V –MONOTONE COCYCLES AND ALMOST PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS DAVID CHEBAN AND CRISTIANA MAMMANA Abstract. In the present paper we consider a special class of equations x ′ = f (t, x) when the function f : R × E → E (E is a finite-dimensional Banach space) is V –monotone with respect to (w.r.t.) x ∈ E, i.e. there exists a continuous non-negative function V : E × E → R + , which equals to zero only on the diagonal, so that the numerical function α(t) := V (x 1 (t),x 2 (t)) is non- increasing w.r.t. t ∈ R + , where x 1 (t) and x 2 (t) are two arbitrary solutions of (1) defined and bounded on R + . The main result of the paper contains the solution of the problem of V.V.Zhikov (1973): every finite-dimensional V -monotone almost periodic dif- ferential equation with bounded solutions admits at least one almost periodic solution. 1. Introduction The problem of the almost periodicity of solutions of non-linear almost periodic ordinary differential equations (1) x ′ = f (t,x) was studied by many authors (see, for example, [3, 4, 5, 6, 7, 8, 14, 20, 21] and the bibliography therein). In the present paper we consider a special class of equations (1), where the function f : R × E → E (E is a finite-dimensional Banach space) is V –monotone with respect to (w.r.t.) x ∈ E, i.e. there exists a continuous non-negative function V : E × E → R + which equals to zero only on the diagonal so that the numerical function α(t) := V (x 1 (t),x 2 (t)) is non-increasing w.r.t. t ∈ R + , where x 1 (t) and x 2 (t) are two arbitrary solutions of (1) defined and bounded on R + . This class of non-linear differential equations (1) is interesting enough and well studied (see, for example, [5, 6, 14, 24] and the bibliography therein). If the function α(t)= V (x 1 (t),x 2 (t)) is strictly decreasing, then equation (1) admits a single almost periodic solution if there exists a bounded solution on R + . Date : September 22, 2003. 1991 Mathematics Subject Classification. primary:34C35, 34D20, 34D40, 34D45, 58F10, 58F12, 58F39; secondary: 35B35, 35B40. Key words and phrases. Non-autonomous dynamical systems, skew-product systems, cocycles, continuous invariant sections of non-autonomous dynamical systems, almost periodic and almost automorphic solutions, V –monotone dynamical systems. 1