Open Access

Intuitionistic Fuzzy Stability of a Quadratic Functional Equation

Fixed Point Theory and Applications20112010:107182

https://doi.org/10.1155/2010/107182

Received: 6 October 2010

Accepted: 23 December 2010

Published: 18 January 2011

Abstract

We consider the intuitionistic fuzzy stability of the quadratic functional equation by using the fixed point alternative, where is a positive integer.

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers's theorem was generalized by Aoki [3] for additive mappings. In 1978, Rassias [4] generalized Hyers theorem by obtaining a unique linear mapping near an approximate additive mapping.

Assume that and are real-normed spaces with complete, is a mapping such that for each fixed , the mapping is continuous on , and there exist and such that
(1.1)
for all . Then there is a unique linear mapping such that
(1.2)

for all .

The paper of Rassias has provided a lot of influence in the development of what we called the generalized Hyers-Ulam-Rassias stability of functional equations. In 1990, Rassias [5] asked whether such a theorem can also be proved for . In 1991, Gajda [6] gave an affirmative solution to this question when , but it was proved by Gajda [6] and Rassias and Semrl [7] that one cannot prove an analogous theorem when . In 1994, Gavruta [8] provided a generalization of Rassias theorem in which he replaced the bound by a general control function . Since then several stability problems for various functional equations have been investigated by many mathematicians [9, 10].

In the following, we first recall some fundamental results in the fixed point theory.

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 the following theorem of Diaz and Margolis [11].

Theorem 1.1 (see [11]).

Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each , either
(1.3)

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

(1) for all ;

(2)the sequence converges to a fixed point of ;

(3) is the unique fixed point of J in the set ;

(4) for all .

In 2003, Cadariu and Radu used the fixed-point method to the investigation of the Jensen functional equation (see [12, 13]) for the first time. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors.

Using the idea of intuitionistic fuzzy metric spaces introduced by Park [14] and Saadati and Park [15, 16], a new notion of intuitionistic fuzzy metric spaces with the help of the notion of continuous t-representable was introduced by Shakeri [17]. We refer to [17] for the notions appeared below.

Consider the set and the order relation defined by
(1.4)

Then is a complete lattice [18, 19].

A binary operation is said to be a continuous -norm if it satisfies the following conditions: (a) is associative and commutative; (b) is continuous; (c) for all ; (d) whenever and for each .

An intuitionistic fuzzy set in a universal set is an object , where, for all , and are called the membership degree and the nonmembership degree, respectively, of and, furthermore, they satisfy .

A triangular norm ( -norm) on is a mapping satisfying the following conditions: , (a) (boundary condition); (b) (commutativity); (c) (associativity); (d) and (monotonicity).

If is an abelian topological monoid with unit , then is said to be a continuous -norm.

The definitions of an intuitionistic fuzzy normed space is given below (see [17]).

Definition 1.2.

Let and be the membership and the nonmembership degree of an intuitionistic fuzzy set from to such that for all and . The triple is said to be an intuitionistic fuzzy normed space (briefly IFN-space) if is a vector space, is a continuous -representable, and is a mapping satisfying the following conditions: for all and ,

(a) ;

(b) if and only if ;

(c) for all ;

(d) .

In this case, is called an intuitionistic fuzzy norm. Here, .

Throughout this paper, we assume that is a fixed positive integer. The functional equation
(1.5)

was considered in [20]. Suppose and are vector spaces. It is proved in [20] that a mapping satisfies (1.5) if and only if it satisfies .

In this short note, we show the intuitionistic fuzzy stability of the functional equation (1.5) by using the fixed point alternative.

2. Main Results

For a given mapping , we define
(2.1)

for all .

Theorem 2.1.

Let be a linear space, an IFN-space, and a function such that for some ,
(2.2)
(2.3)
for all and . Let be a complete IFN-space. If is a mapping such that ,
(2.4)
and , then there is a unique quadratic mapping such that
(2.5)

Proof.

Put in (2.4), we have
(2.6)
for all and . Consider the set and define a generalized metric on by
(2.7)
It is easy to show that is complete. Define by for all . It is not difficult to see that
(2.8)
for all . It follows from (2.6) that
(2.9)
It follows from Theorem 1.1 that has a fixed point in the set . Let be the fixed point of . It follows from that
(2.10)
for all . Since ,
(2.11)
It follows from (2.4) that we have
(2.12)

It follows from (2.3) and [20] that is a quadratic mapping.

The uniqueness of follows from the fact that is the unique fixed point of with the property that
(2.13)

This completes the proof.

Corollary 2.2.

Let . Let be a linear space, an IFN-space, and a complete IFN-space. Suppose . If is a mapping such that ,
(2.14)
and , then there is a unique quadratic mapping such that
(2.15)

Proof.

Let
(2.16)

for all . The result follows from Theorem 2.1 with .

Theorem 2.3.

Let be a linear space, an IFN-space, and a function such that for some ,
(2.17)
for all and . Let be a complete IFN-space. If is a mapping such that ,
(2.18)
and , then there is a unique quadratic mapping such that
(2.19)

Proof.

The proof is similar to that of Theorem 2.1 and we omit it.

Corollary 2.4.

Let . Let be a linear space, an IFN-space, and a complete IFN-space. If is a mapping such that ,
(2.20)
and , then there is a unique quadratic mapping such that
(2.21)

Proof.

The proof is similar to that of Corollary 2.2.

Declarations

Acknowledgment

This work was supported by the Scientific Research Fund of the Shandong Provincial Education Department (J08LI15).

Authors’ Affiliations

(1)
School of Mathematical Sciences, Qufu Normal University

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© Liguang Wang. 2010

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