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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 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 is bounded (uniformly with respect to the parameter μ ∈ ℝ) for each vector b ∈ X if and only if the Poincare map U(q,0) is stable.