Accurate solution estimates for nonlinear nonautonomous vector difference equations (Q1773541)

The authors consider the vector discrete dynamical system \(x_{k+1}=A_kx_k+f_k(x_k)\). A well-known result from Perron states that this system is asymptotically stable if \(A_k=A=\) constant is stable and \(f_k(x)\equiv\tilde f(x)=o(|| x||)\) as \(x\to 0\), although this result does not give any information about the size of the region of asymptotic stability and norms of solutions. The authors give accurate estimates for the norms of the solutions. These estimates give stability conditions for the equation \(x_{k+1}=A_kx_k+f_k(x_k)\) and bounds for the region of attraction of the stationary solution. The approach is based on the ``freezing'' method for difference equations and on recent estimates for the powers of a constant matrix. The authors also discuss applications of their main result to partial reaction-diffusion difference equations.