Intuitionistic Fuzzy Modular Spaces
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
Tayebe Lal Shateri
1
(Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.)
کلید واژه: Fuzzy set, Modular space, Fuzzy modular space, Intuitionistic fuzzy modular space.,
چکیده مقاله :
After the introduction of the concept of fuzzy sets by Zadeh, several researches were conducted on the generalizations of the notion of fuzzy sets. There are many viewpoints on the notion of metric space in fuzzy topology. One of the most important problems in fuzzy topology is obtaining an appropriate concept of fuzzy metric space. This problem has been investigated by many authors from different points of view. Atanassov gives the concept of the intuitionistic fuzzy set as a generalization of the fuzzy set. Park introduced the notion of intuitionistic fuzzy metric space as a natural generalization of fuzzy metric spaces due to George and Veeramani. This paper introduces the concept of intuitionistic fuzzy modular space. Afterward, a Hausdorff topology induced by a δ-homogeneous intuitionistic fuzzy modular is defined and some related topological properties are also examined. After giving the fundamental definitions and the necessary examples, we introduce the definitions of intuitionistic fuzzy boundedness, intuitionistic fuzzy compactness, and intuitionistic fuzzy convergence, and obtain several preservation properties and some characterizations concerning them. Also, we investigate the relationship between an intuitionistic fuzzy modular and an intuitionistic fuzzy metric. Finally, we prove some known results of metric spaces including Baires theorem and the Uniform limit theorem for intuitionistic fuzzy modular spaces.
After the introduction of the concept of fuzzy sets by Zadeh, several researches were conducted on the generalizations of the notion of fuzzy sets. There are many viewpoints on the notion of metric space in fuzzy topology. One of the most important problems in fuzzy topology is obtaining an appropriate concept of fuzzy metric space. This problem has been investigated by many authors from different points of view. Atanassov gives the concept of the intuitionistic fuzzy set as a generalization of the fuzzy set. Park introduced the notion of intuitionistic fuzzy metric space as a natural generalization of fuzzy metric spaces due to George and Veeramani. This paper introduces the concept of intuitionistic fuzzy modular space. Afterward, a Hausdorff topology induced by a δ-homogeneous intuitionistic fuzzy modular is defined and some related topological properties are also examined. After giving the fundamental definitions and the necessary examples, we introduce the definitions of intuitionistic fuzzy boundedness, intuitionistic fuzzy compactness, and intuitionistic fuzzy convergence, and obtain several preservation properties and some characterizations concerning them. Also, we investigate the relationship between an intuitionistic fuzzy modular and an intuitionistic fuzzy metric. Finally, we prove some known results of metric spaces including Baires theorem and the Uniform limit theorem for intuitionistic fuzzy modular spaces.
1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst, 20 (1986), 87-96.
[2] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets Syst, 88 (1997), 81-89.
[3] M. S. El Naschie, On the uncertainty of Cantorian geometry and two-slit experiment, Chaos, Soliton & Fractals, 9(3) (1998), 517-529.
[4] M. S. El Naschie, On the verifications of heterotic strings theory and ϵ(∞)theory, Chaos, Soliton & Fractals, 11(2) (2000), 2397-2407.
[5] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst, 27 (1988), 385-389.
[6] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst, 64 (1994), 395-399.
[7] S. J. Kilmer, W. M. Kozlowski and G. Lewicki, Best approximants in modular function spaces, J. Approximation Theory, 63(3) (1990), 338-367.
[8] W. M. Kozlowski, Notes on modular function spaces I, Comment. Math, 28(1) (1988), 87-100.
[9] W. M. Kozlowski, Notes on modular function spaces II, Comment. Math, 28(1) (1988), 101-116.
[10] W. M. Kozlowski and G. Lewicki, Analyticity and polynomial approximation in modular function spaces, J. Approximation Theory, 58(1) (1989), 15-35.
[11] I. Kramosil and J. Mich´ alek, Fuzzy metric and statistical metric spaces, Kybernetika, 11(5) (1975), 336-344.
[12] W. A. Luxemburg, Banach function spaces, Ph.D. thesis, Delft University of Technology, Delft, The Netherlands, (1959).
[13] J. Musielak and W. Orlicz, On modular Spaces, Studia Math, 18 (1959), 49-56.
[14] H. Nakano, Modulared Semi-Ordered Linear Spaces, Tokyo Math. Book Ser, 1, Maruzen Co. Tokyo, (1950).
[15] K. Nourouzi, Probabilistic modular spaces, Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, (2007).
[16] K. Nourouzi, Baire’s theorem in probabilistic modular spaces, Proceedings of the World Congress on Engineering (WCE 08), 2 (2008), 916-917.
[17] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 22 (2004), 1039-1046.
[18] B. Schweizer and A. Sklar, Statistical metric spaces, Pac. J. Math, 10 (1960), 314-334.
[19] Y. Shen and W. Chen, On fuzzy modular spaces, J. Appl. Math, Article ID 576237, 2013 (2013), 1-8.
[20] L. A. Zadeh, Fuzzy sets, Inf. Cont, 8 (1965), 338-353.
