# Generalized Ulam-Hyers Stability of Jensen Functional Equation in Šerstnev PN Spaces

## Abstract

We establish a generalized Ulam-Hyers stability theorem in a Šerstnev probabilistic normed space (briefly, Šerstnev PN-space) endowed with Open image in new window . In particular, we introduce the notion of approximate Jensen mapping in PN-spaces and prove that if an approximate Jensen mapping in a Šerstnev PN-space is continuous at a point then we can approximate it by an everywhere continuous Jensen mapping. As a version of a theorem of Schwaiger, we also show that if every approximate Jensen type mapping from the natural numbers into a Šerstnev PN-space can be approximated by an additive mapping, then the norm of Šerstnev PN-space is complete.

## Keywords

Additive Mapping Unique Additive Mapping Probabilistic Normed Space Fuzzy Normed Space Random Normed Space## 1. Introduction and Preliminaries

Mengerproposed 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 notion of a probabilistic normed space was introduced by Šerstnev [1]. Alsina, Schweizer, and Skalar gave a general definition of probabilistic normed space based on the definition of Meneger for probabilistic metric spaces in [2, 3].

Ulam propounded the first stability problem in Open image in new window [4]. Hyers gave a partial affirmative answer to the question of Ulam in the next year [5].

Theorem 1.1 (see [6]).

for all Open image in new window .

Hyers' theorem was generalized by Aoki [7] for additive mappings and by Th. M. Rassias [8] for linear mappings by considering an unbounded Cauchy difference. For some historical remarks see [9].

Theorem 1.2 (see [10]).

for all Open image in new window . Moreover, if Open image in new window is continuous in Open image in new window for each fixed Open image in new window , then the function Open image in new window is linear.

Theorem 1.2was later extended for all Open image in new window . The stability phenomenon that was presented by Rassias is called the generalized Ulam-Hyers stability. In 1982, Rassias [11] gave a further generalization of the result of Hyers and proved the following theorem using weaker conditions controlled by a product of powers of norms.

Theorem 1.3 (25).

for all Open image in new window .

The above mentioned stability involving a product of powers of norms is called Ulam-Gavruta-Rassias stability by various authors (see [12, 13, 14, 15, 16, 17, 18, 19, 20, 21]). In the last two decades, several forms of mixed type functional equations and their Ulam-Hyers stability are dealt with in various spaces like fuzzy normed spaces, random normed spaces, quasi-Banach spaces, quasi-normed linear spaces, and Banach algebras by various authors like in [6, 9, 14, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].

Let Open image in new window be a mapping between linear spaces. The Jensen functional equation is

It is easy to see that f with Open image in new window satisfies the Jensen equation if and only if it is additive; compare for [39, Theorem Open image in new window ]. Stability of Jensen equation has been studied at first by Kominek [36] and then by several other mathematicians example, (see [10, 33, 40, 41, 42] and references therein).

PN spaces were first defined by Šerstnev in1963(see [1]). Their definition was generalized in [2]. We recall and apply the definition of probabilistic space briefly as given in [43], together with the notation that will be needed (see [43]). A distance distribution function (briefly, a d.d.f.) is a nondecreasing function Open image in new window from Open image in new window into Open image in new window that satisfies Open image in new window and Open image in new window , and is left-continuous on Open image in new window ; here as usual, Open image in new window . The space of d.d.f.'s will be denoted by Open image in new window and the set of all Open image in new window in Open image in new window for which Open image in new window by Open image in new window . The space Open image in new window is partially ordered by the usual pointwise ordering of functions, that is, Open image in new window if and only if Open image in new window for all Open image in new window in Open image in new window . For any Open image in new window , Open image in new window is the d.d.f. given by

The space Open image in new window can be metrized in several ways [43], but we shall here adopt the Sibley metric Open image in new window . If Open image in new window are d.f.'s and Open image in new window is in Open image in new window , let Open image in new window denote the condition

Then the Sibley metric Open image in new window is defined by

In particular, under the usual pointwise ordering of functions, Open image in new window is the maximal element of Open image in new window . A triangle function is a binary operation on Open image in new window , namely, a function Open image in new window that is associative, commutative, nondecreasing in each place, and has Open image in new window as identity, that is, for all Open image in new window and Open image in new window in Open image in new window :

(TF1) Open image in new window ,

(TF2) Open image in new window ,

(TF3) Open image in new window ,

(TF4) Open image in new window .

Moreover, a triangle function is *continuous* if it is continuous in the metric space Open image in new window .

Typical continuous triangle functions are Open image in new window and Open image in new window . Here Open image in new window is a continuous t-norm, that is, a continuous binary operation on Open image in new window that is commutative, associative, nondecreasing in each variable, and has 1 as identity; Open image in new window is a continuous Open image in new window -conorm, namely, a continuous binary operation on Open image in new window which is related to the continuous Open image in new window -norm Open image in new window through Open image in new window For example, Open image in new window and Open image in new window or Open image in new window and Open image in new window .

Note that Open image in new window for Open image in new window and Open image in new window .

Definition 1.4.

A *Probabilistic Normed space* (briefly, PN space) is a quadruple Open image in new window , where Open image in new window is a real vector space, Open image in new window and Open image in new window are continuous triangle functions with Open image in new window and Open image in new window is a mapping (the *probabilistic norm*) from Open image in new window into Open image in new window such that for every choice of Open image in new window and Open image in new window in Open image in new window the following hold:

(N1) Open image in new window if and only if Open image in new window ( Open image in new window is the null vector in Open image in new window ),

(N2) Open image in new window ,

(N3) Open image in new window ,

(N4) Open image in new window for every Open image in new window .

A PN space is called a Šerstnev space if it satisfies (N1), (N3) and the following condition:

holds for every Open image in new window and Open image in new window . When here is a continuous Open image in new window -norm Open image in new window such that Open image in new window and Open image in new window , the PN space Open image in new window is called Meneger PN space (briefly, MPN space), and is denoted by Open image in new window

Let Open image in new window be an MPN space and let Open image in new window be a sequence in Open image in new window . Then Open image in new window is said to be convergent if there exists Open image in new window such that

for all Open image in new window . In this case Open image in new window is called the limit of Open image in new window .

The sequence Open image in new window in MPN Space Open image in new window is called Cauchy if for each Open image in new window and Open image in new window there exist some Open image in new window such that Open image in new window for all Open image in new window .

Clearly, every convergent sequence in an MPN space is Cauchy. If each Cauchy sequence is convergent in an MPN space Open image in new window , then Open image in new window is called Meneger Probabilistic Banach space (briefly, MPB space).

## 2. Stability of Jensen Mapping in Šerstnev MPN Spaces

In this section, we provide a generalized Ulam-Hyers stability theorem in a Šerstnev MPN space.

Theorem 2.1.

Proof.

for each Open image in new window and Open image in new window . Thus Open image in new window satisfies the Jensen equation and so it is additive.

Next, we approximate the difference between Open image in new window and Open image in new window in the Šerstnev MPN space Open image in new window . For every Open image in new window and Open image in new window , by (2.15), for large enough Open image in new window , we have

Hence the right-hand side of the above inequality tends to Open image in new window as Open image in new window . It follows that Open image in new window for all Open image in new window .

Remark 2.2.

One can prove a similar result for the case that Open image in new window . In this case, the additive mapping Open image in new window is defined by Open image in new window .

Now we examine some conditions under which the additive mapping found in Theorem 2.1 is to be continuous. We use a known strategy of Hyers [5] (see also [44]).

Theorem 2.3.

Moreover, if Open image in new window is a Šerstnev MPN space and Open image in new window is continuous at a point, then Open image in new window is continuous on Open image in new window .

Proof.

which contradicts (2.29).

## 3. Completeness of Šerstnev MPN Spaces

This section contains two results concerning the completeness of a Šerstnev MPN space. Those are versions of a theorem of Schwaiger [45] stating that a normed space Open image in new window is complete if, for each Open image in new window whose Cauchy difference Open image in new window is bounded for all Open image in new window , there exists an additive mapping Open image in new window such that Open image in new window is bounded for all Open image in new window .

Definition 3.1.

for some Open image in new window and all Open image in new window .

In order to prove our next results, we need to put the following conditions on an MPN space.

Definition 3.2.

*pseudodefinite*if for each Open image in new window the following condition holds:

Clearly a definite MPN space is pseudodefinite.

Theorem 3.3.

for all Open image in new window . Then Open image in new window is a Šerstnev MPB-space.

Proof.

Definition 3.4.

for each Open image in new window . Then Open image in new window is said to be an approximately Jensen-type mapping.

Theorem 3.5.

for each Open image in new window . Then Open image in new window is a Šerstnev MPB-space.

Proof.

Hence the subsequence Open image in new window of the Cauchy sequence Open image in new window converges to Open image in new window . Hence Open image in new window also converges to Open image in new window .

## 4. Conclusions

In this work, we have analyzed a generalized Ulam-Hyers theorem in Šerstnev PN spaces endowed with Open image in new window . We have proved that if an approximate Jensen mapping in a Šerstnev PN space is continuous at a point then we can approximate it by an anywhere continuous Jensen mapping. Also, as a version of Schwaiger, we have showed that if every approximate Jensen-type mapping from natural numbers into a Šerstnev PN-space can be approximate by an additive mapping then the norm of Šerstnev PN-space is complete.

## Notes

### Acknowledgments

The authors are grateful to the referees for their helpful comments. The fourth author was supported by National Research Foundation of Korea (NRF-2009-0070788).

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