Intuitionistic Fuzzy Stability of a Quadratic Functional Equation
© Liguang Wang. 2010
Received: 6 October 2010
Accepted: 23 December 2010
Published: 18 January 2011
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.
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.
We recall the following theorem of Diaz and Margolis .
Theorem 1.1 (see ).
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.
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 ,
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
It follows from (2.3) and  that is a quadratic mapping.
This completes the proof.
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).
- Ulam SM: A Collection of Mathematical Problems. Interscience Publishers, New York, NY, USA; 1960:xiii+150.MATHGoogle Scholar
- Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
- Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064MathSciNetView ArticleMATHGoogle Scholar
- Rassias TM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar
- Rassias TM: Functional Equations, Inequalities and Applications. Kluwer Academic, Dodrecht, The Netherlands; 2003:x+224.View ArticleMATHGoogle Scholar
- Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar
- Rassias TM, Semrl P: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proceedings of the American Mathematical Society 1992,114(4):989–993. 10.1090/S0002-9939-1992-1059634-1MathSciNetView ArticleMATHGoogle Scholar
- Gavruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar
- Czerwik S: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Palm Harbor, Fla, USA; 2003.MATHGoogle Scholar
- Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications. Birkhäuser, Boston, Mass, USA; 1998:vi+313.View ArticleMATHGoogle Scholar
- Diaz JB, Margolis B: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bulletin of the American Mathematical Society 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleMATHGoogle Scholar
- Cadariu I, Radu V: Fixed points and the stability of Jensen's functional equation. Journal of Inequalities in Pure and Applied Mathematics 2003, 4: 1–7.MathSciNetMATHGoogle Scholar
- Cadariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. In Iteration Theory (ECIT 02), Grazer Mathematische Berichte. Volume 346. Karl-Franzens-Universitat Graz, Graz, Austria; 2004:43–52.Google Scholar
- Park JH: Intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 2004,22(5):1039–1046. 10.1016/j.chaos.2004.02.051MathSciNetView ArticleMATHGoogle Scholar
- Saadati R, Park JH: Intuitionistic fuzzy Euclidean normed spaces. Communications in Mathematical Analysis 2006,1(2):85–90.MathSciNetMATHGoogle Scholar
- Saadati R, Park JH: On the intuitionistic fuzzy topological spaces. Chaos, Solitons and Fractals 2006,27(2):331–344. 10.1016/j.chaos.2005.03.019MathSciNetView ArticleMATHGoogle Scholar
- Shakeri S: Intuitionistic fuzzy stability of Jensen type mapping. Journal of Nonlinear Science and its Applications 2009,2(2):105–112.MathSciNetMATHGoogle Scholar
- Atanassov KT: Intuitionistic fuzzy sets. Fuzzy Sets and Systems 1986, 20: 87–96. 10.1016/S0165-0114(86)80034-3MathSciNetView ArticleMATHGoogle Scholar
- Deschrijver G, Kerre EE: On the relationship between some extensions of fuzzy set theory. Fuzzy Sets and Systems 2003,133(2):227–235. 10.1016/S0165-0114(02)00127-6MathSciNetView ArticleMATHGoogle Scholar
- Lee JR, An JS, Park C: On the stability of quadratic functional equations. Abstract and Applied Analysis 2008, 2008:-8.Google Scholar
This article is published under license to BioMed Central Ltd. 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.