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On the Stability of Generalized Quartic Mappings in Quasi-
-Normed Spaces
Journal of Inequalities and Applications volume 2010, Article number: 198098 (2010)
Abstract
We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi--normed spaces and then the stability by using a subadditive function for the generalized quartic function
such that
, where
,
,
, for all
.
1. Introduction
One of the interesting questions concerning the stability problems of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation? Such an idea was suggested in 1940 by Ulam [1] as follows. Letbe a group and let
be a metric group with the metric
. Given
, does there exist a
such that if a function
satisfies the inequality
for all
, then there is a homomorphism
with
for all
? In other words, we are looking for situations when the homomorphisms are stable; that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it. In 1941, Hyers [2] considered the case of approximately additive mappings in Banach spaces and satisfying the well-known weak Hyers inequality controlled by a positive constant. The famous Hyers stability result that appeared in [2] was generalized in the stability involving a sum of powers of norms by Aoki [3]. In 1978, Rassias [4] provided a generalization of Hyers Theorem which allows the Cauchy difference to be unbounded. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors [5–10]. In particular, Rassias [11] introduced the quartic functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ1_HTML.gif)
It is easy to see that is a solution of (1.1) by virtue of the identity
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ2_HTML.gif)
For this reason, (1.1) is called a quartic functional equation. Also Chung and Sahoo [12] determined the general solution of (1.1) without assuming any regularity conditions on the unknown function. In fact, they proved that the function is a solution of (1.1) if and only if
where the function
is symmetric and additive in each variable. Lee and Chung [13] introduced a quartic functional equation as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ3_HTML.gif)
for fixed integer with
Let be a real number with
and let
be either
or
We will consider the definition and some preliminary results of a quasi-
-norm on a linear space.
Definition 1.1.
Let be a linear space over a field
A quasi-
-norm
is a real-valued function on
satisfying the followings.
(1) for all
and
if and only if
(2) for all
and all
(3) There is a constant such that
for all
The pair is called a quasi-
-normed space if
is a quasi-
-norm 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
for all
In this case, a quasi-
-Banach space is called a
-Banach space; see [14–16].
In this paper, we consider the following the generalized quartic functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ4_HTML.gif)
for fixed integers and
such that
for all
We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi-
-normed spaces and then the stability by using a subadditive function for the generalized quartic function
satisfying (1.4).
For the same reason as (1.1) and (1.2), we call (1.4) generalized quartic functional equation.
2. Quartic Functional Equations
Let be real vector spaces. In this section, we will investigate that the functional equation (1.1) is equivalent to the presented functional equation (1.4).
Lemma 2.1.
A mapping satisfies the functional equation (1.1) if and only if
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ5_HTML.gif)
where for all
Proof.
We will show it by induction on Assume that it holds for all less than equal
Now, letting
be
in (2.1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ6_HTML.gif)
and also replacing by
in (2.1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ7_HTML.gif)
for all Adding (2.2) and (2.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ8_HTML.gif)
for all By induction steps, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ9_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ10_HTML.gif)
for all Thus they are equivalent.
Theorem 2.2.
If a mapping satisfies the functional equation (1.4), then
satisfies the functional equation (2.1).
Proof.
By letting in (2.1), we have
Since
and
Putting
in (2.1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ11_HTML.gif)
Now, replacing by
in (2.7),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ12_HTML.gif)
By (2.7) and (2.8), we have that is,
Hence
is even. This implies that
that is,
for all
Now, we will show that (2.1) implies (1.4). By letting
in (2.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ13_HTML.gif)
Switching and
in the previous equation,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ14_HTML.gif)
By (2.1) with , the previous equation implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ15_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ16_HTML.gif)
for all
Corollary 2.3.
If a mapping satisfies the functional equation (1.1), then
satisfies the functional equation (1.4).
3. Stabilities
Throughout this section, let be a quasi-
-normed space and let
be a quasi-
-Banach space with a quasi-
-norm
. Let
be the modulus of concavity of
. We will investigate the generalized Hyers-Ulam-Rassias stability problem for the functional equation (1.4). After then we will study the stability by using a subadditive function. For a given mapping
and all fixed integers
and
with
let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ17_HTML.gif)
Theorem 3.1.
Suppose that there exists a mapping for which a mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ18_HTML.gif)
and the series converges for all
Then there exists a unique generalized quartic mapping
which satisfies (1.4) and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ19_HTML.gif)
for all
Proof.
By letting in the inequality (3.2), since
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ20_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ21_HTML.gif)
for all Now, putting
and multiplying
in the inequality (3.5), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ22_HTML.gif)
for all Combining (3.5) and (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ23_HTML.gif)
for all Inductively, since
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ24_HTML.gif)
for all For all
and
with
and switching
and
and multiplying
in the inequality (3.5), inductively,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ25_HTML.gif)
for all Since the right-hand side of the previous inequality tends to 0 as
hence
is a Cauchy sequence in the quasi-
-Banach space
Thus we may define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ26_HTML.gif)
for all Since
replacing
and
by
and
, respectively, and dividing by
in the inequality (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ27_HTML.gif)
for all By taking
the definition of
implies that
satisfies (1.4) for all
that is,
is the generalized quartic mapping. Also, the inequality (3.8) implies the inequality (3.3). Now, it remains to show the uniqueness. Assume that there exists
satisfying (1.4) and (3.3). It is easy to show that for all
and
as in the proof of Theorem 2.2. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ28_HTML.gif)
for all By letting
we immediately have the uniqueness of
.
Theorem 3.2.
Suppose that there exists a mapping for which a mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ29_HTML.gif)
and the series converges for all
Then there exists a unique generalized quartic mapping
which satisfies (2.1) and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ30_HTML.gif)
for all
Proof.
If is replaced by
in the inequality (3.5), then the proof follows from the proof of Theorem 3.1.
Now we will recall a subadditive function and then investigate the stability under the condition that the space is a
-Banach space. The basic definitions of subadditive functions follow from [16].
A function having a domain
and a codomain
that are both closed under addition is called
(1)a subadditive function if
(2)a contractively subadditive function if there exists a constant with
such that
(3)an expansively superadditive function if there exists a constant with
such that
for all
Theorem 3.3.
Suppose that there exists a mapping for which a mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ31_HTML.gif)
for all and the map
is contractively subadditive with a constant
such that
Then there exists a unique generalized quartic mapping
which satisfies (1.4) and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ32_HTML.gif)
for all
Proof.
By the inequalities (3.5) and (3.9) of the proof of Theorem 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ33_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ34_HTML.gif)
for all and for all
and
with
Hence
is a Cauchy sequence in the space
Thus we may define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ35_HTML.gif)
for all Now, we will show that the map
is a generalized quartic mapping. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ36_HTML.gif)
for all Hence the mapping
is a generalized quartic mapping. Note that the inequality (3.18) implies the inequality (3.16) by letting
and taking
Assume that there exists
satisfying (1.4) and (3.16). We know that
for all
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ37_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ38_HTML.gif)
for all By letting
we immediately have the uniqueness of
.
Theorem 3.4.
Suppose that there exists a mapping for which a mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ39_HTML.gif)
for all and the map
is expansively superadditive with a constant
such that
Then there exists a unique generalized quartic mapping
which satisfies (1.4) and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ40_HTML.gif)
for all
Proof.
By letting in (3.23), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ41_HTML.gif)
and then replacing by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ42_HTML.gif)
for all For all
and
with
inductively we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F198098/MediaObjects/13660_2009_Article_2081_Equ43_HTML.gif)
for all The remains follow from the proof of Theorem 3.3.
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Acknowledgments
The author would like to thank referees for their valuable suggestions and comments. The present research was conducted by the research fund of Dankook University in 2009.
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Kang, D. On the Stability of Generalized Quartic Mappings in Quasi--Normed Spaces.
J Inequal Appl 2010, 198098 (2010). https://doi.org/10.1155/2010/198098
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DOI: https://doi.org/10.1155/2010/198098