1. Introduction

In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems, containing the stability problem of homomorphisms as follows

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 is a homomorphism with for all ?

Hyers [2] proved the stability problem of additive mappings in Banach spaces. Rassias [3] provided a generalization of Hyers theorem which allows the Cauchy difference to be unbounded: let be a mapping from a normed vector space into a Banach space subject to the inequality

(1.1)

for all , where and are constants with and . The above inequality provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept. Gvruţa [4] provided a further generalization of Hyers-Ulam theorem. A square norm on an inner product space satisfies the important parallelogram equality:

(1.2)

The functional equation

(1.3)

is called the quadratic functional equation whose solution is said to be a quadratic mapping. A generalized stability problem for the quadratic functional equation was proved by Skof [5] for mappings , where is a normed space and is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [7] proved the generalized stability of the quadratic functional equation, and Park [8] proved the generalized stability of the quadratic functional equation in Banach modules over a -algebra.

Throughout this paper, let and be vector spaces.

Definition 1.1.

A mapping is called bi-quadratic if satisfies the system of the following equations:

(1.4)

When , the function given by is a solution of (1.4).

For a mapping , consider the functional equation:

(1.5)

Definition 1.2 (see [9, 10]).

Let be a real linear space. A quasinorm is real-valued function on satisfying the following

(i) for all and if and only if .

(ii) for all and all .

(iii)There is a constant such that for all .

It follows from the condition (iii) that

(1.6)

for all and all .

The pair is called a quasinormed space if is a quasinorm on . The smallest possible is called the modulus of concavity of . A quasi-Banach space is a complete quasinormed space. A quasinorm is called a -norm () if

(1.7)

for all . In this case, a quasi-Banach space is called a -Banach space.

Given a -norm, the formula gives us a translation invariant metric on . By the Aoki-Rolewicz theorem [10] (see also [9]), each quasinorm is equivalent to some -norm. Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms. In [11], Tabor has investigated a version of Hyers-Rassias-Gajda theorem (see also [3, 12]) in quasi-Banach spaces. Since then, the stability problems have been investigated by many authors (see [13–18]).

The authors [19] solved the solutions of (1.4) and (1.5) as follows.

Theorem A.

A mapping satisfies (1.4) if and only if there exist a multi-additive mapping such that and for all .

Theorem B.

A mapping satisfies (1.4) if and only if it satisfies (1.5).

In this paper, we investigate the generalized Hyers-Ulam stability of (1.4) and (1.5) in quasi-Banach spaces.

2. Stability of (1.4) and (1.5) in Quasi-normed Spaces

Throughout this section, assume that is a quasinormed space with quasinorm and that is a -Banach space with -norm . Let be the modulus of concavity of .

Let and be two functions such that

(2.1)

for all .

Let be two functions satisfying

(2.2)
(2.3)

for all .

Theorem 2.1.

Let be a mapping such that

(2.4)
(2.5)

and let and for all . Then there exist two bi-quadratic mappings such that

(2.6)
(2.7)

for all .

Proof.

Letting in (2.4), we get

(2.8)

for all . Thus we have

(2.9)

for all . Replacing by in the above inequality, we obtain

(2.10)

for all . Since is a -Banach space, for given integers , we see that

(2.11)

for all . By (2.2) and (2.11), the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Define by

(2.12)

for all . Putting and taking in (2.11), one can obtain the inequality (2.6). By (2.4) and (2.5), we get

(2.13)

for all and all . Letting in the above two inequalities and using (2.1), is bi-quadratic.

Next, setting in (2.5),

(2.14)

for all . By the same method as above, define by for all . By the same argument as above, is a bi-quadratic mapping satisfying (2.7).

Corollary 2.2.

Let be a mapping such that

(2.15)

and let and for all . Then there exist two bi-quadratic mappings such that

(2.16)

for all .

Proof.

In Theorem 2.1, putting and for all , we get the desired result.

From now on, let be a function such that

(2.17)
(2.18)

for all .

We will use the following lemma in order to prove Theorem 2.4.

Lemma 2.3 (see [20]).

Let and let be nonnegative real numbers. Then

(2.19)

Theorem 2.4.

Let be a mapping such that

(2.20)

and let and for all . Then there exists a unique bi-quadratic mapping such that

(2.21)

for all .

Proof.

Letting and in (2.20), we have

(2.22)

for all . Thus we obtain

(2.23)

for all and all . Replacing by in the above inequality, we see that

(2.24)

for all and all . By Lemma 2.3, for given integers , we get

(2.25)

for all . By (2.18) and (2.25), the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Define by

(2.26)

for all .

By (2.20), we have

(2.27)

for all and all . Letting and using (2.17), we see that satisfies (1.5). By Theorem B, we obtain that is bi-quadratic. Setting and taking in (2.25), one can obtain the inequality (2.21). If is another bi-quadratic mapping satisfying (2.21), we obtain

(2.28)

for all . Hence the mapping is the unique bi-quadratic mapping, as desired.

Corollary 2.5.

Let be a nonnegative real number. Let be a mapping such that

(2.29)

and let and for all . Then there exists a unique bi-quadratic mapping such that

(2.30)

for all .

Proof.

In Theorem 2.4, putting for all , we get the desired result.