In this paper we discuss the problem of determining a T-periodic solution x∗(·, λ) of the differential equation x =A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space Rk. The customary approach assumes that λ = λ(ϵ), ϵ ϵ R. One then establishes the existence of an ϵ0 > 0 such that the differential equation has a T-periodic solution x∗(·, λ(ϵ)) for all ϵ satisfying 0 < ϵ < ϵ0. More specifically it is usually assumed that λ(ϵ) has the form λ(ϵ) = ϵλ0 where λ0 is a fixed vector in Rk.