Acta Applicandae Mathematicae

, Volume 110, Issue 2, pp 797–803 | Cite as

The Stability of the Quartic Functional Equation in Random Normed Spaces

  • D. MiheţEmail author
  • R. Saadati
  • S. M. Vaezpour


The main problem analyzed in this paper consists in showing that, under some conditions, every almost quartic mapping from a linear space to a random normed space under the Łukasiewicz t-norm can be suitably approximated by a quartic function, which is unique.


Generalized stability Functional equation Quartic functional equation Random normed space 

Mathematics Subject Classification (2000)

46S40 54E40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alsina, C., Schweizer, B., Sklar, A.: On the definition of a probabilistic normed space. Aequ. Math. 46, 91–98 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bag, T., Samanta, S.K.: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11(3), 687–705 (2003) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bourgin, D.G.: Classes of transformations and bordering transformations. Bull. Am. Math. Soc. 57, 223–237 (1951) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cădariu, L., Radu, V.: A Hyers–Ulam–Rassias stability theorem for a quartic functional equation. Autom. Comput. Appl. Math. 13(1), 31–39 (2004) Google Scholar
  6. 6.
    Chang, S.S., Cho, Y.J., Kang, S.M.: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science, New York (2001) zbMATHGoogle Scholar
  7. 7.
    Cheng, S.C., Mordeson, J.N.: Fuzzy linear operator and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 86, 429–436 (1994) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Chung, J.K., Sahoo, P.K.: On the general solution of a quartic functional equation. Bull. Korean Math. Soc. 40(4), 565–576 (2003) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Felbin, C.: Finite dimensional fuzzy normed linear space. Fuzzy Sets Syst. 48, 239–248 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Găvruţă, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hadžić, O., Pap, E.: Fixed Point Theory in PM Spaces. Kluwer Academic, Dordrecht (2001) Google Scholar
  12. 12.
    Hadžić, O., Pap, E., Budincević, M.: Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces. Kybernetica 38(3), 363–381 (2002) Google Scholar
  13. 13.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Kaleva, O., Seikala, S.: On fuzzy metric spaces. Fuzzy Sets Syst. 12, 215–229 (1984) zbMATHCrossRefGoogle Scholar
  15. 15.
    Katsaras, A.K.: Fuzzy topological vector spaces II. Fuzzy Sets Syst. 12, 143–154 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Trends in Logic, vol. 8. Kluwer Academic, Dordrecht (2000) zbMATHGoogle Scholar
  17. 17.
    Kramosil, I., Michalek, J.: Fuzzy metric and statistical metric spaces. Kybernetika 11, 326–334 (1975) MathSciNetGoogle Scholar
  18. 18.
    Lee, S.H., Kim, S.M., Hwang, I.S.: Quartic functional equations. J. Math. Anal. Appl. 307, 387–394 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Miheţ, D.: The probabilistic stability for a functional equation in a single variable. Acta Math. Hung. (2008). doi: 10.1007/s10474-008-8101-y Google Scholar
  20. 20.
    Miheţ, D.: The fixed point method for fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst. (2008). doi: 10.1016/j.fss.2008.06.014 Google 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) zbMATHMathSciNetGoogle Scholar
  22. 22.
    Mirmostafee, A.K.: A fixed point approach to almost quartic mappings in quasi fuzzy normed spaces. Fuzzy Sets Syst. (2009). doi: 10.1016/j.fss.2009.01.011 Google Scholar
  23. 23.
    Mirmostafaee, A.K., Moslehian, M.S.: Fuzzy approximately cubic mappings. Inf. Sci. (2008). doi: 10.1016/j.ins.2008.05.032 MathSciNetGoogle Scholar
  24. 24.
    Mirmostafaee, M., Mirzavaziri, M., Moslehian, M.S.: Fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst. 159, 730–738 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Mushtari, D.H.: On the linearity of isometric mappings on random normed spaces. Kazan Gos. Univ. Uch. Zap. 128, 86–90 (1968) Google Scholar
  26. 26.
    Najati, A.: On the stability of a quartic functional equation. J. Math. Anal. Appl. 340, 569–574 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Radu, V.: Some remarks on quasi-normed and random normed structures. In: Seminar on Probability Theory and Applications (STPA), vol. 159. West Univ. of Timişoara (2003) Google Scholar
  28. 28.
    Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Rassias, Th.M.: On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 62(1), 23–130 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York (1983) zbMATHGoogle Scholar
  31. 31.
    Sherstnev, A.N.: On the notion of a random normed space. Dokl. Akad. Nauk SSSR 149, 280–283 (1963) (in Russian) MathSciNetGoogle Scholar
  32. 32.
    Ulam, S.M.: Problems in Modern Mathematics. Science Editions. Wiley, New York (1964). (Chap. VI, Some Questions in Analysis: Sect. 1, Stability) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceWest University of TimişoaraTimişoaraRomania
  2. 2.Faculty of SciencesUniversity of ShomalAmolIran
  3. 3.Department of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran

Personalised recommendations