## Abstract and Applied Analysis

### Ulam Stability of a Quartic Functional Equation

#### Abstract

The oldest quartic functional equation was introduced by J. M. Rassias in (1999), and then was employed by other authors. The functional equation $f(2x+y)+f(2x-y)=4f(x+y)+4f(x-y)+24f(x)-6f(y)$ is called a quartic functional equation, all of its solution is said to be a quartic function. In the current paper, the Hyers-Ulam stability and the superstability for quartic functional equations are established by using the fixed-point alternative theorem.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 232630, 9 pages.

Dates
First available in Project Euclid: 7 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1399487784

Digital Object Identifier
doi:10.1155/2012/232630

Mathematical Reviews number (MathSciNet)
MR2914887

Zentralblatt MATH identifier
1237.39026

#### Citation

Bodaghi, Abasalt; Alias, Idham Arif; Ghahramani, Mohammad Hosein. Ulam Stability of a Quartic Functional Equation. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 232630, 9 pages. doi:10.1155/2012/232630. https://projecteuclid.org/euclid.aaa/1399487784

#### References

• J. Baker, “The stability of the cosine equation,” Proceedings of the American Mathematical Society, vol. 80, no. 3, pp. 411–416, 1980.
• S. M. Ulam, Problems in Modern Mathematics, 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, pp. 222–224, 1941.
• T. 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.
• I. S. Chang, K. W. Jun, and Y. S. Jung, “The modified Hyers-Ulam-Rassias stability of a cubic type functional equation,” Mathematical Inequalities & Applications, vol. 8, no. 4, pp. 675–683, 2005.
• S. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59–64, 1992.
• M. Eshaghi Gordji and A. Bodaghi, “On the Hyers-Ulam-Rassias stability problem for quadratic functional equations,” East Journal on Approximations, vol. 16, no. 2, pp. 123–130, 2010.
• K. W. Jun and H. M. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,” Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 867–878, 2002.
• J. Lee, J. An, and C. Park, “On the stability of quadratic functional equations,” Abstract and Applied Analysis, vol. 2008, Article ID 628178, 8 pages, 2008.
• S. H. Lee, S. M. Im, and I. S. Hwang, “Quartic functional equations,” Journal of Mathematical Analysis and Applications, vol. 307, no. 2, pp. 387–394, 2005.
• T. Zhou Xu, J. M. Rassias, and W. Xin Xu, “A fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces,” International Journal of the Physical SciencesInt, vol. 6, no. 2, pp. 313–324, 2011.
• T. Z. Xu, J. M. Rassias, and W. X. Xu, “Generalized Ulam-Hyers stability of a general mixed AQCQ-functional equation in multi-Banach spaces: a fixed point approach,” European Journal of Pure and Applied Mathematics, vol. 3, no. 6, pp. 1032–1047, 2010.
• A. Bodaghi, I. A. Alias, and M. Eshaghi Gordji, “On the stability of quadratic double centralizers and quadratic multipliers: a fixed point approach,” Journal of Inequalities and Applications, vol. 2011, Article ID 957541, 9 pages, 2011.
• A. Bodaghi, I. A. Alias, and M. H. Ghahramani, “Approximately cubic functional equations and cubic multipliers,” Journal of Inequalities and Applications, vol. 2011, 53 pages, 2011.
• J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968.
• A. Najati, “On the stability of a quartic functional equation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 569–574, 2008.