where u, g, and / are Λ-vectors. We will establish the existence of T-periodic solutions to (1.1) for large classes of nonlinearities g and forcing terms /. In particular we include cases where g is bounded, sublinear, or superlinear in u9 with arbitrary growth in the other arguments of g in the latter case. We also include cases in which g is mildly singular in t or vanishes at an endpoint of the interval. The second order scalar version of (1.1) has received attention recently with new and interesting results (see, e.g., [2] and [3]). The second order vector case of (1.1) has also been investigated by several authors recently with interesting results, often based on the sign of x g (see, e.g., [1], [5], [6]). Recently the second author in [8] obtained results which apply to higher order vector equations which lead to operators nonnegative outside a large ball. Our results here do not depend on sign conditions in the sense that if the function g in (1.1) satisfies our conditions so does — g. Also our results do not depend on the existence of a Nagumo function for g. Roughly speaking we assume that the nonlinearity g is either sublinear at infinity (see Cor. 2.2) or superlinear at the origin (see Cor. 2.3). Although we see our results as being of principle interest for higher order scalar and vector equations we obtain results which appear to be new even in the second order scalar case (see Example 3.3). We wish also to remark that while our results are all stated for periodic boundary conditions, our methods could be applied equally well to some other boundary value problems, e.g., the Neumann problem x'(0) = x'(T) = 0 for second order equations. For the proof of our result we will rely on an abstract result of Mawhin [4] which is an extension of the Leray-Schauder continuation theorem. Let X and Z be normed vector spaces, L: D(L) Q X -> Z a linear