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

  • I. E. Vitrychenko

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 \(\left| {\lambda _n } \right| = 1 + o(1), n \to + \infty \) .
Published
25.12.1999
How to Cite
VitrychenkoI. E. “A Critical Case of Stability of One Quasilinear Difference Equation of the Second Order”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 51, no. 12, Dec. 1999, pp. 1593–1603, https://umj.imath.kiev.ua/index.php/umj/article/view/4764.
Section
Research articles