Higher Approximations of the Method of Averaging

Further, we suppose that the vector-functions Xj(t, x), j = 1, 2, . . . are almost periodic in t uniformly with respect to x ∈ D. However, essential limitations need to be imposed on the character of almost periodicity of these vectorfunctions. The thing is that to construct higher approximations of the method of averaging we use a close to identical, nonlinear transformation. Then we shall need to find almost periodic solutions of a system of differential equations with right-hand sides almost periodically dependent on t. It is possible to find these solutions if the right-hand sides of the system are the correct almost periodic functions. Recall that the almost periodic function f(t) with the Fourier series

f(t) ∼ ∑

aνe iνt

is called correct if the following formula is valid ∫

f(t)dt = a0t + g(t),

where a0 is the mean value of the function f(t), and g(t) is an almost periodic function with the Fourier series

∑

aν iν

eiνt.