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Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces
Journal of Inequalities and Applications volume 2009, Article number: 527462 (2009)
Abstract
We obtain 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 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
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%2F527462/MediaObjects/13660_2009_Article_1973_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%2F527462/MediaObjects/13660_2009_Article_1973_Equ2_HTML.gif)
for all Moreover if
is continuous in
for each fixed
then
is linear. In
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]).
Jun and Kim [13] introduced the following cubic functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ3_HTML.gif)
and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.3) The function
satisfies the functional equation (1.3)
which is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic function. Jun and Kim proved that a function
between real vector spaces X and Y is a solution of (1.3) 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.
Park and Bea [14] introduced the following quartic functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ4_HTML.gif)
In fact they proved that a function between real vector spaces
and
is a solution of (1.4) if and only if there exists a unique symmetric multiadditive function
such that
for all
(see also [15–18]). It is easy to show that the function
satisfies the functional equation (1.4)
which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic function.
In the sequel we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [19–21]. Throughout this paper, is the space of distribution functions that is, the space of all mappings
, such that
is leftcontinuous 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
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%2F527462/MediaObjects/13660_2009_Article_1973_Equ5_HTML.gif)
Definition 1.1 (see [20]).
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 [22, 23]) that if
is a
-norm and
is a given sequence of numbers in
,
is defined recurrently by
and
for
is defined as
It is known [23] that for the Lukasiewicz
-norm the following implication holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ6_HTML.gif)
Definition 1.2 (see [21]).
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:
for all
if and only if
;
for all
,
;
for all
and
Every normed spaces defines a random normed space
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ7_HTML.gif)
for all 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
in
if, for every
and
, there exists positive integer
such that
whenever
.
(2)A sequence in
is called Cauchy sequence if, for every
and
, there exists positive integer
such that
whenever
.
(3)A 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 [20]).
If is an RN-space and
is a sequence such that
, then
almost everywhere.
The generalized Hyers-Ulam-Rassias stability of different functional equations in random normed spaces has been recently studied in [24–29]. Recently, Eshaghi Gordji et al. [30] established the stability of mixed type cubic and quartic functional equations (see also [31]). In this paper we deal with the following functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ8_HTML.gif)
on random normed spaces. It is easy to see that the function is a solution of the functional equation (1.8)
In the present paper we establish the stability of the functional equation (1.8) in random normed spaces.
2. Main Results
From now on, we suppose that is a real linear space,
is a complete RN-space, and
is a function with
for which there is
(
denoted by
) with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ9_HTML.gif)
for all and all
Theorem 2.1.
Let be odd and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ10_HTML.gif)
for all and all
, then there exists a unique cubic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ11_HTML.gif)
for all and all
Proof.
Setting in (2.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ12_HTML.gif)
for all If we replace
in (2.4) by
and divide both sides of (2.4) by 3, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ13_HTML.gif)
for all and all
Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ14_HTML.gif)
for all and all
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ15_HTML.gif)
for all and all
Therefore we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ16_HTML.gif)
for all and all
As
by the triangle inequality it follows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ17_HTML.gif)
for all and
In order to prove the convergence of the sequence
, we replace
with
in (2.9) to find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ18_HTML.gif)
Since the right-hand side of the inequality tends to as
and
tend to infinity, the sequence
is a Cauchy sequence. Therefore, we may define
for all
. Now, we show that
is a cubic map. Replacing
with
and
respectively in (2.1)
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ19_HTML.gif)
Taking the limit as , we find that
satisfies (1.8) for all
Therefore the mapping
is cubic.
To prove (2.3) take the limit as
in (2.9)
Finally, to prove the uniqueness of the cubic function
subject to (2.3)
let us assume that there exists a cubic function
which satisfies (2.3)
Since
and
for all
and
from (2.3) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ20_HTML.gif)
for all and all
. By letting
in above inequality, we find that
.
Theorem 2.2.
Let be even and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ21_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%2F527462/MediaObjects/13660_2009_Article_1973_Equ22_HTML.gif)
for all and all
Proof.
By putting in (2.1)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ23_HTML.gif)
for all Replacing
in (2.15) by
to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ24_HTML.gif)
for all and all
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ25_HTML.gif)
for all and all
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ26_HTML.gif)
for all and all
So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ27_HTML.gif)
for all and all
As
by the triangle inequality it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ28_HTML.gif)
for all and
We replace
with
in (2.20) to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ29_HTML.gif)
Since the right-hand side of the inequality tends to as
and
tend to infinity, the sequence
is a Cauchy sequence. Therefore, we may define
for all
. Now, we show that
is a quartic map. Replacing
with
and
respectively, in (2.1)
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ30_HTML.gif)
Taking the limit as , we find that
satisfies (1.8) for all
Hence, the mapping
is quartic.
To prove (2.14) take the limit as
in (2.20)
Finally, to prove the uniqueness property of
subject to (2.14)
let us assume that there exists a quartic function
which satisfies (2.14)
Since
and
for all
and
from (2.14) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ31_HTML.gif)
for all and all
. Taking the limit as
, we find that
.
Theorem 2.3.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ32_HTML.gif)
for all and all
, then there exist a unique cubic mapping
and a unique quartic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ33_HTML.gif)
for all and all
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ34_HTML.gif)
for all Then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ35_HTML.gif)
for all Hence, in view of Theorem 2.1, there exists a unique quartic function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ36_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ37_HTML.gif)
for all Then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ38_HTML.gif)
for all From Theorem 2.2, it follows that there exists a unique cubic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F527462/MediaObjects/13660_2009_Article_1973_Equ39_HTML.gif)
Obviously, (2.25) follows from (2.28) and (2.31).
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The second author would like to thank the Office of Gifted Students at Semnan University for its financial support.
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Eshaghi Gordji, M., Savadkouhi, M.B. Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces. J Inequal Appl 2009, 527462 (2009). https://doi.org/10.1155/2009/527462
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DOI: https://doi.org/10.1155/2009/527462