Probabilistic Normed Spaces Research Articles

Convex sets in probabilistic normed spaces

In this paper we obtain some results on convexity in a probabilistic normed space. We also investigate the concept of CSN-closedness and CSN-compactness in a probabilistic normed space and generalize the corresponding results of normed spaces.

Fixed point theorems in generating spaces of quasi-norm family and applications

Some new concepts of generating spaces of quasi-norm family are introduced and their linear topological structures are studied. These spaces are not necessarily locally convex. By virtue of some properties in these spaces, several Schauder-type fixed point theorems are proved, which include the corresponding theorems in locally convex spaces as their special cases. As applications, some new fixed point theorems in Menger probabilistic normed spaces and fuzzy normed spaces are obtained.

D-boundedness andD-compactness in finite dimensional probabilistic normed spaces

In this paper, we prove that in a finite dimensional probabilistic normed space, every two probabilistic norms are equivalent and we study the notion ofD-compactness and D-boundedness in probabilistic normed spaces.

Fixed and periodic points in the probabilistic normed and metric spaces

Abstract In this paper, a fixed point theorem is proved, i.e., if A is a C -contraction in the Menger space ( S , F ) and E ⊂ S be such that A ( E ) ¯ is compact, then A has a fixed point. In addition, under the same condition, the existence of a periodic point of A is proved. Finally, a fixed point theorem in probabilistic normed spaces is proved.

Topological degree theory and fixed point theorems in fuzzy normed space

In this paper, the Leray–Schauder topological degree theory is developed in a fuzzy normed space. Since the linear topology on this fuzzy normed space is not necessarily locally convex, and since each Menger probabilistic normed space can be considered as a special fuzzy normed space, the degree theory in this paper is different from the degree theory in locally convex linear topological space presented by Nagumo (Amer. J. Math. 73 (1951) 497–511), and it also is an extension of the degree theory in Menger probabilistic normed space studied by Zhang and Chen (Appl. Math. Mech. 10(6) (1989) 477–486). Applying this degree theory, some fixed point theorems for operators are given in fuzzy normed spaces, and some former corresponding results are extended and improved.

Separated Presheaves of Normed Spaces and -valued Norms

This paper lays down the axioms of Ω-valued normed spaces and shows their invariance under the so-called tilde-construction. Important examples are given by separated presheaves of normed spaces and by probabilistic normed spaces. Finally, there exists a natural Ω-valued topologization of Ω-valued normed spaces which makes it possible to study continuous linear operators between them.

On nonlinear equations for fuzzy mappings in probabilistic normed spaces

In this paper, we introduce the concept of probabilistic Ψ-contractor couple, which simplifies and weakens that of probabilistic Φ- g-contractor given by Cho et al. (Fuzzy Sets and Systems 110 (2000) 115–122). By using this concept, we study the existence and iteration convergence of solutions for set-valued nonlinear operator equations and nonlinear equations of fuzzy mappings in Menger PN-spaces. The results presented in this paper improve and extend the corresponding results of Cho et al. in a sense.

A Characterization for an L(μ, K)-Topological Module to Admit Enough Canonical Module Homomorphisms

Let K be the scalar field of all real or complex numbers, let (Ω,A,μ) be a σ-finite measure space, and let L(μ,K) be the algebra of the μ-equivalence classes of all K-valued μ-measurable functions defined on (Ω,A,μ). L(μ,K) is a topological algebra over K when endowed with the topology of convergence locally in measure; topological modules over this topological algebra L(μ,K) (briefly, L(μ,K)-topological modules) are an extensive class of topological modules, which arise naturally in the course of the study of the theory of probabilistic normed spaces. The purpose of this paper is to show that an arbitrary regular L(μ,K)-topological module admits enough canonical module homomorphisms if and only if all of its quasi-free submodules of finite rank are complemented in the sense of topological modules.

Nonlinear equations for fuzzy mappings in probabilistic normed spaces

In this paper, we introduce the concept of probabilistic contractor couple in probabilistic normed spaces and discuss the existence of solution for nonlinear equations of fuzzy mappings and the convergence of sequences generated by the algorithms in Menger probabilistic normed spaces.

Probabilistic norm of operators and resonance theorems

In this paper, a simplified definition of boundedness of the sets in probabilistic normed linear space was introduced. By means of the probabilistic norm of linear operators, the linear operator theory on probabilistic normed linear space was further studied. On probabilistic normed linear operator space, some resonance theorems dealing with probabilistic bounded sets, probabilistic semi-bounded sets, and probabilistic non-unbounded sets are obtained.

A Study of Boundedness in Probabilistic Normed Spaces

Probabilistic normed spaces have been redefined by Alsina, Schweizer, and Sklar. We give a detailed analysis of various boundedness notions for linear operators between such spaces and we study the relationship among them and also with the notion of continuity.

Nonlinear semigroups and differential inclusions in probabilistic normed spaces

The purpose of this paper is to introduce and study the semi-groups of nonlinear contractions in probabilistic normed spaces and to establish the Crandall-Liggett's exponential formula for some kind of accretive mappings in probabilistic normed spaces. As applications, these results are utilized to study the Cauchy problem for a kind of differential inclusions with accretive mappings in probabilistic normed spaces.

Differential equations on closed subsets of a probabilistic normed space

This paper is concerned with the problem of existence of solutions to the initial value problemu′(t) = A(t,u(t)), u(a) = z in a probabilistic normed space, whereA : [a, b) × D → E is continuous,D is a closed subset of a probabilistic normed spaceE, andz ∃ D. With a dissipative type condition onA, we estabilish sufficient conditions for this initial value problem to have a solution.

Bounded linear transformations between probabilistic normed vector spaces

We introduce a property of Q-boundedness on linear transformations between probabilistic T-normed vector spaces, where Q and T are lower semicontinuous triangular norms on the unit interval. We introduce a property of metric Q-continuity on functions between probabilistic T-metric spaces. We show that Q-boundedness and Q-continuity coincide on the domain of definition of the former property. We show that the already existing fuzzy topological theories for probabilistic T-metric spaces correspond either to T m-continuity or to Min-continuity. By investigating the Q-continuity of vector space operations in probabilistic T-normed spaces, we find that the condition Q ⩽ T is a useful one. This indicates that the class of probabilistic T-metric spaces may be the largest possible setting for a rich theory of metric T-continuity; a fuzzy topological accommodation of which is, at present, available only when T = T m and when T = Min.

Completion of probabilistic normed spaces

We prove that every probabilistic normed space, either according to the original definition given by erstnev, or according to the recent one introduced by Alsina, Schweizer and Sklar, has a completion.

On the definition of a probabilistic normed space

In this paper we give a new definition of a probabilistic normed space. This definition, which is based on a characterization of normed spaces by means of a betweenness relation, includes the earlier definition of A. N. Serstnev as a special case and leads naturally to the definition of the principal class of probabilistic normed spaces, the Menger spaces.

On the existence and uniqueness problems of solutions for set‐valued and single‐valued nonlinear operator equations in probabilistic normed spaces

In this paper, we introduce the concept of more general probabilistic contractors in probabilistic normed spaces and show the existence and uniqueness of solutions for set‐valued and single‐valued nonlinear operator equations in Menger probabilistic normed spaces.

On the definition of a probabilistic normed space

On the definition of a probabilistic normed space

Probabilistic contractor couple and solutions for a system of nonlinear equations in non-archimedean menger probabilistic normed spaces

The Purpose of this paper is to introduce the concept of probabilistic contractor couple in non-Archimedean probabilistic normed spaces and to study the existence and uniqueness of solutions for a system of nonlinear operator equations with probabilistic contractor couples in non-Archimedean probabilistic normed spaces. The results presented in this paper improve and extend the corresponding results in [1–5].

On the existence of solutions for equations with accretive mappings in probabilistic normed spaces

The purpose of this paper is to introduce the concept of accretive mapping in probabilistic normed space (PN space, in short) and to study the existence problem of solutions for equations with accretive mappings in PN space and some existence theorems are shown.