- Research Article
- Open Access
Fuzzy Stability of Quadratic Functional Equations
- JungRye Lee^{1},
- Sun-Young Jang^{2},
- Choonkil Park^{3} and
- DongYun Shin^{4}Email author
https://doi.org/10.1155/2010/412160
© Jung Rye Lee et al. 2010
- Received: 10 February 2010
- Accepted: 11 April 2010
- Published: 17 May 2010
Abstract
The fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated by Moslehian et al. In this paper, we prove the generalized Hyers-Ulam stability of the following quadratic functional equations and in fuzzy Banach spaces.
Keywords
- Functional Equation
- Quadratic Mapping
- Unique Mapping
- Quadratic Functional Equation
- Fuzzy Normed Space
1. Introduction and Preliminaries
Katsaras [1] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [2–4]. In particular, Bag and Samanta [5], following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [7]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [8].
in the fuzzy normed vector space setting, where are nonzero real numbers with .
Definition 1.1 (see [5, 9, 10]).
Let be a real vector space. A function is called a fuzzy norm on if, for all and all ,
is a nondecreasing function of and ,
The pair is called a fuzzy normed vector space.
The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [9, 10].
Definition 1.2 (see [5, 9, 10]).
Let be a fuzzy normed vector space. A sequence in is said to be convergent or converges if there exists an such that for all . In this case, is called the limit of the sequence and we denote it by - .
Definition 1.3 (see [5, 9, 10]).
Let be a fuzzy normed vector space. A sequence in is called Cauchy if for each and each there exists an such that, for all and all , we have .
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.
We say that a mapping between fuzzy normed vector spaces and is continuous at a point if, for each sequence converging to in , the sequence converges to . If is continuous at each , then is said to be continuous on (see [8]).
The stability problem of functional equations is originated from a question of Ulam [11] concerning the stability of group homomorphisms. Hyers [12] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [13] for additive mappings and by Th. M. Rassias [14] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [14] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [15] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [16] for mappings , where is a normed space and is a Banach space. Cholewa [17] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. In [18], Czerwik proved the generalized Hyers-Ulam stability of the quadratic functional equation. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [19–31]).
This paper is organized as follows. In Section 2, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.1) in fuzzy Banach spaces. In Section 3, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.2) in fuzzy Banach spaces.
Throughout this paper, assume that is a vector space and that is a fuzzy Banach space. Let be nonzero real numbers with .
2. Generalized Hyers-Ulam Stability of the Quadratic Functional Equation (1.1)
In this section, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.1) in fuzzy Banach spaces.
Theorem 2.1.
Proof.
for all and all . So we get (2.7) for .
for all and all . Thus the sequence is Cauchy in . Since is a fuzzy Banach space, the sequence converges to some . So we can define a mapping by - ; namely, for each and , .
for all . Since for all , by , for all . Thus the mapping is quadratic, that is, for all .
It follows that for all . Thus for all .
Corollary 2.2.
Proof.
Define and apply Theorem 2.1 to get the result.
Similarly, we can obtain the following. We will omit the proof.
Theorem 2.3.
Corollary 2.4.
Proof.
3. Generalized Hyers-Ulam Stability of the Quadratic Functional Equation (1.2)
In this section, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.2) in fuzzy Banach spaces.
Lemma 3.1.
Proof.
Theorem 3.2.
Proof.
for all and all . So we get (3.15) for .
for each and all . Thus the sequence is Cauchy in . Since is a fuzzy Banach space, the sequence converges to some . So we can define a mapping by - ; namely, for each and , .
for all . Since for all , by , for all . By Lemma 3.1, the mapping is quadratic.
It follows that for all . Thus for all .
Corollary 3.3.
Proof.
Authors’ Affiliations
References
- Katsaras AK: Fuzzy topological vector spaces. II. Fuzzy Sets and Systems 1984,12(2):143-154. 10.1016/0165-0114(84)90034-4MathSciNetView ArticleMATHGoogle Scholar
- Felbin C: Finite-dimensional fuzzy normed linear space. Fuzzy Sets and Systems 1992,48(2):239-248. 10.1016/0165-0114(92)90338-5MathSciNetView ArticleMATHGoogle Scholar
- Krishna SV, Sarma KKM: Separation of fuzzy normed linear spaces. Fuzzy Sets and Systems 1994,63(2):207-217. 10.1016/0165-0114(94)90351-4MathSciNetView ArticleMATHGoogle Scholar
- Xiao J-Z, Zhu X-H: Fuzzy normed space of operators and its completeness. Fuzzy Sets and Systems 2003,133(3):389-399. 10.1016/S0165-0114(02)00274-9MathSciNetView ArticleMATHGoogle Scholar
- Bag T, Samanta SK: Finite dimensional fuzzy normed linear spaces. Journal of Fuzzy Mathematics 2003,11(3):687-705.MathSciNetMATHGoogle Scholar
- Cheng SC, Mordeson JN: Fuzzy linear operators and fuzzy normed linear spaces. Bulletin of the Calcutta Mathematical Society 1994,86(5):429-436.MathSciNetMATHGoogle Scholar
- Kramosil I, Michálek J: Fuzzy metrics and statistical metric spaces. Kybernetika 1975,11(5):336-344.MathSciNetMATHGoogle Scholar
- Bag T, Samanta SK: Fuzzy bounded linear operators. Fuzzy Sets and Systems 2005,151(3):513-547. 10.1016/j.fss.2004.05.004MathSciNetView ArticleMATHGoogle Scholar
- Mirmostafaee AK, Mirzavaziri M, Moslehian MS: Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 2008,159(6):730-738. 10.1016/j.fss.2007.07.011MathSciNetView ArticleMATHGoogle Scholar
- Mirmostafaee AK, Moslehian MS: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets and Systems 2008,159(6):720-729. 10.1016/j.fss.2007.09.016MathSciNetView ArticleMATHGoogle Scholar
- Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.Google 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 ArticleGoogle 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 ThM: 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
- Găvruţa 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
- Skof F: Proprietà locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano 1983,53(1):113-129. 10.1007/BF02924890MathSciNetView ArticleMATHGoogle Scholar
- Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1-2):76-86. 10.1007/BF02192660MathSciNetView ArticleMATHGoogle Scholar
- Czerwik S: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1992, 62: 59-64. 10.1007/BF02941618MathSciNetView ArticleMATHGoogle Scholar
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.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
- Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.Google Scholar
- Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
- Rassias ThM: New characterizations of inner product spaces. Bulletin des Sciences Mathématiques 1984,108(1):95-99.MATHGoogle Scholar
- Rassias ThM: On the stability of the quadratic functional equation and its applications. Studia Universitatis Babeş-Bolyai. Mathematica 1998,43(3):89-124.MATHGoogle Scholar
- Rassias ThM: The problem of S. M. Ulam for approximately multiplicative mappings. Journal of Mathematical Analysis and Applications 2000,246(2):352-378. 10.1006/jmaa.2000.6788MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000,251(1):264-284. 10.1006/jmaa.2000.7046MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000,62(1):23-130. 10.1023/A:1006499223572MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: Problem 16; 2, report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae 1990, 39: 292-293.Google Scholar
- Rassias ThM, Šemrl 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
- Rassias ThM, Šemrl P: On the Hyers-Ulam stability of linear mappings. Journal of Mathematical Analysis and Applications 1993,173(2):325-338. 10.1006/jmaa.1993.1070MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM, Shibata K: Variational problem of some quadratic functionals in complex analysis. Journal of Mathematical Analysis and Applications 1998,228(1):234-253. 10.1006/jmaa.1998.6129MathSciNetView ArticleMATHGoogle Scholar
Copyright
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.