Approximately Quadratic Mappings on Restricted Domains
© Abbas Najati and Soon-Mo Jung. 2010
Received: 16 September 2010
Accepted: 20 December 2010
Published: 23 December 2010
We introduce a generalized quadratic functional equation , where , are nonzero real numbers with . We show that this functional equation is quadratic if , are rational numbers. We also investigate its stability problem on restricted domains. These results are applied to study of an asymptotic behavior of these generalized quadratic mappings.
Under what conditions does there exist a group homomorphism near an approximate group homomorphism? This question concerning the stability of group homomorphisms was posed by Ulam . The case of approximately additive mappings was solved by Hyers  on Banach spaces. In 1950 Aoki  provided a generalization of the Hyers' theorem for additive mappings and in 1978 Th. M. Rassias  generalized the Hyers' theorem for linear mappings by allowing the Cauchy difference to be unbounded (see also ). The result of Rassias' theorem has been generalized by Găvruţa  who permitted the Cauchy difference to be bounded by a general control function. This stability concept is also applied to the case of other functional equations. For more results on the stability of functional equations, see [7–24]. We also refer the readers to the books in [25–29].
where , are nonzero real numbers with . So, it is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known that a function between real vector spaces and is quadratic if and only if there exists a unique symmetric biadditive function such that for all (see [13, 25, 27]).
We prove that the functional equations (1.1) and (1.2) are equivalent if , are nonzero rational numbers. The functional equation (1.1) is a spacial case of (1.2). Indeed, for the case in (1.2), we get (1.1).
In 1983 Skof  was the first author to solve the Hyers-Ulam problem for additive mappings on a restricted domain (see also [31–33]). In 1998 Jung  investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains (see also [35–37]). J. M. Rassias  investigated the Hyers-Ulam stability of mixed type mappings on restricted domains.
2. Solutions of (1.2)
In this section we show that the functional equation (1.2) is equivalent to the quadratic equation (1.1). That is, every solution of (1.2) is a quadratic function. We recall that , are nonzero real numbers with .
for all , in . It easily follows from (2.6) that is additive, that is, for all . So if is a rational number, then for all in . Therefore, it follows from (2.1) that for all in . Since , are nonzero, we infer that .
for all . Therefore is an inner product space (see ).
3. Stability of (1.2) on Restricted Domains
In this section, we investigate the Hyers-Ulam stability of the functional equation (1.2) on a restricted domain. As an application we use the result to the study of an asymptotic behavior of that equation. It should be mentioned that Skof  was the first author who treats the Hyers-Ulam stability of the quadratic equation. Czerwik  proved a Hyers-Ulam-Rassias stability theorem on the quadratic equation. As a particular case he proved the following theorem.
The result follows from Theorems 3.1 and 3.3.
Skof  has proved an asymptotic property of the additive mappings and Jung  has proved an asymptotic property of the quadratic mappings (see also ). We prove such a property also for the quadratic mappings.
for all . Since is a monotonically decreasing sequence, the quadratic mapping satisfies (3.20) for all . The uniqueness of implies for all . Hence, by letting in (3.20), we conclude that is quadratic.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2010-0007143).
- 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
- Bourgin DG: Classes of transformations and bordering transformations. Bulletin of the American Mathematical Society 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7MathSciNetView 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
- Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984, 27(1–2):76–86.MathSciNetView 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
- Faĭziev VA, Rassias ThM, Sahoo PK: The space of -additive mappings on semigroups. Transactions of the American Mathematical Society 2002, 354(11):4455–4472. 10.1090/S0002-9947-02-03036-2MathSciNetView ArticleMATHGoogle Scholar
- Forti GL: An existence and stability theorem for a class of functional equations. Stochastica 1980, 4(1):23–30. 10.1080/17442508008833155MathSciNetView ArticleMATHGoogle Scholar
- Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequationes Mathematicae 1995, 50(1–2):143–190. 10.1007/BF01831117MathSciNetView ArticleMATHGoogle Scholar
- Grabiec A: The generalized Hyers-Ulam stability of a class of functional equations. Publicationes Mathematicae Debrecen 1996, 48(3–4):217–235.MathSciNetMATHGoogle Scholar
- Kannappan P: Quadratic functional equation and inner product spaces. Results in Mathematics 1995, 27(3–4):368–372.MathSciNetView ArticleMATHGoogle Scholar
- Jordan P, von Neumann J: On inner products in linear, metric spaces. Annals of Mathematics 1935, 36(3):719–723. 10.2307/1968653MathSciNetView ArticleMATHGoogle Scholar
- Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Mathematicae 1992, 44(2–3):125–153. 10.1007/BF01830975MathSciNetView 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
- Jun K-W, Lee Y-H: On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality. Mathematical Inequalities & Applications 2001, 4(1):93–118.MathSciNetView ArticleMATHGoogle Scholar
- Najati A: Hyers-Ulam stability of an -Apollonius type quadratic mapping. Bulletin of the Belgian Mathematical Society. Simon Stevin 2007, 14(4):755–774.MathSciNetMATHGoogle Scholar
- Najati A, Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation. Journal of Mathematical Analysis and Applications 2007, 335(2):763–778. 10.1016/j.jmaa.2007.02.009MathSciNetView ArticleMATHGoogle Scholar
- Najati A, Park C: The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms between -algebras. Journal of Difference Equations and Applications 2008, 14(5):459–479. 10.1080/10236190701466546MathSciNetView ArticleMATHGoogle Scholar
- Park C-G: On the stability of the linear mapping in Banach modules. Journal of Mathematical Analysis and Applications 2002, 275(2):711–720. 10.1016/S0022-247X(02)00386-4MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: On a modified Hyers-Ulam sequence. Journal of Mathematical Analysis and Applications 1991, 158(1):106–113. 10.1016/0022-247X(91)90270-AMathSciNetView 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: 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
- Aczél J, Dhombres J: Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications. Volume 31. Cambridge University Press, Cambridge, UK; 1989:xiv+462.View ArticleMATHGoogle Scholar
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.View 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.MATHGoogle 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 (Ed): Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.MATHGoogle Scholar
- Skof F: Sull' approssimazione delle applicazioni localmente -additive. Atti della Accademia delle Scienze di Torino 1983, 117: 377–389.MathSciNetGoogle Scholar
- Hyers DH, Isac G, Rassias ThM: On the asymptoticity aspect of Hyers-Ulam stability of mappings. Proceedings of the American Mathematical Society 1998, 126(2):425–430. 10.1090/S0002-9939-98-04060-XMathSciNetView ArticleMATHGoogle Scholar
- Jung S-M: Hyers-Ulam-Rassias stability of Jensen's equation and its application. Proceedings of the American Mathematical Society 1998, 126(11):3137–3143. 10.1090/S0002-9939-98-04680-2MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M, Moslehian MS, Sahoo PK: Stability of a generalized Jensen equation on restricted domains. Journal of Mathematical Inequalities 2010, 4: 191–206.MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M: On the Hyers-Ulam stability of the functional equations that have the quadratic property. Journal of Mathematical Analysis and Applications 1998, 222(1):126–137. 10.1006/jmaa.1998.5916MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M: Stability of the quadratic equation of Pexider type. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 2000, 70: 175–190. 10.1007/BF02940912View ArticleMathSciNetMATHGoogle Scholar
- Jung S-M, Kim B: On the stability of the quadratic functional equation on bounded domains. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1999, 69: 293–308. 10.1007/BF02940881MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M, Sahoo PK: Hyers-Ulam stability of the quadratic equation of Pexider type. Journal of the Korean Mathematical Society 2001, 38(3):645–656.MathSciNetMATHGoogle Scholar
- Rassias JM: On the Ulam stability of mixed type mappings on restricted domains. Journal of Mathematical Analysis and Applications 2002, 276(2):747–762. 10.1016/S0022-247X(02)00439-0MathSciNetView ArticleMATHGoogle Scholar
- Skof F: Proprieta' locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53(1):113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle 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.