Stability for planar linear discrete Hamiltonian systems with perturbations

In this article, we will give several conditions under which the following planar linear discrete Hamiltonian system with perturbations {Delta x(n) = [alpha(n) + alpha(1)(n)]x(n + 1) + [beta(n) + beta(0)(n)] y(n) + f(1)(n, x(n), y(n)), Delta y(n) = -[gamma(n) + gamma(0)(n)]x(n + 1) - [alpha(n) + alpha(2)(n)] y(n) + f(2)(n, x(n), y(n)) has the same stability as the corresponding linear system Delta x(n) = alpha(n)x(n + 1) + beta(n)y(n),Delta y(n) = -gamma(n)x(n + 1) - alpha(n) y(n). Moreover, these conditions are shown to be necessary and sharp by examples.