Given two positive continuous functions α and β, necessary and sufficient conditions are given for the system u''(t) = f(t) + A(t)u(t) to have an α -bounded solution u for each β -bounded forcing function f. Applications are given to a nonlinear perturbation problem: u''(t) = A(t)u(t) + F(t, u(t)). Indications are given on how to extend these ideas to {n<SUP>th</SUP>} order equations.