A Fixed Point Approach to Stability of Quintic Functional Equations in Modular Spaces

  • Received : 2012.08.03
  • Accepted : 2013.08.02
  • Published : 2015.06.23


In this paper, we present a fixed point method to prove generalized Hyers-Ulam stability of the systems of quadratic-cubic functional equations with constant coefficients in modular spaces.


stability;quintic functional equation;fixed point;modular space


  1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989.
  2. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan., 2(1950), 64-66.
  3. P. W. Cholewa, Remarks on the stability of functional equations, Aequat. Math., 27(1984), 76-86.
  4. J. K. Chung and P. K. Sahoo, On the general solution of a quartic functional equation, Bull. Korean Math. Soc., 40(2003), 565-576.
  5. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg., 62(1992), 59-64.
  6. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, London, 2002.
  7. A. Ebadian, A. Najati and M. E. Gordji, On approximate additive-quartic and quadratic-cubic functional equations in two variables on abelian groups, Results. Math., DOI 10.1007/s00025-010-0018-4 (2010)
  8. A. Ebadian, N. Ghobadipour and M. E. Gordji, A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in $C^*$-ternary algebras, J. Math. Phys., 51 (2010), 10 pages, doi:10.1063/1.3496391.
  9. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci., 14(1991), 431-434.
  10. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184(1994), 431-436.
  11. P. Gavruta and L. Gavruta, A new method for the generalized Hyers-Ulam-Rassias stability, Int. J. Nonlinear Anal. Appl., 1(2010), 11-18.
  12. M. B. Ghaemi, M. E. Gordji and H. Majani, Approximately quintic and sextic mappings on the probabilistic normed spaces, Bull. Korean Math. Soc., 47(2)(2012), 339-352.
  13. M. E. Gordji, Stability of a functional equation deriving from quartic and additive functions, Bull. Korean Math. Soc., 47(2010), 491-502.
  14. M. Eshaghi Gordji, Y. J. Cho, M. B. Ghaemi and H. Majani, Approximately quintic and sextic mappings from r-divisible groups into Serstnev probabilistic Banach spaces: fixed point method, Discrete Dynamics in Nature and Society, 2011(2011), Article ID 572062, 16 pages.
  15. M. E. Gordji and M. B. Savadkouhi, Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces, Appl. Math. Lett., 23(2010), 1198-1202.
  16. M. E. Gordji, M. B. Ghaemi, S. K. Gharetapeh, S. Shams and A. Ebadian, On the stability of $J^*$-derivations, J. Geom. Phys., 60(2010), 454-459.
  17. M. E. Gordji, S. K. Gharetapeh, C. Park and S. Zolfaghri, Stability of an additive-cubic-quartic functional equation, Advances in Differ. Equat., Vol. 2009, Article ID 395693, 20. pages.
  18. M. E. Gordji, S. K. Gharetapeh, J. M. Rassias and S. Zolfaghari, Solution and stability of a mixed type additive, quadratic and cubic functional equation, Advances in differ. equat., Vol. 2009, Article ID 826130, 17 pages, doi:10.1155/2009/826130.
  19. M. E. Gordji and H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Anal., 71(2009), 5629-5643.
  20. M. E. Gordji and H. Khodaei, On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations, Abstr. Appl. Anal., Vol. 2009, Article ID 923476, 11 pages.
  21. M. E. Gordji and H. Khodaei, The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces, Discr. Dynam. in Nature and Soc., Vol. 2010, Article ID 140767, 15 pages, doi:10.1155/2010/140767.
  22. M. E. Gordji, H. Khodaei and R. Khodabakhsh, General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces, U. P. B. Sci. Bull., Series A, 72(2010), 69-84.
  23. M. E. Gordji and A. Najati, Approximately $J^*$-homomorphisms: A fixed point approach, J. Geom. Phys., 60(2010), 809-814.
  24. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27(1941), 222-224.
  25. D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aeq. Math., 44(1992), 125-153.
  26. D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
  27. G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of ${\psi}$-additive mappings, J. Approx. Theory, 72(1993), 131-137.
  28. G. Isac and Th. M. Rassias, Stability of ${\psi}$-additive mappings : Applications to non linear analysis, Internat. J. Math. and Math. Sci., 19(2)(1996), 219-228.
  29. K. W. Jun and H. M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274(2002), 867-878.
  30. K. W. Jun, H. M. Kim and I. S. Chang, On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation, J. Comput. Anal. Appl., 7(2005), 21-33.
  31. S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press Inc., Palm Harbor, Florida, 2001.
  32. S. M. Jung, Hyers-Ulam-Rassias stability of Jensen's equation and its application, Proc. Amer. Math. Soc., 126(1998), 3137-3143.
  33. S. M. Jung, Stability of the quadratic equation of Pexider type, Abh. Math. Sem. Univ. Hamburg, 70(2000), 175-190.
  34. P. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27(1995), 368-372.
  35. M. A. Khamsi, Quasicontraction Mapping in modular spaces without ${\Delta}_2$-condition, Fixed Point Theory and Applications Volume (2008), Artical ID 916187, 6 pages.
  36. H. Khodaei and Th. M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl., 1(2010), 22-41.
  37. S. Koshi, T. Shimogaki, On F-norms of quasi-modular spaces , J. Fac. Sci. Hokkaido Univ. Ser. I, 15(3)(1961), 202-218.
  38. M. Krbec, Modular interpolation spaces, Z. Anal. Anwendungen, 1(1982), 25-40.
  39. S. H. Lee, S. M. Im and I. S. Hawng, Quartic functional equation , J. Math. Anal. Appl., 307(2005), 387-394.
  40. W. A. Luxemburg, Banach function spaces, Ph. D. thesis, Delft Univrsity of technology, Delft, The Netherlands, 1959.
  41. L. Maligranda, Orlicz Spaces and Interpolation, in: Seminars in Math., Vol. 5, Univ. of Campinas, Brazil, 1989.
  42. J. Musielak, Orlicz Spaces and Modular Spaces, in: Lecture Notes in Math. Vol. 1034, Springer-verlag, Berlin, 1983.
  43. A. Najati, Hyers-Ulam-Rassias stability of a cubic functional equation, Bull. Korean Math. Soc., 44(2007), 825-840.
  44. H. Nakano, Modulared Semi-Ordered Linear Spaces, in: Tokyo Math. Book Ser., Vol. 1, Maruzen Co., Tokyo, 1950.
  45. W. Orlicz, Collected Papers, Vols. I, II, PWN, Warszawa, 1988.
  46. C. Park, On an approximate automorphism on a $C^*$-algebra, Proc. Amer. Math. Soc., 132(2004), 1739-1745.
  47. C. Park and M. E. Gordji, Comment on Approximate ternary Jordan derivations on Banach ternary algebras [Bavand Savadkouhi et al. J. Math. Phys. 50, 042303 (2009)], J. Math. Phys. 51, 044102 (2010), 7 pages.
  48. C. Park and A. Najati, Generalized additive functional inequalities in Banach algebras, Int. J. Nonlinear Anal. Appl., 1(2010), 54-62.
  49. C. Park and Th. M. Rassias, Isomorphisms in unital $C^*$-algebras, Int. J. Nonlinear Anal. Appl., 1(2010), 1-10.
  50. C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in $C^*$-algebras: a fixed point approach, Abstr. Appl. Anal. Vol. 2009, Article ID 360432, 17 pages.
  51. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(1978), 297-300.
  52. Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251(1)(2000), 264-284.
  53. Th. M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., 114(1992), 989-993.
  54. Gh. Sadeghi, A fixed point approach to stability of functional equations in modular spaces, Bull. Malays. Math. Sci. Soc., to appear.
  55. Gh. Sadeghi, On the orthogonal stability of the pexiderized quadratic equations in modular spaces, preprint.
  56. F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano., 53(1983), 113-129.
  57. Ph. Turpin, Fubini inequalities and bounded multiplier property in generalized modular spaces, Comment. Math., Tomus specialis in honorem Ladislai Orlicz I (1978), 331-353.
  58. S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Sci. Ed., Wiley, New York, 1964.
  59. S. Yamamuro, On conjugate spaces of Nakano spaces, Trans. Amer. Math. Soc., 90(1959), 291-311.