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On Stability of the Cauchy Equation on Solvable Groups

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The notion of (ψ, γ)-stability was introduced in [V. A. Faiziev, Th. M. Rassias, and P. K. Sahoo, Trans. Amer. Math. Soc., 354, 4455 (2002)]. It was shown that the Cauchy equation f(xy) = f(x) + f(y) is (ψ, γ)-stable both on any Abelian group and on any meta-Abelian group. In [V. A. Faiziev and P. K. Sahoo, Publ. Math. Debrecen, 75, 6 (2009)], it was proved that the Cauchy equation is (ψ, γ)-stable on step-two solvable groups and step-three nilpotent groups. In the present paper, we prove a more general result and show that the Cauchy equation is (ψ, γ)-stable on solvable groups.

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References

  1. J. Acz’el and J. Dhombres, “Functional equations in several variables,” Encycl. Math. Its Appl., Cambridge Univ. Press, Cambridge (1989).

  2. V. A. Faiziev, “Pseudocharacters on the free product of semigroups,” Funkts. Analiz Prilozh., 21, 86–87 (1987).

    MathSciNet  Google Scholar 

  3. V. A. Faiziev, “On the stability of a functional equation on groups,” Usp. Mat. Nauk, 48, 193–194 (1993).

    MathSciNet  Google Scholar 

  4. V. A. Faiziev and P. K. Sahoo, “On (ψ, γ)-stability of Cauchy equation on some noncommutative groups,” Publ. Math. Debrecen, 75, 67–83 (2009).

    MathSciNet  MATH  Google Scholar 

  5. V. A. Faiziev, Th. M. Rassias, and P. K. Sahoo, “The space of (ψ, γ)-additive mappings on semigroups,” Trans. Amer. Math. Soc., 354, 4455–4472 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  6. G. L. Forti, “Remark 11,” Rept of 22nd Internat. Symp. on Funct. Equat.: Aequat. Math., 29, 90–91 (1985).

  7. G. L. Forti, “Hyers–Ulam stability of functional equations in several variables,” Aequat. Math., 50, 143–190 (1995).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton (2011).

  11. S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York (1960).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Tver State Agricultural Academy, Tver, Russia

    V. A. Faiziev

  2. University of Louisville, Louisville, USA

    P. K. Sahoo

Authors
  1. V. A. Faiziev
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  2. P. K. Sahoo
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 7, pp. 1000–1005, July, 2015.

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Faiziev, V.A., Sahoo, P.K. On Stability of the Cauchy Equation on Solvable Groups. Ukr Math J 67, 1126–1132 (2015). https://doi.org/10.1007/s11253-015-1140-y

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  • DOI: https://doi.org/10.1007/s11253-015-1140-y

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