Open Access

# A Fixed Point Approach to the Fuzzy Stability of an Additive-Quadratic-Cubic Functional Equation

Fixed Point Theory and Applications20092009:918785

DOI: 10.1155/2009/918785

Accepted: 23 October 2009

Published: 19 November 2009

## Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic 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 [24]. 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 for the functional equation

(1.1)

in the fuzzy normed vector space setting.

Definition 1.1 (see [5, 911]).

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, 911]).

Let be a fuzzy normed vector space. A sequence in is said to be convergent orconverge if there exists an such that for all . In this case, is called thelimit of the sequence and we denote it by - .

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 becomplete 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 , then the sequence converges to . If is continuous at each , then is said to be continuous on (see [8]).

In 1940, Ulam [13] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

We are given a group and a metric group with metric . Given , does there exist a such that if satisfies for all , then a homomorphism exists with for all ?

By now an affirmative answer has been given in several cases, and some interesting variations of the problem have also been investigated. We will call such an an approximate homomorphism.

In 1941, Hyers [14] considered the case of approximately additive mappings , where and are Banach spaces and satisfies the Hyers inequality

(1.2)

for all . It was shown that the limit

(1.3)

exists for all and that is the unique additive mapping satisfying

(1.4)

for all .

No continuity conditions are required for this result, but if is continuous in the real variable for each fixed , then is -linear, and if is continuous at a single point of , then is also continuous.

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 Găvruţa [17] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.

In 1982–1994, a generalization of the Hyers's result was proved by J. M. Rassias. He introduced the following weaker condition:

(1.5)

for all , controlled by a product of different powers of norms, where and real numbers , and retained the condition of continuity of in for each fixed . Besides he investigated that it is possible to replace in the above Hyers inequality by a nonnegative real-valued function such that the pertinent series converges and other conditions hold and still obtain stability results. In all the cases investigated in these results, the approach to the existence question was to prove asymptotic type formulas of the form

(1.6)

Theorem 1.3 (see [1823]).

Let be a real normed linear space and a real Banach space. Assume that is an approximately additive mapping for which there exist constants and such that and satisfies the Cauchy-Rassias inequality
(1.7)
for all . Then there exists a unique additive mapping satisfying
(1.8)

for all . If, in addition, is a mapping such that is continuous in for each fixed , then is an -linear mapping.

The functional equation

(1.9)

is called aquadratic 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 [24] for mappings , where is a normed space and is a Banach space. Cholewa [25] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [26] 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 [2769]).

In [70], Jun and Kim considered the following cubic functional equation:

(1.10)

It is easy to show that the function satisfies the functional (1.10), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic 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.

Theorem 1.4 (see [71, 72]).

Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either
(1.11)

for all nonnegative integers or there exists a positive integer such that

(1) ;
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, Isac and Th. M. Rassias [73] 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 [7478]).

This paper is organized as follows. In Section 2, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubic functional (1.1) in fuzzy Banach spaces for an odd case. In Section 3, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubic functional (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-cubic mapping, that is,

(2.1)

It was shown in [79, Lemma ] that and are cubic and additive, respectively, and that .

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

(2.2)

For a given mapping , we define

(2.3)

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
(2.4)
for all . Let be an odd mapping satisfying
(2.5)
for all and all . Then
(2.6)
exists for each and defines a cubic mapping such that
(2.7)

for all and all .

Proof.

Letting in (2.5), we get
(2.8)

for all and all .

Replacing by in (2.5), we get

(2.9)

for all and all .

By (2.8) and (2.9),

(2.10)
for all and all . Letting and for all , we get
(2.11)

for all and all .

Consider the set

(2.12)
and introduce the generalized metric on :
(2.13)

where, as usual, . It is easy to show that is complete. (See the proof of Lemma of [80].)

Now we consider the linear mapping such that

(2.14)

for all .

Let be given such that . Then

(2.15)
for all and all . Hence
(2.16)
for all and all . So implies that . This means that
(2.17)

for all .

It follows from (2.11) that

(2.18)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

is a fixed point of , that is,

(2.19)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(2.20)
This implies that is a unique mapping satisfying (2.19) such that there exists a satisfying
(2.21)

for all and all .

as . This implies the equality

(2.22)

for all .

, which implies the inequality

(2.23)

This implies that inequality (2.7) holds.

By (2.5),

(2.24)
for all , all and all . So
(2.25)
for all , all and all . Since for all and all ,
(2.26)

for all and all . Thus the mapping is cubic, 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
(2.27)
for all and all . Then
(2.28)
exists for each and defines a cubic mapping such that
(2.29)

for all and all .

Proof.

The proof follows from Theorem 2.1 by taking
(2.30)

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
(2.31)
for all . Let be an odd mapping satisfying (2.5). Then
(2.32)
exists for each and defines a cubic mapping such that
(2.33)

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

(2.34)

for all .

Let be given such that . Then

(2.35)
for all and all . Hence
(2.36)
for all and all . So implies that . This means that
(2.37)

for all .

It follows from (2.11) that

(2.38)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

is a fixed point of , that is,

(2.39)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(2.40)
This implies that is a unique mapping satisfying (2.39) such that there exists a satisfying
(2.41)

for all and all .

as . This implies the equality

(2.42)

for all .

, which implies the inequality

(2.43)

This implies that the inequality (2.33) holds.

The rest of the proof is similar to that of 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 (2.27). Then
(2.44)
exists for each and defines a cubic mapping such that
(2.45)

for all and all .

Proof.

The proof follows from Theorem 2.3 by taking
(2.46)

for all . Then we can choose and we get the desired result.

Theorem 2.5.

Let be a function such that there exists an with
(2.47)
for all . Let be an odd mapping satisfying (2.5). Then
(2.48)
exists for each and defines an additive mapping such that
(2.49)

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

Letting and for all in (2.10), we get

(2.50)

for all and all .

Now we consider the linear mapping such that

(2.51)

for all .

Let be given such that . Then

(2.52)
for all and all . Hence
(2.53)
for all and all . So implies that . This means that
(2.54)

for all .

It follows from (2.50) that

(2.55)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following

is a fixed point of , that is,

(2.56)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(2.57)
This implies that is a unique mapping satisfying (2.56) such that there exists a satisfying
(2.58)

for all and all .

as . This implies the equality

(2.59)

for all ;

, which implies the inequality

(2.60)

This implies that inequality (2.49) holds.

The rest of the proof is similar to that of the proof of Theorem 2.1.

Corollary 2.6.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (2.27). Then
(2.61)
exists for each and defines an additive mapping such that
(2.62)

for all and all .

Proof.

The proof follows from Theorem 2.5 by taking
(2.63)

for all . Then we can choose and we get the desired result.

Theorem 2.7.

Let be a function such that there exists an with
(2.64)
for all . Let be an odd mapping satisfying (2.5). Then
(2.65)
exists for each and defines an additive mapping such that
(2.66)

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

(2.67)

for all .

Let be given such that . Then

(2.68)
for all and all . Hence
(2.69)
for all and all . So implies that . This means that
(2.70)

for all .

It follows from (2.50) that

(2.71)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

is a fixed point of , that is,

(2.72)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(2.73)
This implies that is a unique mapping satisfying (2.72) such that there exists a satisfying
(2.74)

for all and all .

as . This implies the equality

(2.75)

for all .

, which implies the inequality

(2.76)

This implies that inequality (2.66) holds.

The rest of the proof is similar to that of the proof of Theorem 2.1.

Corollary 2.8.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (2.27). Then
(2.77)
exists for each and defines an additive mapping such that
(2.78)

for all and all .

Proof.

The proof follows from Theorem 2.7 by taking
(2.79)

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
(3.1)
for all . Let be an even mapping satisfying and (2.5). Then
(3.2)
exists for each and defines a quadratic mapping such that
(3.3)

for all and all .

Proof.

Replacing by in (2.5), we get
(3.4)

for all and all .

It follows from (3.4) that

(3.5)

for all and all .

Consider the set

(3.6)
and introduce the generalized metric on :
(3.7)

where, as usual, . It is easy to show that is complete. (See the proof of Lemma of [80].)

Now we consider the linear mapping such that

(3.8)

for all .

Let be given such that . Then

(3.9)
for all and all . Hence
(3.10)
for all and all . So implies that . This means that
(3.11)

for all .

It follows from (3.5) that .

By Theorem 1.4, there exists a mapping satisfying the following:

is a fixed point of , that is,

(3.12)
for all . Since is even, is an even mapping. The mapping is a unique fixed point of in the set
(3.13)
This implies that is a unique mapping satisfying (3.12) such that there exists a satisfying
(3.14)

for all and all .

as . This implies the equality

(3.15)

for all .

, which implies the inequality

(3.16)

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 (2.27). Then
(3.17)
exists for each and defines a quadratic mapping such that
(3.18)

for all and all .

Proof.

The proof follows from Theorem 3.1 by taking
(3.19)

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
(3.20)
for all . Let be an even mapping satisfying and (2.5). Then
(3.21)
exists for each and defines a quadratic mapping such that
(3.22)

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

(3.23)

for all .

Let be given such that . Then

(3.24)
for all and all . Hence
(3.25)
for all and all . So implies that . This means that
(3.26)

for all .

It follows from (3.4) that

(3.27)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

is a fixed point of , that is,

(3.28)
for all . Since is even, is an even mapping. The mapping is a unique fixed point of in the set
(3.29)
This implies that is a unique mapping satisfying (3.28) such that there exists a satisfying
(3.30)

for all and all .

as . This implies the equality

(3.31)

for all .

, which implies the inequality

(3.32)

This implies that inequality (3.22) 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 (2.27). Then
(3.33)
exists for each and defines a quadratic mapping such that
(3.34)

for all and all .

Proof.

The proof follows from Theorem 3.3 by taking
(3.35)

for all . Then we can choose and we get the desired result.

## Declarations

### Acknowledgment

This work 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).

## Authors’ Affiliations

(1)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University

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