- Research Article
- Open access
- Published:
Quadratic-Quartic Functional Equations in RN-Spaces
Journal of Inequalities and Applications volume 2009, Article number: 868423 (2009)
Abstract
We obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary -norms
.
1. Introduction
The stability problem of functional equations originated from a question of Ulam [1] in 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 exists 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. 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%2F868423/MediaObjects/13660_2009_Article_2022_Equ1_HTML.gif)
for all and some
Then there exists a unique additive mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ2_HTML.gif)
for all Moreover, if
is continuous in
for each fixed
then
is
-linear. In 1978, Rassias [3] provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. In
Gajda [4] answered the question for the case
, which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [5–12]). The functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ3_HTML.gif)
is related to a symmetric biadditive mapping. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic functional 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 exits a unique symmetric biadditive mapping
such that
for all
(see [5, 13]). The biadditive mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ4_HTML.gif)
The Hyers-Ulam-Rassias 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 [14]). Cholewa [15] noticed that the theorem of Skof is still true if relevant domain
is replaced an abelian group. In [16], Czerwik proved the Hyers-Ulam-Rassias stability of the functional equation (1.3). Grabiec [17] has generalized the results mentioned above.
In [18], Park and Bae considered the following quartic functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_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 the function
satisfies the functional equation (1.5), which is called a quartic functional equation (see also [19]). In addition, Kim [20] has obtained the Hyers-Ulam-Rassias stability for a mixed type of quartic and quadratic functional equation.
The Hyers-Ulam-Rassias stability of different functional equations in random normed and fuzzy normed spaces has been recently studied in [21–26]. It should be noticed that in all these papers the triangle inequality is expressed by using the strongest triangular norm .
The aim of this paper is to investigate the stability of the additive-quadratic functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary continuous -norms.
In this sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [22, 23, 27–29]. Throughout this paper, is the space of distribution functions, that is, the space of all mappings
such that
is left-continuous and nondecreasing on
and
. Also,
is a subset of
consisting of all functions
for which
, where
denotes the left limit of the function
at the point
, that is,
. The space
is partially ordered by the usual point-wise ordering of functions, that is,
if and only if
for all
in
. The maximal element for
in this order is the distribution function
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ6_HTML.gif)
Definition 1.1 (see [28]).
A mapping is a continuous triangular norm (briefly, a continuous
-norm) if
satisfies the following conditions:
(a) is commutative and associative;
(b) is continuous;
(c) for all
;
(d) whenever
and
for all
.
Typical examples of continuous -norms are
,
and
(the Lukasiewicz
-norm). Recall (see [30, 31]) that if
is a
-norm and
is a given sequence of numbers in
, then
is defined recurrently by
and
for
.
is defined as
It is known [31] that for the Lukasiewicz
-norm, the following implication holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ7_HTML.gif)
Definition 1.2 (see [29]).
A random normed space (briefly, RN-space) is a triple , where
is a vector space,
is a continuous
-norm, and
is a mapping from
into
such that the following conditions hold:
(RN1) for all
if and only if
;
(RN2) for all
,
;
(RN3) for all
and
Every normed space defines a random normed space
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ8_HTML.gif)
for all and
is the minimum
-norm. This space is called the induced random normed space.
Definition 1.3.
Let be an RN-space.
A sequence in
is said to be convergent to
in
if, for every
and
, there exists a positive integer
such that
whenever
.
A sequence in
is called a Cauchy sequence if, for every
and
, there exists a positive integer
such that
whenever
.
An RN-space is said to be complete if and only if every Cauchy sequence in
is convergent to a point in
.
Theorem 1.4 (see [28]).
If is an RN-space and
is a sequence such that
, then
almost everywhere.
Recently, Gordji et al. establish the stability of cubic, quadratic and additive-quadratic functional equations in RN-spaces (see [32, 33]).
In this paper, we deal with the following functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ9_HTML.gif)
on RN-spaces. It is easy to see that the function is a solution of (1.9).
In Section 2, we investigate the general solution of the functional equation (1.9) when is a mapping between vector spaces and in Section 3, we establish the stability of the functional equation (1.9) in RN-spaces.
2. General Solution
We need the following lemma for solution of (1.9). Throughout this section, and
are vector spaces.
Lemma 2.1.
If a mapping satisfies (1.9) for all
then
is quadratic-quartic.
Proof.
We show that the mappings defined by
and
defined by
are quadratic and quartic, respectively.
Letting in (1.9), we have
. Putting
in (1.9), we get
. Thus the mapping
is even. Replacing
by
in (1.9), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ10_HTML.gif)
for all . Interchanging
with
in (1.9), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ11_HTML.gif)
for all . Since
is even, by (2.2), one gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ12_HTML.gif)
for all It follows from (2.1) and (2.3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ13_HTML.gif)
for all . This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ14_HTML.gif)
for all . Therefore, 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%2F868423/MediaObjects/13660_2009_Article_2022_Equ15_HTML.gif)
for all Since
is even, the mapping
is even. Now if we interchange
with
in the last equation, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ16_HTML.gif)
for all . Thus, it is enough to prove that
satisfies (2.7). Replacing
and
by
and
in (1.9), respectively, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ17_HTML.gif)
for all . Since
for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ18_HTML.gif)
for all . By (2.8) and (2.9), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ19_HTML.gif)
for all . By multiplying both sides of (1.9) by
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ20_HTML.gif)
for all . If we subtract the last equation from (2.10), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ21_HTML.gif)
for all .
Therefore, the mapping is quartic. This completes the proof of the lemma.
Theorem 2.2.
A mapping satisfies (1.9) for all
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%2F868423/MediaObjects/13660_2009_Article_2022_Equ22_HTML.gif)
for all .
Proof.
Let satisfy (1.9) and assume that
are mappings defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ23_HTML.gif)
for all By Lemma 2.1, we obtain that the mappings
and
are quadratic and quartic, respectively, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ24_HTML.gif)
for all
Therefore, there exist a unique symmetric multiadditive mapping and a unique symmetric bi-additive mapping
such that
and
for all
[5, 18]. So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ25_HTML.gif)
for all The proof of the converse is obvious.
3. Stability
Throughout this section, assume that is a real linear space and
is a complete RN-space.
Theorem 3.1.
Let be a mapping with
for which there is
(
is denoted by
) with the property:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ26_HTML.gif)
for all and all
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ27_HTML.gif)
for all and all
, then there exists a unique quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ28_HTML.gif)
for all and all
Proof.
Putting in (3.1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ29_HTML.gif)
for all and all
Letting
in (3.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ30_HTML.gif)
for all and all
Putting
in (3.1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ31_HTML.gif)
for all and all
Replacing
by
in (3.6), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ32_HTML.gif)
for all and all
It follows from (3.5) and (3.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ33_HTML.gif)
for all and all
If we add (3.4) to (3.8), then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ34_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ35_HTML.gif)
for all and all
. Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ36_HTML.gif)
for all and all
Let
be a mapping defined by
. Then we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ37_HTML.gif)
for all and all
. Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ38_HTML.gif)
for all and all
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ39_HTML.gif)
for all , all
and all
This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ40_HTML.gif)
for all all
and all
By the triangle inequality, from
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ41_HTML.gif)
for all and all
In order to prove the convergence of the sequence
, we replace
with
in (3.16) to obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ42_HTML.gif)
Since the right-hand side of the inequality (3.17) tends to as
and
tend to infinity, the sequence
is a Cauchy sequence. Thus we may define
for all
.
Now we show that is a quadratic mapping. Replacing
with
and
in (3.1), respectively, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ43_HTML.gif)
Taking the limit as , we find that
satisfies (1.9) for all
. By Lemma 2.1, the mapping
is quadratic.
Letting the limit as in (3.16), we get (3.3) by (3.10).
Finally, to prove the uniqueness of the quadratic mapping subject to (3.3), let us assume that there exists another quadratic mapping
which satisfies (3.3). Since
for all
and all
from (3.3), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ44_HTML.gif)
for all and all
. Letting
in (3.19), we conclude that
, as desired.
Theorem 3.2.
Let be a mapping with
for which there is
(
is denoted by
) with the property:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ45_HTML.gif)
for all and all
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ46_HTML.gif)
for all and all
, then there exists a unique quartic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ47_HTML.gif)
for all and all
Proof.
Putting in (3.20), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ48_HTML.gif)
for all and all
. Letting
in (3.20), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ49_HTML.gif)
for all and all
. Putting
in (3.20), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ50_HTML.gif)
for all and all
. Replacing
by
in (3.25), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ51_HTML.gif)
for all and all
. It follows from (3.5) and (3.26) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ52_HTML.gif)
for all and all
. If we add (3.23) to (3.27), then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ53_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ54_HTML.gif)
for all and all
. Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ55_HTML.gif)
for all and all
Let
be a mapping defined by
. Then we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ56_HTML.gif)
for all and all
. Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ57_HTML.gif)
for all and all
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ58_HTML.gif)
for all , all
and all
This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ59_HTML.gif)
for all all
and all
By the triangle inequality, from
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ60_HTML.gif)
for all and all
In order to prove the convergence of the sequence
, we replace
with
in (3.35) to obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ61_HTML.gif)
Since the right-hand side of (3.36) tends to as
and
tend to infinity, the sequence
is a Cauchy sequence. Thus we may define
for all
.
Now we show that is a quartic mapping. Replacing
with
and
in (3.20), respectively, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ62_HTML.gif)
Taking the limit as , we find that
satisfies (1.9) for all
. By Lemma 2.1 we get that the mapping
is quartic.
Letting the limit as in (3.35), we get (3.22) by (3.29).
Finally, to prove the uniqueness of the quartic mapping subject to
let us assume that there exists a quartic mapping
which satisfies (3.22). Since
and
for all
and all
from (3.22), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ63_HTML.gif)
for all and all
. Letting
in (3.38), we get that
, as desired.
Theorem 3.3.
Let be a mapping with
for which there is
(
is denoted by
) with the property:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ64_HTML.gif)
for all and all
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ65_HTML.gif)
for all and 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%2F868423/MediaObjects/13660_2009_Article_2022_Equ66_HTML.gif)
for all and all
Proof.
By Theorems 3.1 and 3.2, there exist a quadratic mapping and a quartic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ67_HTML.gif)
for all and all
. It follows from the last inequalities that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F868423/MediaObjects/13660_2009_Article_2022_Equ68_HTML.gif)
for all and all
. Hence we obtain (3.41) by letting
and
for all
The uniqueness property of
and
is trivial.
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Acknowledgment
C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).
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Gordji, M.E., Savadkouhi, M.B. & Park, C. Quadratic-Quartic Functional Equations in RN-Spaces. J Inequal Appl 2009, 868423 (2009). https://doi.org/10.1155/2009/868423
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DOI: https://doi.org/10.1155/2009/868423