, no. 1, pp. 101-116, 1988. · Zbl 0747.46022
[5] 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. · Zbl 0692.41010 · doi:10.1016/0021-9045(89)90004-X
[6] 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. · Zbl 0718.41049 · doi:10.1016/0021-9045(90)90126-B
[7] 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. · Zbl 0714.47040 · doi:10.1016/0362-546X(90)90111-S
[8] 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. · Zbl 1001.47039 · doi:10.1016/S0362-546X(00)00117-6
[9] 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. · Zbl 1285.47063 · doi:10.1186/1687-1812-2011-75
[10] 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. · Zbl 1062.46011 · doi:10.1016/j.jmaa.2004.02.047
[11] 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. · Zbl 1229.47079 · doi:10.1016/j.na.2010.06.061
[12] 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. · Zbl 1221.47093 · doi:10.1016/j.jmaa.2011.03.031
[13] 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. · Zbl 1251.47050 · doi:10.1016/j.aml.2011.11.027
[14] K. Nourouzi, “Probabilistic modular spaces,” in Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, 2007. · Zbl 1167.46013
[15] 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.
[16] K. Fallahi and K. Nourouzi, “Probabilistic modular spaces and linear operators,” Acta Applicandae Mathematicae, vol. 105, no. 2, pp. 123-140, 2009. · Zbl 1191.46008 · doi:10.1007/s10440-008-9267-6
[17] A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395-399, 1994. · Zbl 0843.54014 · doi:10.1016/0165-0114(94)90162-7
[18] B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313-334, 1960. · Zbl 0096.33203 · doi:10.2140/pjm.1960.10.673
[19] P. Klement and R. Mesiar, “Triangular norms,” Tatra Mountains Mathematical Publications, vol. 13, pp. 169-193, 1997. · Zbl 0915.04002
[20] 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.
[21] H. Dutta and F. Ba\csar, “A generalization of Orlicz sequence spaces by Cesàro mean of order one,” Acta Mathematica Universitatis Comenianae, vol. 80, no. 2, pp. 185-200, 2011. · Zbl 1265.46004
[22] V. Karakaya and H. Dutta, “On some vector valued generalized difference modular sequence spaces,” Filomat, vol. 25, no. 3, pp. 15-27, 2011. · Zbl 1265.46009 · doi:10.2298/FIL1103015K
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.