Intuitionistic Fuzzy Stability of a Quadratic Functional Equation
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Research Article Intuitionistic Fuzzy Stability of a Quadratic Functional Equation Liguang Wang School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China Correspondence should be addressed to Liguang Wang, [email protected] Received 6 October 2010; Accepted 23 December 2010 Academic Editor: B. Rhoades Copyright q 2010 Liguang Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the intuitionistic fuzzy stability of the quadratic functional equation fkx yfkx − y 2k 2 fx 2fy by using the fixed point alternative, where k 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 E1 and E2 are real-normed spaces with E2 complete, f : E1 → E2 is a mapping such that for each fixed x ∈ E1 , the mapping t → ftx is continuous on R, and there exist ε > 0 and p ∈ 0, 1 such that f x y − fx − f y ≤ ε xp yp
1.1
for all x, y ∈ E1 . Then there is a unique linear mapping T : E1 → E2 such that fx − T x ≤ for all x ∈ E1 .
2 xp |2 − 2p |
1.2
2
Fixed Point Theory and Applications
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 p ≥ 1. In 1991, Gajda 6 gave an affirmative solution to this question when p > 1, but it was proved by Gajda 6 and Rassias and Semrl 7 that one cannot prove an analogous theorem when p 1. In 1994, Gavruta 8 provided a generalization of Rassias theorem in which he replaced the bound εxp yp by a general control function φx, y. 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 X be a set. A function d : X × X → 0, ∞ is called a generalized metric on X if d satisfies 1 dx, y 0 if and only if x y; 2 dx, y dy, x for all x, y ∈ X; 3 dx, z ≤ dx, y dy, z for all x, y, z ∈ X. We recall the following theorem of Diaz and Margolis 11. Theorem 1.1 see 11. Let X, d be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant 0 < α < 1. Then for each x ∈ X, either d J n x, J n1 x ∞
1.3
for all nonnegative integers n or there exists a nonnegative integer n0 such that 1 dJ n x, J n1 x < ∞ fo
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