PN Spaces Research Articles

New Classes of Generalized PN Spaces and Their Normability

In this paper, we obtain some properties of invertible operators; convex, balanced, absorbing sets; and $\mathcal {D}$ -boundedness in Serstnev spaces. We prove that some PN spaces (V,ν,τ,τ ∗), which are not Serstnev spaces, in which the triangle function τ ∗ is not Archimedean can be endowed with a structure of a topological vector space, and we give suitable example to illustrate this result. Also, we show that the topological spaces obtained in such a manner are normable under certain given conditions: some examples are given.

APPROXIMATELY QUINTIC AND SEXTIC MAPPINGS ON THE PROBABILISTIC NORMED SPACES

We prove the stability for the systems of quadratic-cubic and additive-quadrat functional equations with constant coefficients on the probabilistic normed spaces (briey PN spaces).

On ideal convergence of double sequences in probabilistic normed spaces

The notion of ideal convergence is a generalization of statistical convergence which has been intensively investigated in last few years. For an admissible ideal ∮ ⊂ ℔ × ℔, the aim of the present paper is to introduce the concepts of ∮-convergence and ∮⋆-convergence for double sequences on probabilistic normed spaces (PN spaces for short). We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide. We also define ∮ -Cauchy and ∮⋆-Cauchy double sequences on PN spaces and show that ∮ -convergent double sequences are ∮-Cauchy on these spaces. We establish example which shows that our method of convergence for double sequences on PN spaces is more general.

Some Nonlinear Problems in M-PN Spaces

Some new bounded conditions are constructed in Menger PN spaces,meanwhile,solutions of some nonlinear operator equations are obtained by virtue of topological degree.The main results obtained in this paper improve the recent ones.Furthermore,the new conclusions are applied to a special equation.

On α-Šerstnev probabilistic normed spaces

In this article, the condition α-Š is defined for α ∈]0, 1[∪]1, +∞[and several classes of α-Šerstnev PN spaces, the relationship between α-simple PN spaces and α-Šerstnev PN spaces and a study of PN spaces of linear operators which are α-Šerstnev PN spaces are given.2000 Mathematical Subject Classification: 54E70; 46S70.

Topological degree for 1-set-contractive fields in M-PN spaces and its applications

In this paper, we define the topological degree for 1-set-contractive fields in PN spaces. Based on this, we obtain some new fixed point theorems for 1-set-contractive operators. As an application, we study the existence of solutions for a kind of nonlinear Volterra integral equations in Z-M-PN space.

Strong Probabilistic Contractor Couple and Nonlinear Operator Equations in Menger PN Space

The purpose of the paper is to introduce the concept of strong probabilistic contractor couple in Menger PN spaces and to study the existence and uniqueness of solutions for a system of nonlinear operator equations with probabilistic contractor couples in Menger PN spaces non-Archimedean Menger PN spaces.

A study of boundedness in probabilistic normed spaces

It was shown in Lafuerza-Guillén, Rodríguez-Lallena and Sempi (1999) [8] that uniform boundedness in a Šerstnev PN space ( V , ν , τ , τ ∗ ) , (named boundedness in the present setting) of a subset A ⊂ V with respect to the strong topology is equivalent to the fact that the probabilistic radius R A of A is an element of D + . Here we extend the equivalence just mentioned to a larger class of PN spaces, namely those PN spaces that are topological vector spaces (briefly TV spaces), but are not Šerstnev PN spaces. We present a characterization of those PN spaces, whether they are TV spaces or not, in which the equivalence holds. Then, a characterization of the Archimedeanity of triangle functions τ ∗ of type τ T , L is given. This work is a partial solution to a problem of comparing the concepts of distributional boundedness ( D -bounded in short) and that of boundedness in the sense of associated strong topology.

Some properties of continuous linear operators in topological vector PN-spaces

The notion of a probabilistic metric space corresponds to thesituations when we do not know exactly the distance. Probabilistic Metric space was introduced by Karl Menger. Alsina, Schweizer and Sklar gave a general definition of probabilistic normed space based on the definition of Menger [1]. In this note we study the PN spaces which are topological vector spaces and the open mapping and closed graph Theorems in this spaces are proved.

Pointwise convergence and Ascoli theorems for nearness spaces

We first study subspaces and product spaces in the context of nearness spaces and prove that U-N spaces, C-N spaces, PN spaces and totally bounded nearness spaces are nearness hereditary; T-N spaces and compact nearness spaces are N-closed hereditary. We prove that N2 plus compact implies N-closed subsets. We prove that totally bounded, compact and N2 are productive. We generalize the concepts of neighborhood systems into the nearness spaces and prove that the nearness neighborhood systems are consistent with existing concepts of neighborhood systems in topological spaces, uniform spaces and proximity spaces respectively when considered in the respective sub-categories. We prove that a net of functions is convergent under the pointwise convergent nearness structure if and only if its cross-section at each point is convergent. We have also proved two Ascoli-Arzela type of theorems.

Invariant and semi-invariant probabilistic normed spaces

Probabilistic metric spaces were introduced by Karl Menger. Alsina, Schweizer and Sklar gave a general definition of probabilistic normed space based on the definition of Menger [1]. We introduce the concept of semi-invariance among the PN spaces. In this paper we will find a sufficient condition for some PN spaces to be semi-invariant. We will show that PN spaces are normal spaces. Urysohn’s lemma, and Tietze extension theorem for them are proved.

On the normability of generalized Šerstnev PN spaces

Probabilistic Normed spaces have been redefined by C. Alsina, B. Schweizer and A. Sklar. But even today, whether the generalized Šerstnev PN space is normable or not is still an open question. In this paper, through applying Kolmogorov's theorem, we will give several sufficient conditions, under which many generalized Šerstnev PN spaces are normable. As the application of our results, we will give two examples which are normable generalized Šerstnev PN space, but not Šerstnev space.

Quotient probabilistic normed spaces and completeness results

We introduce the concept of quotient in PN spaces and give some examples. We prove some theorems with regard to the completeness of a quotient.

A Short and Partial History of Probabilistic Normed Spaces

This contribution aims at presenting a survey of a portion of the theory of Probabilistic Normed spaces. No result will be proved, so that the reader is referred to the original sources for the proofs. The theory of PN spaces has many facets and touches on many branches of mathematics, for instance, geometry, functional analysis, topology, probability. This justifies the adjective “partial” that appears in the title. Therefore, it is perhaps better to declare from the start what one may expect from this survey. It is only natural to investigate which “classical”properties of normed spaces are preserved in the new setting. But it is probably more interesting to look for those properties that pertain to the new theory and which have no corresponding analogue in the classical theory. It must also be added that Probabilistic Normed spaces may be approached from different standpoints: they may be studied for their own sake, as a special subject in functional analysis or in topology that is worth investigating simply because it is there, or because it provides a tool to approach open problems or, again, to shed light on topics that one thought had been thoroughly investigated. In this, one thinks immediately of the possible applications in probability and statistics.