Asymptotic behavior results for solutions to some nonlinear difference equations

We give asymptotic results for convergent solutions {xn}{xn} of (real or complex) difference equations xn+1=Jxn+fn(xn), where xnxn is an m-vector, J   is a constant m×mm×m matrix and fn(y)fn(y) is a vector valued function which is continuous in y for fixed n  , and where fn(y)fn(y) is small in a sense. In addition, we obtain asymptotic results for solutions {xn}{xn} of the Poincaré difference equation xn+1=(A+Bn)xn where BnBn satisfies ‖Bn‖=O(ηn)‖Bn‖=O(ηn) with η∈(0,1)η∈(0,1). An application illustrates the results.