Best approximation in quotient probabilistic normed space

In this article, we study the best approximation in quotient probabilistic normed space. We define the notion of quotient space of a probabilistic normed space, then prove some theorems of approximation in quotient space are extended to quotient probabilistic normed space.

Compactness and D-Boundedness in Menger’s 2-probabilistic normed spaces

The idea of convex sets and various related results in 2-Probabilistic normed spaces were established in [7]. In this paper, we obtain the concepts of convex series closedness, convex series compactness, boundedness and their interrelationships in Menger?s 2-probabilistic normed space. Finally, the idea of D-Boundedness in Menger?s 2-probabilistic normed spaces and Menger?s Generalized 2-Probabilistic Normed spaces are discussed.

On Asymptotically Double Lacunary Statistical Equivalent Sequences in Probabilistic Normed Space

In this paper we study the concept of asymptotically double lacunary statistical convergent sequences in probabilistic normed spaces and prove some basic properties.

On ideal convergence in probabilistic normed spaces

An interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.: I-Convergence, Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I-convergence, I*-convergence, I-limit points and I-cluster points in probabilistic normed space. We discuss the relationship between I-convergence and I*-convergence, i.e. we show that I*-convergence implies the I-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I*-convergence in probabilistic normed space.

$\bar \lambda $-statistically convergent double sequences in probabilistic normed spaces

The purpose of this paper is to introduce and study the concepts of double $\bar \lambda $-statistically convergent and double $\bar \lambda $-statistically Cauchy sequences in probabilistic normed space.

Generalized Hyers-Ulam-Rassias Theorem in Menger Probabilistic Normed Spaces

We introduce two reasonable versions of approximately additive functions in a Šerstnev probabilistic normed space endowed with triangle function. More precisely, we show under some suitable conditions that an approximately additive function can be approximated by an additive mapping in above mentioned spaces.

Some remarks on functions with values in probabilistic normed spaces

In this paper we consider an enlargement of the notion of the probabilistic normed space. For this new class of probabilistic normed spaces we give some topological properties. By using properties of the probabilistic norm we prove some differential and integral properties of functions with values into probabilistic normed spaces. As special cases, results for deterministic and random functions can be obtained.