A Fixed Point Approach to the Stability of a Quadratic Functional Equation in -Algebras
© Mohammad B. Moghimi et al. 2009
Received: 18 May 2009
Accepted: 31 July 2009
Published: 19 August 2009
We use a fixed point method to investigate the stability problem of the quadratic functional equation in -algebras.
1. Introduction and Preliminaries
for all . Aoki  and Th. M. Rassias  provided a generalization of the Hyers' theorem for additive mappings and for linear mappings, respectively, by allowing the Cauchy difference to be unbounded (see also ).
Theorem 1.1 (Th. M. Rassias).
for all . If then inequality (1.2) holds for and (1.4) for . Also, if for each the mapping is continuous in , then is -linear.
The Hyers-Ulam stability problem for the quadratic functional equation (1.5) was studied 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 we replace by an Abelian group. Czerwik  proved the generalized Hyers-Ulam stability of the quadratic functional equation (1.5). Grabiec  has generalized these results mentioned above. Jun and Lee  proved the generalized Hyers-Ulam stability of a Pexiderized quadratic functional equation.
Let be a set. A function is called a generalized metric on if satisfies
if and only if ;
for all ;
We recall the following theorem by Margolis and Diaz.
Theorem 1.2 (see ).
for all nonnegative integers or there exists a non-negative integer such that
for all ;
the sequence converges to a fixed point of ;
is the unique fixed point of in the set ;
for all .
2. Solutions of (1.8)
Let be a linear space. If a mapping satisfies and the functional equation (1.8), then is quadratic.
for all Hence is quadratic.
A quadratic mapping does not satisfy (1.8) in general. Let be the mapping defined by for all It is clear that is quadratic and that does not satisfy (1.8).
Let be a linear space. If a mapping satisfies the functional equation (1.8), then there exists a symmetric biadditive mapping such that for all
3. Generalized Hyers-Ulam Stability of (1.8) in -Algebras
for all where is a linear space.
Moreover, if is continuous in for each fixed , then is -quadratic, that is, for all and all
is a generalized complete metric space .
Let be the mapping defined by
for all By Theorem 2.1, the function is quadratic.
Moreover, if is continuous in for each fixed , then by the same reasoning as in the proof of  is -quadratic.
for all . Moreover, if is continuous in for each fixed , then is -quadratic.
The following theorem is an alternative result of Theorem 3.1 and we will omit the proof.
for all , where is defined as in Theorem 3.1. Moreover, if is continuous in for each fixed , then is -quadratic.
for all . Moreover, if is continuous in for each fixed , then is -quadratic.
which contradicts (3.33).
The third author was supported by Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00041).
- Ulam SM: A Collection of Mathematical Problems. Interscience Publishers, New York, NY, USA; 1960:xiii+150.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 ArticleGoogle 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/00210064MATHMathSciNetView ArticleGoogle 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/S0002-9939-1978-0507327-1MATHMathSciNetView ArticleGoogle Scholar
- Bourgin DG: Classes of transformations and bordering transformations. Bulletin of the American Mathematical Society 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7MATHMathSciNetView ArticleGoogle 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.1994.1211MATHMathSciNetView ArticleGoogle 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, Graz, Austria; 2004:43–52.Google Scholar
- Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1–2):76–86.MATHMathSciNetView ArticleGoogle 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/BF02941618MATHMathSciNetView ArticleGoogle 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/S0002-9947-02-03036-2MATHMathSciNetView ArticleGoogle Scholar
- Grabiec A: The generalized Hyers-Ulam stability of a class of functional equations. Publicationes Mathematicae Debrecen 1996,48(3–4):217–235.MATHMathSciNetGoogle Scholar
- Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Mathematicae 1992,44(2–3):125–153. 10.1007/BF01830975MATHMathSciNetView ArticleGoogle 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/S0161171296000324MATHMathSciNetView ArticleGoogle 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.MATHMathSciNetView ArticleGoogle 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.MATHMathSciNetGoogle Scholar
- Kannappan Pl: Quadratic functional equation and inner product spaces. Results in Mathematics 1995,27(3–4):368–372.MATHMathSciNetView ArticleGoogle 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-zMATHMathSciNetView ArticleGoogle 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)00386-4MATHMathSciNetView ArticleGoogle 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-AMATHMathSciNetView ArticleGoogle 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.2000.7046MATHMathSciNetView ArticleGoogle 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 ArticleGoogle Scholar
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.MATHView ArticleGoogle 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.Google 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: Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.MATHView ArticleGoogle 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. 10.2307/1968653MATHMathSciNetView ArticleGoogle Scholar
- Skof F: Local properties and approximation of operators. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890MATHMathSciNetView ArticleGoogle 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/S0002-9904-1968-11933-0MATHMathSciNetView ArticleGoogle Scholar
- Hyers DH, Isac G, Rassias ThM: Topics in Nonlinear Analysis & Applications. World Scientific, River Edge, NJ, USA; 1997:xiv+699.MATHView ArticleGoogle Scholar
- Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431–434. 10.1155/S016117129100056XMATHMathSciNetView 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.