Abstract and Applied Analysis

On the Stability of Quadratic Functional Equations

Jung Rye Lee, Jong Su An, and Choonkil Park

Full-text: Open access

Abstract

Let X , Y be vector spaces and k a fixed positive integer. It is shown that a mapping f ( k x + y ) + f ( k x - y ) = 2 k 2 f ( x ) + 2 f ( y ) for all x , y X if and only if the mapping f : X Y satisfies f ( x + y ) + f ( x - y ) = 2 f ( x ) + 2 f ( y ) for all x , y X . Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 628178, 8 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969154

Digital Object Identifier
doi:10.1155/2008/628178

Mathematical Reviews number (MathSciNet)
MR2393119

Zentralblatt MATH identifier
1146.39045

Citation

Lee, Jung Rye; An, Jong Su; Park, Choonkil. On the Stability of Quadratic Functional Equations. Abstr. Appl. Anal. 2008 (2008), Article ID 628178, 8 pages. doi:10.1155/2008/628178. https://projecteuclid.org/euclid.aaa/1220969154


Export citation

References

  • S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1960.
  • 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.
  • Th. M. Rassias, “Problem 16; 2, Report of the 27th International Symposium on Functional Equations,” Aequationes Mathematicae , vol. 39, no. 2-3, pp. 292–293, 309, 1990.
  • Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences , vol. 14, no. 3, pp. 431–434, 1991.
  • Th. M. Rassias and P. Šemrl, “On the Hyers-Ulam stability of linear mappings,” Journal of Mathematical Analysis and Applications , vol. 173, no. 2, pp. 325–338, 1993.
  • J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques , vol. 108, no. 4, pp. 445–446, 1984.
  • F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano , vol. 53, pp. 113–129, 1983.
  • P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae , vol. 27, no. 1, 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, pp. 59–64, 1992.
  • K.-W. Jun and Y.-H. Lee, “On the Hyers-Ulam-Rassias stability of a pexiderized quadratic inequality,” Mathematical Inequalities & Applications , vol. 4, no. 1, pp. 93–118, 2001.
  • S.-M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic property,” Journal of Mathematical Analysis and Applications , vol. 222, no. 1, pp. 126–137, 1998.
  • 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 approximation of approximately quadratic mappings by quadratic mappings,” Annales Mathematicae Silesianae , no. 15, pp. 67–78, 2001.
  • M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,” Bulletin of the Brazilian Mathematical Society , vol. 37, no. 3, pp. 361–376, 2006.
  • Th. M. Rassias, “On the stability of the quadratic functional equation and its applications,” Studia Universitatis Babeş-Bolyai , vol. 43, no. 3, pp. 89–124, 1998.