Hyers-Ulam stability of nonhomogeneous linear differential equations of second order.

*(English)*Zbl 1181.34014Summary: The aim of this paper is to prove the stability in the sense of Hyers-Ulam of the differential equation

\[ y''+p(x)y'+q(x)y+r(x)=0. \]

That is, if \(f\) is an approximate solution of thay equation then there exists an exact solution of this equation near to \(f\).

\[ y''+p(x)y'+q(x)y+r(x)=0. \]

That is, if \(f\) is an approximate solution of thay equation then there exists an exact solution of this equation near to \(f\).

##### MSC:

34A30 | Linear ordinary differential equations and systems |

34D99 | Stability theory for ordinary differential equations |

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\textit{Y. Li} and \textit{Y. Shen}, Int. J. Math. Math. Sci. 2009, Article ID 576852, 7 p. (2009; Zbl 1181.34014)

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##### References:

[1] | S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960. · Zbl 0086.24101 |

[2] | D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222-224, 1941. · Zbl 0061.26403 |

[3] | Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. · Zbl 0398.47040 |

[4] | Y.-H. Lee and K.-W. Jun, “A generalization of the Hyers-Ulam-Rassias stability of Jensen’s equation,” Journal of Mathematical Analysis and Applications, vol. 238, no. 1, pp. 305-315, 1999. · Zbl 0933.39053 |

[5] | C.-G. Park, “On the stability of the linear mapping in Banach modules,” Journal of Mathematical Analysis and Applications, vol. 275, no. 2, pp. 711-720, 2002. · Zbl 1021.46037 |

[6] | C. Alsina and R. Ger, “On some inequalities and stability results related to the exponential function,” Journal of Inequalities and Applications, vol. 2, no. 4, pp. 373-380, 1998. · Zbl 0918.39009 |

[7] | T. Miura, S.-E. Takahasi, and H. Choda, “On the Hyers-Ulam stability of real continuous function valued differentiable map,” Tokyo Journal of Mathematics, vol. 24, no. 2, pp. 467-476, 2001. · Zbl 1002.39039 |

[8] | T. Miura, “On the Hyers-Ulam stability of a differentiable map,” Scientiae Mathematicae Japonicae, vol. 55, no. 1, pp. 17-24, 2002. · Zbl 1025.47041 |

[9] | S.-E. Takahasi, T. Miura, and S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued differential equation y\(^{\prime}\)=\lambda y,” Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp. 309-315, 2002. · Zbl 1011.34046 |

[10] | T. Miura, S. Miyajima, and S.-E. Takahasi, “A characterization of Hyers-Ulam stability of first order linear differential operators,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 136-146, 2003. · Zbl 1045.47037 |

[11] | S.-E. Takahasi, H. Takagi, T. Miura, and S. Miyajima, “The Hyers-Ulam stability constants of first order linear differential operators,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 403-409, 2004. · Zbl 1074.47022 |

[12] | S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. II,” Applied Mathematics Letters, vol. 19, no. 9, pp. 854-858, 2006. · Zbl 1125.34328 |

[13] | S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. III,” Journal of Mathematical Analysis and Applications, vol. 311, no. 1, pp. 139-146, 2005. · Zbl 1087.34534 |

[14] | G. Wang, M. Zhou, and L. Sun, “Hyers-Ulam stability of linear differential equations of first order,” Applied Mathematics Letters, vol. 21, no. 10, pp. 1024-1028, 2008. · Zbl 1159.34041 |

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