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

Анотація

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 Waźewski 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 function V(t,x), such that lim_||x||→∞V(t,x)=∞ for all t⊂R, 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.