Some Topological Properties and Schauder Basis in Fuzzy Normed Linear Spaces

Finite dimensional fuzzy normed linear space

In this paper we will highlight some properties of fuzzy bounded sets in fuzzy normed linear spaces. More precisely, we will prove that any finite union or sum of fuzzy bounded sets is fuzzy bounded. We show that the closure of a fuzzy bounded set is also fuzzy bounded. We give a characterization for fuzzy bounded sets and finally we obtain that the Cartesian product of two fuzzy bounded sets is also fuzzy bounded.

In this paper, we use the denition of fuzzy normed spaces given by Bag and Samanta and the behaviors of solutions of the additive functional equation are described. The Hyers-Ulam stability problem of this equation is discussed and theorems concerning the Hyers-Ulam-Rassias stability of the equation are proved on fuzzy normed linear space.

In this article we define the notion of fuzzy normed linear space valued sequence space lFp (X), 1 ≤ p < ∞ in a fuzzy normed linear space X and discuss some of its properties like completeness, monotone, solidness, convergence free and symmetricity. Also we prove some inclusion results.

Fuzzy bounded linear operators

This paper deals with few spectral properties for elements from a fuzzy normed algebra. It is introduced the definition of the spectral radius of an element from the fuzzy normed linear space (X, N, Λ) and some basic properties of it are shown. More precisely, we will prove that almost all the properties of this spectrum from some particular case of locally convex spaces are also true for this setup.

In this paper, we aim to present some properties of the space of all weakly fuzzy bounded linear operators, with the Bag and Samanta's operator norm on fuzzy normed linear spaces. We introduced the natural linear injection of a fuzzy normed linear space X in to its second dual spaceX".

In this study, first, lacunary convergence of double sequences is introduced in fuzzy normed spaces, and basic definitions and theorems about lacunary convergence for double sequences are given in fuzzy normed spaces. Then, we introduce the concept of lacunary ideal convergence of double sequences in fuzzy normed spaces, and the relation between lacunary convergence and lacunary ideal convergence is investigated for double sequences in fuzzy normed spaces. Finally, in fuzzy normed spaces, we give the concept of limit point and cluster point for double sequences, and the theorems related to these concepts are given.

In this paper, the authors have developed and established the solution of the n-dimensional quadratic functional equation within the context of a vector space, specifically focusing on its properties and behavior in fuzzy normed spaces. They not only provide an explicit form of the solution but also investigate the stability of this solution under various perturbations. By extending the classical stability results to the setting of fuzzy normed spaces, the authors explore how uncertainties, represented by fuzzy norms, affect the solution’s stability.

In this paper, we introduce the definition of weakly convergent sequence in fuzzy normed spaces. We investigate relationship between convergent sequence and weakly convergent sequence in fuzzy normed spaces. We also study the dual spaces of a standard fuzzy normed space and $01$-fuzzy normed space.

In this paper, we introduce a new type of convergence for a sequence of function, namely, λ-statistically convergent sequences of functions in fuzzy 2-normed space, which is a natural generalization of convergence in fuzzy 2-normed space. In particular, we introduce the concepts of uniform λ-statistical convergence and pointwise λ-statistical convergence in the topology induced by fuzzy 2-normed spaces, and give some basic properties of these concepts.

In this paper, we delve into the exploration of ideal convergence within the framework of triple sequences on L -fuzzy normed spaces. Our primary focus is to establish a comprehensive characterization of ideal convergence for these triple sequences, particularly in relation to their convergence in the classical sense. Through rigorous analysis, we demonstrate that the notion of ideal convergence, as developed in this context, exhibits a weaker form of convergence compared to the traditional convergence criteria applied to triple sequences in L - fuzzy normed spaces. This weaker form of convergence, while more generalized, retains significant applicability and provides a broader understanding of the behavior of sequences within these structured spaces. The results presented herein offer new insights into the subtleties of sequence convergence in fuzzy normed spaces, paving the way for further advancements in this area of mathematical analysis.

We establish some stability results concerning the general cubic functional equation f(x + ky) - kf(x + y) + kf(x - y) - f(x - ky) = 2k(k2 - 1)f(y) for fixed k ℕ\{1} in the fuzzy normed spaces. More precisely, we show under some suitable conditions that an approximately cubic function can be approximated by a cubic mapping in a fuzzy sense and we establish that the existence of a solution for any approximately cubic mapping guarantees the completeness of the fuzzy normed spaces

The fixed point method for fuzzy stability of the Jensen functional equation

We obtain the general solution of the following functional equation f(kx1+x2+???+xk)+f(x1+kx2+???+xk)+???+f(x1+x2+???+kxk)+f(x1)+ f(x2)+???+ f(xk)= 2kf(x1+ x2+???+xk), k ? 2. We establish the Hyers-Ulam-Rassias stability of the above functional equation in the fuzzy normed spaces. More precisely, we show under suitable conditions that a fuzzy q-almost affine mapping can be approximated by an affine mapping. Further, we determine the stability of same functional equation by using fixed point alternative method in fuzzy normed spaces.