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

We introduce two reasonable versions of approximately additive 
functions in a Serstnev probabilistic normed space endowed 
with Π𝑀 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.


Introduction and Preliminaries
Menger proposed transferring the probabilistic notions of quantum mechanic from physics to the underlying geometry.The theory of probabilistic normed spaces briefly, PN spaces is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations.The theory of probabilistic metric spaces introduced in 1942 by Menger 1 , as well as by the authors in 2, 3 .The notion of a probabilistic normed space was introduced by Šerstnev 4 .Alsina, Schweizer and Skalar gave a general definition of probabilistic normed space based on the definition of Menger for probabilistic metric spaces in 5, 6 .
It can be defined, in some way, the class of approximate solutions of the given functional equation one can ask whether each mapping from this class can be somehow approximated by an exact solution of the considered equation.Such a problem was formulated by Ulam in 1940 cf., 7 and solved the next year for the Cauchy functional equation by Hyers 8 .In 1950, Aoki 9 and in 1978, Rassias 10 proved a generalization of Hyers' theorem for additive and linear mappings, respectively.Theorem 1.1.Let f be an approximately additive mapping from a normed vector space E into a Banach space F, that is, f satisfies the inequality for all x, y ∈ E, where ε and p are constants with ε > 0 and 0 ≤ p < 1.Then the mapping L : E → F defined by L x lim n → ∞ 2 −n f 2 n x is the unique additive mapping which satisfies The result of Rassias has influenced the development of what is now called the Hyers-Ulam-Rassias stability theory for functional equations.In 1994, a generalization of Rassias' theorem was obtained by Gȃvrut ¸a 11 by replacing the bound ε x p y p by a general control function ϕ x, y .Several stability results have been recently obtained for various equations, also for mapping with more general domains and ranges see 12-18 .PN spaces were first defined by Šerstnev in 1962 see 4 .Their definition was generalized in 5 .We recall and apply the definition of probabilistic space briefly as given in 2 , together with the notation that will be needed see 2 .A distance distribution function briefly, a d.d.f. is a nondecreasing function F from R into 0, 1 that satisfies F 0 0 and F ∞ 1, and is left-continuous on 0, ∞ ; here as usual, R : 0, ∞ .The space of d.d.f.'s will be denoted by Δ ; and the set of all F in Δ for which lim t → ∞ − F t 1 by D .The space Δ is partially ordered by the usual pointwise ordering of functions, that is, F ≤ G if and only if F x ≤ G x for all x in R .For any a ≥ 0, ε a is the d.d.f.given by

1.3
The space Δ can be metrized in several ways 2 , but we will here adopt the Sibley metric d S .If F, G are d.f.'s and h is in 0, 1 , let F, G; h denote the condition: Then the Sibley metric d S is defined by In particular, under the usual pointwise ordering of functions, ε 0 is the maximal element of Δ .A triangle function is a binary operation on Δ , namely a function τ : Δ × Δ → Δ that is associative, commutative, nondecreasing in each place and has ε 0 as identity, that is, for all F, G and H in Δ : Moreover, a triangle function is continuous if it is continuous in the metric space Δ , d S .
Typical continuous triangle functions are Π T F, G x sup s t x T F s , G t , and Here T is a continuous t-norm, that is, a continuous binary operation on 0, 1 that is commutative, associative, nondecreasing in each variable and has 1 as identity; T * is a continuous t-conorm, namely a continuous binary operation on 0, 1 which is related to the continuous t-norm T through T * x, y 1 − T 1 − x, 1 − y .For example T x, y min x, y M x, y and T * x, y max x, y or T x, y π x, y xy and T * x, y π * x, y x y − xy.
Definition 1.2.A Probabilistic Normed space briefly, PN space is a quadruple X, ν, τ, τ * , where X is a real vector space, τ and τ * are continuous triangle functions with τ ≤ τ * and ν is a mapping the probabilistic norm from V into Δ , such that for every choice of p and q in V the following hold: N1 ν p ε 0 if and only if p θ θ is the null vector in X ; N2 ν −p ν p ; N3 ν p q ≥ τ ν p , ν q ; N4 ν p ≤ τ * ν λp , ν 1−λ p for every λ ∈ 0, 1 .
A PN space is called a Šerstnev space if it satisfies N1 , N3 and the following condition: holds for every α / 0 ∈ R and x > 0. When here is a continuous t-norm T such that τ Π T and τ * Π T * , the PN space X, ν, τ, τ * is called Menger PN space briefly, MPN space , and is denoted by X, ν, τ .Let X, ν, τ be an MPN space let {x n } be a sequence in X.Then {x n } is said to be convergent if there exists x ∈ X such that for all t > 0. In this case x is called the limit of {x n }.
The sequence x n in MPN space X, ν, τ is called Cauchy if for each ε > 0 and δ > 0, there exists some n 0 such that ν x n − x m δ > 1 − ε for all m, n ≥ n 0 .
Clearly, every convergent sequence in a MPN space is Cauchy.If each Cauchy sequence is convergent in a MPN space X, ν, τ , then X, ν, τ is called Menger probabilistic Banach space briefly, MPB space .
Recently, the stability of functional equations on PN spaces and MPN spaces have been investigated by some authors; see 19-23 and references therein.In this paper, we investigate the stability of additive functional equations on Šerstnev probabilistic normed space endowed with Π M triangle function.

Main Results
We begin our work with uniform version of the Hyers-Ulam-Rassias stability in a Šerstnev PN space in which we uniformly approximate a uniform approximate additive mapping.
Theorem 2.1.Let X be a linear space and Y, ν, Π M be a Šerstnev PB space.Let ϕ : X×X → 0, ∞ be a control function such that converges to zero.Let f : X → Y be a uniformly approximately additive function with respect to ϕ in the sense that uniformly on X × X.Then T x : lim n → ∞ 2 −n f 2 n x for any x ∈ X exists and defines an additive mapping T : X → Y such that if for some δ > 0, α > 0 Proof.Given ε > 0, by 2.2 , we can choose some t 0 such that for all x, y ∈ X and all t ≥ t 0 .Putting y x in 2.5 we get and by replacing x by 2 n x, we obtain By passing to a nonincreasing subsequence, if necessary, we may assume that {2 −n−1 ϕ 2 n x, 2 n y } is nonincreasing.
Thus for each n > m we have The convergence of 2.1 implies that for given δ > 0 there is n 0 ∈ N such that t 0 2 −n−1 ϕ 2 n x, 2 n x < δ ∀n ≥ n 0 .

2.9
Thus by 2.8 we deduce that for each n ≥ n 0 .Hence 2 −n f 2 n x is a Cauchy sequence in Y .Since Y, ν, Π M is complete, this sequence converges to some T x ∈ Y .Therefore, we can define a mapping T : X → Y by T x : lim n → ∞ 2 −n f 2 n x , namely, for each t > 0, and x ∈ X, ν T x − 2 −n f 2 n x t 1.