In the paper, the periodic BVP on [0,p] \((p>0)\) for the n-th order differential equation in \({\mathbb{R}}^ m\) (1) \(x^{(n)}+Ag(t,x)=h(t)\) is considered, A being a constant matrix and g and h continuous functions. Some existence theorems are given, under suitable hypotheses on the interaction of the nonlinear field g with the spectrum of the linear differential operator \(x\to x^{(n)}\) (with periodic boundary conditions on [0,p]). Precisely, a (uniform) growth restriction on g of the type \((x| g)\geq a| g|^ 2+b| g| +c\) (a,b,c real constants) is coupled with a semi-abstract sign condition which contains various geometrical assumptions on nonlinear fields previously considered in the literature (see the Appendix of the paper). In particular, for n even, the growth restriction permits us to extend the classical nonresonance condition, for scalar equations \((A=(-1)^{(n+2)/2}), g(t,x)/x\leq k<\omega^ n:=(2\pi /p)^ n,\) to \(g(t,x)/x\uparrow \omega^ n\), provided not ”too quickly”. Other applications of the approach presented here are given elsewhere, the authors, Existence results for forced nonlinear periodic BVPs at resonance, Ann. Mat. Pura Appl. (to appear), dealing with the equation \(x''+k^ 2\omega^ 2x+Ag(t,x)=h(t).\) Recently, extensions of the results have been obtained, by the authors, for Liénard systems, relaxing also the uniform growth restriction to a nonuniform one.