Robustness of nonuniform polynomial dichotomies for difference equations (Q648541)

In the present paper the authors extend existing results on exponential dichotomy (stability) roughness to polynomial dichotomy (contraction). They consider the difference equation \[ x_{m+1}=(A_m+B_m) x_m,\quad m\in \mathbb N, \tag{\(B\)} \] where \(A_m, B_m\) are bounded linear operators for each \(m \in \mathbb N\). The following hypotheses are used in the main results. {1.} The system below admits the nonuniform polynomial dichotomy (contraction). \[ x_{m+1}=A_m x_m,\quad m\in \mathbb N. \tag{\(A\)} \] {2.} There exist \(\eta, \rho\) such that \(\|B_m\| \leq \eta m^{-\rho}\) for every \(m \in \mathbb N\). They show that for the sufficiently small parameter \(\eta\) the perturbed system \((B)\) still admits the nonuniform polynomial dichotomy (contraction). Construction of bounded solutions of the system \((B)\) is done by using Contraction Principle.