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On the stability of a mixed functional equation deriving from additive, quadratic and cubic mappings

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Abstract

In this paper, we investigate the general solution and the Hyers-Ulam stability of the following mixed functional equation

$f(2x + y) + f(2x - y) = 2f(2x) + 2f(x + y) + 2f(x - y) - 4f(x) - f(y) - f( - y)$

deriving from additive, quadratic and cubic mappings on Banach spaces.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Cieplinski, K.: Applications of fixed point theorems to Hyers-Ulam stability of functional equation — a survey. Ann. Funct. Anal., 3(1), 151–164 (2012)

    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 Publ. Co., Singapore, New Jersey, London, 2002

    Book  MATH  Google Scholar 

  5. Eskandani, G. Z., Gavruta, P., Rassias, J. M., et al.: Generalized Hyers-Ulam stability for a general mixed functional equation in quasi-β-normed spaces. Mediterr. J. Math., 8, 331–348 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gavruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mapings. J. Math. Anal. Appl., 184, 431–436 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Jung, S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications, 48, Springer, New York, 2011

    MATH  Google Scholar 

  9. Jun, K.-W., Kim, H.-M.: The generalized Hyers-Ulam-Rassias stability of a cubic function equation. J. Math. Anal. Appl., 274(2), 267–278 (2002)

    Article  MathSciNet  Google Scholar 

  10. Kim, H.-M., Rassias, J. M.: Generalization of Ulam stability problem for Euler-Lagrange quadratic mappings. J. Math. Anal. Appl., 336, 277–296 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Li, Y., Hua, L.: Hyers-Ulam stability of a polynomial equation. Banach J. Math. Anal., 3(2), 86–90 (2009)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Park, C.: On the Hyers-Ulam-Rassias stability of generalized quadratic mapping in Banach modules. J. Math. Anal. Appl., 291(1), 214–223 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Rassias, Th. M., Moslehian, M. S.: Stability of functional equations in non-Archimedean spaces. Appl. Anal. Discrete Math., 1(2), 325–334 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rassias, Th. M., Semrl, P.: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc. Amer. Math. Soc., 114, 989–993 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  18. Skof, F.: Local properties and approximations of operators. Rend. Sem. Mat. Fis. Milano, 53, 113–129 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ulam, S. M.: A Collection of Mathematical Problems, Interscience Publ., New York, 1960

    MATH  Google Scholar 

  20. Wang, L., Liu, B.: The Hyers-Ulam stability of a functional equation deriving from quadratic and cubic functions in quasi-β-normed spaces. Acta Math. Sin., Engl. Ser., 26(12), 2335–2348 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Xu, T., Rassias, J. M., Xu, W.: Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces. Acta Math. Sin., Engl. Ser., 28(3), 529–560 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, P. R. China

    Li Guang Wang & Qiu Wen Liu

  2. Zaozhuang Economics School, Zaozhuang, 277101, P. R. China

    Kun Peng Xu

Authors
  1. Li Guang Wang
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  2. Kun Peng Xu
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  3. Qiu Wen Liu
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Corresponding authors

Correspondence to Li Guang Wang or Kun Peng Xu.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 11371222), Natural Science Foundation of Shandong Province (Grant No. ZR2012AM024) and China Scholarship Council

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Wang, L.G., Xu, K.P. & Liu, Q.W. On the stability of a mixed functional equation deriving from additive, quadratic and cubic mappings. Acta. Math. Sin.-English Ser. 30, 1033–1049 (2014). https://doi.org/10.1007/s10114-014-3335-9

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  • DOI: https://doi.org/10.1007/s10114-014-3335-9

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