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.
Similar content being viewed by others
References
Alsina C., Ger R.: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 2, 373–380 (1998)
Aoki T.: On the stability of the linear transformations in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)
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
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)
Hyers D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)
Hyers D.H., Isac G., Rassias Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)
Jung, S.-M., Min, S.: On approximate Euler differential equations. Abstr. Appl. Anal. 2009, 8 (2009) (Article ID 537963). doi:10.1155/2009/537963
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)
Lungu N., Ciplea S.: Ulam stability of some integral equations from economic dynamics. Autom. Comput. Appl. Math. 19, 289–295 (2010)
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)
Miura T., Miyajima S., Takahasi S.E.: Hyers–Ulam stability of linear differential operator with constant coefficients. Math. Nachr. 258, 90–96 (2003)
Mortici C., Rassias Th.M., Jung S.M.: The inhomogeneous Euler equation and its Hyers–Ulam stability. Appl. Math. Lett. 40, 23–28 (2014)
Obloza M.: Hyers stability of the linear differential equation. RocznikNauk-Dydakt. Prace Mat. 13, 259–270 (1993)
Obloza M.: Connections between Hyers and Lyapunov stability of the ordinary differenatial equations. RovcznikNauk-Dydakt. Prace Mat. 14, 141–146 (1997)
Popa D., Raşa I.: On the Hyers–Ulam stability of the linear differential equation. J. Math. Anal. Appl. 381, 530–537 (2011)
Popa D., Raşa I.: Hyers–Ulam stability of the linear differential operator with non-constant coefficients. Appl. Math. Comput. 219, 1562–1568 (2012)
Rus I.A.: Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 10, 305–320 (2009)
Rus I.A.: Ulam stability of ordinary differential equations. Studia Univ. Babeş-Bolyai Math. 54, 125–134 (2009)
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)
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)
Ulam S.M.: A Collection of Mathematical Problems. Interscience, New York (1960)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-015-0465-z