- Research Article
- Open access
- Published:
Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces
Journal of Inequalities and Applications volume 2010, Article number: 403101 (2010)
Abstract
We achieve the general solution and the generalized stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary t-norms: , where
for any fixed integer
with
.
1. Introduction
In 1940, the stability problem of functional equations originated from a question given by Ulam [1], that is, the stability of group homomorphisms.
Let be a group and
be a metric group with the metric
. For any
, does there exist
, such that, if a mapping
satisfies 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.
In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces.
Let be a mapping between Banach spaces
and
such that, for some
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ1_HTML.gif)
Then there exists a unique additive mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ2_HTML.gif)
Moreover, if is continuous in
for any fixed
then
is linear.
In 1978, Rassias [3] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. In 1991, Gajda [4] answered the question for the case , which was raised by Rassias. This new concept is known as the Hyers-Ulam-Rassias stability of functional equations (see [5–17]).
The functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ3_HTML.gif)
is related to symmetric biadditive mapping. It is natural that this 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
and
is quadratic if and only if there exits a unique symmetric biadditive mapping
such that
for all
(see [5, 18]). The biadditive mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ4_HTML.gif)
Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof [19] for a mapping , where
is a normed space and
is a Banach space. Cholewa [20] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an abelian group. In [21], Czerwik proved the Hyers-Ulam-Rassias stability of (1.3) and Grabiec [22] generalized these results mentioned above.
Recently, Jun and Kim [23] introduced the following cubic functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ5_HTML.gif)
where is a mapping from a real vector space
into a real vector space
and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.5). The function
satisfies the functional equation (1.5)
which is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic function. Also, Jun and Kim proved that a function
between real vector spaces
and
is a solution of (1.5) if and only if there exits a unique function
such that
for all
and
is symmetric for each fixed one variable and is additive for fixed two variables.
In the sequel, we adopt the usual terminology, notations and conventions of the theory of random normed spaces as in [24–28]. 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
.
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 pointwise ordering of functions, that is,
if and only if
for all
. The maximal element for
in this order is the distribution function
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ6_HTML.gif)
Definition 1.1 (see [27]).
A mapping is called 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 [29, 30]) that, if is a
-norm and
is a given sequence of numbers in
, then
is defined recurrently by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ7_HTML.gif)
and is defined as
It is known [30] that, for the Lukasiewicz -norm, the following implication holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ8_HTML.gif)
Definition 1.2 (see [28]).
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
satisfying the following conditions:
for all
if and only if
;
for all
, and
with
;
for all
and
Every normed spaces defines a random normed space
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ9_HTML.gif)
and is the minimum
-norm. This space is called the induced random normed space.
Definition 1.3.
Let be a RN-space.
(1)A sequence in
is said to be convergent to a point
if, for any
and
, there exists a positive integer
such that
for all
.
(2)A sequence in
is called a Cauchy sequence if, for any
and
, there exists a positive integer
such that
for all
.
(3)A RN-space is said to be complete if every Cauchy sequence in
is convergent to a point in
.
Theorem 1.4 (see [27]).
If is a RN-space and
is a sequence in
such that
, then
almost everywhere.
The stability of different functional equations in fuzzy normed spaces and random normed spaces has been studied in [13, 31–42].
In this paper, we deal with the following functional equation for fixed integers with
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ10_HTML.gif)
where on random normed spaces. It is easy to see that the mapping
is a solution of the functional equation (1.10). In Section 2, we investigate the general solution of functional equation (1.10) when
is a mapping between vector spaces and, in Section 3, we establish the stability of the functional equation (1.10) in RN-spaces.
2. General Solutions
Before proceeding to the proof of Theorem 2.3 which is the main result in this section, we need the following two lemmas.
Lemma 2.1.
If an even function with
satisfies (1.10), then
is quadratic.
Proof.
Setting in (1.10), by the evenness of
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ11_HTML.gif)
Interchanging with
in (2.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ12_HTML.gif)
Letting in (1.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ13_HTML.gif)
It follows from (2.2) and (2.3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ14_HTML.gif)
According to (2.2) (2.4) and using the evenness of
, (1.10) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ15_HTML.gif)
Replacing by
in (2.5) and then using (2.2), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ16_HTML.gif)
Interchanging with
in (2.5), by the evenness of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ17_HTML.gif)
But, since , it follows from (2.6) and (2.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ18_HTML.gif)
which shows that is quadratic. This completes the proof.
Lemma 2.2.
If an odd function satisfies (1.10), then
is cubic.
Proof.
Letting in (1.10), by the oddness of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ19_HTML.gif)
Setting in (1.10) and then using (2.9), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ20_HTML.gif)
According to (2.9), (2.10) and using the oddness of , (1.10) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ21_HTML.gif)
Letting in (2.11) and using (2.9), by the oddness of
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ22_HTML.gif)
Replacing by
in (2.11), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ23_HTML.gif)
Now, replacing by
in (2.11) and using (2.12), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ24_HTML.gif)
Substituting with
in (2.11) and then
with
in (2.11), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ26_HTML.gif)
If we subtract (2.16) from (2.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ27_HTML.gif)
Interchanging with
in (2.17) and using the oddness of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ28_HTML.gif)
Thus it follows from (2.14) and (2.18) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ29_HTML.gif)
Again, substituting with
in (2.11) and then
with
in (2.11), we get, by the oddness of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ31_HTML.gif)
Then, by adding (2.20) to (2.21) and then using (2.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ32_HTML.gif)
Finally, if we compare (2.19) with (2.22), then we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ33_HTML.gif)
Therefore, is a cubic function. This completes the proof.
Theorem 2.3.
A function with
satisfies (1.10) for all
if and only if there exist functions
and
such that
for all
where the function
is symmetric for each fixed one variable and is additive for fixed two variables and
is symmetric biadditive.
Proof.
Let be a mapping with
satisfies (1.10). We decompose
into the even part and odd part by putting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ34_HTML.gif)
It is clear that for all
and it is easy to show that the functions
and
satisfy (1.10). Hence, by Lemmas 2.1 and 2.2, we know that the functions
and
are quadratic and cubic, respectively. Thus there exist a symmetric biadditive function
such that
for all
and the function
such that
for all
where the function
is symmetric for each fixed one variable and is additive for fixed two variables. Therefore, we get
for all
Conversely, let for all
where the function
is symmetric for each fixed one variable and is additive for fixed two variables and
is biadditive. By a simple computation, one can show that the functions
and
satisfy the functional equation (1.10). Thus the function
satisfies (1.10). This completes the proof.
3. Stability Problems
From now on, we suppose that is a real linear space,
is a complete RN-space and
is a function with
for which there exists a mapping
(
is denoted by
) with the property:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ35_HTML.gif)
where for all
.
Theorem 3.1.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ36_HTML.gif)
Suppose that an even function with
satisfies inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ37_HTML.gif)
where for all
. Then there exists a unique quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ38_HTML.gif)
Proof.
It follows from (3.3) and the evenness of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ39_HTML.gif)
Setting in (3.5), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ40_HTML.gif)
If we replace by
in (3.6) and divide both sides of (3.6) by 2, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ41_HTML.gif)
In other words, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ42_HTML.gif)
Therefore, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ43_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ44_HTML.gif)
This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ45_HTML.gif)
Since by the triangle inequality, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ46_HTML.gif)
In order to prove the convergence of the sequence , we replace
with
in (3.12) to find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ47_HTML.gif)
Since the right hand side of inequality (3.13) tends to as
, the sequence
is a Cauchy sequence. Therefore, we may define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ48_HTML.gif)
Now, we show that is a quadratic mapping. In fact, replacing
with
and
, respectively, in (3.3) and then taking the limit as
, we find that
satisfies (1.10) for all
Therefore, the mapping
is quadratic.
To prove (3.4) taking the limit as
in (3.12), we get (3.4).
Finally, to prove the uniqueness of the quadratic function subject to (3.4), assume that there exists a quadratic function
which satisfies (3.4). Since
and
for all
and
it follows from (3.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ49_HTML.gif)
for all and
. Therefore, by letting
in inequality (3.15), we find that
. This completes the proof.
Theorem 3.2.
Let be an odd mapping with
satisfies inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ50_HTML.gif)
where for all
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ51_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ52_HTML.gif)
then there exists a unique cubic mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ53_HTML.gif)
for all and
.
Proof.
It follows from (3.15) and the oddness of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ54_HTML.gif)
By putting in (3.20) and using
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ55_HTML.gif)
If we replace by
in (3.21) and divide both sides of (3.21) by
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ56_HTML.gif)
Letting in (3.20), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ57_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ58_HTML.gif)
It follows from (3.22) and (3.24) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ59_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ60_HTML.gif)
Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ61_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ62_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ63_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ64_HTML.gif)
Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ65_HTML.gif)
Since by the triangle inequality, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ66_HTML.gif)
In order to prove the convergence of the sequence , we replace
with
in (3.32) to find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ67_HTML.gif)
Since the right hand side of inequality (3.33) tends to as
, the sequence
is a Cauchy sequence. Therefore, we can define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ68_HTML.gif)
Now, we show that is a cubic mapping. In fact, replacing
with
and
in (3.15), respectively, and then taking the limit
, we find that
satisfies (1.10) for all
Therefore, the mapping
is cubic.
Letting the limit in (3.32), we get inequality (3.18) by (3.26).
Finally, to prove the uniqueness of the cubic function subject to inequality (3.19), assume that there exists a cubic function
which satisfies inequality (3.19). Since
and
for all
and
it follows from (3.19) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ69_HTML.gif)
for all and
. By letting
in inequality (3.35), we know that
. This completes the proof.
Theorem 3.3.
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ70_HTML.gif)
for all and
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ71_HTML.gif)
then there exist a unique quadratic mapping and a unique cubic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ72_HTML.gif)
Proof.
If for all
Then
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ73_HTML.gif)
where for all
Hence, in view of Theorem 3.1, there exists a unique quadratic function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ74_HTML.gif)
Let for all
Then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ75_HTML.gif)
where for all
From Theorem 3.2, it follows that there exists a unique cubic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F403101/MediaObjects/13660_2010_Article_2143_Equ76_HTML.gif)
Obviously, (3.38) follows from (3.40) and (3.42). This completes the proof.
References
Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1964:xvii+150.
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 TM: 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
Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991, 14(3):431–434. 10.1155/S016117129100056X
Aczél J, Dhombres J: Functional Equations in Several Variables, Encyclopedia of Mathematics and its Applications. Volume 31. Cambridge University Press, Cambridge, UK; 1989:xiv+462.
Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064
Bourgin DG: Classes of transformations and bordering transformations. Bulletin of the American Mathematical Society 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7
Gordji ME, Kaboli Gharetapeh S, Rashidi E, Karimi T, Aghaei M: Ternary Jordan derivations in -ternary algebras. Journal of Computational Analysis and Applications 2010, 12(2):463–470.
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
Găvruta P, Găvruta L: A new method for the generalized Hyers-Ulam-Rassias stability. International Journal of Nonlinear Analysis and Applications 2010, 1(2):11–18.
Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. Birkhäuser Boston, Boston, Mass, USA; 1998:vi+313.
Isac G, Rassias TM: On the Hyers-Ulam stability of -additive mappings. Journal of Approximation Theory 1993, 72(2):131–137. 10.1006/jath.1993.1010
Khodaei H, Rassias ThM: Approximately generalized additive functions in several variables. International Journal of Nonlinear Analysis and Applications 2010, 1: 22–41.
Park C, Najati A: Generalized additive functional inequalities in Banach algebras. International Journal of Nonlinear Analysis and Applications 2010, 1: 54–62.
Park C, Gordji ME: Comment on approximate ternary Jordan derivations on Banach ternary algebras [ Bavand Savadkouhi et al. J. Math. Phys. 50, 042303 (2009)]. Journal of Mathematical Physics 2010, 51:-7.
Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000, 62(1):23–130. 10.1023/A:1006499223572
Rassias TM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000, 251(1):264–284. 10.1006/jmaa.2000.7046
Kannappan P: Quadratic functional equation and inner product spaces. Results in Mathematics. Resultate der Mathematik 1995, 27(3–4):368–372.
Skof F: Local properties and approximation of operators. 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 St: 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.
Jun K-W, Kim H-M: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. Journal of Mathematical Analysis and Applications 2002, 274(2):267–278.
Chang S-S, Cho YJ, Kang SM: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science Publishers, Huntington, NY, USA; 2001:x+338.
Gordji ME, Ebadian A, Zolfaghari S: Stability of a functional equation deriving from cubic and quartic functions. Abstract and Applied Analysis 2008, 2008:-17.
Gordji ME, Khodaei H: Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces. Nonlinear Analysis 2009, 71(11):5629–5643. 10.1016/j.na.2009.04.052
Schweizer B, Sklar A: Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics. North-Holland Publishing, New York, NY, USA; 1983:xvi+275.
Sherstnev AN: On the notion of a random normed space. Doklady Akademii Nauk SSSR 1963, 149: 280–283.
Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications. Volume 536. Kluwer Academic, Dordrecht, The Netherlands; 2001:x+273.
Hadžić O, Pap E, Budinčević M: Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces. Kybernetika 2002, 38(3):363–382.
Baktash E, Cho YJ, Jalili M, Saadati R, Vaezpour SM: On the stability of cubic mappings and quadratic mappings in random normed spaces. Journal of Inequalities and Applications 2008, 2008:-11.
Gordji ME, Ghaemi MB, Majani H: Generalized Hyers-Ulam-Rassias theorem in Menger probabilistic normed spaces. Discrete Dynamics in Nature and Society 2010, 2010:-11.
Khodaei H, Kamyar M: Fuzzy approximately additive mappings. International Journal of Nonlinear Analysis and Applications 2010, 1: 44–53.
Mohamadi M, Cho YJ, Park C, Vetro F, Saadati R: Random stability on an additive-quadratic-quartic functional equation. Journal of Inequalities and Applications 2010, 2010:-18.
Miheţ D, Saadati R, Vaezpour SM: The stability of the quartic functional equation in random normed spaces. Acta Applicandae Mathematicae 2010, 110(2):797–803. 10.1007/s10440-009-9476-7
Miheţ D, Saadati R, Vaezpour SM: The stability of an additive functional equation in Menger probabilistic -normed spaces. Mathematics, Slovak. In press
Park C: Fuzzy stability of an additive-quadratic-quartic functional equation. Journal of Inequalities and Applications 2010, 2010:-22.
Park C: Fuzzy stability of additive functional inequalities with the fixed point alternative. Journal of Inequalities and Applications 2009, 2009:-17.
Park C: A fixed point approach to the fuzzy stability of an additive-quadratic-cubic functional equation. Fixed Point Theory and Applications 2009, 2009:-24.
Saadati R, Vaezpour SM, Cho YJ: Erratum: a note to paper "On the stability of cubic mappings and quartic mappings in random normed spaces". Journal of Inequalities and Applications 2009, 2009:-6.
Shakeri S, Saadati R, Park C: Stability of the quadratic functional equation in non-Archimedean −fuzzy normed spaces. International Journal of Nonlinear Analysis and Applications 2010, 1: 72–83.
Zhang S-S, Rassias JM, Saadati R: Stability of a cubic functional equation in intuitionistic random normed spaces. Applied Mathematics and Mechanics. English Edition 2010, 31(1):21–26. 10.1007/s10483-010-0103-6
Acknowledgment
This paper was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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Cho, Y., Gordji, M. & Zolfaghari, S. Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces. J Inequal Appl 2010, 403101 (2010). https://doi.org/10.1155/2010/403101
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DOI: https://doi.org/10.1155/2010/403101