In this note we establish the existence theorem of almost periodic solution for the almost periodic quasi-linear system dx/dt = A(t)x + f(t) + μF(X,t,μ)(1) where X is n-vector, ‖X‖ ≤ K, |t| < ∞,0 ≤μ≤μ, moreover suppose that (a) A(t) is an almost periodic and continuous n × n-matrix function, f(t) is an almost periodic and continuous vector function, and F(X, t,μ) is continuous in (X, t, μ), almost periodic in t, uniformly on ‖X‖≤K,0≤μ≤μ, and Lipshitzian in X; (b) A(t) is of class C~1, ‖A (t)‖≤K_o,‖dA(t)/dt‖≤K_1.The characteristic roots of A(t) are ρ1(t),ρ2(t),…,ρn(t), |Re (ρi(t) + ρi(t))|≥2δ_o>0,i,j = 1,2,3,…,n, and K_1≤1/2(1/K~o)~2(K_o/1+K_o)~((n-1)),K~o=2~(n-1)/2δ_o(1+K_o/2δ_o)~((n-1)(n+2))/2, where K_o,K_1, and δ_o are constants.At the same time we also give conditions for the existence of exponential dichotomies for linear differential equations. Now let us detail our results as follows:1. The system dX/dt=A(t)X(2)admits an exponential dichotomy, i. e. there exist positive constants α,β such that ‖Y_1(t)Z_1(τ)‖≤βexp(-α(t - τ)), t ≥τ, ‖Y_2(t)Z_2(τ)‖≤βexp(α(t - τ)),t ≤τ. where Y_1(t)+ Y_2(t)= Y(t) is the fundamental matrix of (1), and Z_1(t)+ Z_2(t)=Y~(-1)(t), iff there exists the quadric form V(X,t)=XG(t)X, where G(t) is bounded, symmetric and regular matrix, and total time derivative of V(X,t) with respect to the system (2) is positive definite (for the "if" part we require the matrix A(t) to be bounded).2. If A(t) satisfies the condition (b),then there is a quadric form V(X,t)=XG(t)X,such that (1) G(t) is a regular and symmetric matrix; (2) The total time derivative ofV(X,t) with respect to the system (2)is positive definite.3. The existence theorem: If (1) satisfies the condition (a) and (b), then there existsthe unique almost periodic solution of (1) for every μ in the interval 0≤μ≤μ_o for someu_o>0.
