ABSTRACT
Definition 3.1. We shall call operator L regular, if for every almost periodic function f(t), a system of differential equations
Lx = f(t)
has a unique almost periodic solution x(t).
Due to Banach’s Inverse Mapping Theorem (see Appendix C) the regularity of the operator L implies the existence of a continuous inverse L−1:
x(t) = L−1f(t)
in the space Bn of almost periodic vector-functions. Theorem 2.1 says that a system
dx
dt = Ax + f(t)
has a unique solution x(t) ∈ Bn for any given f(t) ∈ Bn, if all eigenvalues of the matrix A have non-zero real parts. Thus, if A satisfies this condition, the operator
Lx = dx
dt −Ax
is regular. The inverse L−1 is given by
x(t) = L−1f(t) =
∞∫
−∞ G(t− s)f(s)ds,
where G(t) is Green’s function for the problem of bounded solutions. Similarly, one can define a regular operator
Lx = dx
dt + A(t)x,
where A(t) is a matrix whose elements are T -periodic functions. In this case, one needs to require that for any given T -periodic vector function f(t), the system
Lx = f(t)
would have a unique T -periodic solution x(t). Using Remark 2.1 we obtain that the operator
Lx = dx
dt −Ax
is regular if the condition Π is satisfied, i.e., the matrix A has neither zero nor imaginary eigenvalues of the form i2πT k, where k is an integer.
