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We establish a number of new sufficient conditions for the existence ofglobal (defined on the entire time axis) solutions of nonlinear nonautonomoussystems by means of the Wazewski topological principle. The systems underconsideration are characterized by the monotonicity property with respect to acertain auxiliary guiding function W(t,x) depending on time and phasecoordinates. Another auxiliary spatially coercive function V(t,x) is used toestimate the location of global solutions in the extended phase space. Theapproach developed is applied to Lagrangian systems, and in particular, toestablish new sufficient conditions for the existence of almost periodicsolutions.

Existence of V-bounded solutions for nonautonomous nonlinear systems via the Wazewski topological principle