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A critical case of stability of one quasilinear difference equation of the second order

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Abstract

We obtain sufficient conditions for the Perron stability of the trivial solution of a real difference equation of the form

$$y_{n + 1} - 2\lambda _n y_n + y_{n - 1} = F(n,y_n ,\Delta y_{n - 1} ), n \in N$$

where\(y_n \in \left] { - 1,1} \right[,\left| {F(n,y_n ,\Delta y_{n - 1} )} \right| \le L_n \left( {\left| {y_n \left| + \right|\Delta y_{n - 1} } \right|} \right)^{1 + \alpha } ,L_n \ge 0\) and\(\alpha \in \left] {0, + \infty } \right[\). The resuits obtained are valid for the case where\(\left| {\lambda _n } \right| = 1 + o(1), n \to + \infty \).

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Authors and Affiliations

  1. Odessa University, Odessa

    I. E. Vitrichenko

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  1. I. E. Vitrichenko
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 12, pp. 1593–1603, December, 1999.

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Vitrichenko, I.E. A critical case of stability of one quasilinear difference equation of the second order. Ukr Math J 51, 1799–1812 (1999). https://doi.org/10.1007/BF02525138

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  • DOI: https://doi.org/10.1007/BF02525138

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