Pergamon Nonlinear Anulysis, Theory, Methods & Applications, Vol. 25, No. 4, pp. 409-415, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/95 $9.50+ .Ml 0362-546X(94)00153-7 REMARKS ON THE EXISTENCE OF ALMOST PERIODIC SOLUTIONS OF SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS SHIGEO KATO and MASATO IMAI Kitami Institute of Technology, Kitami, Hokkaido, Japan (Received 30 August 1993; received in revised form 20 January 1994; received for publication 4 July 1994) Key words and phrases: Dissipative type condition, almost periodic solution. 1. INTRODUCTION We denote by R the real line and by R” the Euclidean n-space. For x E R”, llxll be any convenient norm of x. In this paper we study the problem of existence and uniqueness of almost periodic solutions of a differential equation of the form x’ = F(t, x, x) + G(t, x), (D.E) where F(t, u, x) E C(R x R” x R”; R”) and G(t, x) E C(R x R”; R”). In [l] Kartsatos uses a result due to Medvedev [2] to get conditions sufficient for the existence of an almost periodic solution of (D.E). Uniqueness of the almost periodic solution of (D.E), however, is not implied in the result of [l]. Seifert [3] points out this fact and gives sufficient conditions to guarantee both the existence and uniqueness of almost periodic solutions of (D.E). All the results obtained in [l, 31 are essentially based on the result of [2]. Recently, we gave an improvement of the result of [2], and proved the existence and uniqueness of almost period solutions of a certain class of almost periodic systems. The purpose of this paper is to extend the results of [l, 31 by using some dissipative type conditions for F(t, u, x) and G(t, x). 2. PRELIMINARIES We define the functional [ , 1: R” x R” + R by kY1 = ;i:+ Wlx + hYll - Ilxllh The following lemma on the functional [ , ] is well known (see, for instance, [4]). LEMMA 2.1. Let x, y and z be in R”. Then the functional [ , ] has the following properties: (1) LGYI = pw + hYll - Ilxll); (2) I[-%Y1l 22 IIYII; (3) Lx, Y + 21 5 Ix, Yl + 1x9 21; (4) Let u be a function from a real interval J into R” such that u’(t,) exists for an interior point t, of J. Then ~+IJu(t,)(l exists and D+ II aJII = W,h W,)l~ where D+IIu(t,,)JI denotes the right derivative of Ilu(t at tO. 409