Random Normed Spaces Research Articles

Stability of a pexiderial functional equation in random normed spaces

The concept of Hyers-Ulam-Rassias stability originated from Th.M. Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72:297–300, 1978. Recently, the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation $$f(x+y)+f(x-y)=2f(x)+2f(y)$$ proved in the earlier work. In this paper, using direct method we prove the generalized Hyers-Ulam stability of the following Pexiderial functional equation $$f(x+y)+f(x-y)=2g(x)+2g(y)$$ in random normed space.

On the stability of the additive Cauchy functional equation in random normed spaces

In this paper, we prove a stability result for the additive Cauchy functional equation in random normed spaces, related to the main theorem from the paper [D. Miheţ, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008) 567–572].

Solution and stability of mixed type functional equation in non-Archimedean random normed spaces

The generalized stability of the Euler-Lagrange quadratic mappings in the framework of non-Archimedean random normed spaces is proved. The interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, and the theory of functional equations is presented.

HYERS-ULAM-RASSIAS STABILITY OF THE APOLLONIUS TYPE QUADRATIC MAPPING IN RN-SPACES

Recently, in [5], Najati and Moradlou proved Hyers-Ulam-Rassias stability of the following quadratic mapping of Apollonius type \[Q(z - x) + Q(z - y) =\frac{ 1}{ 2}Q(x - y) + 2Q ( z -\frac{ x + y}{ 2})\] in non-Archimedean space. In this paper we establish Hyers-Ulam-Rassias stability of this functional equation in random normed spaces by direct method and fixed point method. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

Stability of a mixed type additive, quadratic and cubic functional equation in random normed spaces

In this paper, we obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary t-norms f(x+3y)+f(x?3y)= 9(f(x+y)+f(x?y))?16f(x).

On Random Topological Structures

We present some topics about random spaces. The main purpose of this paper is to study topological structure of random normed spaces and random functional analysis. These subjects are important to the study of nonlinear analysis in random normed spaces.

Nonlinear -Random Stability of an ACQ Functional Equation

We prove the generalized Hyers-Ulam stability of the following additive-cubic-quartic functional equation: in complete latticetic random normed spaces.

The stability of the quartic functional equation in various spaces

The purpose of this paper is first to introduce the notation of intuitionistic random normed spaces, and then by virtue of this notation to study the stability of a quartic functional equation in the setting of these spaces under arbitrary triangle norms. Then we prove the stability of above quartic functional equation in non-Archimedean random normed spaces. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, the theory of intuitionistic spaces and the theory of functional equations are also presented in the paper.

On statistical convergence in random 2-normed spaces

The idea of statistical convergence was introduced in [1] and [17] and since then several generalizations and applications of this concept have been investigated by various authors. Recently Karakus [7] and Gürdal and S. Pehlivan [6] studied statistical convergence in probabilistic normed space and 2-Banach space, respectively. In this paper we propose to study statistical convergence in random 2-normed space which seems to be a quite new and interesting idea.

A FIXED POINT APPROACH TO GENERALIZED STABILITY OF A MIXED TYPE FUNCTIONAL EQUATION IN RANDOM NORMED SPACES

In this note, by using the fixed point method, we prove the generalized stability for a mixed type functional equation in random normed spaces of which the general solution is either cubic or quadratic.

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

We achieve the general solution and the generalized stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary t-norms: , where for any fixed integer with .

STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS IN RANDOM NORMED SPACES

In this paper, we prove the generalized Hyers-Ulam sta- bility of the following quadratic functional equations in random normed spaces.

Approximately generalized additive functions in several variables

The goal of this paper is to investigate the solutionand stability in random normed spaces, in non--Archimedean spacesand also in $p$--Banach spaces and finally the stability using thealternative fixed point of generalized additive functions inseveral variables.

On the Stability of a General Mixed Additive-Cubic Functional Equation in Random Normed Spaces

We prove the generalized Hyers-Ulam stability of the following additive-cubic equation in the setting of random normed spaces.

Stability of a cubic functional equation in intuitionistic random normed spaces

In this paper, the stability of a cubic functional equation in the setting of intuitionistic random normed spaces is proved. We first introduce the notation of intuitionistic random normed spaces. Then, by virtue of this notation, we study the stability of a cubic functional equation in the setting of these spaces under arbitrary triangle norms. Furthermore, we present the interdisciplinary relation among the theory of random spaces, the theory of intuitionistic spaces, and the theory of functional equations.

Random Stability of an Additive-Quadratic-Quartic Functional Equation

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-quartic functional equation in complete random normed spaces.

The Stability of the Quartic Functional Equation in Random Normed Spaces

The main problem analyzed in this paper consists in showing that, under some conditions, every almost quartic mapping from a linear space to a random normed space under the Łukasiewicz t-norm can be suitably approximated by a quartic function, which is unique.

A Mazur–Ulam theorem for probabilistic normed spaces

A classical theorem of S. Mazur and S. Ulam asserts that any surjective isometry between two normed spaces is an affine mapping. D. Mushtari proved in 1968 the same result in the case of random normed spaces in the sense of A. Sherstnev. The aim of the present paper is to show that the result holds also for the probabilistic normed spaces as defined by C. Alsina, B. Schweizer and A. Sklar, Aequationes Math. 46 (1993), 91–98.

A Note to Paper “On the Stability of Cubic Mappings and Quartic Mappings in Random Normed Spaces” (Erratum)

Recently, Baktash et al. (2008) proved the stability of the cubic functional equation and the quartic functional equation in random normed spaces. In this note, we correct the proofs.

Fixed Points and Stability for Functional Equations in Probabilistic Metric and Random Normed Spaces

AbstractWe prove a general Ulam-Hyers stability theorem for a nonlinear equation in probabilistic metric spaces, which is then used to obtain stability properties for different kinds of functional equations (linear functional equations, generalized equation of the square root, spiral generalized gamma equations) in random normed spaces. As direct and natural consequences of our results, we obtain general stability properties for the corresponding functional equations in (deterministic) metric and normed spaces.