Research Article | Open Access
Generalized Hyers-Ulam-Rassias Theorem in Menger Probabilistic Normed Spaces
We introduce two reasonable versions of approximately additive functions in a Šerstnev probabilistic normed space endowed with 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.
1. 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 , as well as by the authors in [2, 3]. The notion of a probabilistic normed space was introduced by Šerstnev . 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 (cf., ) and solved the next year for the Cauchy functional equation by Hyers . In , Aoki  and in , Rassias  proved a generalization of Hyers' theorem for additive and linear mappings, respectively.
Theorem 1.1. Let be an approximately additive mapping from a normed vector space into a Banach space , that is, satisfies the inequality for all , where and are constants with and . Then the mapping defined by is the unique additive mapping which satisfies for all .
The result of Rassias has influenced the development of what is now called the Hyers-Ulam-Rassias stability theory for functional equations. In , a generalization of Rassias' theorem was obtained by Gvruţa  by replacing the bound by a general control function . 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 (see ). Their definition was generalized in . We recall and apply the definition of probabilistic space briefly as given in , together with the notation that will be needed (see ). A distance distribution function (briefly, a d.d.f.) is a nondecreasing function from into that satisfies and , and is left-continuous on ; here as usual, . The space of d.d.f.'s will be denoted by ; and the set of all in for which by . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all in . For any , is the d.d.f. given by
The space can be metrized in several ways , but we will here adopt the Sibley metric . If are d.f.'s and is in , let denote the condition:
Then the Sibley metric is defined by
In particular, under the usual pointwise ordering of functions, is the maximal element of . A triangle function is a binary operation on , namely a function that is associative, commutative, nondecreasing in each place and has as identity, that is, for all and in :(TF1),(TF2),(TF3),(TF4).
Moreover, a triangle function is continuous if it is continuous in the metric space .
Typical continuous triangle functions are and . Here is a continuous -norm, that is, a continuous binary operation on that is commutative, associative, nondecreasing in each variable and has 1 as identity; is a continuous -conorm, namely a continuous binary operation on which is related to the continuous -norm through For example and or and .
Definition 1.2. A Probabilistic Normed space (briefly, PN space) is a quadruple , where is a real vector space, and are continuous triangle functions with and is a mapping (the probabilistic norm) from into , such that for every choice of and in the following hold:(N1) if and only if ( is the null vector in );(N2);(N3);(N4) for every .
A PN space is called a Šerstnev space if it satisfies (N1), (N3) and the following condition:
holds for every and . When here is a continuous -norm such that and , the PN space is called Menger PN space (briefly, MPN space), and is denoted by
Let be an MPN space let be a sequence in . Then is said to be convergent if there exists such that
for all . In this case is called the limit of .
The sequence in MPN space is called Cauchy if for each and , there exists some such that for all .
Clearly, every convergent sequence in a MPN space is Cauchy. If each Cauchy sequence is convergent in a MPN space , then 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 triangle function.
2. 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 be a linear space and be a Šerstnev PB space. Let be a control function such that converges to zero. Let be a uniformly approximately additive function with respect to in the sense that uniformly on . Then for any exists and defines an additive mapping such that if for some then
Proof. Given , by (2.2), we can choose some such that
for all and all . Putting in (2.5) we get
and by replacing by , we obtain
By passing to a nonincreasing subsequence, if necessary, we may assume that is nonincreasing.
Thus for each we have The convergence of (2.1) implies that for given there is such that Thus by (2.8) we deduce that for each . Hence is a Cauchy sequence in . Since is complete, this sequence converges to some . Therefore, we can define a mapping by , namely, for each , and , Next, let . Temporarily fix and . Since converges to zero, there is some such that for all . Hence for each , we have but we have and by (2.5) and for large enough , we have Thus It follows that for all and by (N1), we have .
To end the proof, let, for some positive and , (2.3) hold. Let . Putting and in (2.10), we get for all positive integers . Thus for large enough , we have therefore
Corollary 2.2. Let be a linear space and be a Šerstnev PB space. Let be a control function satisfying (2.2). Let be a uniformly approximately additive function with respect to . Then there is a unique additive mapping such that uniformly on .
Proof. The existence of uniform limit (2.19) immediately follows from Theorem 2.1. It remains to prove the uniqueness assertion.
Let be another additive mapping satisfying (2.19). Fix . Given , by (2.19) for and , we can find some such that for all and . Fix some and find some integer such that Then we have It follows that for all . Thus for all .
Considering the control function for some , we obtain the following.
Corollary 2.3. Let be a normed linear space. Let be a Šerstnev PB space. Let and . Suppose that is a function such that uniformly on . Then there is a unique additive mapping such that uniformly on .
We are ready to give our nonuniform version of the Hyers-Ulam-Rassias theorem in Šerstnev PN spaces.
Theorem 2.4. Let be a linear space. Let be a Šerstnev MPN space. Let be a function such that for some , for all and . Let be a Šerstnev PB space and let be a -approximately additive mapping in the sense that for each and . Then there exists unique additive mapping such that where and .
Proof. Put in (2.26) to obtain Using (2.25) and induction on , one can verify that for all and . Replacing by in (2.28) and using (2.29) we get for all and . It follows from (2.30) that whence for all , and . So whence for all , and . Fix . By we deduce that is a Cauchy sequence in . Since is complete, this sequence converges to some point . It follows from (2.26) that whence We have By (2.37) and the fact that for all and , each term on the right-hand side tends to as . Hence By (N1), it means that Furthermore, let and . Using (2.34) with we obtain Hence The uniqueness of can be proved in a similar fashion as in the proof of Corollary 2.2.
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