- Research Article
- Open Access
On the Stability of a Generalized Quadratic and Quartic Type Functional Equation in Quasi-Banach Spaces
© M. Eshaghi Gordji et al. 2009
- Received: 31 May 2009
- Accepted: 9 September 2009
- Published: 12 October 2009
- Banach Space
- Functional Equation
- Real Vector Space
- Nonnegative Real Number
- Quadratic Functional Equation
for all Moreover, if is continuous in for each fixed then is -linear. In 1978, Th. M. Rassias  provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. The functional equation
The generalized Hyers-Ulam stability problem for the quadratic functional equation (1.3) was proved by Skof for mappings , where is a normed space and is a Banach space (see ). Cholewa  noticed that the theorem of Skof is still true if relevant domain is replaced by an abelian group. In  , Czerwik proved the generalized Hyers-Ulam stability of the functional equation (1.3). Grabiec  has generalized these results mentioned above.
In , Park and Bae considered the following quartic functional equation:
In fact, they proved that a mapping between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric multiadditive mapping such that for all . It is easy to show that satisfies the functional equation (1.5), which is called a quartic functional equation (see also ).
In addition, Kim  has obtained the generalized Hyers-Ulam stability for a mixed type of quartic and quadratic functional equation between two real linear Banach spaces. Najati and Zamani Eskandani  have established the general solution and the generalized Hyers-Ulam stability for a mixed type of cubic and additive functional equation, whenever is a mapping between two quasi-Banach spaces (see also [16, 17]).
in quasi-Banach spaces. It is easy to see that the function is a solution of the functional equation (1.6). In the present paper we investigate the general solution of the functional equation (1.6) when is a mapping between vector spaces, and we establish the generalized Hyers-Ulam stability of this functional equation whenever is a mapping between two quasi-Banach spaces.
We recall some basic facts concerning quasi-Banach space and some preliminary results.
Given a -norm, the formula gives us a translation invariant metric on X. By the Aoki-Rolewicz theorem  (see also ), each quasi-norm 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 [3, 21]) in quasi-Banach spaces.
If a mapping satisfies the functional equation (1.6), then is a quadratic and quartic mapping.
for all where the mapping is symmetric multi-additive and is bi-additive. By a simple computation, one can show that the mappings and satisfy the functional equation (1.6), so the mapping f satisfies (1.6).
From now on, let and be a quasi-Banach space with quasi-norm and a -Banach space with -norm , respectively. Let be the modulus of concavity of . In this section, using an idea of G vruta , we prove the stability of (1.6) in the spirit of Hyers, Ulam, and Rassias. For convenience we use the following abbreviation for a given mapping :
Lemma 3.1 (see ).
The proof is similar to the proof of Theorem 3.2.
The proof is similar to the proof of Theorem 3.6.
The third and corresponding author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). Also, the second author would like to thank the office of gifted students at Semnan University for its financial support.
- Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1940.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
- Aczèl J, Dhombres J: Functional Equations in Several Variables. Volume 31. Cambridge University Press, Cambridge, UK; 1989.View ArticleMATHGoogle Scholar
- Amir D: Characterizations of Inner Product Spaces. Volume 20. Birkhäuser, Basel, Switzerland; 1986.MATHGoogle Scholar
- Jordan P, von Neumann J: On inner products in linear, metric spaces. Annals of Mathematics. Second Series 1935,36(3):719–723. 10.2307/1968653MathSciNetView ArticleMATHGoogle Scholar
- Kannappan Pl: Quadratic functional equation and inner product spaces. Results in Mathematics 1995,27(3–4):368–372.MathSciNetView ArticleMATHGoogle Scholar
- Skof F: Proprietà locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle 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
- Grabiec A: The generalized Hyers-Ulam stability of a class of functional equations. Publicationes Mathematicae Debrecen 1996,48(3–4):217–235.MathSciNetMATHGoogle 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
- Chung JK, Sahoo PK: On the general solution of a quartic functional equation. Bulletin of the Korean Mathematical Society 2003,40(4):565–576.MathSciNetView ArticleMATHGoogle Scholar
- Kim H-M: On the stability problem for a mixed type of quartic and quadratic functional equation. Journal of Mathematical Analysis and Applications 2006,324(1):358–372. 10.1016/j.jmaa.2005.11.053MathSciNetView 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
- Eshaghi Gordji M: Stability of a functional equation deriving from quartic and additive functions. to appear in Bulletin of the Korean Mathematical Society to appear in Bulletin of the Korean Mathematical SocietyGoogle Scholar
- Eshaghi Gordji M, Ebadian A, Zolfaghari S: Stability of a functional equation deriving from cubic and quartic functions. Abstract and Applied Analysis 2008, 2008:-17.Google Scholar
- Benyamini Y, Lindenstrauss J: Geometric Nonlinear Functional Analysis. Vol. 1. Volume 48. American Mathematical Society, Providence, RI, USA; 2000.MATHGoogle Scholar
- Rolewicz S: Metric Linear Spaces. 2nd edition. PWN—Polish Scientific, Warsaw, Poland; 1984.MATHGoogle 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
- 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
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.