Stability of a pexiderial functional equation in random normed spaces

  • Hassan Azadi KenaryEmail author


The concept of Hyers-Ulam-Rassias stability originated from Th.M. Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72:297–300, 1978. Recently, the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation
proved in the earlier work. In this paper, using direct method we prove the generalized Hyers-Ulam stability of the following Pexiderial functional equation
in random normed space.


Hyers-Ulam stability Random normed spaces 

Mathematics Subject Classification (2000)

39B22 39B52 


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  1. 1.
    Azadi Kenary, H., Cho, Y.J.: Stability of mixed additive-quadratic Jensen type functional equation in various spaces. Comput. Math. Appl. (2011). doi: 10.1016/j.camwa.2011.03.024 zbMATHGoogle Scholar
  2. 2.
    Cholewa, P.W.: Remarks on the stability of functional equations. Aequ. Math. 27(1–2), 76–86 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 62, 59–64 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Gavruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184(3), 431–436 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of functional equations in several variables. In: Progress in Nonlinear Differential Equations and their Applications, vol. 34. Birkhäuser, Basel (1998) Google Scholar
  7. 7.
    Hyers, D.H., Isac, G., Rassias, Th.M.: On the asymptoticity aspect of Hyers-Ulam stability of mappings. Proc. Am. Math. Soc. 126(2), 425–430 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Hyers, D.H., Rassias, Th.M.: Approximate homomorphisms. Aequ. Math. 44(2–3), 125–153 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Jordan, P., von Neumann, J.: On inner products in linear metric spaces. Ann. Math. 36(3), 719–723 (1935) zbMATHCrossRefGoogle Scholar
  10. 10.
    Jung, S.-M.: On the Hyers-Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl. 222(1), 126–137 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Jung, S.-M.: On the Hyers-Ulam-Rassias stability of a quadratic functional equation. J. Math. Anal. Appl. 232(2), 384–393 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Jung, S.-M.: Stability of the quadratic equation of Pexider type. Abh. Math. Semin. Univ. Hamb. 70, 175–190 (2000) zbMATHCrossRefGoogle Scholar
  13. 13.
    Kannappan, P.: Quadratic functional equation and inner product spaces. Results Math. 27(3–4), 368–372 (1995) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Najati, A., Park, C.: On the stability of a cubic functional equation. Acta Math. Sin. Engl. Ser. (2011, to appear) Google Scholar
  15. 15.
    Park, C.: Universal Jensen’s equations in Banach modules over a C -algebra and its unitary group. Acta Math. Sin. Engl. Ser. 20(6), 1047–1056 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Park, C., Hou, J., Oh, S.: Homomorphisms between JC -algebras and Lie C -algebras. Acta Math. Sin. Engl. Ser. 21(6), 1391–1398 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Park, C., Rassias, Th.M.: The N-isometric isomorphisms in linear N-normed C -algebras. Acta Math. Sin. Engl. Ser. 22(6), 1863–1890 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72(2), 297–300 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251(1), 264–284 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Saadati, R., Vaezpour, M., Cho, Y.J.: A note to paper “On the stability of cubic mappings and quartic mappings in random normed spaces”. J. Inequal. Appl. 2009, 214530 (2009) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Schewizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland, New York (1983) Google Scholar
  22. 22.
    Skof, F.: Local properties and approximation of operators. Rend. Semin. Mat. Fis. Milano 53, 113–129 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Ulam, S.M.: A collection of mathematical problems. Intersci. Tracts Pure Appl. Math. (8) (1960) Google Scholar

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesYasouj UniversityYasoujIran

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