Fuzzy Stability of an Additive-Quartic Functional Equation: A Fixed Point Approach

Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 52)

Abstract

Mirmostafaee, Mirzavaziri and Moslehian have investigated the fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces. Using the fixed point method, we prove the generalized Hyers–Ulam stability of the following additive-quartic functional equation
$$\begin{array}{rcl} f(2x + y) + f(2x - y) =& & 2f(x + y) + 2f(-x - y) + 2f(x - y) + 2f(y - x) \\ & & +14f(x) + 10f(-x) - 3f(y) - 3f(-y) \\ \end{array}$$
in fuzzy Banach spaces.

Keywords

Fuzzy Banach space Fixed point Generalized Hyers–Ulam stability Quartic mapping Additive mapping 

References

  1. 1.
    Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bag, T., Samanta, S.K.: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11, 687–705 (2003)MathSciNetMATHGoogle Scholar
  3. 3.
    Bag, T., Samanta, S.K.: Fuzzy bounded linear operators. Fuzzy Sets and Systems 151, 513–547 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003)Google Scholar
  5. 5.
    Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346, 43–52 (2004)MATHGoogle Scholar
  6. 6.
    Cădariu, L., Radu, V.: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, Art. ID 749392 (2008)Google Scholar
  7. 7.
    Cheng, S.C., Mordeson, J.M.: Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 86, 429–436 (1994)MathSciNetMATHGoogle Scholar
  8. 8.
    Cholewa, P.W.: Remarks on the stability of functional equations. Aequationes Math. 27, 76–86 (1984)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg 62, 59–64 (1992)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Diaz, J., Margolis, B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 74, 305–309 (1968)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Felbin, C.: Finite dimensional fuzzy normed linear spaces. Fuzzy Sets and Systems 48, 239–248 (1992)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Găvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27, 222–224 (1941)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hyers, D.H., Isac G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)MATHCrossRefGoogle Scholar
  15. 15.
    Isac, G., Rassias, Th.M.: Stability of ψ-additive mappings: Appications to nonlinear analysis. Int. J. Math. Math. Sci. 19, 219–228 (1996)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Jung, S.M.: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press lnc., Palm Harbor, Florida (2001)MATHGoogle Scholar
  17. 17.
    Katsaras, A.K.: Fuzzy topological vector spaces II. Fuzzy Sets and Systems 12, 143–154 (1984)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kramosil, I., Michalek, J.: Fuzzy metric and statistical metric spaces. Kybernetika 11, 326–334 (1975)MathSciNetGoogle Scholar
  19. 19.
    Krishna, S.V., Sarma, K.K.M.: Separation of fuzzy normed linear spaces. Fuzzy Sets and Systems 63, 207–217 (1994)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Lee, S., Im S., Hwang, I.: Quartic functional equations. J. Math. Anal. Appl. 307, 387–394 (2005)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Miheţ, D., Radu, V.: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, 567–572 (2008)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Mirmostafaee, A.K., Mirzavaziri, M., Moslehian, M.S.: Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 159, 730–738 (2008)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Mirmostafaee, A.K., Moslehian, M.S.: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets and Systems 159 (2008), 720–729.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Mirmostafaee, A.K. Moslehian, M.S.: Fuzzy approximately cubic mappings. Inform. Sci. 178, 3791–3798 (2008)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Mirzavaziri, M., Moslehian, M.S.: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc. 37, 361–376 (2006)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Park, C.: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl. 2007, Art. ID 50175 (2007)Google Scholar
  27. 27.
    Park, C.: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl. 2008, Art. ID 493751 (2008)Google Scholar
  28. 28.
    Park, C., Cho, Y., Han, M.: Functional inequalities associated with Jordan-von Neumann type additive functional equations. J. Inequal. Appl. 2007, Art. ID 41820 (2007)Google Scholar
  29. 29.
    Park, C., Cui, J.: Generalized stability of C -ternary quadratic mappings. Abstr. Appl. Anal. 2007, Art. ID 23282 (2007)Google Scholar
  30. 30.
    Park, C., Najati, A.: Homomorphisms and derivations in C -algebras. Abstr. Appl. Anal. 2007, Art. ID 80630 (2007)Google Scholar
  31. 31.
    Park, C., Park, W., Najati, A.: Functional equations related to inner product spaces. Abstr. Appl. Anal. 2009, Art. ID 907121 (2009)Google Scholar
  32. 32.
    Radu, V.: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4, 91–96 (2003)MathSciNetMATHGoogle Scholar
  33. 33.
    Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Rassias, Th.M.: Problem 16; 2. In: Report of the 27th International Symposium on Functional Equations, Aequationes Math. 39, 309 (1990)Google Scholar
  35. 35.
    Rassias, Th.M.: On the stability of the quadratic functional equation and its applications. Stud. Univ. Babeş-Bolyai Math. 43, 89–124 (1998)MATHGoogle Scholar
  36. 36.
    Rassias, Th.M.: The problem of S.M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 246, 352–378 (2000)Google Scholar
  37. 37.
    Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251, 264–284 (2000)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Rassias, Th.M.: On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62, 23–130 (2000)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Rassias, Th.M., Šemrl, P.: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc. Amer. Math. Soc. 114, 989–993 (1992)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Rassias, Th.M., Šemrl, P.: On the Hyers-Ulam stability of linear mappings. J. Math. Anal. Appl. 173, 325–338 (1993)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Rassias, Th.M., Shibata, K.: Variational problem of some quadratic functionals in complex analysis. J. Math. Anal. Appl. 228, 234–253 (1998)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Skof, F.: Proprietà locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53, 113–129 (1983)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Ulam, S.M.: A Collection of the Mathematical Problems. Interscience Publ., New York (1960)Google Scholar
  44. 44.
    Xiao, J.Z., Zhu, X.H.: Fuzzy normed spaces of operators and its completeness. Fuzzy Sets and Systems 133, 389–399 (2003)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsHanyang UniversitySeoulRepublic of Korea
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

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