Differential and Integral Equations, Volume 6, Number 5, September 1993, pp. 1119-1123. HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS GABRIELE BONANNO Dipartimento di Matematica, Universita di Messina, 98166 Sant'Agata-Messina, Italy and Universita di Reggio Calabria, via E. Cuzzocrea 48, 89128 Reggio Calabria, Italy SALVATORE A. MARANO Dipartimento di Matematica, Citta Universitaria, Viale A. Doria 6, 95125 Catania, Italy (Submitted by: M.Z. Nashed) Introduction. Let [a, b] be a compact real interval, n a non-negative integer, the real Euclidean (n + 1)-space, f a continuous real function defined on [a, b] x Then it is well-known that there exist a real number (3, with a < (3 :::; b, and a function u E cn+I ([a, (3]) such that u(i-l)(a) = 0 fori= 1, 2, ... , n + 1 and u(n+l)(t) = f(t,u(t),u'(t), ... ,u(n)(t)) for every t E [a,(3]. In general, one has (3 < b, even if the function f does not depend on t and is a polynomial. This happens, for example, when [a, b] = [0, 1], n = 0 and f(t, x) = (x + 1) 2 for every (t, x) E [0, 1] x for instance, [3], pp. 13-14). The aim of the present paper is to point out the following (as far as we know, new) problem: find, if possible, a positive integer v 2: n + 1 such that for every k 2: v and every h, t 2, .•• , tk E [a, b] there exists u E Ck([a, b]) such that u(k)(t) = f(t, u(t), u'(t), ... , u(n)(t)) for every t E [a, b], u(i-l)(ti) = 0 fori= 1, 2, ... , k. We show that if b- a < 1r /2 then this is possible (in fact, our answer is given in a more general setting; see Theorem 1.2 below). In this way, we obtain for instance that if 0 < b < 1r /2, then the problem { x(k) = (x + 1) 2 x(i-l)(ti) = 0, i = 1, 2, ... , k has at least one solution u E ck([O, b]) for any k 2: 2 + ['Y] and any t1, t2, ... , tk E [0, b], where')'= and, as usual, b] denotes the greatest integer which is less than or equal to ')' (see Remark 1.1 below). At present, we do not know what happens when b- a 2: 1r /2. So, our paper must be understood only as a first contribution to the study of the above-mentioned problem. Received February 1992. AMS Subject Classification: 34810, 34815. 1119