- Research Article
- Open Access
Fixed Points and Stability of a Generalized Quadratic Functional Equation
© A. Najati and C. Park. 2009
Received: 26 November 2008
Accepted: 11 February 2009
Published: 24 February 2009
Theorem 1.1 (Th. M. Rassias ).
In 1994, a generalization of Theorems 1.1 and 1.2 was obtained by Găvruţa , who replaced the bounds and by a general control function .
In addition, J. M. Rassias  generalized the Euler-Lagrange quadratic mapping (1.12) and investigated its stability problem. The Euler-Lagrange quadratic mapping (1.12) has provided a lot of influence in the development of general Euler-Lagrange quadratic equations (mappings) which is now known as Euler-Lagrange-Rassias quadratic functional equations (mappings).
Jun and Lee  proved the generalized Hyers-Ulam stability of a pexiderized quadratic 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 [8, 20–47]). We also refer the readers to the books [48–51].
We recall the following theorem by Margolis and Diaz.
Theorem 1.3 (see ).
Throughout this paper, we assume that are nonzero rational numbers with and that is a unital Banach algebra with unit , norm , and . Assume that is a normed left -module and is a (unit linked) Banach left -module. A quadratic mapping is called -quadratic if for all and all .
and using the fixed point method (see [24, 25, 38, 53–55]), we prove the generalized Hyers-Ulam stability of -quadratic mappings in Banach -modules associated with the functional equation (1.14). In 1996, Isac and Th. M. Rassias  were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications.
2. Fixed Points and Stability of the Generalized Quadratic Functional Equation (1.14)
It is easy to show that is a generalized complete metric space .
For the case , we have the following counterexample which is a modification of the example of Czerwik .
which contradicts (2.88).
The authors would like to thank the referees for bringing some useful references to their attention. The second author was supported by Korea Research Foundation Grant KRF-2008-313-C00041.
- 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(4):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
- Rassias JM: On approximation of approximately linear mappings by linear mappings. Journal of Functional Analysis 1982,46(1):126–130. 10.1016/0022-1236(82)90048-9MathSciNetView ArticleMATHGoogle Scholar
- Rassias JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences Mathématiques 1984,108(4):445–446.MathSciNetMATHGoogle Scholar
- Rassias JM: Solution of a problem of Ulam. Journal of Approximation Theory 1989,57(3):268–273. 10.1016/0021-9045(89)90041-5MathSciNetView ArticleMATHGoogle Scholar
- Găvruţa P: On the stability of some functional equations. In Stability of Mappings of Hyers-Ulam Type, Hadronic Press Collection of Original Articles. Hadronic Press, Palm Harbor, Fla, USA; 1994:93–98.Google Scholar
- Aczél J, Dhombres J: Functional Equations in Several Variables with Applications to Mathematics, Information Theory and to the Natural and Social Sciences, Encyclopedia of Mathematics and Its Applications. Volume 31. Cambridge University Press, Cambridge, UK; 1989:xiv+462.MATHGoogle Scholar
- Amir D: Characterizations of Inner Product Spaces, Operator Theory: Advances and Applications. Volume 20. Birkhäuser, Basel, Switzerland; 1986:vi+200.View ArticleGoogle 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
- Kannappan P: Quadratic functional equation and inner product spaces. Results in Mathematics 1995,27(3–4):368–372.MathSciNetView ArticleMATHGoogle Scholar
- Skof F: Local properties and approximation of operators. 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.MathSciNetView ArticleMATHGoogle Scholar
- Rassias JM: On the stability of the Euler-Lagrange functional equation. Chinese Journal of Mathematics 1992,20(2):185–190.MathSciNetMATHGoogle Scholar
- Czerwik S: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1992,62(1):59–64. 10.1007/BF02941618MathSciNetView 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
- Rassias JM: On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces. Journal of Mathematical and Physical Sciences 1994,28(5):231–235.MathSciNetMATHGoogle Scholar
- Rassias JM: On the stability of the general Euler-Lagrange functional equation. Demonstratio Mathematica 1996,29(4):755–766.MathSciNetMATHGoogle 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
- 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
- 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
- Cădariu L, Radu V: Remarks on the stability of monomial functional equations. Fixed Point Theory 2007,8(2):201–218.MathSciNetMATHGoogle Scholar
- Cădariu 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-Universität Graz, Graz, Austria; 2004:43–52.Google 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, article 4):1–7.MATHGoogle Scholar
- Forti GL: On an alternative functional equation related to the Cauchy equation. Aequationes Mathematicae 1982,24(2–3):195–206.MathSciNetView ArticleMATHGoogle Scholar
- Forti GL: The stability of homomorphisms and amenability, with applications to functional equations. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1987,57(1):215–226. 10.1007/BF02941612MathSciNetView 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
- 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
- Găvruţa P: On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings. Journal of Mathematical Analysis and Applications 2001,261(2):543–553. 10.1006/jmaa.2001.7539MathSciNetView ArticleMATHGoogle Scholar
- Găvruţa P: On the Hyers-Ulam-Rassias stability of the quadratic mappings. Nonlinear Functional Analysis and Applications 2004,9(3):415–428.MathSciNetMATHGoogle Scholar
- Jun K-W, Kim H-M: On the Hyers-Ulam stability of a difference equation. Journal of Computational Analysis and Applications 2005,7(4):397–407.MathSciNetMATHGoogle Scholar
- Jun K-W, Kim H-M: Stability problem of Ulam for generalized forms of Cauchy functional equation. Journal of Mathematical Analysis and Applications 2005,312(2):535–547. 10.1016/j.jmaa.2005.03.052MathSciNetView ArticleMATHGoogle Scholar
- Jun K-W, Kim H-M: Stability problem for Jensen-type functional equations of cubic mappings. Acta Mathematica Sinica 2006,22(6):1781–1788. 10.1007/s10114-005-0736-9MathSciNetView ArticleMATHGoogle Scholar
- Jun K-W, Kim H-M: Ulam stability problem for a mixed type of cubic and additive functional equation. Bulletin of the Belgian Mathematical Society. Simon Stevin 2006,13(2):271–285.MathSciNetMATHGoogle Scholar
- Jun K-W, Kim H-M, Rassias JM: Extended Hyers-Ulam stability for Cauchy-Jensen mappings. Journal of Difference Equations and Applications 2007,13(12):1139–1153. 10.1080/10236190701464590MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M, Kim T-S: A fixed point approach to the stability of the cubic functional equation. Boletín de la Sociedad Matemática Mexicana 2006,12(1):51–57.MathSciNetMATHGoogle Scholar
- Jung S-M, Rassias JM: A fixed point approach to the stability of a functional equation of the spiral of Theodorus. Fixed Point Theory and Applications 2008, 2008:-7.Google 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
- Moslehian MS, Rassias ThM: Stability of functional equations in non-Archimedean spaces. Applicable Analysis and Discrete Mathematics 2007,1(2):325–334. 10.2298/AADM0702325MMathSciNetView 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: On the stability of a quartic functional equation. Journal of Mathematical Analysis and Applications 2008,340(1):569–574. 10.1016/j.jmaa.2007.08.048MathSciNetView ArticleMATHGoogle Scholar
- Najati A, Moghimi MB: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. Journal of Mathematical Analysis and Applications 2008,337(1):399–415. 10.1016/j.jmaa.2007.03.104MathSciNetView ArticleMATHGoogle 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: 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
- Ravi K, Arunkumar M, Rassias JM: Ulam stability for the orthogonally general Euler-Lagrange type functional equation. International Journal of Mathematics and Statistics 2008,3(A08):36–46.MathSciNetMATHGoogle 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
- 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
- Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003,4(1):91–96.MathSciNetMATHGoogle Scholar
- Rassias JM: Alternative contraction principle and Ulam stability problem. Mathematical Sciences Research Journal 2005,9(7):190–199.MathSciNetGoogle Scholar
- Rassias JM: Alternative contraction principle and alternative Jensen and Jensen type mappings. International Journal of Applied Mathematics & Statistics 2006,4(M06):1–10.MathSciNetGoogle 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
- Kurepa S: On the quadratic functional. Publications de l'Institut Mathématique de l'Académie Serbe des Sciences 1961, 13: 57–72.MathSciNetMATHGoogle 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.