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 . 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.

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 [4]. Hyers gave a partial affirmative answer to the question of Ulam in the next year [5].

Theorem 1.1 (see [6]).

Let be Banach spaces and let be a mapping satisfying

(1.1)

for all . Then the limit

(1.2)

exists for all and is the unique additive mapping satisfying

(1.3)

for all .

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]).

Let and be two Banach spaces. Let and let . If a function satisfies the inequality

(1.4)

for all , then there exists a unique linear mapping such that

(1.5)

for all . Moreover, if is continuous in for each fixed , then the function is linear.

Theorem 1.2was later extended for all . 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 be a mapping from a normed vector space into a Banach space subject to the inequality

(1.6)

for all , where and are constants with and . Then the limit

(1.7)

exists for all and is the unique additive mapping which satisfies

(1.8)

for all .

The above mentioned stability involving a product of powers of norms is called Ulam-Gavruta-Rassias stability by various authors (see [12–21]). 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, 22–38].

Let be a mapping between linear spaces. The Jensen functional equation is

(1.9)

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

(1.10)

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

(1.11)

Then the Sibley metric is defined by

(1.12)

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 t-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 .

Note that for and .

Definition 1.4.

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:

(1.13)

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

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

(1.14)

for all . In this case is called the limit of .

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

Clearly, every convergent sequence in an MPN space is Cauchy. If each Cauchy sequence is convergent in an MPN space , then 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 be a real linear space and let be a mapping from to a Šerstnev MPB space such that . Suppose that is a mapping from into a Šerstnev MPN space such that

(2.1)

for all and positive real number . If for some real number with , then there is a unique additive mapping such that and

(2.2)

where

(2.3)

Proof.

Without loss of generality we may assume that . Replacing by in (2.1) we get

(2.4)

and replacing by and by in (2.1), we obtain

(2.5)

Thus

(2.6)

and so

(2.7)

By our assumption, we have

(2.8)

Replacing by in (2.7) and applying (2.8), we get

(2.9)

Thus for each , we have

(2.10)

Let and be given. Since

(2.11)

there is some such that . Since

(2.12)

there is some such that for all . Thus for all we have

(2.13)

This shows that is a Cauchy sequence in . Since is complete, converges to some . Thus we can well define a mapping by

(2.14)

Moreover, if we put in (2.10), then we obtain

(2.15)

Next we will show that is additive. Let . Then we have

(2.16)

But we have

(2.17)

and by (2.1) we have

(2.18)

which tends to as . Therefore

(2.19)

for each and . Thus satisfies the Jensen equation and so it is additive.

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

(2.20)

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

(2.21)

Therefore by the additivity of and ,

(2.22)

for all , and . Since ,

(2.23)

Hence the right-hand side of the above inequality tends to as . It follows that for all .

Remark 2.2.

One can prove a similar result for the case that . In this case, the additive mapping is defined by .

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 be a linear space. Let be a Šerstnev MPN space and let be a mapping with . Suppose that is a positive real number and is a fixed vector in a Šerstnev MPN space such that

(2.24)

for all and positive real number . Then there is a unique additive mapping such that

(2.25)

Moreover, if is a Šerstnev MPN space and is continuous at a point, then is continuous on .

Proof.

Using Theorem 2.1 with , we deduce the existence of the required additive mapping . Let us put . Suppose that is continuous at a point . If were not continuous at a point, then there would be a sequence in such that

(2.26)

By passing to a subsequence if necessary, we may assume that

(2.27)

and there are and such that

(2.28)

Since , there is such that . There is a positive integer such that . We have

(2.29)

On the other hand

(2.30)

By (2.25) we have

(2.31)

and we have

(2.32)

Therefore for sufficiently large ,

(2.33)

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 is complete if, for each whose Cauchy difference is bounded for all , there exists an additive mapping such that is bounded for all .

Definition 3.1.

Let be an MPN space and let . A mapping is said to be -approximately Jensen-type if

(3.1)

for some and all .

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

Definition 3.2.

An MPN space is called definite if

(3.2)

holds. It is called pseudodefinite if for each the following condition holds:

(3.3)

Clearly a definite MPN space is pseudodefinite.

Theorem 3.3.

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

(3.4)

for all . Then is a Šerstnev MPB-space.

Proof.

Let be a Cauchy sequence in . Temporarily fix . There is an increasing sequence of positive integers such that and

(3.5)

Put and define by . Then by (3.5) we have

(3.6)

for each . Thus is -approximately Jensen-type. By our assumption, there exist numbers , and an additive mapping such that

(3.7)

for all . Since is additive, . Hence

(3.8)

Let . Then there is some such that

(3.9)

for all . Take some such that and . It follows that . Let , then, for large enough ,

(3.10)

for each . By (3.3), . Put . Then for each and ,

(3.11)

for sufficiently large . This means that

(3.12)

Definition 3.4.

Let be a Šerstnev MPN space and let be a mapping. Assume that, for each , there are numbers and such that

(3.13)

for each . Then is said to be an approximately Jensen-type mapping.

Theorem 3.5.

Let be a Šerstnev MPN space such that for every approximately Jensen-type mapping there is an additive mapping such that

(3.14)

for each . Then is a Šerstnev MPB-space.

Proof.

Let be a Cauchy sequence in . Take a sequence in interval such that increasingly tends to . For each one can find some such that

(3.15)

for each . Let for each . Define by , for . If , take some such that and let . Then for each , we have

(3.16)

Therefore is an approximately Jensen-type mapping. By our assumption, there is an additive mapping such that

(3.17)

This means that

(3.18)

Hence the subsequence of the Cauchy sequence converges to . Hence also converges to .

4. Conclusions

In this work, we have analyzed a generalized Ulam-Hyers theorem in Šerstnev PN spaces endowed with . 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.