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
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
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.
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.
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
We will use the following lemma in order to prove Theorem 2.4.
Lemma 2.3 (see ).
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