- Research Article
- Open Access
Fuzzy Stability of an Additive-Quadratic-Quartic Functional Equation
- Choonkil Park^{1}Email author
https://doi.org/10.1155/2010/253040
© Choonkil Park. 2010
- Received: 27 August 2009
- Accepted: 30 November 2009
- Published: 12 January 2010
Abstract
Keywords
- Functional Equation
- Unique Mapping
- Additive Mapping
- Unique Fixed Point
- Normed Vector 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].
We use the definition of fuzzy normed spaces given in [5, 9, 10] to investigate a fuzzy version of the generalized Hyers-Ulam stability forthe following functional equation
in the fuzzy normed vector space setting.
Definition 1.1 (see [5, 9–11]).
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, 12].
Definition 1.2 (see [5, 9–11]).
Let be a fuzzy normed vector space. A sequence in is said to be convergent or converge if there exists an such that for all . In this case, is called thelimit 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 wellknown 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 originated from a question of Ulam [13] concerning the stability of group homomorphisms. Hyers [14] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [15] for additive mappings and by Th. M. Rassias [16] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [16] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by G vruta [17] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias' approach.
The functional equation
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be aquadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [18] for mappings , where is a normed space and is a Banach space. Cholewa [19] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [20] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [16, 21–39]).
In [40], Lee et al. considered the following quartic functional equation:
It is easy to show that the function satisfies the functional equation (1.3), which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping.
Let be a set. A function is called a generalized metric on if satisfies
We recall a fundamental result in fixed point theory.
for all nonnegative integers or there exists a positive integer such that
(2)the sequence converges to a fixed point of ;
(3) is the unique fixed point of in the set ;
In 1996, G. Isac and Th. M. Rassias [43] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [12, 44–48]).
This paper is organized as follows. In Section 2, we prove the generalized Hyers-Ulam stability of the additive-quadratic-quartic functional equation (1.1) in fuzzy Banach spaces for an odd case. In Section 3, we prove the generalized Hyers-Ulam stability of the additive-quadratic-quartic functional equation (1.1) in fuzzy Banach spaces for an even case.
Throughout this paper, assume that is a vector space and that is a fuzzy Banach space.
2. Generalized Hyers-Ulam Stability of the Functional Equation (1.1): An Odd Case
One can easily show that an odd mapping satisfies (1.1) if and only if the odd mapping mapping is an additive mapping, that is,
One can easily show that an even mapping satisfies (1.1) if and only if the even mapping is a quadratic-quartic mapping, that is,
It was shown in [49, Lemma 2.1] that and are quartic and quadratic, respectively, and that .
For a given mapping , we define
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in fuzzy Banach spaces: an odd case.
Theorem 2.1.
Proof.
Consider the set
where, as usual, . It is easy to show that is complete. (see the proof of Lemma 2.1 of [50].)
Now we consider the linear mapping such that
It follows from (2.8) that
This implies that inequality (2.7) holds.
By (2.5),
for all and all . Thus the mapping is additive, as desired.
Corollary 2.2.
Proof.
for all . Then we can choose and we get the desired result.
Theorem 2.3.
Proof.
Let be the generalized metric space defined in the proof of Theorem 2.1.
Consider the linear mapping such that
It follows from (2.8) that
This implies that the inequality (2.30) holds.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.4.
Proof.
3. Generalized Hyers-Ulam Stability of the Functional Equation (1.1):An Even Case
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in fuzzy Banach spaces: an even case.
Theorem 3.1.
Proof.
By (3.4) and (3.5),
Consider the set
where, as usual, . It is easy to show that is complete. (see the proof of Lemma 2.1 of [50]).
Now we consider the linear mapping such that
It follows from (3.7) that
This implies that inequality (3.3) holds.
The rest of the proof is similar to that of the proof of Theorem 2.1.
Corollary 3.2.
Proof.
for all . Then we can choose and we get the desired result.
Theorem 3.3.
Proof.
Let be the generalized metric space defined in the proof of Theorem 3.1.
Consider the linear mapping such that
It follows from (3.7) that
This implies that inequality (3.25) holds.
The rest of the proof is similar to that of the proof of Theorem 2.1.
Corollary 3.4.
Proof.
for all . Then we can choose and we get the desired result.
Theorem 3.5.
Proof.
Let be the generalized metric space defined in the proof of Theorem 3.1.
Letting for all in (3.6), we get
Now we consider the linear mapping such that
It follows from (3.42) that
This implies that inequality (3.41) holds.
The rest of the proof is similar to that of the proof of Theorem 2.1.
Corollary 3.6.
Proof.
for all . Then we can choose and we get the desired result.
Theorem 3.7.
Proof.
Let be the generalized metric space defined in the proof of Theorem 3.1.
Consider the linear mapping such that
It follows from (3.42) that
This implies that inequality (3.58) holds.
The rest of the proof is similar to that of the proof of Theorem 2.1.
Corollary 3.8.
Proof.
Declarations
Acknowledgment
This work was supported by the Hanyang University in 2009.
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
- Mirmostafaee AK, Moslehian MS: Fuzzy approximately cubic mappings. Information Sciences 2008, 178(19):3791–3798. 10.1016/j.ins.2008.05.032MathSciNetView ArticleMATHGoogle Scholar
- Mirzavaziri M, Moslehian MS: A fixed point approach to stability of a quadratic equation. Bulletin of the Brazilian Mathematical Society 2006, 37(3):361–376. 10.1007/s00574-006-0016-zMathSciNetView 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 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 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: Local properties and approximation of operators. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleMATHGoogle Scholar
- Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984, 27(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar
- Czerwik St: 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
- Baktash E, Cho YJ, Jalili M, Saadati R, Vaezpour SM: On the stability of cubic mappings and quadratic mappings in random normed spaces. Journal of Inequalities and Applications 2008, 2008:-11.Google Scholar
- Eshaghi-Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghri S: Stability of an additive-cubic-quartic functional equation. Advances in Difference Equations to appear to appearGoogle Scholar
- Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.MATHGoogle Scholar
- Jun K-W, Kim H-M: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. Journal of Mathematical Analysis and Applications 2002, 274(2):267–278.MathSciNetView ArticleGoogle Scholar
- Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
- Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. Bulletin des Sciences Mathématiques 2008, 132(2):87–96.View ArticleMATHMathSciNetGoogle Scholar
- Park C, Cui J: Generalized stability of -ternary quadratic mappings. Abstract and Applied Analysis 2007, 2007:-6.Google Scholar
- Park C, Najati A: Homomorphisms and derivations in -algebras. Abstract and Applied Analysis 2007, 2007:-12.Google Scholar
- Rassias JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences Mathématiques 1984, 108(4):445–446.MATHMathSciNetGoogle Scholar
- Rassias JM: Refined Hyers-Ulam approximation of approximately Jensen type mappings. Bulletin des Sciences Mathématiques 2007, 131(1):89–98.View ArticleMATHMathSciNetGoogle Scholar
- Rassias JM, Rassias MJ: Asymptotic behavior of alternative Jensen and Jensen type functional equations. Bulletin des Sciences Mathématiques 2005, 129(7):545–558.View ArticleMATHMathSciNetGoogle Scholar
- Rassias ThM: Problem 16; 2, report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae 1990, 39(2–3):292–293.Google Scholar
- Rassias ThM: On the stability of the quadratic functional equation and its applications. Studia Universitatis Babes-Bolyai 1998, 43(3):89–124.MATHMathSciNetGoogle 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, 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
- Lee SH, Im SM, Hwang IS: Quartic functional equations. Journal of Mathematical Analysis and Applications 2005, 307(2):387–394. 10.1016/j.jmaa.2004.12.062MathSciNetView ArticleMATHGoogle Scholar
- Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. Journal of Inequalities in Pure and Applied Mathematics 2003., 4(1):Google 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
- Isac G, Rassias ThM: Stability of -additive mappings: applications to nonlinear analysis. International Journal of Mathematics and Mathematical Sciences 1996, 19(2):219–228. 10.1155/S0161171296000324MathSciNetView ArticleMATHGoogle Scholar
- Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. In Iteration Theory, Grazer Mathematische Berichte. Volume 346. Karl-Franzens-Universitaet, Graz, Austria; 2004:43–52.Google Scholar
- Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Applications 2008, 2008:-15.Google Scholar
- Park C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory and Applications 2007, 2007:-15.Google Scholar
- Park C: Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach. Fixed Point Theory and Applications 2008, 2008:-9.Google Scholar
- Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4(1):91–96.MathSciNetMATHGoogle Scholar
- Eshaghi-Gordji M, Abbaszadeh S, Park C: On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces. Journal of Inequalities and Applications 2009, 2009:-26.Google Scholar
- Miheţ D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. Journal of Mathematical Analysis and Applications 2008, 343(1):567–572.MathSciNetView 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.