Existence and uniqueness of C(n)C(n)-almost periodic solutions to some ordinary differential equations

In this paper we prove the existence and uniqueness of C(n)C(n)-almost periodic solutions to the ordinary differential equation x′(t)=A(t)x(t)+f(t),t∈R, where the matrix A(t):R→Mk(C)A(t):R→Mk(C) is ττ-periodic and f:R→Ckf:R→Ck is C(n)C(n)-almost periodic. We also prove the existence and uniqueness of an ultra-weak C(n)C(n)-almost periodic solution in the case when A(t)=AA(t)=A is independent of tt. Finally we prove also the existence and uniqueness of a mild C(n)C(n)-almost periodic solution of the semilinear hyperbolic equation x′(t)=Ax(t)+f(t,x)x′(t)=Ax(t)+f(t,x) considered in a Banach space, assuming f(t,x)f(t,x) is C(n)C(n)-almost periodic in tt for each x∈Xx∈X, satisfies a global Lipschitz condition and takes values in an extrapolation space FA−1FA−1 associated to AA.