- Research Article
- Open Access
Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces
© W.-G. Park and J.-H. Bae. 2010
- Received: 27 April 2010
- Accepted: 18 June 2010
- Published: 5 July 2010
We obtain the generalized Hyers-Ulam stability of the bi-quadratic functional equation in quasinormed spaces.
- Banach Space
- Functional Equation
- Stability Problem
- Additive Mapping
- Product Space
In 1940, Ulam  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 ?
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  for mappings , where is a normed space and is a Banach space. Cholewa  noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik  proved the generalized stability of the quadratic functional equation, and Park  proved the generalized stability of the quadratic functional equation in Banach modules over a -algebra.
Throughout this paper, let and be vector spaces.
When , the function given by is a solution of (1.4).
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 .
for all and all .
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  (see also ), 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 , 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  solved the solutions of (1.4) and (1.5) as follows.
A mapping satisfies (1.4) if and only if there exist a multi-additive mapping such that and for all .
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.
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 .
for all .
for all .
for all .
for all and all . Letting in the above two inequalities and using (2.1), is bi-quadratic.
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).
for all .
In Theorem 2.1, putting and for all , we get the desired result.
for all .
We will use the following lemma in order to prove Theorem 2.4.
Lemma 2.3 (see ).
for all .
for all .
for all . Hence the mapping is the unique bi-quadratic mapping, as desired.
for all .
In Theorem 2.4, putting for all , we get the desired result.
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