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Stabilities of Cubic Mappings in Fuzzy Normed Spaces
Advances in Difference Equations volume 2010, Article number: 150873 (2010)
Abstract
Rassias(2001) introduced the pioneering cubic functional equation in the history of mathematical analysis: and solved the pertinent famous Ulam stability problem for this inspiring equation. This Rassias cubic functional equation was the historic transition from the following famous Euler-Lagrange-Rassias quadratic functional equation:
to the cubic functional equations. In this paper, we prove the Ulam-Hyers stability of the cubic functional equation:
in fuzzy normed linear spaces. We use the definition of fuzzy normed linear spaces to establish a fuzzy version of a generalized Hyers-Ulam-Rassias stability for above equation in the fuzzy normed linear space setting. The fuzzy sequentially continuity of the cubic mappings is discussed.
1. Introduction
Studies on fuzzy normed linear spaces are relatively recent in the field of fuzzy functional analysis. The notion of fuzzyness has a wide application in many areas of science. In 1984, Katsaras [1] first introduced a definition of fuzzy norm on a linear space. Later, several notions of fuzzy norm have been introduced and discussed from different points of view [2, 3]. Concepts of sectional fuzzy continuous mappings and strong uniformly convex fuzzy normed linear spaces have been introduced by Bag and Samanta [4]. Bag and Samanta [5] introduced a notion of boundedness of a linear operator between fuzzy normed spaces, and studied the relation between fuzzy continuity and fuzzy boundedness. They studied boundedness of linear operators over fuzzy normed linear spaces such as fuzzy continuity, sequential fuzzy continuity, weakly fuzzy continuity and strongly fuzzy continuity.
The problem of stability of functional equation originated from a question of Ulam [6] concerning the stability of group homomorphism in 1940. Hyers gave a partial affirmative answer to the question of Ulam for Banach spaces in the next year [7]. Let and
be Banach spaces. Assume that
satisfies
for all
and some
. Then, there exists a unique additive mapping
such that
for all
.
theorem was generalized by Aoki [8] for additive mappings. In 1978, a generalized solution for approximately linear mappings was given by Th. M. Rassias [9]. He considered a mapping
satisfying the condition

for all , where
and
. This result was later extended to all
.
In 1982, J. M. Rassias [10] gave a further generalization of the result of Hyers and prove the following theorem using weaker conditions controlled by a product of powers of norms. Let be a mapping from a normed vector space
into a Banach space
subject to the inequality

for all , where
and
. Then there exists a unique additive mapping
which satisfies

for all . The above mentioned stability involving a product of powers of norms is called Ulam–Gavruta–Rassias stability by various authors [11–25].
In 2008, J. M. Rassias [26] generalized even further the above two stabilities via a new stability involving a mixed product-sum of powers of norms, called JMRassias stability by several authors [27–30].
In the last two decades, several form of mixed type functional equation and its Ulam–Hyers stability are dealt in various spaces like Fuzzy normed spaces, Random normed spaces, Quasi–Banach spaces, Quasinormed linear spaces and Banach algebra by various authors like [31–40].
In 1994, Cheng and Mordeson [2] introduced an idea of a fuzzy norm on a linear space whose associated metric is Kramosil and Michálek type [41]. Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view [42–44].
In 2001, J. M. Rassias [45] introduced the pioneering cubic functional equation in history of mathematical analysis, as follows:

and solved the famous Ulam stability problem for this inspiring functional equation. Note that this cubic functional equation (*) was the historic transition from the following famous Euler-Lagrange quadratic functional equation:

to the cubic functional equation (*).
The notion of fuzzy stability of the functional equations was initiated by Mirmostafaee and Moslehian in [46]. Later, several various fuzzy versions of stability were investigated [47, 48]. Now, let us introduce the following functional equation:

Since the cubic function satisfies in this equation, so we promise that (1.5) is called a cubic functional equation and every solution will be called a cubic function. The stability problem for the cubic functional equation was proved by Wiwatwanich and Nakmahachalasint [49] for mapping
, where
and
are real Banach spaces. A number of mathematicians worked on the stability of some types of the cubic equation [45, 50–54]. In [55], Park and Jung introduced a cubic functional equation different from (1.5) as follows:

and investigated the generalized Hyers-Ulam-Rassias stability for this equation on abelian groups. They also obtained results in sense of Hyers-Ulam stability and Hyers-Ulam-Rassias stability. A number of results concerning the stability of different functional equations can be found in [23, 56–59].
In this paper, we prove the Hyers-Ulam-Rassias stability of the cubic functional equation (1.5) in fuzzy normed spaces. Later, we will show that there exists a close relationship between the fuzzy sequentially continuity behavior of a cubic function, control function and the unique cubic mapping which approximates the cubic map.
2. Notation and Preliminary Results
In this section some definitions and preliminary results are given which will be used in this paper. Following [48], we give the following notion of a fuzzy norm.
Definition 2.1.
Let be a linear space. A fuzzy subset
of
into
is called a fuzzy norm on
if for every
and
(N1) for
,
(N2) if and only if
for all
,
(N3) if
,
(N4) ,
(N5) is a non-decreasing function on
and
.
The pair will be referred to as a fuzzy normed linear space. One may regard
as the truth value of the statement "the norm of
is less than or equal to the real number r ". Let
be a normed linear space. One can be easily verify that

is a fuzzy norm on . Other examples of fuzzy normed linear spaces are considered in the main text of this paper.
Note that the fuzzy normed linear space is exactly a Menger probabilistic normed linear space
where
[60].
Definition 2.2.
A sequence in a fuzzy normed space
converges to
(one denote
) if for every
and
, there exists a positive integer
such that
whenever
.
Recall that, a sequence in
is called Cauchy if for every
and
, there exists a positive integer
such that for all
and all
, we have
. It is known that every convergent sequence in a fuzzy normed space is Cauchy. The fuzzy normed space
is said to be fuzzy Banach space if every Cauchy sequence in
is convergent to a point in
[46].
3. Main Results
We will investigate the generalized Hyers-Ulam type theorem of the functional equation (1.5) in fuzzy normed spaces. In the following theorem, we will show that under special circumstances on the control function , every
-almost cubic mapping
can be approximated by a cubic mapping
.
Theorem 3.1.
Let . Let
be a linear space, and let
be a fuzzy normed space. Suppose that an even function
satisfies
for all
and for all
. Suppose that
is a fuzzy Banach space. If a function
satisfies

for all and
, then there exists a unique cubic function
which satisfies (1.5) and the inequality

holds for all and
.
Proof.
We have the following two cases.
Case 1 ().
Replacing by
in (3.1) and summing the resulting inequality with (3.1), we get

Since (3.1) and (3.3) hold for any , let us fix
for convenience. By (
), we have

Replacing by
in (3.4). By (
), we have

Replacing by
in (3.5), we get

It follows from

and last inequality that

In order to prove convergence of the sequence , we replace
by
to find that for
,

Replacing by
in last inequality to get

For every and
, we put

Replacing by
in last inequality, we observe that

Let and
be given. Since
, there is some
such that
for every
. The convergence of the series
gives some
such that
for every
and
. For every
and
, we have

This shows that is a Cauchy sequence in the fuzzy Banach space
, therefore this sequence converges to some point
. Fix
and put
in (3.13) to obtain

For every ,

The first two terms on the right hand side of the above inequality tend to 1 as . Therefore we have

for large enough. By last inequality, we have

Now, we show that is cubic. Use inequality (3.1) with
replaced by
and
by
to find that

On the other hand , hence by (
)

We conclude that fulfills (1.5). It remains to prove the uniqueness assertion. Let
be another cubic mapping satisfying (3.17). Fix
. Obviously

for all . For every
, we can write

Since , we have

Therefore for all
, whence
.
Case 2 ().
We can state the proof in the same pattern as we did in the first case. Replace ,
by
and
, respectively in (3.4) to get

We replace and
by
and
in last inequality, respectively, we find that

For each , one can deduce

where . It is easy to see that
is a Cauchy sequence in (
). Since
is a fuzzy Banach space, this sequence converges to some point
, that is,

Moreover, satisfies (1.5) and

The proof for uniqueness of for this case proceeds similarly to that in the previous case, hence it is omitted.
We note that need not be equal to 27. But we do not guarantee whether the cubic equation is stable in the sense of Hyers, Ulam and Rassias if
is assumed in Theorem 3.1.
Remark 3.2.
Let . Suppose that the mapping
from
into
is right continuous. Then we get a fuzzy approximation better than (3.17) as follows.
For every , we have

for large enough . It follows that

Tending to zero we infer

From Theorem 3.1, we obtain the following corollary concerning the stability of (1.5) in the sense of the JMRassias stability of functional equations controlled by the mixed product-sum of powers of norms introduced by J. M. Rassias [26] and called JMRassias stability by several authors [27–30].
Corollary 3.3.
Let be a Banach space and let
be a real number. Suppose that a function
satisfies

for all where
. Then there exists a unique cubic function
which satisfying (1.5) and the inequality

for all . The function
is given by
for all
Proof.
Define by

It is easy to see that is a fuzzy Banach space. Denote by
the map sending each
to
. By assumption,

Note that given by

is a fuzzy norm on . By Theorem 3.1, there exists a unique cubic function
satisfies (1.5) and inequality

for all and
. Consequently,
.
Definition 3.4.
Let be a mapping where
and
are fuzzy normed spaces.
is said to be sequentially fuzzy continuous at
if for any
satisfying
implies
. If
is sequentially fuzzy continuous at each point of
, then
is said to be sequentially fuzzy continuous on
.
For the various definitions of continuity and also defining a topology on a fuzzy normed space we refer the interested reader to [61, 62]. Now we examine some conditions under which the cubic mapping found in Theorem 3.1 to be continuous. In the following theorem, we investigate fuzzy sequentially continuity of cubic mappings in fuzzy normed spaces. Indeed, we will show that under some extra conditions on Theorem 3.1, the cubic mapping is fuzzy sequentially continuous.
Theorem 3.5.
Denote the fuzzy norm obtained as Corollary 3.3 on
. Suppose that conditions of Theorem 3.1 hold. If for every
the mappings
(from
into
and
(from
into
are sequentially fuzzy continuous, then the mapping
is sequentially continuous and
for all
.
Proof.
We have the following case.
Case 1 ().
Let be a sequence in
that converges to some
, and let
. Let
be given. Since
,

there is such that

It follows form (3.17) and (3.38) that

By the sequentially fuzzy continuity of maps and
, we can find some
such that for any
,

and

Hence by last inequality and (3.38), we get

On the other hand,

Hence by last inequality and (3.42), we obtain

Therefore it follows from (3.44), (3.40) and (3.39) that for every ,

Therefore for every choice ,
and
, we can find some
such that
for every
. This shows that
.
The proof for proceeds similarly to that in the previous case.
It is not hard to see that for every rational number
. Since
is a fuzzy sequentially continuous map, by the same reasoning as the proof of [46], the cubic function
satisfies
for every
.
The following corollary is the Hyers-Ulam stability [7] of (1.5).
Corollary 3.6.
Let be a Banach space, and let
be a real number. Suppose that a function
satisfies

for all . Then there exists a unique cubic function
which satisfies (1.5) and the inequality

for all . Moreover, if for each fixed
the mapping
from
to
is fuzzy sequentially continuous, then
for all
.
Proof.
Denote and
the fuzzy norms obtained as Corollary 3.3 on
and
, respectively. This time we choose
. By Theorem 3.1, there exists a unique cubic function
which satisfies the inequality

for all . It follows that
. The rest of proof is an immediate consequence of Theorem 3.5.
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Ghaffari, A., Alinejad, A. Stabilities of Cubic Mappings in Fuzzy Normed Spaces. Adv Differ Equ 2010, 150873 (2010). https://doi.org/10.1155/2010/150873
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DOI: https://doi.org/10.1155/2010/150873
Keywords
- Banach Space
- Functional Equation
- Linear Space
- Cauchy Sequence
- Ulam Stability