Using the Integral Manifolds to Solvability of Boundary Value Problems *

Acyclic solution sets of quasi-linearized differential systems with constraints are employed for solving nonlinear boundary value problems. A special attention is paid to periodic and anti-periodic solutions. The sufficient criteria are obtained in terms of (not necessarily smooth) bounding functions. The intersection of sublevel sets of these Liapunov-like functions forms a desired bound set with a transversality behaviour on its boundary.