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Fuzzy Stability of an Additive-Quadratic-Quartic Functional Equation
Journal of Inequalities and Applications volume 2010, Article number: 253040 (2010)
Abstract
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-quartic functional equation: in fuzzy Banach spaces.
1. Introduction and Preliminaries
Katsaras [1] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [2–4]. In particular, Bag and Samanta [5], following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [7]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [8].
We use the definition of fuzzy normed spaces given in [5, 9, 10] to investigate a fuzzy version of the generalized Hyers-Ulam stability forthe following functional equation

in the fuzzy normed vector space setting.
Definition 1.1 (see [5, 9–11]).
Let be a real vector space. A function
is called a fuzzy norm on
if for all
and all
,
for
;
if and only if
for all
;
if
;
;
is a nondecreasing function of
and
;
for ,
is continuous on
.
The pair is called a fuzzy normed vector space.
The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [9, 12].
Definition 1.2 (see [5, 9–11]).
Let be a fuzzy normed vector space. A sequence
in
is said to be convergent or converge if there exists an
such that
for all
. In this case,
is called thelimit of the sequence
and we denote it by
-
.
Definition 1.3 (see [5, 9, 10]).
Let be a fuzzy normed vector space. A sequence
in
is called Cauchy if for each
and each
there exists an
such that for all
and all
, we have
.
It is wellknown that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.
We say that a mapping between fuzzy normed vector spaces
and
is continuous at a point
if for each sequence
converging to
in
, the sequence
converges to
. If
is continuous at each
, then
is said to be continuous on
(see [8]).
The stability problem of functional equations originated from a question of Ulam [13] concerning the stability of group homomorphisms. Hyers [14] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [15] for additive mappings and by Th. M. Rassias [16] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [16] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Gvruta [17] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias' approach.
The functional equation

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be aquadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [18] for mappings , where
is a normed space and
is a Banach space. Cholewa [19] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an Abelian group. Czerwik [20] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [16, 21–39]).
In [40], Lee et al. considered the following quartic functional equation:

It is easy to show that the function satisfies the functional equation (1.3), which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping.
Let be a set. A function
is called a generalized metric on
if
satisfies
(1) if and only if
;
(2) for all
;
(3) for all
.
We recall a fundamental result in fixed point theory.
Let be a complete generalized metric space and let
be a strictly contractive mapping with Lipschitz constant
. Then for each given element
, either

for all nonnegative integers or there exists a positive integer
such that
(1);
(2)the sequence converges to a fixed point
of
;
(3) is the unique fixed point of
in the set
;
(4) for all
.
In 1996, G. Isac and Th. M. Rassias [43] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [12, 44–48]).
This paper is organized as follows. In Section 2, we prove the generalized Hyers-Ulam stability of the additive-quadratic-quartic functional equation (1.1) in fuzzy Banach spaces for an odd case. In Section 3, we prove the generalized Hyers-Ulam stability of the additive-quadratic-quartic functional equation (1.1) in fuzzy Banach spaces for an even case.
Throughout this paper, assume that is a vector space and that
is a fuzzy Banach space.
2. Generalized Hyers-Ulam Stability of the Functional Equation (1.1): An Odd Case
One can easily show that an odd mapping satisfies (1.1) if and only if the odd mapping mapping
is an additive mapping, that is,

One can easily show that an even mapping satisfies (1.1) if and only if the even mapping
is a quadratic-quartic mapping, that is,

It was shown in [49, Lemma  2.1] that and
are quartic and quadratic, respectively, and that
.
For a given mapping , we define

for all .
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in fuzzy Banach spaces: an odd case.
Theorem 2.1.
Let be a function such that there exists an
with

for all . Let
be an odd mapping satisfying

for all and all
. Then

exists for each and defines an additive mapping
such that

for all and all
.
Proof.
Letting in (2.5), we get

for all and all
.
Consider the set

and introduce the generalized metric on

where, as usual, . It is easy to show that
is complete. (see the proof of Lemma  2.1 of [50].)
Now we consider the linear mapping such that

for all .
Let be given such that
. Then

for all and all
. Hence

for all and all
. So
implies that
. This means that

for all .
It follows from (2.8) that

for all and all
. So
.
By Theorem 1.4, there exists a mapping satisfying the following.
-
(1)
is a fixed point of
, that is,

for all . Since
is odd,
is an odd mapping. The mapping
is a unique fixed point of
in the set

This implies that is a unique mapping satisfying (2.16) such that there exists a
satisfying

for all and all
.
-
(2)
as
. This implies the equality

for all ;
-
(3)
, which implies the inequality

This implies that inequality (2.7) holds.
By (2.5),

for all , all
and all
. So

for all , all
and all
. Since
for all
and all
,

for all and all
. Thus the mapping
is additive, as desired.
Corollary 2.2.
Let and let
be a real number with
. Let
be a normed vector space with norm
. Let
be an odd mapping satisfying

for all and all
. Then

exists for each and defines an additive mapping
such that

for all and all
.
Proof.
The proof follows from Theorem 2.1 by taking

for all . Then we can choose
and we get the desired result.
Theorem 2.3.
Let be a function such that there exists an
with

for all . Let
be an odd mapping satisfying (2.5). Then

exists for each and defines an additive mapping
such that

for all and all
.
Proof.
Let be the generalized metric space defined in the proof of Theorem 2.1.
Consider the linear mapping such that

for all .
Let be given such that
. Then

for all and all
. Hence

for all and all
. So
implies that
. This means that

for all .
It follows from (2.8) that

for all and all
. So
.
By Theorem 1.4, there exists a mapping satisfying the following.
-
(1)
is a fixed point of
, that is,

for all . Since
is odd,
is an odd mapping. The mapping
is a unique fixed point of
in the set

This implies that is a unique mapping satisfying (2.36) such that there exists a
satisfying

for all and all
.
-
(2)
as
. This implies the equality

for all ;
-
(3)
, which implies the inequality

This implies that the inequality (2.30) holds.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.4.
Let and let
be a real number with
. Let
be a normed vector space with norm
. Let
be an odd mapping satisfying (2.24). Then

exists for each and defines an additive mapping
such that

for all and all
.
Proof.
The proof follows from Theorem 2.3 by taking

for all . Then we can choose
and we get the desired result.
3. Generalized Hyers-Ulam Stability of the Functional Equation (1.1):An Even Case
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in fuzzy Banach spaces: an even case.
Theorem 3.1.
Let be a function such that there exists an
with

for all . Let
be an even mapping satisfying
and (2.5). Then

exists for each and defines a quartic mapping
such that

for all and all
.
Proof.
Letting in (2.5), we get

for all and all
.
Replacing by
in (2.5), we get

for all and all
.
By (3.4) and (3.5),

for all and all
. Letting
for all
, we get

for all and all
.
Consider the set

and introduce the generalized metric on

where, as usual, . It is easy to show that
is complete. (see the proof of Lemma  2.1 of [50]).
Now we consider the linear mapping such that

for all .
Let be given such that
. Then

for all and all
. Hence

for all and all
. So
implies that
. This means that

for all .
It follows from (3.7) that

for all and all
. So
.
By Theorem 1.4, there exists a mapping satisfying the following.
-
(1)
is a fixed point of
, that is,

for all . Since
is even,
is an even mapping. The mapping
is a unique fixed point of
in the set

This implies that is a unique mapping satisfying (3.15) such that there exists a
satisfying

for all and all
.
-
(2)
as
. This implies the equality

for all .
-
(3)
, which implies the inequality

This implies that inequality (3.3) holds.
The rest of the proof is similar to that of the proof of Theorem 2.1.
Corollary 3.2.
Let and let
be a real number with
. Let
be a normed vector space with norm
. Let
be an even mapping satisfying
and (2.24). Then

exists for each and defines a quartic mapping
such that

for all and all
.
Proof.
The proof follows from Theorem 3.1 by taking

for all . Then we can choose
and we get the desired result.
Theorem 3.3.
Let be a function such that there exists an
with

for all . Let
be an even mapping satisfying
and (2.5). Then

exists for each and defines a quartic mapping
such that

for all and all
.
Proof.
Let be the generalized metric space defined in the proof of Theorem 3.1.
Consider the linear mapping such that

for all .
Let be given such that
. Then

for all and all
. Hence

for all and all
. So
implies that
. This means that

for all .
It follows from (3.7) that

for all and all
. So
.
By Theorem 1.4, there exists a mapping satisfying the following.
-
(1)
is a fixed point of
, that is,

for all . Since
is even,
is an even mapping. The mapping
is a unique fixed point of
in the set

This implies that is a unique mapping satisfying (3.31) such that there exists a
satisfying

for all and all
.
-
(2)
as
. This implies the equality

for all ;
-
(3)
, which implies the inequality

This implies that inequality (3.25) holds.
The rest of the proof is similar to that of the proof of Theorem 2.1.
Corollary 3.4.
Let and let
be a real number with
. Let
be a normed vector space with norm
. Let
be an even mapping satisfying
and (2.24). Then

exists for each and defines a quartic mapping
such that

for all and all
.
Proof.
The proof follows from Theorem 3.3 by taking

for all . Then we can choose
and we get the desired result.
Theorem 3.5.
Let be a function such that there exists an
with

for all . Let
be an even mapping satisfying
and (2.5). Then

exists for each and defines a quadratic mapping
such that

for all and all
.
Proof.
Let be the generalized metric space defined in the proof of Theorem 3.1.
Letting for all
in (3.6), we get

for all and all
.
Now we consider the linear mapping such that

for all .
Let be given such that
. Then

for all and all
. Hence

for all and all
. So
implies that
. This means that

for all .
It follows from (3.42) that

for all and all
. So
.
By Theorem 1.4, there exists a mapping satisfying the following.
-
(1)
is a fixed point of
, that is,

for all . Since
is even,
is an even mapping. The mapping
is a unique fixed point of
in the set

This implies that is a unique mapping satisfying (3.48) such that there exists a
satisfying

for all and all
.
-
(2)
as
. This implies the equality

for all .
-
(3)
, which implies the inequality

This implies that inequality (3.41) holds.
The rest of the proof is similar to that of the proof of Theorem 2.1.
Corollary 3.6.
Let and let
be a real number with
. Let
be a normed vector space with norm
. Let
be an even mapping satisfying
and (2.24). Then

exists for each and defines a quadratic mapping
such that

for all and all
.
Proof.
The proof follows from Theorem 3.5 by taking

for all . Then we can choose
and we get the desired result.
Theorem 3.7.
Let be a function such that there exists an
with

for all . Let
be an even mapping satisfying
and (2.5). Then

exists for each and defines a quadratic mapping
such that

for all and all
.
Proof.
Let be the generalized metric space defined in the proof of Theorem 3.1.
Consider the linear mapping such that

for all .
Let be given such that
. Then

for all and all
. Hence

for all and all
. So
implies that
. This means that

for all .
It follows from (3.42) that

for all and all
. So
.
By Theorem 1.4, there exists a mapping satisfying the following.
-
(1)
is a fixed point of
, that is,

for all . Since
is even,
is an even mapping. The mapping
is a unique fixed point of
in the set

This implies that is a unique mapping satisfying (3.64) such that there exists a
satisfying

for all and all
.
-
(2)
as
. This implies the equality

for all .
-
(3)
, which implies the inequality

This implies that inequality (3.58) holds.
The rest of the proof is similar to that of the proof of Theorem 2.1.
Corollary 3.8.
Let and let
be a real number with
. Let
be a normed vector space with norm
. Let
be an even mapping satisfying
and (2.24). Then

exists for each and defines a quadratic mapping
such that

for all and all
.
Proof.
The proof follows from Theorem 3.7 by taking

for all . Then we can choose
and we get the desired result.
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Acknowledgment
This work was supported by the Hanyang University in 2009.
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Park, C. Fuzzy Stability of an Additive-Quadratic-Quartic Functional Equation. J Inequal Appl 2010, 253040 (2010). https://doi.org/10.1155/2010/253040
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DOI: https://doi.org/10.1155/2010/253040
Keywords
- Functional Equation
- Unique Mapping
- Additive Mapping
- Unique Fixed Point
- Normed Vector Space