On Fuzzy Modular Spaces

Abstract

The concept of fuzzy modular space is first proposed in this paper. Afterwards, a Hausdorff topology induced by aβ-homogeneous fuzzy modular is defined and some related topological properties are also examined. And then, several theorems onμ-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.

The stability of sextic functional equation in fuzzy modular spaces

“…With this property, it is suitable to call this * a strongest triangular norm. As is claimed by Shen and Chen [15], if V is a real vector space equipped with a β-homogeneous fuzzy modular µ and a strongest triangular norm * , then a µ-convergent sequence is µ-Cauchy. The authors also mentioned that if * is not the strongest one, such implementation is not always true.…”

“…Shen and Chen [15] also studied the topological properties of a fuzzy modular space with a special property that for every x ∈ V and a non-zero real λ, the equality µ(λx, t) = µ x, t |λ| β holds for some fixed β ∈ (0, 1]. If the fuzzy modular µ has this property, we shall say that it is β-homogeneous.…”

“…Now, suppose that µ is β-homogeneous for some β ∈ (0, 1]. According to Shen and Chen [15], the family B of all µ-balls forms a base for a first-countable Hausdorff topology, written as T µ . With the notion of the µ-balls, it is easy to see that a sequence (x n ) in V µ-converges (i.e.…”

“…After that, Shen and Chen [15] following the idea of probabilistic modular spaces and the definition of fuzzy metric spaces based on George and Veeramani 's sense [5], applied fuzzy concept to the classical notions of modular and modular spaces, and in 2013, Shen and Chen [15] presented the concept of a fuzzy modular space. After that, Kumam [9,10], Wongkum and et al [18] studied fixed points and some properties in modular or fuzzy modular spaces.…”

Some Analogies of the Banach Contraction Principle in Fuzzy Modular Spaces

“…Nourouzi [8] investigated the continuity and boundedness of linear operators defined between probabilistic modular spaces in the probabilistic sense. After that, Shen and Chen [9] following the idea of probabilistic modular space and the definition of fuzzy metric space in the sense of George and Veeramani [10], applied fuzzy concept to the classical notions of modular and modular spaces, and in 2013, Shen and Chen [9] introduced the concept of a fuzzy modular space.…”