About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 712743, 10 pages
http://dx.doi.org/10.1155/2012/712743
Research Article

Fixed Points and Generalized Hyers-Ulam Stability

Department of Mathematics, “Politehnica” University of Timişoara, Piaţa Victoriei No. 2, 300006 Timişoara, Romania

Received 17 May 2012; Accepted 5 July 2012

Academic Editor: Krzysztof Ciepliński

Copyright © 2012 L. Cădariu 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, Chapter 6, Science Editors, Wiley, 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. 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
  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
  6. P. Găvruţa, M. Hossu, D. Popescu, and C. Căprău, “On the stability of mappings and an answer to a problem of Th. M. Rassias,” Annales Mathématiques Blaise Pascal, vol. 2, no. 2, pp. 55–60, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143–190, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. T. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, New Jersey, NJ, USA, 2002.
  10. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basle, Switzerland, 1998.
  11. 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.
  12. G. L. Forti, “An existence and stability theorem for a class of functional equations,” Stochastica, vol. 4, no. 1, pp. 23–30, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268–273, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. P. Găvruţa, “On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings,” Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 543–553, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. 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
  17. G.-L. Forti, “Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 127–133, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. 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
  19. 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 Zentralblatt MATH
  20. L. Cădariu and V. Radu, “Fixed points and the stability of Jensen's functional equation,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 4, 2003. View at Zentralblatt MATH
  21. L. Cădariu and V. Radu, “Fixed point methods for the generalized stability of functional equations in a single variable,” Fixed Point Theory and Applications, Article ID 749392, 15 pages, 2008. View at Zentralblatt MATH
  22. D. Miheţ, “The Hyers-Ulam stability for two functional equations in a single variable,” Banach Journal of Mathematical Analysis, vol. 2, no. 1, pp. 48–52, 2008. View at Zentralblatt MATH
  23. L. Găvruţa, “Matkowski contractions and Hyers-Ulam stability,” Buletinul Ştiinţific al Universităţii Politehnica din Timişoara. Seria Matematică-Fizică, vol. 53(67), no. 2, pp. 32–35, 2008.
  24. P. Găvruţa and L. Găvruţa, “A new method for the generalization Hyers-Ulam-Rassias stability,” International Journal of Nonlinear Analysis and Applications, vol. 1, no. 2, pp. 11–18, 2010. View at Publisher · View at Google Scholar · View at Scopus
  25. L. Cădariu, L. Găvruţa, and P. Găvruţa, “Weighted space method for the stability of some nonlinear equations,” Applicable Analysis and Discrete Mathematics, vol. 6, pp. 126–139, 2012.
  26. L. Cădariu and V. Radu, “A general fixed point method for the stability of Cauchy functional equation,” in Functional Equations in Mathematical Analysis, M. Th. Rassias and J. Brzdęk, Eds., vol. 52 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011.
  27. L. Cădariu and V. Radu, “A general fixed point method for the stability of the monomial functional equation,” Carpathian Journal of Mathematics, no. 1, pp. 25–36, 2012.
  28. 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.
  29. J. Brzdęk, J. Chudziak, and Z. Páles, “A fixed point approach to stability of functional equations,” Nonlinear Analysis. Theory, Methods and Applications A, vol. 74, no. 17, pp. 6728–6732, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. J. Brzdęk and K. Ciepliński, “A fixed point approach to the stability of functional equations in non-archimedean metric spaces,” Nonlinear Analysis. Theory, Methods and Applications A, vol. 74, no. 18, pp. 6861–6867, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH