About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 198018, 10 pages
http://dx.doi.org/10.1155/2013/198018
Research Article

Stability of a Functional Equation Deriving from Quadratic and Additive Functions in Non-Archimedean Normed Spaces

1Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran
2Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of Korea

Received 19 July 2013; Revised 27 August 2013; Accepted 7 September 2013

Academic Editor: Krzysztof Ciepliński

Copyright © 2013 Abasalt Bodaghi and Sang Og Kim. 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, Science Editions, chapter 6, John Wiley & Sons, New York, NY, USA, 1940. View at MathSciNet
  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 · View at MathSciNet
  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 · View at MathSciNet
  4. 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 · View at MathSciNet
  5. 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 · View at MathSciNet
  6. F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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 · View at MathSciNet
  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 · View at MathSciNet
  9. 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. View at Zentralblatt MATH · View at MathSciNet
  10. M. E. 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. View at MathSciNet
  11. L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” in Iteration theory (ECIT '02), vol. 346 of Grazer Mathematische Berichte, pp. 43–52, Karl-Franzens-Universität Graz, Graz, Austria, 2004. View at Zentralblatt MATH · View at MathSciNet
  12. L. Cǎdariu and V. Radu, “Fixed points and the stability of quadratic functional equations,” Analele Universităţii de Vest din Timişoara, Seria Matematică-Informatică, vol. 41, no. 1, pp. 25–48, 2003. View at Zentralblatt MATH · View at MathSciNet
  13. N. Brillouët-Belluot, J. Brzdęk, and K. Ciepliński, “On some recent developments in Ulam's type stability,” Abstract and Applied Analysis, vol. 2012, Article ID 716936, 41 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. K. Ciepliński, “Applications of fixed point theorems to the Hyers-Ulam stability of functional equations—a survey,” Annals of Functional Analysis, vol. 3, no. 1, pp. 151–164, 2012. View at Zentralblatt MATH · View at MathSciNet
  15. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  16. G. Z. Eskandani, H. Vaezi, and Y. N. Dehghan, “Stability of a mixed additive and quadratic functional equation in non-Archimedean Banach modules,” Taiwanese Journal of Mathematics, vol. 14, no. 4, pp. 1309–1324, 2010. View at Zentralblatt MATH · View at MathSciNet
  17. A. Najati and M. B. Moghimi, “Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 399–415, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. S. Jin and Y. H. Lee, “On the stability of the functional equation deriving from quadratic and additive function in random normed spaces via fixed point method,” Journal of the Chungcheong Mathematical Society, vol. 25, no. 1, pp. 51–63, 2012.
  19. J. Aczel and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  20. K. Hensel, “Uber eine neue Begrndung der Theorie der algebraischen Zahlen,” Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 6, no. 3, pp. 83–88, 1897.
  21. L. M. Arriola and W. A. Beyer, “Stability of the Cauchy functional equation over p-adic fields,” Real Analysis Exchange, vol. 31, no. 1, pp. 125–132, 2005. View at MathSciNet
  22. J. Brzdęk and K. Ciepliński, “A fixed point approach to the stability of functional equations in non-Archimedean metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 18, pp. 6861–6867, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. S. Moslehian and T. M. Rassias, “Stability of functional equations in non-Archimedean spaces,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 2, pp. 325–334, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. J. Schwaiger, “Functional equations for homogeneous polynomials arising from multilinear mappings and their stability,” Annales Mathematicae Silesianae, no. 8, pp. 157–171, 1994. View at Zentralblatt MATH · View at MathSciNet
  25. T. Z. Xu, “Stability of multi-Jensen mappings in non-Archimedean normed spaces,” Journal of Mathematical Physics, vol. 53, no. 2, Article ID 023507, 9 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  26. A. Bodaghi, I. A. Alias, and M. H. Ghahramani, “Ulam stability of a quartic functional equation,” Abstract and Applied Analysis, vol. 2012, Article ID 232630, 9 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. Brzdęk, “Stability of the equation of the p-Wright affine functions,” Aequationes Mathematicae, vol. 85, no. 3, pp. 497–503, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  28. G. Maksa and Z. Páles, “Hyperstability of a class of linear functional equations,” Acta Mathematica, vol. 17, no. 2, pp. 107–112, 2001. View at Zentralblatt MATH · View at MathSciNet
  29. M. Piszczek and J. Szczawińska, “Hyperstability of the drygas functional equation,” Journal of Function Spaces and Applications, vol. 2013, Article ID 912718, 4 pages, 2013. View at Publisher · View at Google Scholar