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

Open Access
Research Article

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]).

Let Open image in new window be Banach spaces and let Open image in new window be a mapping satisfying
for all Open image in new window . Then the limit
exists for all Open image in new window and Open image in new window is the unique additive mapping satisfying

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 , then there exists a unique linear mapping Open image in new window such that

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).

Let Open image in new window be a mapping from a normed vector space Open image in new window into a Banach space Open image in new window subject to the inequality
exists for all Open image in new window and Open image in new window is the unique additive mapping which satisfies

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.

Let Open image in new window be a real linear space and let Open image in new window be a mapping from Open image in new window to a Šerstnev MPB space Open image in new window such that Open image in new window . Suppose that Open image in new window is a mapping from Open image in new window into a Šerstnev MPN space Open image in new window such that
for all Open image in new window and positive real number Open image in new window . If Open image in new window for some real number Open image in new window with Open image in new window , then there is a unique additive mapping Open image in new window such that Open image in new window and

Proof.

Without loss of generality we may assume that Open image in new window . Replacing Open image in new window by Open image in new window in (2.1) we get
By our assumption, we have
Replacing Open image in new window by Open image in new window in (2.7) and applying (2.8), we get
Thus for each Open image in new window , we have
This shows that Open image in new window is a Cauchy sequence in Open image in new window . Since Open image in new window is complete, Open image in new window converges to some Open image in new window . Thus we can well define a mapping Open image in new window by
Moreover, if we put Open image in new window in (2.10), then we obtain
Next we will show that Open image in new window is additive. Let Open image in new window . Then we have
But we have
and by (2.1) we have

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

The uniqueness assertion can be proved by standard fashion. Let Open image in new window be another additive mapping, which satisfies the required inequality. Then for each Open image in new window and Open image in new window ,

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.

Let Open image in new window be a linear space. Let Open image in new window be a Šerstnev MPN space and let Open image in new window be a mapping with Open image in new window . Suppose that Open image in new window is a positive real number and Open image in new window is a fixed vector in a Šerstnev MPN space Open image in new window such that
for all Open image in new window and positive real number Open image in new window . Then there is a unique additive mapping Open image in new window such that

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.

Using Theorem 2.1 with Open image in new window , we deduce the existence of the required additive mapping Open image in new window . Let us put Open image in new window . Suppose that Open image in new window is continuous at a point Open image in new window . If Open image in new window were not continuous at a point, then there would be a sequence Open image in new window in Open image in new window such that
By passing to a subsequence if necessary, we may assume that
On the other hand
By (2.25) we have
and we have
Therefore for sufficiently large Open image in new window ,

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.

Let Open image in new window be an MPN space and let Open image in new window . A mapping Open image in new window is said to be Open image in new window -approximately Jensen-type if

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.

An MPN space Open image in new window is called definite if
holds. It is called pseudodefinite if for each Open image in new window the following condition holds:

Clearly a definite MPN space is pseudodefinite.

Theorem 3.3.

Let Open image in new window be a pseudodefinite Šerstnev MPN space. Suppose that for each Open image in new window and each Open image in new window -approximately Jensen-type Open image in new window there exist numbers Open image in new window , Open image in new window and an additive mapping Open image in new window such that

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

Proof.

Let Open image in new window be a Cauchy sequence in Open image in new window . Temporarily fix Open image in new window . There is an increasing sequence Open image in new window of positive integers such that Open image in new window and
for each Open image in new window . Thus Open image in new window is Open image in new window -approximately Jensen-type. By our assumption, there exist numbers Open image in new window , Open image in new window and an additive mapping Open image in new window such that
for sufficiently large Open image in new window . This means that

Definition 3.4.

Let Open image in new window be a Šerstnev MPN space and let Open image in new window be a mapping. Assume that, for each Open image in new window , there are numbers Open image in new window and Open image in new window such that

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.

Let Open image in new window be a Šerstnev MPN space such that for every approximately Jensen-type mapping Open image in new window there is an additive mapping Open image in new window such that

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

Proof.

Let Open image in new window be a Cauchy sequence in Open image in new window . Take a sequence Open image in new window in interval Open image in new window such that Open image in new window increasingly tends to Open image in new window . For each Open image in new window one can find some Open image in new window such that
Therefore Open image in new window is an approximately Jensen-type mapping. By our assumption, there is an additive mapping Open image in new window such that
This means that

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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Copyright information

© M. Eshaghi Gordji et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsSemnan UniversitySemnanIran
  2. 2.Department of MathematicsIran University of Science and TechnologyNarmakIran
  3. 3.Department of MathematicsHanyang UniversitySeoulSouth Korea

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