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On the stability of higher ring left derivations

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Abstract

In this note, we investigate the Hyers-Ulam, the Isac and Rassias-type stability and the Bourgin-type superstability of a functional inequality corresponding to the following functional equation:

$${h_n}\left( {xy} \right) = \sum\limits_{\begin{array}{*{20}{c}} {i + j = n} \\ {i \leqslant j} \end{array}} {\left[ {{h_i}\left( x \right){h_j}\left( y \right) + {c_{ij}}{h_i}\left( y \right){h_j}\left( x \right)} \right]} $$

, where

$${c_{ij}} = \left\{ {\begin{array}{*{20}{c}} 1&{if\;i \ne j,} \\ 0&{if\;i = j.} \end{array}} \right.$$

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Badora, On approximate ring homomorphisms, J. Math. Anal. Appl., 276(2) (2002), 589–597.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Badora, On approximate derivations, Math. Inequal. Appl., 9(1) (2006), 167–173.

    MathSciNet  MATH  Google Scholar 

  4. D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16 (1949), 385–397.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc., 10 (1990), 7–16.

    MathSciNet  MATH  Google Scholar 

  6. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431–434.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27 (1941), 222–224.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. N. P. Jewell, Continuity of module and higher derivations, Pacific J. Math., 68 (1977), 91–98.

    Article  MathSciNet  MATH  Google Scholar 

  11. B. E. Johnson, Continuity of derivations on commutative Banach algebras, Amer. J. Math., 91 (1969), 1–10.

    Article  MathSciNet  MATH  Google Scholar 

  12. B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math., 90 (1968), 1067–1073.

    Article  MathSciNet  MATH  Google Scholar 

  13. K.-W. Jun and Y.-W. Lee, The image of a continuous strong higher derivation is contained in the radical, Bull. Korean Math. Soc., 33 (1996), 229–233.

    MathSciNet  MATH  Google Scholar 

  14. Y.-S. Jung, On left derivations and derivations of Banach algebras, Bull. Korean Math. Soc., 35(4) (1998), 659–667.

    MathSciNet  MATH  Google Scholar 

  15. Y.-S. Jung, On the generalized Hyers-Ulam stability of module left derivations, J. Math. Anal. Appl., 339 (2008), 108–114.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. R. J. Roy, A class of topological algebras of formal power series, Carleton Mathematical Series, No. 11, November (1969).

    Google Scholar 

  18. I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann., 129 (1955), 260–264.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math., 128(2) (1988), 435–460.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sun Moon University, Asan, Chungnam, 336-708, Republic of Korea

    Yong-Soo Jung

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Correspondence to Yong-Soo Jung.

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Jung, YS. On the stability of higher ring left derivations. Indian J Pure Appl Math 47, 523–533 (2016). https://doi.org/10.1007/s13226-016-0201-8

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  • DOI: https://doi.org/10.1007/s13226-016-0201-8

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