## Abstract and Applied Analysis

### On the Generalized Hyers-Ulam-Rassias Stability of Quadratic Functional Equations

#### Abstract

We achieve the general solution and the generalized Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities for quadratic functional equations $f(ax+by)+f(ax-by)=(b(a+b)/2)f(x+y)+$ $(b(a+b)/2)f(x-y)+$ $(2{a}^{2}-ab-{b}^{2})f(x)+$ $({b}^{2}-ab)f(y)$ where $a, b$ are nonzero fixed integers with $b{\,\neq\,}\pm a,-3a$, and $f(ax+by)+f(ax-by)=2{a}^{2}f(x)+2{b}^{2}f(y)$ for fixed integers $a, b$ with $a,b{\,\neq\,}0$ and $a\pm b{\,\neq\,}0$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 923476, 11 pages.

Dates
First available in Project Euclid: 16 March 2010

https://projecteuclid.org/euclid.aaa/1268745570

Digital Object Identifier
doi:10.1155/2009/923476

Mathematical Reviews number (MathSciNet)
MR2516012

Zentralblatt MATH identifier
1167.39014

#### Citation

Gordji, M. Eshaghi; Khodaei, H. On the Generalized Hyers-Ulam-Rassias Stability of Quadratic Functional Equations. Abstr. Appl. Anal. 2009 (2009), Article ID 923476, 11 pages. doi:10.1155/2009/923476. https://projecteuclid.org/euclid.aaa/1268745570

#### References

• S. M. Ulam, Problems in Modern Mathematics, chapter 6, John Wiley & Sons, New York, NY, USA, 1940.
• D. H. Hyers, On the stability of the linear functional equation,'' Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222--224, 1941.
• T. Aoki, On the stability of the linear transformation in Banach spaces,'' Journal of the Mathematical Society of Japan, vol. 2, pp. 64--66, 1950.
• Th. M. Rassias, On the stability of the linear mapping in Banach spaces,'' Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297--300, 1978.
• J. M. Rassias, On approximation of approximately quadratic mappings by quadratic mappings,'' Annales Mathematicae Silesianae, no. 15, pp. 67--78, 2001.
• J. M. Rassias, On approximation of approximately linear mappings by linear mappings,'' Journal of Functional Analysis, vol. 46, no. 1, pp. 126--130, 1982.
• J. M. Rassias, On approximation of approximately linear mappings by linear mappings,'' Bulletin des Sciences Mathématiques, vol. 108, no. 4, pp. 445--446, 1984.
• J. M. Rassias, On a new approximation of approximately linear mappings by linear mappings,'' Discussiones Mathematicae, vol. 7, pp. 193--196, 1985.
• J. M. Rassias, Solution of a problem of Ulam,'' Journal of Approximation Theory, vol. 57, no. 3, pp. 268--273, 1989.
• J. M. Rassias, Solution of a stability problem of Ulam,'' Discussiones Mathematicae, vol. 12, pp. 95--103, 1992.
• P. Găvruţa, An answer to a question of John M. Rassias concerning the stability of Cauchy equation,'' in Advances in Equations and Inequalities, Hadronic Mathematics, pp. 67--71, Hadronic Press, Palm Harbor, Fla, USA, 1999.
• Z. Gajda, On stability of additive mappings,'' International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431--434, 1991.
• P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,'' Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431--436, 1994.
• K. Ravi, M. Arunkumar, and J. M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation,'' International Journal of Mathematics and Statistics, vol. 3, no. A08, pp. 36--46, 2008.
• K.-W. Jun, H.-M. Kim, and J. M. Rassias, Extended Hyers-Ulam stability for Cauchy-Jensen mappings,'' Journal of Difference Equations and Applications, vol. 13, no. 12, pp. 1139--1153, 2007.
• H.-M. Kim, K.-W. Jun, and J. M. Rassias, Extended stability problem for alternative Cauchy-Jensen mappings,'' Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 4, article 120, pp. 1--17, 2007.
• C.-G. Park and J. M. Rassias, Hyers-Ulam stability of an Euler-Lagrange type additive mapping,'' International Journal of Applied Mathematics & Statistics, vol. 7, no. Fe07, pp. 112--125, 2007.
• J. M. Rassias, Solution of a quadratic stability Hyers-Ulam type problem,'' Ricerche di Matematica, vol. 50, no. 1, pp. 9--17, 2001.
• J. M. Rassias, On the stability of a multi-dimensional Cauchy type functional equation,'' in Geometry, Analysis and Mechanics, pp. 365--376, World Scientific, River Edge, NJ, USA, 1994.
• J. M. Rassias, On the stability of the Euler-Lagrange functional equation,'' Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185--190, 1992.
• J. M. Rassias, Refined Hyers-Ulam approximation of approximately Jensen type mappings,'' Bulletin des Sciences Mathématiques, vol. 131, no. 1, pp. 89--98, 2007.
• J. M. Rassias, Solution of a Cauchy-Jensen stability Ulam type problem,'' Archivum Mathematicum, vol. 37, no. 3, pp. 161--177, 2001.
• J. Aczél and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989.
• Pl. Kannappan, Quadratic functional equation and inner product spaces,'' Results in Mathematics, vol. 27, no. 3-4, pp. 368--372, 1995.
• F. Skof, Proprieta' locali e approssimazione di operatori,'' Rendiconti del Seminario Matemàtico e Fisico di Milano, vol. 53, no. 1, pp. 113--129, 1983.
• P. W. Cholewa, Remarks on the stability of functional equations,'' Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76--86, 1984.
• St. Czerwik, On the stability of the quadratic mapping in normed spaces,'' Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, no. 1, pp. 59--64, 1992.
• A. Grabiec, The generalized Hyers-Ulam stability of a class of functional equations,'' Publicationes Mathematicae Debrecen, vol. 48, no. 3-4, pp. 217--235, 1996.
• J. R. Lee, J. S. An, and C. Park, On the stability of quadratic functional equations,'' Abstract and Applied Analysis, vol. 2008, Article ID 628178, 8 pages, 2008.