# Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces

- YeolJe Cho
^{1}, - MadjidEshaghi Gordji
^{2}Email author and - Somaye Zolfaghari
^{3}

**2010**:403101

https://doi.org/10.1155/2010/403101

© Yeol Je Cho et al. 2010

**Received: **29 July 2010

**Accepted: **31 August 2010

**Published: **7 September 2010

## Abstract

## 1. Introduction

In 1940, the stability problem of functional equations originated from a question given by Ulam [1], that is, the stability of group homomorphisms.

Let be a group and be a metric group with the metric . For any , does there exist , such that, if a mapping satisfies 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.

In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces.

Moreover, if is continuous in for any fixed then is linear.

In 1978, Rassias [3] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. In 1991, Gajda [4] answered the question for the case
, which was raised by Rassias. This new concept is known as the *Hyers-Ulam-Rassias stability of functional equations* (see [5–17]).

*quadratic functional equation*. In particular, every solution of the quadratic equation (1.3) is said to be a

*quadratic mapping*. It is well known that a mapping between real vector spaces and is quadratic if and only if there exits a unique symmetric biadditive mapping such that for all (see [5, 18]). The biadditive mapping is given by

Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof [19] for a mapping , where is a normed space and is a Banach space. Cholewa [20] noticed that the theorem of Skof is still true if the relevant domain is replaced by an abelian group. In [21], Czerwik proved the Hyers-Ulam-Rassias stability of (1.3) and Grabiec [22] generalized these results mentioned above.

where
is a mapping from a real vector space
into a real vector space
and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.5). The function
satisfies the functional equation (1.5)
which is thus called a *cubic functional equation*. Every solution of the cubic functional equation is said to be a *cubic function*. Also, Jun and Kim proved that a function
between real vector spaces
and
is a solution of (1.5) if and only if there exits a unique function
such that
for all
and
is symmetric for each fixed one variable and is additive for fixed two variables.

Definition 1.1 (see [27]).

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

(a) is commutative and associative;

Typical examples of continuous -norms are , and (the Lukasiewicz -norm).

Definition 1.2 (see [28]).

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
satisfying the following conditions:

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

Definition 1.3.

(1)A sequence
in
is said to be *convergent* to a point
if, for any
and
, there exists a positive integer
such that
for all
.

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

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

Theorem 1.4 (see [27]).

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

The stability of different functional equations in fuzzy normed spaces and random normed spaces has been studied in [13, 31–42].

where on random normed spaces. It is easy to see that the mapping is a solution of the functional equation (1.10). In Section 2, we investigate the general solution of functional equation (1.10) when is a mapping between vector spaces and, in Section 3, we establish the stability of the functional equation (1.10) in RN-spaces.

## 2. General Solutions

Before proceeding to the proof of Theorem 2.3 which is the main result in this section, we need the following two lemmas.

Lemma 2.1.

If an even function with satisfies (1.10), then is quadratic.

Proof.

which shows that is quadratic. This completes the proof.

Lemma 2.2.

If an odd function satisfies (1.10), then is cubic.

Proof.

Therefore, is a cubic function. This completes the proof.

Theorem 2.3.

A function with satisfies (1.10) for all if and only if there exist functions and such that for all where the function is symmetric for each fixed one variable and is additive for fixed two variables and is symmetric biadditive.

Proof.

It is clear that for all and it is easy to show that the functions and satisfy (1.10). Hence, by Lemmas 2.1 and 2.2, we know that the functions and are quadratic and cubic, respectively. Thus there exist a symmetric biadditive function such that for all and the function such that for all where the function is symmetric for each fixed one variable and is additive for fixed two variables. Therefore, we get for all

Conversely, let for all where the function is symmetric for each fixed one variable and is additive for fixed two variables and is biadditive. By a simple computation, one can show that the functions and satisfy the functional equation (1.10). Thus the function satisfies (1.10). This completes the proof.

## 3. Stability Problems

Theorem 3.1.

Proof.

Now, we show that is a quadratic mapping. In fact, replacing with and , respectively, in (3.3) and then taking the limit as , we find that satisfies (1.10) for all Therefore, the mapping is quadratic.

To prove (3.4) taking the limit as in (3.12), we get (3.4).

for all and . Therefore, by letting in inequality (3.15), we find that . This completes the proof.

Theorem 3.2.

Proof.

Now, we show that is a cubic mapping. In fact, replacing with and in (3.15), respectively, and then taking the limit , we find that satisfies (1.10) for all Therefore, the mapping is cubic.

Letting the limit in (3.32), we get inequality (3.18) by (3.26).

for all and . By letting in inequality (3.35), we know that . This completes the proof.

Theorem 3.3.

Proof.

Obviously, (3.38) follows from (3.40) and (3.42). This completes the proof.

## Declarations

### Acknowledgment

This paper was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

## Authors’ Affiliations

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