Asymptotic stability and uniform boundedness with respect to parameters for discrete non-autonomous periodic systems

Let X be a complex Banach space and q >= 2 be a fixed integer number. Let U = {U(n, j)}(n >= j >= 0) subset of L(X) be a q-periodic discrete evolution family generated by the L(X)-valued, q-periodic sequence (A(n)). We prove that the solution of the following discrete problem y(n+1) = A(n)y(n) + e(i mu n)b, n is an element of Z(+), y(0) = 0 is bounded (uniformly with respect to the parameter mu is an element of R) for each vector b is an element of X if and only if the Poincare map U(q,0) is stable.