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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 482936, 14 pages
http://dx.doi.org/10.1155/2012/482936
Research Article

A Fixed Point Approach to the Stability of an -Dimensional Mixed-Type Additive and Quadratic Functional Equation

1Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea
2Mathematics Section, College of Science and Technology, Hongik University, Jochiwon 339-701, Republic of Korea

Received 8 October 2011; Accepted 3 December 2011

Academic Editor: Alberto d'Onofrio

Copyright © 2012 Yang-Hi Lee and Soon-Mo Jung. 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, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics No. 8, Interscience, New York, NY ,USA, 1960.
  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. D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223–237, 1951. 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. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002. View at Publisher · View at Google Scholar
  7. 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
  8. D. H. Hyers, G. Isac, and T. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and Their Applications, 34, Birkhäuser, Boston, Mass, USA, 1998.
  9. 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.
  10. G. H. Kim, “On the stability of functional equations with square-symmetric operation,” Mathematical Inequalities & Applications, vol. 4, no. 2, pp. 257–266, 2001. View at Google Scholar · View at Zentralblatt MATH
  11. H.-M. Kim, “On the stability problem for a mixed type of quartic and quadratic functional equation,” Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 358–372, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y.-H. Lee, “On the stability of the monomial functional equation,” Bulletin of the Korean Mathematical Society, vol. 45, no. 2, pp. 397–403, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Y.-H. Lee and K.-W. Jun, “A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation,” Journal of Mathematical Analysis and Applications, vol. 238, no. 1, pp. 305–315, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. Y.-H. Lee and K.-W. Jun, “A generalization of the Hyers-Ulam-Rassias stability of the Pexider equation,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 627–638, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Y.-H. Lee and K.-W. Jun, “A note on the Hyers-Ulam-Rassias stability of Pexider equation,” Journal of the Korean Mathematical Society, vol. 37, no. 1, pp. 111–124, 2000. View at Google Scholar · View at Zentralblatt MATH
  16. Y.-H. Lee and K.-W. Jun, “On the stability of approximately additive mappings,” Proceedings of the American Mathematical Society, vol. 128, no. 5, pp. 1361–1369, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. 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
  18. W. Towanlong and P. Nakmahachalasint, “An n-dimensional mixed-type additive and quadratic functional equation and its stability,” ScienceAsia, vol. 35, pp. 381–385, 2009. View at Google Scholar
  19. B. Margolis and J. B. Diaz, “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
  20. J. A. Baker, “The stability of certain functional equations,” Proceedings of the American Mathematical Society, vol. 112, no. 3, pp. 729–732, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. 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
  22. L. Cădariu and V. Radu, “Fixed points and the stability of Jensen's functional equation,” JIPAM: Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 4, 2003. View at Google Scholar · View at Zentralblatt MATH
  23. L. Cǎdariu and V. Radu, “Fixed points and the stability of quadratic functional equations,” Analele Universitatii Din Timisoara (Seria Matematica-Informatica), vol. 41, no. 1, pp. 25–48, 2003. View at Google Scholar
  24. L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” in Iteration Theory, vol. 346 of Grazer Mathematische Berichte, pp. 43–52, Karl-Franzens-Universitaet Graz, Graz, Austria, 2004. View at Google Scholar · View at Zentralblatt MATH
  25. L. Cădariu and V. Radu, “Fixed point methods for the generalized stability of functional equations in a single variable,” Fixed Point Theory and Applications, vol. 2008, Article ID 749392, 15 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,” Bulletin of the Brazilian Mathematical Society New Series, vol. 37, no. 3, pp. 361–376, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. J. Brzdȩk, J. Chudziak, and Z. Páles, “A fixed point approach to stability of functional equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 17, pp. 6728–6732, 2011. View at Publisher · View at Google Scholar
  28. J. Brzdek and K. Ciepliski, “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
  29. S.-M. Jung, “A fixed point approach to the stability of a Volterra integral equation,” Fixed Point Theory and Applications, vol. 2007, Article ID 57064, 9 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. S.-M. Jung and T.-S. Kim, “A fixed point approach to the stability of the cubic functional equation,” Sociedad Matemática Mexicana Boletín Tercera Serie, vol. 12, no. 1, pp. 51–57, 2006. View at Google Scholar · View at Zentralblatt MATH
  31. S.-M. Jung, T.-S. Kim, and K.-S. Lee, “A fixed point approach to the stability of quadratic functional equation,” Bulletin of the Korean Mathematical Society, vol. 43, no. 3, pp. 531–541, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH