Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2008, Article ID 672618, 13 pages
http://dx.doi.org/10.1155/2008/672618
Research Article

Fixed Points and Stability of an Additive Functional Equation of 𝑛-Apollonius Type in 𝐶-Algebras

1Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
2Department of Mathematics, Hanyang University, Seoul 133–791, South Korea

Received 22 April 2008; Revised 11 June 2008; Accepted 16 July 2008

Academic Editor: John Rassias

Copyright © 2008 Fridoun Moradlou 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, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York, NY, USA, 1960. View at Zentralblatt MATH · 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 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 Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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 · 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. L. Maligranda, “A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions—a question of priority,” Aequationes Mathematicae, vol. 75, no. 3, pp. 289–296, 2008. View at Publisher · View at Google Scholar
  7. 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 · View at MathSciNet
  8. L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” Grazer Mathematische Berichte, vol. 346, pp. 43–52, 2004. View at Google Scholar · View at Zentralblatt MATH
  9. H.-M. Kim and J. M. Rassias, “Generalization of Ulam stability problem for Euler-Lagrange quadratic mappings,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 277–296, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Y.-S. Lee and S.-Y. Chung, “Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions,” Applied Mathematics Letters, vol. 21, no. 7, pp. 694–700, 2008. View at Publisher · View at Google Scholar
  11. P. Nakmahachalasint, “On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 63239, 10 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  12. C.-G. Park, “On the stability of the linear mapping in Banach modules,” Journal of Mathematical Analysis and Applications, vol. 275, no. 2, pp. 711–720, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. C.-G. Park, “On an approximate automorphism on a C-algebra,” Proceedings of the American Mathematical Society, vol. 132, no. 6, pp. 1739–1745, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. C.-G. Park, “Homomorphisms between Poisson JC-algebras,” Bulletin of the Brazilian Mathematical Society, vol. 36, no. 1, pp. 79–97, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. C. Park, “Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras,” Fixed Point Theory and Applications, vol. 2007, Article ID 50175, 15 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. C.-G. Park, “Stability of an Euler-Lagrange-Rassias type additive mapping,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 101–111, 2007. View at Google Scholar · View at MathSciNet
  17. C. Park, “Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach,” Fixed Point Theory and Applications, vol. 2008, Article ID 493751, 9 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. Pietrzyk, “Stability of the Euler-Lagrange-Rassias functional equation,” Demonstratio Mathematica, vol. 39, no. 3, pp. 523–530, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. M. Rassias and M. J. Rassias, “Refined Ulam stability for Euler-Lagrange type mappings in Hilbert spaces,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 126–132, 2007. View at Google Scholar · View at MathSciNet
  20. Th. M. Rassias, “The problem of S. M. Ulam for approximately multiplicative mappings,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 352–378, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. 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 · View at MathSciNet
  22. J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques, vol. 108, no. 4, pp. 445–446, 1984. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. 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 · View at MathSciNet
  25. 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, 7 pages, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. L. Cădariu and V. Radu, “Fixed points and the stability of quadratic functional equations,” Analele Universităţii de Vest din Timişoara, vol. 41, no. 1, pp. 25–48, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. S.-M. Jung and J. M. Rassias, “A fixed point approach to the stability of a functional equation of the spiral of Theodorus,” Fixed Point Theory and Applications. In press.
  28. 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 · View at MathSciNet
  29. J. M. Rassias, “Alternative contraction principle and Ulam stability problem,” Mathematical Sciences Research Journal, vol. 9, no. 7, pp. 190–199, 2005. View at Google Scholar · View at MathSciNet
  30. J. M. Rassias, “Alternative contraction principle and alternative Jensen and Jensen type mappings,” International Journal of Applied Mathematics & Statistics, vol. 4, pp. 1–10, 2006. View at Google Scholar · View at MathSciNet
  31. 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
  32. 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 Collection of Original Articles, pp. 81–91, Hadronic Press, Palm Harbor, Fla, USA, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. C. Borelli and G. L. Forti, “On a general Hyers-Ulam stability result,” International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 2, pp. 229–236, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. K.-W. Jun and Y.-H. Lee, “On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality,” Mathematical Inequalities & Applications, vol. 4, no. 1, pp. 93–118, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. G. H. Kim, “On the stability of the quadratic mapping in normed spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 25, no. 4, pp. 217–229, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. C.-G. Park and Th. M. Rassias, “Hyers-Ulam stability of a generalized Apollonius type quadratic mapping,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 371–381, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. J. Aczél 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 Zentralblatt MATH · View at MathSciNet
  38. A. Najati, “Hyers-Ulam stability of an n-Apollonius type quadratic mapping,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 14, no. 4, pp. 755–774, 2007. View at Google Scholar · View at MathSciNet
  39. C. Park and Th. M. Rassias, “Homomorphisms in C-ternary algebras and JB-triples,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 13–20, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. C. Park, “Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between C-algebras,” Mathematische Nachrichten, vol. 281, no. 3, pp. 402–411, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  41. 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 · View at MathSciNet
  42. P. Ara and M. Mathieu, Local Multipliers of C-Algebras, Springer Monographs in Mathematics, Springer, London, UK, 2003. View at Zentralblatt MATH · View at MathSciNet