2019
Том 71
№ 6

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

Vitrychenko I. E.

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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 \(\left| {\lambda _n } \right| = 1 + o(1), n \to + \infty \) .

English version (Springer): Ukrainian Mathematical Journal 51 (1999), no. 12, pp 1799-1812.

Citation Example: Vitrychenko I. E. A critical case of stability of one quasilinear difference equation of the second order // Ukr. Mat. Zh. - 1999. - 51, № 12. - pp. 1593–1603.

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