Nonautonomous Periodic Boundary-Value Problems in a Special Critical Case

We study the problem of finding conditions for the existence of solutions of weakly nonlinear periodic boundary-value problems for systems of ordinary differential equations and the construction of these solutions. We consider the special critical case where the equation for generating amplitudes of a weakly nonlinear periodic boundary-value problem reduces to the identity. We construct a new classification of critical cases and an iteration algorithm for the construction of solutions of weakly nonlinear periodic boundary-value problems in a special critical case.