We discuss two techniques useful in the investigation of periodic solutions of broad classes of non-linear non-autonomous
ordinary differential equations, namely the trigonometric collocation and the method based upon periodic successive approximations.
Multipoint boundary value problems for degenerate differential-operator equations of arbitrary order are studied. Several
conditions for the separability in Banach-valued Lp-spaces are given. Sharp estimates for the resolvent of the corresponding differential operator are obtained. In particular,
the sectoriality of this operator is established. As applications, the boundary value problems for degenerate quasielliptic
partial differential equations and infinite systems of differential equations on cylindrical domain are studied.
This paper focuses on boundary value problems for anisotropic differential-operator equations of high order with variable coefficients in the half plane. Several conditions are obtained which guarantee the maximal regularity of anisotropic elliptic and parabolic problems in Banach-valued Lp-spaces. Especially, it is shown that this differential operator is R-positive and is a generator of an analytic semigroup. These results are also applied to infinite systems of anisotropic type partial differential equations in the half plane.