Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012, Article ID 926390, 23 pages
http://dx.doi.org/10.1155/2012/926390
Research Article

On the Hyers-Ulam Stability of a General Mixed Additive and Cubic Functional Equation in n-Banach Spaces

1School of Mathematics, Beijing Institute of Technology, Beijing 100081, China
2Pedagogical Department E.E., Section of Mathematics and Informatics, National and Kapodistrian University of Athens, 4 Agamemnonos Street, Aghia Paraskevi, Athens 15342, Greece

Received 7 January 2012; Accepted 15 February 2012

Academic Editor: Krzysztof Cieplinski

Copyright © 2012 Tian Zhou Xu and John Michael Rassias. 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.

Linked References

  1. Z. Moszner, “On the stability of functional equations,” Aequationes Mathematicae, vol. 77, no. 1-2, pp. 33–88, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. S. M. Ulam, A Collection of Mathematical Problems, vol. 8 of Interscience Tracts in Pure and Applied Mathematics, Interscience, New York, NY, USA, 1960.
  3. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. R. P. Agarwal, B. Xu, and W. Zhang, “Stability of functional equations in single variable,” Journal of Mathematical Analysis and Applications, vol. 288, no. 2, pp. 852–869, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. A. Najati and G. Z. Eskandani, “Stability of a mixed additive and cubic functional equation in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 1318–1331, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. W.-G. Park, “Approximate additive mappings in 2-Banach spaces and related topics,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 193–202, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. R. Saadati, Y. J. Cho, and J. Vahidi, “The stability of the quartic functional equation in various spaces,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 1994–2002, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. T. Z. Xu, J. M. Rassias, and W. X. Xu, “Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces,” Acta Mathematica Sinica, English Series, vol. 28, no. 3, pp. 529–560, 2011. View at Publisher · View at Google Scholar
  12. T. Z. Xu, J. M. Rassias, and W. X. Xu, “Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces,” Journal of Mathematical Physics, vol. 51, no. 9, Article ID 093508, 19 pages, 2010. View at Publisher · View at Google Scholar
  13. S. Gähler, “2-metrische Räume und ihre topologische Struktur,” Mathematische Nachrichten, vol. 26, pp. 115–148, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. Gähler, “Lineare 2-normierte Räume,” Mathematische Nachrichten, vol. 28, pp. 1–43, 1964. View at Publisher · View at Google Scholar
  15. Y. J. Cho, P. C. S. Lin, S. S. Kim, and A. Misiak, Theory of 2-Inner Product Spaces, Nova Science, Huntington, NY, USA, 2001.
  16. A. Misiak, “n-inner product spaces,” Mathematische Nachrichten, vol. 140, pp. 299–319, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. X. Y. Chen and M. M. Song, “Characterizations on isometries in linear n-normed spaces,” Nonlinear Analysis, vol. 72, no. 3-4, pp. 1895–1901, 2010. View at Publisher · View at Google Scholar
  18. S. Gähler, “Über 2-Banach-Räume,” Mathematische Nachrichten, vol. 42, pp. 335–347, 1969. View at Publisher · View at Google Scholar
  19. A. G. White, Jr., “2-Banach spaces,” Mathematische Nachrichten, vol. 42, pp. 43–60, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH