- Research Article
- Open access
- Published:
On the Stability of a Generalized Quadratic and Quartic Type Functional Equation in Quasi-Banach Spaces
Journal of Inequalities and Applications volume 2009, Article number: 153084 (2009)
Abstract
We establish the general solution of the functional equation for fixed integers
with
and investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces.
1. Introduction
The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. 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 exists a homomorphism
with
for all
In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let
be a mapping between Banach spaces such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ1_HTML.gif)
for all and for some
Then there exists a unique additive mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ2_HTML.gif)
for all Moreover, if
is continuous in
for each fixed
then
is
-linear. In 1978, Th. M. Rassias [3] provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. The functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ3_HTML.gif)
is related to a symmetric biadditive mapping [4–7]. It is natural that this functional equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.3) is said to be a quadratic mapping. It is well known that a mapping between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive mapping
such that
for all
(see [4, 7]). The biadditive mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ4_HTML.gif)
The generalized Hyers-Ulam stability problem for the quadratic functional equation (1.3) was proved by Skof for mappings , where
is a normed space and
is a Banach space (see [8]). Cholewa [9] noticed that the theorem of Skof is still true if relevant domain
is replaced by an abelian group. In [10] , Czerwik proved the generalized Hyers-Ulam stability of the functional equation (1.3). Grabiec [11] has generalized these results mentioned above.
In [12], Park and Bae considered the following quartic functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ5_HTML.gif)
In fact, they proved that a mapping between two real vector spaces
and
is a solution of (1.5) if and only if there exists a unique symmetric multiadditive mapping
such that
for all
. It is easy to show that
satisfies the functional equation (1.5), which is called a quartic functional equation (see also [13]).
In addition, Kim [14] has obtained the generalized Hyers-Ulam stability for a mixed type of quartic and quadratic functional equation between two real linear Banach spaces. Najati and Zamani Eskandani [15] have established the general solution and the generalized Hyers-Ulam stability for a mixed type of cubic and additive functional equation, whenever is a mapping between two quasi-Banach spaces (see also [16, 17]).
Now we introduce the following functional equation for fixed integers with
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ6_HTML.gif)
in quasi-Banach spaces. It is easy to see that the function is a solution of the functional equation (1.6). In the present paper we investigate the general solution of the functional equation (1.6) when
is a mapping between vector spaces, and we establish the generalized Hyers-Ulam stability of this functional equation whenever
is a mapping between two quasi-Banach spaces.
We recall some basic facts concerning quasi-Banach space and some preliminary results.
Definition 1.1 (See [18, 19]).
Let be a real linear space. A quasinorm is a real-valued function on
satisfying the following.
(1) for all
and
if and only if
(2) for all
and all
(3)There is a constant such that
for all
It follows from the condition (3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ7_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 quasi-normed space.
A quasi-norm is called a
-norm
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ8_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 X. By the Aoki-Rolewicz theorem [19] (see also [18]), each quasi-norm is equivalent to some
-norm. Since it is much easier to work with
-norms, henceforth we restrict our attention mainly to
-norms. In [20], Tabor has investigated a version of Hyers-Rassias-Gajda theorem (see [3, 21]) in quasi-Banach spaces.
2. General Solution
Throughout this section, and
will be real vector spaces. We here present the general solution of (1.6).
Lemma 2.1.
If a mapping satisfies the functional equation (1.6), then
is a quadratic and quartic mapping.
Proof.
Letting in (1.6), we get
. Setting
in (1.6), we get
for all
. So the mapping
is even. Replacing
by
in (1.6) and then
by
in (1.6), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ10_HTML.gif)
for all . Interchanging
and
in (1.6) and using the evenness of
, we get the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ11_HTML.gif)
for all . Replacing
by
in (1.6) and then using (2.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ12_HTML.gif)
for all . If we add (2.1) to (2.2) and use (2.4), then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ13_HTML.gif)
for all . Replacing
by
in (1.6) and then
by
in (1.6) and using the evenness of
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ15_HTML.gif)
for all . Interchanging
with
in (2.6) and (2.7) and using the evenness of
, we get the relations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ17_HTML.gif)
for all . Replacing
by
in (1.6) and then
by
in (1.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ19_HTML.gif)
for all . Replacing
by
in (1.6), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ20_HTML.gif)
for all . Adding (2.10) to (2.11) and using (2.8), (2.9), and (2.12), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ21_HTML.gif)
for all . By (2.5) and (2.13), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ22_HTML.gif)
for all . Interchanging
and
in (2.14) and using the evenness of
, we get the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ23_HTML.gif)
for all .
Now we show that (2.15) is a quadratic and quartic functional equation. To get this, we show that the mappings , defined by
, and
, defined by
, are quadratic and quartic, respectively.
Replacing by
in (2.15) and using the evenness of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ24_HTML.gif)
for all . Interchanging
with
in (2.16) and then using (2.15), we obtain by the evenness of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ25_HTML.gif)
for all . By (2.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ26_HTML.gif)
for all . This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ27_HTML.gif)
for all . Thus the mapping
is quadratic.
To prove that is quartic, we have to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ28_HTML.gif)
for all . Replacing
and
by
and
in (2.15), respectively, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ29_HTML.gif)
for all . Since
for all
and
is a quadratic mapping, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ30_HTML.gif)
for all . So it follows from (2.15), (2.21), and (2.22) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ31_HTML.gif)
for all . Thus
is a quartic mapping.
Theorem 2.2.
A mapping satisfies (1.6) if and only if there exist a unique symmetric multiadditive mapping
and a unique symmetric bi-additive mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ32_HTML.gif)
for all
Proof.
We first assume that the mapping satisfies (1.6). Let
be mappings defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ33_HTML.gif)
for all By Lemma 2.1, we achieve that the mappings
and
are quadratic and quartic, respectively, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ34_HTML.gif)
for all Thus there exist a unique symmetric multiadditive mapping
and a unique symmetric bi-additive mapping
such that
and
for all
(see citead, ki). So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ35_HTML.gif)
for all
Conversely assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ36_HTML.gif)
for all where the mapping
is symmetric multi-additive and
is bi-additive. By a simple computation, one can show that the mappings
and
satisfy the functional equation (1.6), so the mapping f satisfies (1.6).
3. Generalized Hyers-Ulam Stability of (1.6)
From now on, let and
be a quasi-Banach space with quasi-norm
and a
-Banach space with
-norm
, respectively. Let
be the modulus of concavity of
. In this section, using an idea of G
vruta [22], we prove the stability of (1.6) in the spirit of Hyers, Ulam, and Rassias. For convenience we use the following abbreviation for a given mapping
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ37_HTML.gif)
for all . Let
. We will use the following lemma in this section.
Lemma 3.1 (see [15]).
Let and let
be nonnegative real numbers. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ38_HTML.gif)
Theorem 3.2.
Let be a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ39_HTML.gif)
for all and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ40_HTML.gif)
for all and all
Suppose that a mapping
with
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ41_HTML.gif)
for all Then the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ42_HTML.gif)
exists for all and
is a unique quadratic mapping satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ43_HTML.gif)
for all where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ44_HTML.gif)
Proof.
Setting in (3.5) and then interchanging
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ45_HTML.gif)
for all . Replacing
by
,
,
,
and
in (3.5), respectively, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ46_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ47_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ48_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ49_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ50_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ51_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ52_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ53_HTML.gif)
for all . Combining (3.9) and (3.11)–(3.17), respectively, yields the following inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ54_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ55_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ56_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ57_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ58_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ59_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ60_HTML.gif)
for all .
Replacing and
by
and
in (3.5), respectively, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ61_HTML.gif)
for all . Putting
and
instead of
and
in (3.5), respectively, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ62_HTML.gif)
for all . It follows from (3.10), (3.18), (3.19), (3.20), (3.21), and (3.25) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ63_HTML.gif)
for all . Also, from (3.10), (3.18), (3.19), (3.22), (3.23), (3.24), and (3.26), we conclude
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ64_HTML.gif)
for all . Finally, combining (3.27) and (3.28) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ65_HTML.gif)
for all . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ66_HTML.gif)
Then the inequality (3.29) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ67_HTML.gif)
for all
Let be a mapping defined by
for all
From (3.31), we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ68_HTML.gif)
for all If we replace
in (3.32) by
and multiply both sides of (3.32) by
then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ69_HTML.gif)
for all and all nonnegative integers
. Since
is a p-Banach space, the inequality (3.33) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ70_HTML.gif)
for all nonnegative integers and
with
and all
Since
, by Lemma 3.1 and (3.30), we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ71_HTML.gif)
for all Therefore, it follows from (3.4) and (3.35) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ72_HTML.gif)
for all It follows from (3.34) and (3.36) that the sequence
is Cauchy for all
Since
is complete, the sequence
converges for all
So one can define the mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ73_HTML.gif)
for all Letting
and passing the limit
in (3.34), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ74_HTML.gif)
for all Thus (3.7) follows from (3.4) and (3.38).
Now we show that is quadratic. It follows from (3.3), (3.33) and (3.37) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ75_HTML.gif)
for all So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ76_HTML.gif)
for all On the other hand, it follows from (3.3), (3.5), (3.6) and (3.37) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ77_HTML.gif)
for all Hence the mapping
satisfies (1.6). By Lemma 2.1, the mapping
is quadratic. Hence (3.40) implies that the mapping
is quadratic.
It remains to show that is unique. Suppose that there exists another quadratic mapping
which satisfies (1.6) and (3.7). Since
and
for all
, we conclude from (3.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ78_HTML.gif)
for all On the other hand, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ79_HTML.gif)
for all and all
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ80_HTML.gif)
for all . Using (3.44) and (3.42), we get
as desired.
Theorem 3.3.
Let be a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ81_HTML.gif)
for all and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ82_HTML.gif)
for all and all
Suppose that a mapping
with
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ83_HTML.gif)
for all Then the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ84_HTML.gif)
exists for all and
is a unique quadratic mapping satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ85_HTML.gif)
for all x∈ X, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ86_HTML.gif)
Proof.
The proof is similar to the proof of Theorem 3.2.
Corollary 3.4.
Let be nonnegative real numbers such that
or
. Suppose that a mapping
with
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ87_HTML.gif)
for all Then there exists a unique quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ88_HTML.gif)
for all where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ89_HTML.gif)
Proof.
In Theorem 3.2, putting for all
, we get the desired result.
Corollary 3.5.
Let and
be nonnegative real numbers such that
. Suppose that a maping
with
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ90_HTML.gif)
for all Then there exists a unique quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ91_HTML.gif)
for all
Proof.
In Theorem 3.2, putting for all
, we get the desired result.
Theorem 3.6.
Let be a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ92_HTML.gif)
for all and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ93_HTML.gif)
for all and all
Suppose that a mapping
with
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ94_HTML.gif)
for all Then the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ95_HTML.gif)
exists for all and
is a unique quartic mapping satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ96_HTML.gif)
for all where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ97_HTML.gif)
Proof.
Similar to the proof Theorem 3.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ98_HTML.gif)
for all where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ99_HTML.gif)
Let be a mapping defined by
. Then we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ100_HTML.gif)
for all If we replace
in (3.65) by
and multiply both sides of (3.65) by
then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ101_HTML.gif)
for all and all nonnegative integers
. Since
is a
-Banach space, the inequality (3.66) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ102_HTML.gif)
for all nonnegative integers and
with
and all
Since
, by Lemma 3.1, we conclude from (3.64) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ103_HTML.gif)
for all It follows from (3.57) and (3.67) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ104_HTML.gif)
for all Thus we conclude from (3.67) and (3.69) that the sequence
is Cauchy for all
Since
is complete, the sequence
converges for all
So one can define the mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ105_HTML.gif)
for all Letting
and passing the limit
in (3.67), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ106_HTML.gif)
for all Thus (3.60) follows from (3.58) and (3.70).
Now we show that is quartic. From (3.57), (3.66), and (3.70), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ107_HTML.gif)
for all So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ108_HTML.gif)
for all On the other hand, by (3.59), (3.69), and (3.70), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ109_HTML.gif)
for all Hence the mapping
satisfies (1.6). By Lemma 2.1, the mapping
is quartic. Therefore, (3.75) implies that the mapping
is quartic.
To prove the uniqueness property of let
be another quartic mapping satisfying (3.61). Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ110_HTML.gif)
for all and all
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ111_HTML.gif)
for all . It follows from (3.61) and (3.86) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ112_HTML.gif)
for all So
as desired.
Theorem 3.7.
Let be a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ113_HTML.gif)
for all and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ114_HTML.gif)
for all and all
Suppose that a mapping
with
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ115_HTML.gif)
for all Then the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ116_HTML.gif)
exists for all and
is a unique quartic mapping satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ117_HTML.gif)
for all where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ118_HTML.gif)
Proof.
The proof is similar to the proof of Theorem 3.6.
Corollary 3.8.
Let be nonnegative real numbers such that
or
. Suppose that a mapping
with
satisfies the inequality(3.51)for all
Then there exists a unique quartic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ119_HTML.gif)
for all where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ120_HTML.gif)
Proof.
In Theorem 3.6, putting for all
, we get the desired result.
Corollary 3.9.
Let and
be nonnegative real numbers such that
. Suppose that a mapping
with
satisfies the inequality (3.56) for all
Then there exists a unique quartic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ121_HTML.gif)
for all
Proof.
In Theorem 3.6, putting for all
, we get the desired result.
Theorem 3.10.
Let be a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ122_HTML.gif)
for all and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ123_HTML.gif)
for all and all
. Suppose that a mapping
with
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ124_HTML.gif)
for all Then there exist a unique quadratic mapping
and a unique quartic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ125_HTML.gif)
for all where
and
are defined in Theorems 3.2 and 3.7, respectively.
Proof.
By Theorems 3.2 and 3.7, there exist a quadratic mapping and a quartic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ126_HTML.gif)
for all It follows from the last inequalities that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ127_HTML.gif)
for all So we obtain (3.92) by letting
and
for all
To prove the uniqueness property of and
we first show the uniqueness property for
and
and then we conclude the uniqueness property of
and
Let
be another quadratic and quartic mappings satisfying (3.92) and let
,
,
and
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ128_HTML.gif)
for all Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ129_HTML.gif)
for all (3.62) implies that
for all
Thus
But
is only a quartic function and
is only a quadratic function.
Therefore, we have and this completes the uniqueness property of
and
We can prove the other results similarly.
Corollary 3.11.
Let be nonnegative real numbers such that
or
or
. Suppose that a mapping
satisfies the inequality(3.51)for all
Then there exist a unique quadratic mapping
and a unique quartic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ130_HTML.gif)
for all where
, and
are defined as in Corollaries 3.4 and 3.8.
Corollary 3.12.
Let and
be nonnegative real numbers such that
. Suppose that a mapping
satisfies the inequality (3.56) for all
Then there exist a unique quadratic mapping
and a unique quartic function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F153084/MediaObjects/13660_2009_Article_1897_Equ131_HTML.gif)
for all
References
Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1940.
Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1
Aczèl J, Dhombres J: Functional Equations in Several Variables. Volume 31. Cambridge University Press, Cambridge, UK; 1989.
Amir D: Characterizations of Inner Product Spaces. Volume 20. Birkhäuser, Basel, Switzerland; 1986.
Jordan P, von Neumann J: On inner products in linear, metric spaces. Annals of Mathematics. Second Series 1935,36(3):719–723. 10.2307/1968653
Kannappan Pl: Quadratic functional equation and inner product spaces. Results in Mathematics 1995,27(3–4):368–372.
Skof F: Proprietà locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890
Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1–2):76–86.
Czerwik S: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1992, 62: 59–64. 10.1007/BF02941618
Grabiec A: The generalized Hyers-Ulam stability of a class of functional equations. Publicationes Mathematicae Debrecen 1996,48(3–4):217–235.
Park W-G, Bae J-H: On a bi-quadratic functional equation and its stability. Nonlinear Analysis: Theory, Methods & Applications 2005,62(4):643–654. 10.1016/j.na.2005.03.075
Chung JK, Sahoo PK: On the general solution of a quartic functional equation. Bulletin of the Korean Mathematical Society 2003,40(4):565–576.
Kim H-M: On the stability problem for a mixed type of quartic and quadratic functional equation. Journal of Mathematical Analysis and Applications 2006,324(1):358–372. 10.1016/j.jmaa.2005.11.053
Najati A, Eskandani GZ: Stability of a mixed additive and cubic functional equation in quasi-Banach spaces. Journal of Mathematical Analysis and Applications 2008,342(2):1318–1331. 10.1016/j.jmaa.2007.12.039
Eshaghi Gordji M: Stability of a functional equation deriving from quartic and additive functions. to appear in Bulletin of the Korean Mathematical Society to appear in Bulletin of the Korean Mathematical Society
Eshaghi Gordji M, Ebadian A, Zolfaghari S: Stability of a functional equation deriving from cubic and quartic functions. Abstract and Applied Analysis 2008, 2008:-17.
Benyamini Y, Lindenstrauss J: Geometric Nonlinear Functional Analysis. Vol. 1. Volume 48. American Mathematical Society, Providence, RI, USA; 2000.
Rolewicz S: Metric Linear Spaces. 2nd edition. PWN—Polish Scientific, Warsaw, Poland; 1984.
Tabor J: Stability of the Cauchy functional equation in quasi-Banach spaces. Annales Polonici Mathematici 2004,83(3):243–255. 10.4064/ap83-3-6
Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431–434. 10.1155/S016117129100056X
Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431–436. 10.1006/jmaa.1994.1211
Acknowledgment
The third and corresponding author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). Also, the second author would like to thank the office of gifted students at Semnan University for its financial support.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Gordji, M.E., Abbaszadeh, S. & Park, C. On the Stability of a Generalized Quadratic and Quartic Type Functional Equation in Quasi-Banach Spaces. J Inequal Appl 2009, 153084 (2009). https://doi.org/10.1155/2009/153084
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/153084