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

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