About this Journal Submit a Manuscript Table of Contents 
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 763728, 14 pages
doi:10.1155/2012/763728
Research Article

On the Stability Problem in Fuzzy Banach Space

G. Zamani Eskandani,1 P. Găvruţa,2 and Gwang Hui Kim3

1Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
2Department of Mathematics, University Politehnica of Timisoara, Piata Victoriei 2, 300006 Timisoara, Romania
3Department of Mathematics, Kangnam University, Suwon, Kyunggi 449-702, Republic of Korea

Received 6 February 2012; Accepted 17 May 2012

Academic Editor: Nicole Brillouet-Belluot

Copyright © 2012 G. Zamani Eskandani 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.
  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.
  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.
  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
  5. Th. M. Rassias, “Problem 16; 2, report of the 27th International Symposium on functional equations,” Aequationes Mathematicae, vol. 39, pp. 292–293, 1990.
  6. 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
  7. Th. M. Rassias and P. Šemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,” Proceedings of the American Mathematical Society, vol. 114, no. 4, pp. 989–993, 1992. View at Publisher · View at Google Scholar
  8. 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
  9. J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques, vol. 108, no. 4, pp. 445–446, 1984.
  10. 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
  11. 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
  12. P. Găvruţă, “An answer to question of John M. Rassias concerning the stability of Cauchy equation,” Advanced in Equation and Inequality, pp. 67–71, 1999.
  13. 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
  14. G. Isac and Th. M. Rassias, “Functional inequalities for approximately additive mappings,” in Stability of Mappings of Hyers-Ulam Type, Hadronic Press Collection of Original Articles, pp. 117–125, Hadronic Press, Palm Harbor, Fla, USA, 1994.
  15. 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
  16. C. Baak and M. S. Moslehian, “On the stability of J*-homomorphisms,” Nonlinear Analysis, vol. 63, no. 1, pp. 42–48, 2005. View at Publisher · View at Google Scholar
  17. B. Bouikhalene, E. Elqorachi, and J. M. Rassias, “The superstability of d'Alembert's functional equation on the Heisenberg group,” Applied Mathematics Letters, vol. 23, no. 1, pp. 105–109, 2010. View at Publisher · View at Google Scholar
  18. L. Cădariu and V. Radu, “The fixed points method for the stability of some functional equations,” Carpathian Journal of Mathematics, vol. 23, no. 1-2, pp. 63–72, 2007.
  19. G. Z. Eskandani, “On the Hyers-Ulam-Rassias stability of an additive functional equation in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 405–409, 2008. View at Publisher · View at Google Scholar
  20. 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.
  21. G. Z. Eskandani, H. Vaezi, and F. Moradlou, “On the Hyers-Ulam-Rassias stability of functional equations in quasi-Banach spaces,” International Journal of Applied Mathematics & Statistics, vol. 15, pp. 1–15, 2009.
  22. P. Găvruţă, “On the Hyers-Ulam-Rassias stability of mappings,” in Recent Progress in Inequalities, G. V. Milovanovic, Ed., vol. 430, pp. 465–469, Kluwer Academic, Dordrecht, The Netherlands, 1998.
  23. P. Găvruţă and L. Cădariu, “General stability of the cubic functional equation,” Buletinul Stiintific al Universitătii “Politehnica” din Timişoara. Seria Matematica-Fizica, vol. 47(61), no. 1, pp. 59–70, 2002.
  24. 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 Publisher · View at Google Scholar
  25. K. Jun, H. Kim, and J. Rassias, “Extended Hyers-Ulam stability for Cauchy-Jensen mappings,” Journal of Difference Equations and Applications, pp. 1–15, 2007.
  26. S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001.
  27. S. M. Jung, “Asymptotic properties of isometries,” Journal of Mathematical Analysis and Applications, vol. 276, no. 2, pp. 642–653, 2002. View at Publisher · View at Google Scholar
  28. Pl. Kannappan, “Quadratic functional equation and inner product spaces,” Results in Mathematics, vol. 27, no. 3-4, pp. 368–372, 1995.
  29. H. M. Kim, J. M. Rassias, and Y. S. Cho, “Stability problem of Ulam for Euler-Lagrange quadratic mappings,” Journal of Inequalities and Applications, vol. 2007, Article ID 10725, 15 pages, 2007. View at Publisher · View at Google Scholar
  30. Y. S. Lee and S. Y. Chung, “Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions,” Applied Mathematics Letters of Rapid Publication, vol. 21, no. 7, pp. 694–700, 2008. View at Publisher · View at Google Scholar
  31. D. Miheţ, “The fixed point method for fuzzy stability of the Jensen functional equation,” Fuzzy Sets and Systems, vol. 160, no. 11, pp. 1663–1667, 2009. View at Publisher · View at Google Scholar
  32. F. Moradlou, H. Vaezi, and G. Z. Eskandani, “Hyers-Ulam-Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces,” Mediterranean Journal of Mathematics, vol. 6, no. 2, pp. 233–248, 2009. View at Publisher · View at Google Scholar
  33. M. S. Moslehian, “On the orthogonal stability of the Pexiderized quadratic equation,” Journal of Difference Equations and Applications, vol. 11, no. 11, pp. 999–1004, 2005. View at Publisher · View at Google Scholar
  34. 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
  35. C. Park and J. M. Rassias, “Stability of the Jensen-type functional equation in C*-algebras: a fixed point approach,” Abstract and Applied Analysis, vol. 2009, Article ID 360432, 17 pages, 2009. View at Publisher · View at Google Scholar
  36. A. Pietrzyk, “Stability of the Euler-Lagrange-Rassias functional equation,” Demonstratio Mathematica, vol. 39, no. 3, pp. 523–530, 2006.
  37. J. M. Rassias, J. Lee, and H. M. Kim, “Refinned Hyers-Ulam stability for Jensen type mappings,” Journal of the Chungcheong Mathematical Society, vol. 22, pp. 101–116, 2009.
  38. J. M. Rassias, “Complete solution of the multi-dimensional problem of Ulam,” Discussiones Mathematicae, vol. 14, pp. 101–107, 1994.
  39. K. Ravi and M. Arunkumar, “On the Ulam-Gavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 143–156, 2007.
  40. K. Ravi, J. M. Rassias, M. Arunkumar, and R. Kodandan, “Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 4, article 114, pp. 1–29, 2009.
  41. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002. View at Publisher · View at Google Scholar
  42. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications, 34, Birkhäuser, Basle, Switzerland, 1998. View at Publisher · View at Google Scholar
  43. 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
  44. 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.
  45. K. Ravi, M. Arunkumar, and J. M. Rassias, “Ulam stability for the orthogonally general Euler-Lagrange type functional equation,” International Journal of Mathematics and Statistics, vol. 3, no. A08, pp. 36–46, 2008.
  46. H. X. Cao, J. R. Lv, and J. M. Rassias, “Superstability for generalized module left derivations and generalized module derivations on a Banach module. I,” Journal of Inequalities and Applications, vol. 2009, Article ID 718020, 10 pages, 2009. View at Publisher · View at Google Scholar
  47. H. X. Cao, J. R. Lv, and J. M. Rassias, “Superstability for generalized module left derivations and generalized module derivations on a Banach module. II,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 3, article 85, pp. 1–8, 2009.
  48. M. B. Savadkouhi, M. E. Gordji, J. M. Rassias, and N. Ghobadipour, “Approximate ternary Jordan derivations on Banach ternary algebras,” Journal of Mathematical Physics, vol. 50, no. 4, Article ID 042303, pp. 1–9, 2009. View at Publisher · View at Google Scholar
  49. T. Bag and S. K. Samanta, “Finite dimensional fuzzy normed linear spaces,” Journal of Fuzzy Mathematics, vol. 11, no. 3, pp. 687–705, 2003.
  50. 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
  51. A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, “Fuzzy stability of the Jensen functional equation,” Fuzzy Sets and Systems, vol. 159, no. 6, pp. 730–738, 2008. View at Publisher · View at Google Scholar
  52. A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy approximately cubic mappings,” Information Sciences, vol. 178, no. 19, pp. 3791–3798, 2008. View at Publisher · View at Google Scholar
  53. A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy almost quadratic functions,” Results in Mathematics, vol. 52, no. 1-2, pp. 161–177, 2008. View at Publisher · View at Google Scholar