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Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces
Journal of Inequalities and Applications volume 2010, Article number: 472721 (2010)
Abstract
We obtain the generalized Hyers-Ulam stability of the bi-quadratic functional equation in quasinormed spaces.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ1_HTML.gif)
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. G
vruţa [4] provided a further generalization of Hyers-Ulam theorem. A square norm on an inner product space satisfies the important parallelogram equality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ2_HTML.gif)
The functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ3_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ4_HTML.gif)
When , the function
given by
is a solution of (1.4).
For a mapping , consider the functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ8_HTML.gif)
for all .
Let be two functions satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ10_HTML.gif)
for all .
Theorem 2.1.
Let be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ12_HTML.gif)
and let and
for all
. Then there exist two bi-quadratic mappings
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ14_HTML.gif)
for all .
Proof.
Letting in (2.4), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ15_HTML.gif)
for all . Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ16_HTML.gif)
for all . Replacing
by
in the above inequality, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ17_HTML.gif)
for all . Since
is a
-Banach space, for given integers
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ19_HTML.gif)
for all . Putting
and taking
in (2.11), one can obtain the inequality (2.6). By (2.4) and (2.5), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ20_HTML.gif)
for all and all
. Letting
in the above two inequalities and using (2.1),
is bi-quadratic.
Next, setting in (2.5),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ21_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ22_HTML.gif)
and let and
for all
. Then there exist two bi-quadratic mappings
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ24_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ26_HTML.gif)
Theorem 2.4.
Let be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ27_HTML.gif)
and let and
for all
. Then there exists a unique bi-quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ28_HTML.gif)
for all .
Proof.
Letting and
in (2.20), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ29_HTML.gif)
for all . Thus we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ30_HTML.gif)
for all and all
. Replacing
by
in the above inequality, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ31_HTML.gif)
for all and all
. By Lemma 2.3, for given integers
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ32_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ33_HTML.gif)
for all .
By (2.20), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ36_HTML.gif)
and let and
for all
. Then there exists a unique bi-quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F472721/MediaObjects/13660_2010_Article_2159_Equ37_HTML.gif)
for all .
Proof.
In Theorem 2.4, putting for all
, we get the desired result.
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Park, WG., Bae, JH. Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces. J Inequal Appl 2010, 472721 (2010). https://doi.org/10.1155/2010/472721
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DOI: https://doi.org/10.1155/2010/472721