# Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces

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## Abstract

We obtain the generalized Hyers-Ulam stability of the bi-quadratic functional equation Open image in new window in quasinormed spaces.

### Keywords

Banach Space Functional Equation Stability Problem Additive Mapping Product Space## 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* Open image in new window *be a group and let* Open image in new window *be a metric group with the metric* Open image in new window *. Given* Open image in new window *, does there exist a* Open image in new window *such that if a mapping* Open image in new window *satisfies the inequality* Open image in new window *for all* Open image in new window *then there is a homomorphism* Open image in new window *with* Open image in new window *for all* Open image in new window *?*

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 Open image in new window , where Open image in new window is a normed space and Open image in new window is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain Open image in new window 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 Open image in new window -algebra.

Throughout this paper, let Open image in new window and Open image in new window be vector spaces.

Definition 1.1.

*bi-quadratic*if Open image in new window satisfies the system of the following equations:

When Open image in new window , the function Open image in new window given by Open image in new window is a solution of (1.4).

Let Open image in new window be a real linear space. A *quasinorm* is real-valued function on Open image in new window satisfying the following

(i) Open image in new window for all Open image in new window and Open image in new window if and only if Open image in new window .

(ii) Open image in new window for all Open image in new window and all Open image in new window .

(iii)There is a constant Open image in new window such that Open image in new window for all Open image in new window .

for all Open image in new window and all Open image in new window .

*quasinormed space*if Open image in new window is a quasinorm on Open image in new window . The smallest possible Open image in new window is called the

*modulus of concavity*of Open image in new window . A

*quasi-Banach space*is a complete quasinormed space. A quasinorm Open image in new window is called a Open image in new window

*-norm*( Open image in new window ) if

for all Open image in new window . In this case, a quasi-Banach space is called a Open image in new window *-Banach space*.

Given a Open image in new window -norm, the formula Open image in new window gives us a translation invariant metric on Open image in new window . By the Aoki-Rolewicz theorem [10] (see also [9]), each quasinorm is equivalent to some Open image in new window -norm. Since it is much easier to work with Open image in new window -norms, henceforth we restrict our attention mainly to Open image in new window -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, 14, 15, 16, 17, 18]).

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

Theorem A.

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

Theorem B.

A mapping Open image in new window 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 Open image in new window is a quasinormed space with quasinorm Open image in new window and that Open image in new window is a Open image in new window -Banach space with Open image in new window -norm Open image in new window . Let Open image in new window be the modulus of concavity of Open image in new window .

for all Open image in new window .

for all Open image in new window .

Theorem 2.1.

for all Open image in new window .

Proof.

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

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

Corollary 2.2.

for all Open image in new window .

Proof.

In Theorem 2.1, putting Open image in new window and Open image in new window for all Open image in new window , we get the desired result.

for all Open image in new window .

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

Lemma 2.3 (see [20]).

Theorem 2.4.

for all Open image in new window .

Proof.

for all Open image in new window .

for all Open image in new window . Hence the mapping Open image in new window is the unique bi-quadratic mapping, as desired.

Corollary 2.5.

for all Open image in new window .

Proof.

In Theorem 2.4, putting Open image in new window for all Open image in new window , we get the desired result.

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