- Research Article
- Open Access

# Quadratic-Quartic Functional Equations in RN-Spaces

- M. Eshaghi Gordji
^{1}, - M. Bavand Savadkouhi
^{1}and - Choonkil Park
^{2}Email author

**2009**:868423

https://doi.org/10.1155/2009/868423

© M. Eshaghi Gordji et al. 2009

**Received:**20 July 2009**Accepted:**3 November 2009**Published:**1 December 2009

## Abstract

We obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary -norms .

## Keywords

- Banach Space
- Vector Space
- General Solution
- Abelian Group
- Functional Equation

## 1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] in concerning the stability of group homomorphisms. Let be a group and let be a metric group with the metric Given , does there exist a such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all In other words, under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that

for all and some Then there exists a unique additive mapping such that

for all Moreover, if is continuous in for each fixed then is -linear. In 1978, Rassias [3] provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. In Gajda [4] answered the question for the case , which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [5–12]). The functional equation

is related to a symmetric biadditive mapping. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation (1.3) is said to be a quadratic mapping. It is well known that a mapping between real vector spaces is quadratic if and only if there exits a unique symmetric biadditive mapping such that for all (see [5, 13]). The biadditive mapping is given by

The Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof for mappings , where is a normed space and is a Banach space (see [14]). Cholewa [15] noticed that the theorem of Skof is still true if relevant domain is replaced an abelian group. In [16], Czerwik proved the Hyers-Ulam-Rassias stability of the functional equation (1.3). Grabiec [17] has generalized the results mentioned above.

In [18], Park and Bae considered the following quartic functional equation

In fact, they proved that a mapping between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric multiadditive mapping such that for all . It is easy to show that the function satisfies the functional equation (1.5), which is called a quartic functional equation (see also [19]). In addition, Kim [20] has obtained the Hyers-Ulam-Rassias stability for a mixed type of quartic and quadratic functional equation.

The Hyers-Ulam-Rassias stability of different functional equations in random normed and fuzzy normed spaces has been recently studied in [21–26]. It should be noticed that in all these papers the triangle inequality is expressed by using the strongest triangular norm .

The aim of this paper is to investigate the stability of the additive-quadratic functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary continuous -norms.

In this sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [22, 23, 27–29]. Throughout this paper, is the space of distribution functions, that is, the space of all mappings such that is left-continuous and nondecreasing on and . Also, is a subset of consisting of all functions for which , where denotes the left limit of the function at the point , that is, . The space is partially ordered by the usual point-wise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the distribution function given by

Definition 1.1 (see [28]).

A mapping is a continuous triangular norm (briefly, a continuous -norm) if satisfies the following conditions:

(a) is commutative and associative;

(b) is continuous;

(c) for all ;

(d) whenever and for all .

Typical examples of continuous -norms are , and (the Lukasiewicz -norm). Recall (see [30, 31]) that if is a -norm and is a given sequence of numbers in , then is defined recurrently by and for . is defined as It is known [31] that for the Lukasiewicz -norm, the following implication holds:

Definition 1.2 (see [29]).

A *random normed space* (briefly, RN-space) is a triple
, where
is a vector space,
is a continuous
-norm, and
is a mapping from
into
such that the following conditions hold:

(RN1) for all if and only if ;

(RN2) for all , ;

(RN3) for all and

for all and is the minimum -norm. This space is called the induced random normed space.

Definition 1.3.

Let be an RN-space.

A sequence
in
is said to be *convergent* to
in
if, for every
and
, there exists a positive integer
such that
whenever
.

A sequence
in
is called a *Cauchy sequence* if, for every
and
, there exists a positive integer
such that
whenever
.

An RN-space
is said to be *complete* if and only if every Cauchy sequence in
is convergent to a point in
.

Theorem 1.4 (see [28]).

If is an RN-space and is a sequence such that , then almost everywhere.

Recently, Gordji et al. establish the stability of cubic, quadratic and additive-quadratic functional equations in RN-spaces (see [32, 33]).

In this paper, we deal with the following functional equation:

on RN-spaces. It is easy to see that the function is a solution of (1.9).

In Section 2, we investigate the general solution of the functional equation (1.9) when is a mapping between vector spaces and in Section 3, we establish the stability of the functional equation (1.9) in RN-spaces.

## 2. General Solution

We need the following lemma for solution of (1.9). Throughout this section, and are vector spaces.

Lemma 2.1.

If a mapping satisfies (1.9) for all then is quadratic-quartic.

Proof.

We show that the mappings defined by and defined by are quadratic and quartic, respectively.

Letting in (1.9), we have . Putting in (1.9), we get . Thus the mapping is even. Replacing by in (1.9), we get

for all . Therefore, the mapping is quadratic.

To prove that is quartic, we have to show that

for all .

Therefore, the mapping is quartic. This completes the proof of the lemma.

Theorem 2.2.

for all .

Proof.

for all

Therefore, there exist a unique symmetric multiadditive mapping and a unique symmetric bi-additive mapping such that and for all [5, 18]. So

for all The proof of the converse is obvious.

## 3. Stability

Throughout this section, assume that is a real linear space and is a complete RN-space.

Theorem 3.1.

for all and all

Proof.

Since the right-hand side of the inequality (3.17) tends to as and tend to infinity, the sequence is a Cauchy sequence. Thus we may define for all .

Now we show that is a quadratic mapping. Replacing with and in (3.1), respectively, we get

Taking the limit as , we find that satisfies (1.9) for all . By Lemma 2.1, the mapping is quadratic.

Letting the limit as in (3.16), we get (3.3) by (3.10).

Finally, to prove the uniqueness of the quadratic mapping subject to (3.3), let us assume that there exists another quadratic mapping which satisfies (3.3). Since for all and all from (3.3), it follows that

for all and all . Letting in (3.19), we conclude that , as desired.

Theorem 3.2.

for all and all

Proof.

Since the right-hand side of (3.36) tends to as and tend to infinity, the sequence is a Cauchy sequence. Thus we may define for all .

Now we show that is a quartic mapping. Replacing with and in (3.20), respectively, we get

Taking the limit as , we find that satisfies (1.9) for all . By Lemma 2.1 we get that the mapping is quartic.

Letting the limit as in (3.35), we get (3.22) by (3.29).

Finally, to prove the uniqueness of the quartic mapping subject to let us assume that there exists a quartic mapping which satisfies (3.22). Since and for all and all from (3.22), it follows that

for all and all . Letting in (3.38), we get that , as desired.

Theorem 3.3.

for all and all

Proof.

for all and all . Hence we obtain (3.41) by letting and for all The uniqueness property of and is trivial.

## Declarations

### Acknowledgment

C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).

## Authors’ Affiliations

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