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Hyers–Ulam Stability of Euler’s Differential Equation

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Abstract

We obtain a result on generalized Hyers–Ulam stability for Euler’s differential equation in Banach spaces. Our result extends and improves some recent results of Mortici, Jung and Rassias concerning the stability of Euler’s equation on a bounded domain.

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References

  1. Alsina C., Ger R.: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 2, 373–380 (1998)

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  3. Brzdek, J., Brillouët-Bellout, N., Ciepliński, K.: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012, 41 (2012) (Article ID 716936). doi:10.115/2012/716936

  4. Cîmpean D.S., Popa D.: On the stability of the linear differential equation of higher order with constant coefficients. Appl. Math. Comput. 217, 4141–4146 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  7. Jung, S.-M., Min, S.: On approximate Euler differential equations. Abstr. Appl. Anal. 2009, 8 (2009) (Article ID 537963). doi:10.1155/2009/537963

  8. Jung S.-M.: Hyers–Ulam stability of a system of first order linear differential equations with constant coefficients. J. Math. Anal. Appl. 320, 549–561 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lungu N., Ciplea S.: Ulam stability of some integral equations from economic dynamics. Autom. Comput. Appl. Math. 19, 289–295 (2010)

    MathSciNet  Google Scholar 

  10. Miura T., Miyajima S., Takahasi S.E.: A characterization of Hyers–Ulam stability of first order linear differential operators. J. Math. Anal. Appl. 286, 136–146 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Miura T., Miyajima S., Takahasi S.E.: Hyers–Ulam stability of linear differential operator with constant coefficients. Math. Nachr. 258, 90–96 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mortici C., Rassias Th.M., Jung S.M.: The inhomogeneous Euler equation and its Hyers–Ulam stability. Appl. Math. Lett. 40, 23–28 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Obloza M.: Hyers stability of the linear differential equation. RocznikNauk-Dydakt. Prace Mat. 13, 259–270 (1993)

    MathSciNet  MATH  Google Scholar 

  14. Obloza M.: Connections between Hyers and Lyapunov stability of the ordinary differenatial equations. RovcznikNauk-Dydakt. Prace Mat. 14, 141–146 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Popa D., Raşa I.: On the Hyers–Ulam stability of the linear differential equation. J. Math. Anal. Appl. 381, 530–537 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Popa D., Raşa I.: Hyers–Ulam stability of the linear differential operator with non-constant coefficients. Appl. Math. Comput. 219, 1562–1568 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Rus I.A.: Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 10, 305–320 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Rus I.A.: Ulam stability of ordinary differential equations. Studia Univ. Babeş-Bolyai Math. 54, 125–134 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Takahasi S.E., Miura T., Miyajima S.: On the Hyers–Ulam stability of the Banach space-valued differential equation \({y^{\prime}=\lambda y}\). Bull. Korean Math. Soc. 39, 309–315 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Takahasi S.E., Takagi H., Miura T., Miyajima S.: The Hyers–Ulam stability constants of first order linear differential operators. J. Math. Anal. Appl. 296, 403–409 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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Correspondence to Dorian Popa.

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Popa, D., Pugna, G. Hyers–Ulam Stability of Euler’s Differential Equation. Results. Math. 69, 317–325 (2016). https://doi.org/10.1007/s00025-015-0465-z

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  • DOI: https://doi.org/10.1007/s00025-015-0465-z

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