Intuitionistic Fuzzy Stability of a Quadratic Functional Equation

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.