Stability of abstract difference equations

We suppose that M is a closed subspace of l∞(J, X), the space of all bounded sequences {x(n)}n∈J ⊂ X, where J ∈ {Z+, Z} and X is a complex Banach space. Consider the operator difference equation , where ψ ∈ l∞(Z, X) such that the restriction ψ|J on J belongs to M, and A is the generator of a C0-semigroup of linear bounded operators (T(t))t≥0 on X. Certain conditions will be imposed to guarantee the existence of solutions in the class M. We deduce that the zero equilibrium of the homogeneous equation , is globally asymptotically stable iff every solution is bounded.