# Wa\'zewski Topological Principle and V-bounded Solutions of Nonlinear Systems

Research paper by Volodymyr Lagoda, Igor Parasyuk

Indexed on: 02 Jan '09Published on: 02 Jan '09Published in: Mathematics - Classical Analysis and ODEs

#### Abstract

We use the Wa\'zewski topological principle to establish a number of new sufficient conditions for the existence of proper (defined on the entire time axis) solutions of essentially nonlinear nonautonomous systems. 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 $V(t,x)$, which is positively defined in the phase variables $x$ for any $t$, is used to estimate the deviation of the proper solutions from the origin.