V –MONOTONE COCYCLES AND ALMOST PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS

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 (x1(t), x2(t)) is non-increasing w.r.t. t ∈ R+, where x1(t) and x2(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