- Research Article
- Open Access
Intuitionistic Fuzzy Stability of a Quadratic Functional Equation
© Liguang Wang. 2010
- Received: 6 October 2010
- Accepted: 23 December 2010
- Published: 18 January 2011
We consider the intuitionistic fuzzy stability of the quadratic functional equation by using the fixed point alternative, where is a positive integer.
- Functional Equation
- Linear Space
- Stability Problem
- Fixed Point Theory
- Unique Fixed Point
The stability problem of functional equations originated from a question of Ulam  concerning the stability of group homomorphisms. Hyers  gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers's theorem was generalized by Aoki  for additive mappings. In 1978, Rassias  generalized Hyers theorem by obtaining a unique linear mapping near an approximate additive mapping.
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  asked whether such a theorem can also be proved for . In 1991, Gajda  gave an affirmative solution to this question when , but it was proved by Gajda  and Rassias and Semrl  that one cannot prove an analogous theorem when . In 1994, Gavruta  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 .
Theorem 1.1 (see ).
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  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 . We refer to  for the notions appeared below.
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 ).
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 ,
(b) if and only if ;
(c) for all ;
In this case, is called an intuitionistic fuzzy norm. Here, .
In this short note, we show the intuitionistic fuzzy stability of the functional equation (1.5) by using the fixed point alternative.
for all .
It follows from (2.3) and  that is a quadratic mapping.
This completes the proof.
for all . The result follows from Theorem 2.1 with .
The proof is similar to that of Theorem 2.1 and we omit it.
The proof is similar to that of Corollary 2.2.
This work was supported by the Scientific Research Fund of the Shandong Provincial Education Department (J08LI15).
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