- Research Article
- Open Access

# Random Stability of an Additive-Quadratic-Quartic Functional Equation

- M Mohamadi
^{1}, - YJ Cho
^{2}, - C Park
^{3}Email author, - P Vetro
^{4}and - R Saadati
^{5}Email author

**2010**:754210

https://doi.org/10.1155/2010/754210

© M. Mohamadi et al. 2010

**Received:**7 December 2009**Accepted:**8 February 2010**Published:**14 February 2010

## Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-quartic functional equation in complete random normed spaces.

## Keywords

- Functional Equation
- Linear Space
- Quadratic Mapping
- Unique Fixed Point
- Normed Vector Space

## 1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call *generalized Hyers-Ulam stability* or as *Hyers-Ulam-Rassias stability* of functional equations. A generalization of the Th. M. Rassias theorem was obtained by G
vru
a [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.

The functional equation

is called a *quadratic functional equation*. In particular, every solution of the quadratic functional equation is said to be a *quadratic mapping*. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings
, where
is a normed space and
is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an Abelian group. Czerwik [8] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [4, 9–26]).

In [27], Lee et al. considered the following quartic functional equation

It is easy to show that the function
satisfies the functional equation (1.2), which is called a *quartic functional equation* and every solution of the quartic functional equation is said to be a *quartic mapping*.

Let
be a set. A function
is called a *generalized metric* on
if
satisfies

(1) if and only if ,

(2) for all

(3) for all .

We recall a fundamental result in fixed point theory.

for all nonnegative integers or there exists a positive integer such that

(1)

(2)the sequence converges to a fixed point of ,

(3) is the unique fixed point of in the set ,

(4) for all .

In 1996, Isac and Th. M. Rassias [30] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [31–36]).

## 2. Preliminaries

Definition 2.1 ([40]).

A mapping is a continuous triangular norm (briefly, a -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 ukasiewicz -norm).

Recall (see [42, 43]) that if is a -norm and is a given sequence of numbers in , is defined recurrently by and for . is defined as .

It is known ([43]) that for the ukasiewicz -norm the following implication holds:

Definition 2.2 ([41]).

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 .

Definition 2.3.

Let be a RN-space.

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

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

(3)A RN-space
is said to be *complete* if and only if every Cauchy sequence in
is convergent to a point in
. A complete RN-space is said to be random Banach space.

Theorem 2.4 ([40]).

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

The theory of random normed spaces (RN-spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The RN-spaces may also provide us the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physics. The generalized Hyers-Ulam stability of different functional equations in random normed spaces, RN-spaces and fuzzy normed spaces has been recently studied in, Alsina [44], Mirmostafaee, Mirzavaziri and Moslehian [33, 45–47], Mihe and Radu [38, 39, 48, 49], Mihe , Saadati and Vaezpour [50, 51], Baktash et al. [52] and Saadati et al. [53].

## 3. Generalized Hyers-Ulam Stability of the Functional Equation : An Odd Case

One can easily show that an odd mapping satisfies if and only if the odd mapping mapping is an additive mapping, that is,

One can easily show that an even mapping satisfies if and only if the even mapping is a quadratic-quartic mapping, that is,

It was shown in [54, Lemma ] that and are quartic and quadratic, respectively, and that .

For a given mapping , we define

for all .

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in complete RN-spaces: an odd case.

Theorem 3.1.

for all and all .

Proof.

for all and all .

where, as usual, . It is easy to show that is complete. (See the proof of Lemma in [38].)

for all and we prove that is a strictly contractive mapping with the Lipschitz constant .

for all .

- (1)

This implies that the inequality (3.7) holds.

for all , all and all .

for all , all and all .

for all and all . Thus the mapping is additive, as desired.

Corollary 3.2.

for all and all .

Proof.

for all . Then we can choose and we get the desired result.

Similarly, we can obtain the following. We will omit the proof.

Theorem 3.3.

for all and all .

Corollary 3.4.

for all and all .

Proof.

for all . Then we can choose and we get the desired result.

## 4. Generalized Hyers-Ulam Stability of the Functional Equation : An Even Case

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in random Banach spaces: an even case.

Theorem 4.1.

for all and all .

Proof.

for all and all .

for all and all .

for all and all .

Let be the generalized metric space defined in the proof of Theorem 3.1.

for all . It is easy to see that is a strictly contractive self-mapping on with the Lipschitz constant .

- (1)

This implies that the inequality (4.3) holds.

Proceeding as in the proof of Theorem 3.1, we obtain that the mapping satisfies .

for every . Since the mapping is quartic (see [54, Lemma ]), we get that the mapping is quartic.

Corollary 4.2.

for all and all .

Proof.

for all . Then we can choose and we get the desired result.

Similarly, we can obtain the following. We will omit the proof.

Theorem 4.3.

for all and all .

Corollary 4.4.

for all and all .

Proof.

for all . Then we can choose and we get the desired result.

Theorem 4.5.

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 4.1.

for all and all .

for all , is a strictly contractive self-mapping with the Lipschitz constant .

- (1)

This implies that the inequality (4.28) holds.

Proceeding as in the proof of Theorem 4.1, we obtain that the mapping satisfies .

for every . Since the mapping is quadratic (see [54, Lemma ]), we get that the mapping is quadratic.

Corollary 4.6.

for all and all .

Proof.

for all . Then we can choose and we get the desired result.

Similarly, we can obtain the following. We will omit the proof.

Theorem 4.7.

for all and all .

Corollary 4.8.

for all and all .

Proof.

for all . Then we can choose and we get the desired result.

## Declarations

### Acknowledgments

The authors would like to thank referees for giving useful suggestions for the improvement of this paper. The first author is supported by Islamic Azad University-Ayatollah Amoli Branch, Amol, Iran. The second author was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050) and the third author 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). The fourth author is supported by Università degli Studi di Palermo, R.S. ex 60%.

## Authors’ Affiliations

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