A GLOBAL DESCRIPTION OF THE PERIODIC SOLUTIONS OF SOME ORDINARY DIFFERENTIAL EQUATIONS P. M. FITZPATRICK, I. MASSABO AND J. PEJSACHOWICZ 1. The object of this paper is to prove precise results on the size of the set of solutions of the periodic boundary value problem x(t) + j(u(t,x(t))^J + A(x(t)) = 0, 0 ^ t ^ 2n, (1.1) X(0)-X(2TT) = X(0)-X(2TT) = 0, where x : [0, 2TT] -> U", U : U x U" -> U" is continuously differentiate with U{t, s) = U{t + 2n, s) when {t, s ) e U x W, and A is a constant nxn matrix. The problem of determining the existence of at least one solution of equation (1.1) has been considered by a number of authors (see [2, 3, 10, 11] and the references contained therein). On the other hand, if m denotes the nullity of A, m > 0, and U has the form U(t, s) = B(s) + g(t), where B is a constant nxn matrix and g is a given function, then, whenever x 0 is a solution of equation (1.1), we also have the m-dimensional plane of solutions consisting of {x o + A|A is a constant function whose value is in the null space of A}. The four results which we shall prove consist of descriptions of the manner in which this m-dimensional aspect of the solutions of equation (1.1) persists in the global description of the set of solutions of the above equation when U is allowed to be nonlinear. Since, when U is nonlinear, we cannot expect the solution of equation (1.1) to have any linear structure, when we use the term 'dimension' we shall mean the natural extension of the linear concept of dimension, namely, the Lebesgue covering dimension (see [8]). Our basic method of proof is to reformulate equation (1.1) as an operator equation of the form (1.2) x = F(X, x), {X, x) e 0 £ U m x X, where X is a Banach space, 0 is an open subset of R m xX, and F: 0 -* X is a compact operator. Then, under hypotheses on U and A which we specify in Section 2, we are able to apply recent abstract functional analytic results (Alexander and Yorke [1], Holm and Spanier [7], Massabo and Pejsachowicz [12], the authors [4]) to obtain our global description of the solutions of equation (1.1). In the case when Received 14 July, 1982; revised 13 March, 1984. J. London Math. Soc. (2), 29 (1984), 499-508