- Research Article
- Open Access
Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces
© W.-G. Park and J.-H. Bae. 2010
- Received: 27 April 2010
- Accepted: 18 June 2010
- Published: 5 July 2010
- Banach Space
- Functional Equation
- Stability Problem
- Additive Mapping
- Product Space
In 1940, Ulam  gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems, containing the stability problem of homomorphisms as follows
is called the quadratic functional equation whose solution is said to be a quadratic mapping. A generalized stability problem for the quadratic functional equation was proved by Skof  for mappings , where is a normed space and is a Banach space. Cholewa  noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik  proved the generalized stability of the quadratic functional equation, and Park  proved the generalized stability of the quadratic functional equation in Banach modules over a -algebra.
Given a -norm, the formula gives us a translation invariant metric on . By the Aoki-Rolewicz theorem  (see also ), each quasinorm is equivalent to some -norm. Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms. In , Tabor has investigated a version of Hyers-Rassias-Gajda theorem (see also [3, 12]) in quasi-Banach spaces. Since then, the stability problems have been investigated by many authors (see [13–18]).
The authors  solved the solutions of (1.4) and (1.5) as follows.
In this paper, we investigate the generalized Hyers-Ulam stability of (1.4) and (1.5) in quasi-Banach spaces.
We will use the following lemma in order to prove Theorem 2.4.
Lemma 2.3 (see ).
- Ulam SM: A Collection of Mathematical Problems. Interscience, New York, NY, USA; 1968.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
- 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
- Park C: On the stability of the quadratic mapping in Banach modules. Journal of Mathematical Analysis and Applications 2002, 276(1):135–144. 10.1016/S0022-247X(02)00387-6MathSciNetView ArticleMATHGoogle Scholar
- Benyamini Y, Lindenstrauss J: Geometric Nonlinear Functional Analysis. 1, American Mathematical Society Colloquium Publications. Volume 48. American Mathematical Society, Providence, RI, USA; 2000:xii+488.MATHGoogle Scholar
- Rolewicz S: Metric Linear Spaces. 2nd edition. PWN-Polish Scientific, Warsaw, Poland; 1984:xi+459.Google Scholar
- Tabor J: Stability of the Cauchy functional equation in quasi-Banach spaces. Annales Polonici Mathematici 2004, 83(3):243–255. 10.4064/ap83-3-6MathSciNetView 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
- 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
- Najati A: Homomorphisms in quasi-Banach algebras associated with a Pexiderized Cauchy-Jensen functional equation. Acta Mathematica Sinica 2009, 25(9):1529–1542. 10.1007/s10114-009-7648-zMathSciNetView ArticleMATHGoogle Scholar
- Najati A, Eskandani GZ: Stability of a mixed additive and cubic functional equation in quasi-Banach spaces. Journal of Mathematical Analysis and Applications 2008, 342(2):1318–1331. 10.1016/j.jmaa.2007.12.039MathSciNetView ArticleMATHGoogle Scholar
- Najati A, Moradlou F: Stability of a quadratic functional equation in quasi-Banach spaces. Bulletin of the Korean Mathematical Society 2008, 45(3):587–600. 10.4134/BKMS.2008.45.3.587MathSciNetView 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
- Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. Bulletin des Sciences Mathématiques 2008, 132(2):87–96.View ArticleMathSciNetMATHGoogle Scholar
- Park W-G, Bae J-H: On a bi-quadratic functional equation and its stability. Nonlinear Analysis: Theory, Methods & Applications 2005, 62(4):643–654. 10.1016/j.na.2005.03.075MathSciNetView 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
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.