Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2011, Article ID 536520, 11 pages
http://dx.doi.org/10.1155/2011/536520
Research Article

Stability of an Additive-Cubic-Quartic Functional Equation in Multi-Banach Spaces

1School of Science, Hubei University of Technology, Wuhan, Hubei 430068, China
2Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China
3Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

Received 5 August 2011; Accepted 30 September 2011

Academic Editor: Narcisa C. Apreutesei

Copyright © 2011 Zhihua Wang et al. 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. S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964.
  2. 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
  3. 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
  4. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. D. Miheţ and V. Radu, “On the stability of the additive Cauchy functional equation in random normed spaces,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 567–572, 2008. View at Google Scholar · View at Zentralblatt MATH
  6. A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy versions of Hyers-Ulam-Rassias theorem,” Fuzzy Sets and Systems, vol. 159, no. 6, pp. 720–729, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76–86, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. S. Czerwik, “The stability of the quadratic functional equation,” in Stability of Mappings of Hyers-Ulam Type, Th. M. Rassias and J. Tabor, Eds., Hadronic Press Collect. Orig. Artic., pp. 81–91, Hadronic Press, Palm Harbor, Fla, USA, 1994. View at Google Scholar
  10. S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Fla, USA, 2003.
  11. V. Faĭziev and Th. M. Rassias, “The space of (ψ, γ)-pseudocharacters on semigroups,” Nonlinear Functional Analysis and Applications, vol. 5, no. 1, pp. 107–126, 2000. View at Google Scholar
  12. V. A. Faĭziev, Th. M. Rassias, and P. K. Sahoo, “The space of (ψ, γ)-additive mappings on semigroups,” Transactions of the American Mathematical Society, vol. 354, no. 11, pp. 4455–4472, 2002. View at Publisher · View at Google Scholar
  13. P. Găvruţă, S.-M. Jung, and Y. Li, “Hyers-Ulam stability of mean value points,” Annals of Functional Analysis, vol. 1, no. 2, pp. 68–74, 2010. View at Google Scholar · View at Zentralblatt MATH
  14. A. Gilányi, K. Nagatou, and P. Volkmann, “Stability of a functional equation coming from the characterization of the absolute value of additive functions,” Annals of Functional Analysis, vol. 1, no. 2, pp. 1–6, 2010. View at Google Scholar · View at Zentralblatt MATH
  15. D. H. Hyers and Th. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125–153, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. G. Isac and Th. M. Rassias, “On the Hyers-Ulam stability of ψ-additive mappings,” Journal of Approximation Theory, vol. 72, no. 2, pp. 131–137, 1993. View at Publisher · View at Google Scholar
  17. G. Isac and Th. M. Rassias, “Stability of ψ-additive mappings: applications to nonlinear analysis,” International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 2, pp. 219–228, 1996. View at Publisher · View at Google Scholar
  18. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001.
  19. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011.
  20. 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. 267–278, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. Th. M. Rassias, “On a modified Hyers-Ulam sequence,” Journal of Mathematical Analysis and Applications, vol. 158, no. 1, pp. 106–113, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264–284, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. Th. M. Rassias, “On the stability of functional equations originated by a problem of Ulam,” Mathematica, vol. 44(67), no. 1, pp. 39–75, 2002. View at Google Scholar · View at Zentralblatt MATH
  25. Th. M. Rassias, “On the stability of minimum points,” Mathematica, vol. 45(68), no. 1, pp. 93–104, 2003. View at Google Scholar · View at Zentralblatt MATH
  26. Th. M. Rassias, Ed., Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. View at Zentralblatt MATH
  27. V. Radu, “The fixed point alternative and the stability of functional equations,” Fixed Point Theory, vol. 4, no. 1, pp. 91–96, 2003. View at Google Scholar · View at Zentralblatt MATH
  28. L. P. Castro and A. Ramos, “Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations,” Banach Journal of Mathematical Analysis, vol. 3, no. 1, pp. 36–43, 2009. View at Google Scholar
  29. C. Park and Th. M. Rassias, “Fixed points and generalized Hyers-Ulam stability of quadratic functional equations,” Journal of Mathematical Inequalities, vol. 1, no. 4, pp. 515–528, 2007. View at Google Scholar · View at Zentralblatt MATH
  30. H. G. Dales and M. E. Polyakov, “Multi-normed spaces and multi-Banach algebras,” preprint.
  31. H. G. Dales and M. S. Moslehian, “Stability of mappings on multi-normed spaces,” Glasgow Mathematical Journal, vol. 49, no. 2, pp. 321–332, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. M. S. Moslehian, “Superstability of higher derivations in multi-Banach algebras,” Tamsui Oxford Journal of Mathematical Sciences, vol. 24, no. 4, pp. 417–427, 2008. View at Google Scholar · View at Zentralblatt MATH
  33. M. S. Moslehian, K. Nikodem, and D. Popa, “Asymptotic aspect of the quadratic functional equation in multi-normed spaces,” Journal of Mathematical Analysis and Applications, vol. 355, no. 2, pp. 717–724, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. M. Eshaghi-Gordji, S. Kaboli-Gharetapeh, C. Park, and S. Zolfaghari, “Stability of an additive-cubic-quartic functional equation,” Advances in Difference Equations, vol. 2009, Article ID 395693, 20 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. O. Hadžić, E. Pap, and V. Radu, “Generalized contraction mapping principles in probabilistic metric spaces,” Acta Mathematica Hungarica, vol. 101, no. 1-2, pp. 131–148, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH