Probabilistic Modular Spaces and Linear Operators

Abstract

In this paper, we study the continuity and boundedness of linear operators defined between probabilistic modular spaces in the probabilistic sense.

Fixed Point Theorems for Contractive Mappings in Probabilistic Modular Spaces

“…Definition 2.2. [13] A probabilistic modular space (briefly, PM-space) is a pair (X, µ) in which X is a real vector space and µ is a mapping from X into ∆ (for x ∈ X, the function µ(x) is denoted by µ x , and µ x (t) is the value of µ x at t ∈ R) satisfying the following conditions:…”

“…Many authors have worked ever since on the fixed point theory in modular spaces, see [10], [11], [12]. In 2009, Nourouzi introduced in [13] the notion of probabilistic modular space according to Menger's probabilistic approach [6].…”

On Fuzzy Modular Spaces

“…In [16], Fallahi and Nourouzi proved that everyconvergent sequence is a -Cauchy sequence in thehomogeneous F-modular space. Here we will propose a similar result in F-modular space.…”

“…Nourouzi [15] extended the wellknown Baire's theorem to probabilistic modular spaces by using a special condition. Fallahi and Nourouzi [16] investigated the continuity and boundedness of linear operators defined between probabilistic modular spaces in the probabilistic sense.…”

Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras

“…In 2007, Nourouzi [12] presented probabilistic modular spaces related to the theory of modular spaces. Fallahi and Nourouzi [13,14] investigated the continuity and boundedness of linear operators defined between probabilistic modular spaces in the probabilistic sense. After then, 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 [16], applied fuzzy concept to the classical notions of modular spaces.…”