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On Approximate Additive–Quartic and Quadratic–Cubic Functional Equations in Two Variables on Abelian Groups

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Abstract

We establish the generalized Hyers–Ulam stability of the additive–quartic and quadratic–cubic mappings with two variables on abelian groups.

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References

  1. Aoki T.: On the stability of the linear transformationin Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baak C., Moslehian M.S.: On the stability of orthogonally cubic functional equations. Kyungpook Math. J. 47(1), 69–76 (2007)

    MATH  MathSciNet  Google Scholar 

  3. Czerwik S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg 62, 59–64 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Czerwik S.: Functional Equations and Inequalities in Several Variables. World Scientific, London (2002)

    Book  MATH  Google Scholar 

  5. Cholewa P.W.: Remarks on the stability of functional equations. Aequationes Math. 27, 76–86 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  6. Eshaghi Gordji M., Khodaei H.: Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces. Nonlinear Anal. 71(11), 5629–5643 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Eshaghi Gordji M., Kaboli S., Zolfaghari S., Rassias J.M.: Stability of a mixed type quadratic, cubic and quartic functional equation (preprint)

  8. Eshaghi M., Moslehian M.S., Kaboli S., Zolfaghari S. et al.: Stability of a mixed type additive, quadratic, cubic and quartic functional equation. In: Khan, A. (eds) Nonlinear Analysis and Variational Problems, pp. 65–80. Springer, Berlin (2009)

    Google Scholar 

  9. Gajda Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Găvruta P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hyers D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

  12. Hyers D.H., Isac G., Rassias T.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  13. Isac G., Rassias T.M.: On the Hyers–Ulam stability of ψ-additive mappings. J. Approx. Theory 72, 131–137 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Isac, G., Rassias, T.M.: Functional inequalities for approximately additive mappings. Stability of mappings of Hyers–Ulam type, pp. 117–125. Hadronic Press Collection of Original Articles. Hadronic Press, Palm Harbor (1994)

  15. Jun K.W., Kim H.M.: The generalized Hyers–Ulam–Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274, 267–278 (2002)

    Article  MathSciNet  Google Scholar 

  16. Jung S.-M.: Stability of the quadratic equation of Pexider type. Abh. Math. Sem. Univ. Hamburg 70, 175–190 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jung S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)

    MATH  Google Scholar 

  18. Kannappan P.: Quadratic functional equation and inner product spaces. Results Math. 27, 368–372 (1995)

    MathSciNet  Google Scholar 

  19. Mirzavaziri M., Moslehian M.S.: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc. 37(3), 361–376 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Moslehian M.S.: On the orthogonal stability of the Pexiderized quadratic equation. J. Differ. Equ. Appl. 11, 999–1004 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Moslehian M.S., Rassias T.M.: Stability of functional equations in non-Archimedian spaces. Appl. Anal. Disc. Math. 1(2), 325–334 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Moslehian M.S., Nikodem K., Popa D.: Asymptotic aspect of the quadratic functional equation in multi-normed spaces. J. Math. Anal. Appl. 355(2), 717–724 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Najati A., Park C.: On the stability of an n-dimensional functional equation originating from quadratic forms. Taiwanese J. Math. 12(7), 1609–1624 (2008)

    MATH  MathSciNet  Google Scholar 

  24. Najati A., Moradlou F.: Stability of a quadratic functional equation in quasi-Banach spaces. Bull. Korean Math. Soc. 45(3), 587–600 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Najati A.: On the stability of a quartic functional equation. J. Math. Anal. Appl. 340(1), 569–574 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Najati A.: Hyers–Ulam–Rassias stability of a cubic functional equation. Bull. Korean Math. Soc. 44(4), 825–840 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Najati A., Park C.: On the stability of a cubic functional equation. Acta Math. Sin. (Engl. Ser.) 24(12), 1953–1964 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Najati A.: On the stability of a quartic functional equation. J. Math. Anal. Appl. 340(1), 569–574 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Najati A., Rahimi A.: A fixed point approach to the stability of a generalized Cauchy functional equation. Banach J. Math. Anal. 2(1), 105–112 (2008)

    MATH  MathSciNet  Google Scholar 

  30. Nikodem K.: On some properties of quadratic stochastic processes. Ann. Math. Silesianae 3(15), 59–69 (1990)

    MathSciNet  Google Scholar 

  31. Park C.: On the stability of the orthogonally quartic functional equation. Bull. Iranian Math. Soc. 31(1), 63–70 (2005)

    MATH  MathSciNet  Google Scholar 

  32. Park W.-G., Bae J.-H.: On a bi-quadratic functional equation and its stability. Nonlinear Anal. 62(4), 643–654 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  33. Rassias T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    MATH  MathSciNet  Google Scholar 

  34. Rassias, T.M.: Problem 16 ; 2, Report of the 27th International symposium on functional equations. Aequationes Math. vol. 39, pp. 292–293, 309 (1990)

  35. Rassias, T.M.: (ed.) Functional equations, inequalities and applications. Kluwer, Dordrecht (2003)

    MATH  Google Scholar 

  36. Rassias T.M., Šemrl P.: On the behaviour of mappings which do not satisfy Hyers–Ulam stability. Proc. Am. Math. Soc. 114, 989–993 (1992)

    MATH  Google Scholar 

  37. Sahoo P.K.: A generalized cubic functional equation. Acta Math. Sin. (Engl. Ser.) 21(5), 1159–1166 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  38. Skof F.: Proprietà locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53, 113–129 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  39. Ulam S.M.: Problems in Modern Mathematics, Chapter VI, Science Editions. Wiley, New York (1964)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Urmia University, Urmia, Iran

    A. Ebadian

  2. Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran

    A. Najati

  3. Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran

    M. Eshaghi Gordji

Authors
  1. A. Ebadian
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  2. A. Najati
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  3. M. Eshaghi Gordji
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Correspondence to A. Ebadian.

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Ebadian, A., Najati, A. & Gordji, M.E. On Approximate Additive–Quartic and Quadratic–Cubic Functional Equations in Two Variables on Abelian Groups. Results. Math. 58, 39–53 (2010). https://doi.org/10.1007/s00025-010-0018-4

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  • DOI: https://doi.org/10.1007/s00025-010-0018-4

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