Menger Probabilistic Normed Spaces Research Articles

Stability of preserving lattice cubic functional equation in Menger probabilistic normed Riesz spaces

The aim of this paper is to investigate Hyers–Ulam–Rassias stability of preserving lattice functional equation in various spaces. First, we prove stability of generalized preserving lattice functional equation in Banach lattices. Next, we show stability of preserving lattice cubic functional equation in Menger probabilistic normed Riesz spaces.

Some fixed point theorems in generating spaces of quasi-metric family

The aim of this paper is to introduce the concepts of compatible mappings and compatible mappings of type $(R)$ in non-Archimedean Menger probabilistic normed spaces and to study the existence problems of common fixed points for compatible mappings of type $(R)$, also, we give an applications by using the main theorems.

Fróchet differentiation between Menger probabilistic normed spaces

In this paper, we define and study Menger weakly and strongly P-convergent sequences and then Menger probabilistic continuity. We also display Frechet differentiation of nonlinear operators between Menger probabilistic normed spaces.

Solutions and Stability of Generalized Mixed Type QCA-Functional Equations in Random Normed Spaces

Abstract In this paper, we investigate the general solution and the generalized stability for the quartic, cubic and additive functional equation (briefly, QCA-functional equation) for any k∈ℤ-{0,±1} in Menger probabilistic normed spaces.

Fixed Point And Hyers-ulam-rassias Stability Of A Quadratic Functional Equation In Menger Probabilistic Normed Spaces

In this paper, using the fixed point alternative approach, we investigate the Hyers Ulam-Rassias stability of the quadratic functional equation \[f(x+y)+f(x-y)=2f(x)+2f(y)\] in Menger probabilistic normed spaces.

HYERS-ULAM-RASSIAS STABILITY OF FUNCTIONAL EQUATIONS IN MENGER PROBABILISTIC NORMED SPACES

We introduce the approximately quadratic functional equation in Menger probabilistic normed spaces. More precisely, we show under some suitable conditions that an approximately quadratic functional equation can be approximated by a quadratic function in above mentioned spaces. Also we consider the stability problem for approximately pexiderized functional equation in Menger probabilistic normed spaces.

Approximation of Mixed-Type Functional Equations in Menger PN-Spaces

LetXandYbe vector spaces. We show that a functionf:X→Ywithf(0)=0satisfiesΔf(x1,…,xn)=0for allx1,…,xn∈X, if and only if there exist functionsC:X×X×X→Y,B:X×X→YandA:X→Ysuch thatf(x)=C(x,x,x)+B(x,x)+A(x)for allx∈X, where the functionCis symmetric for each fixed one variable and is additive for fixed two variables,Bis symmetric bi-additive,Ais additive andΔf(x1,…,xn)=∑k=2n(∑i1=2k∑i2=i1+1k+1⋯∑in-k+1=in-k+1n)f(∑i=1,i≠i1,…,in-k+1nxi-∑r=1n-k+1xir)+f(∑i=1nxi)-2n-2∑i=2n(f(x1+xi)+f(x1-xi))+2n-1(n-2)f(x1)(n∈N,n≥3) for allx1,…,xn∈X. Furthermore, we solve the stability problem for a given functionfsatisfyingΔf(x1,…,xn)=0, in the Menger probabilistic normed spaces.

Generalization by Optimization in Menger Probabilistic Normed Spaces with Application on Portfolio in Insurance

The purpose of this paper is to show that the category of normed spaces can be embedded in the category of Menger probabilistic normed spaces, and that C( Ω ) is probabilistic normable, whereas it is not normable in the classical case, when Ω is an open subset of Rn. So, the spectrum of the category of Menger probabilistic normed spaces is broader than the category of classical normed spaces. Therefore, it can be a meaningful replacement in some model of security markets. As our model is suitably generalized, that fact can help us to adapt and improve within natural problems of finance, especially, in portfolio optimization of insurance.

On the stability of general cubic-quartic functional equations in Menger probabilistic normed spaces

In this paper, the stability of the general cubic-quartic functional equation, f(x+ky)+f(x−ky)=k2(f(x+y)+f(x−y))+2(1−k2)f(x)+[(k4−k2)/4](f(2y)−8f(y))+f̃(2x)−16f̃(x), where f̃(x)≔f(x)+f(−x) in the setting of Menger probabilistic normed spaces, is proved.

Stability of the Cubic Functional Equation in Menger Probabilistic Normed Spaces

Stability of the Cubic Functional Equation in Menger Probabilistic Normed Spaces

A NEW APPROACH FOR PROBABILISTICALLY BOUNDED SETS IN MENGER PROBABILISTIC NORMED SPACES

The main purpose of this paper is to introduce a new topological structure which is called fuzzifying topological structure determined by the method of co-tower topologies in Menger probabilistic normed spaces. We prove that the fuzzifying topological structure is compatible with the vector structure. A characterization of probabilistically bounded sets in Menger probabilistic normed spaces is obtained.

Resonance theorems for family of quasi-homogeneous operators in fuzzy normed linear spaces

In this paper, we establish some resonance theorems for families of point-wise bounded (semi-bounded and non-unbounded) quasi-homogeneous and | ∥ · ∥ | -subadditive operators in fuzzy normed linear spaces. As their consequences, we obtain the corresponding resonance theorems in classical normed spaces and Menger probabilistic normed spaces.

Level topologies of fuzzifying topological linear spaces and their applications

In this paper, some properties of level topologies of fuzzifying topological linear spaces are discussed. Then the characterizations of fuzzy boundedness and fuzzy precompactness are obtained. They may be considered as reply to the problem suggested by Qiu [Fuzzifying topological linear spaces, Fuzzy Sets and Systems 147 (2004) 249–272]. As applications, in Menger probabilistic normed spaces with respect to t-norm minimum, unified fuzzifying topological characterizations of probabilistic bounded, probabilistic semi-bounded, probabilistic unbounded sets, and probabilistic precompact, probabilistic semi-precompact, probabilistic unprecompact sets are proved, respectively.

On set-valued nonlinear equations in Menger probabilistic normed spaces

We introduce the concept of probabilistic ( ϕ , ψ ) -contractor couple, which allows us to consider set-valued nonlinear operator equations in a large class of probabilistic normed spaces. Our results complete those from Cho et al. [Nonlinear equations for fuzzy mappings in probabilistic normed spaces, Fuzzy Sets and Systems 110 (2000) 115–122] and Fang [On nonlinear equations for fuzzy mappings in probabilistic normed spaces, Fuzzy Sets and Systems 131 (2002) 357–364].

The topological structure of a certain Menger space

In this paper we will reconsider the topological structure of Menger probabilistic normed spaces (briefly PN-spaces) under the t-norm M. We will prove that this topology is compatible with the topology induced by a countable and separating family of semi-norms, and hence the well-known theorems of classical functional analysis (such as the principle of uniform boundedness, open mapping and closed graph theorems) are valid in this context also. We will meanwhile obtain a method by which one may construct easily a large class of PN-spaces. Finally, using this method, we see that a certain subspace of bounded linear operators between PN-spaces, i.e. the class of strongly bounded linear operators, has a natural PN structure.

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.

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.

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.

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].