  We establish a number of new sufficient conditions for the existence of
global (defined on the entire time axis) solutions of nonlinear nonautonomous
systems by means of the Wazewski topological principle. The systems under
consideration are characterized by the monotonicity property with respect to a
certain auxiliary guiding function W(t,x) depending on time and phase
coordinates. Another auxiliary spatially coercive function V(t,x) is used to
estimate the location of global solutions in the extended phase space. The
approach developed is applied to Lagrangian systems, and in particular, to
establish new sufficient conditions for the existence of almost periodic
solutions.
