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

TL;DR

This paper applies the Ważewski topological principle to derive new sufficient conditions for the existence of global solutions in nonlinear nonautonomous systems characterized by monotonicity and auxiliary functions.

Contribution

It introduces novel criteria based on auxiliary functions to ensure the existence of solutions defined on the entire time axis for nonlinear systems.

Findings

Established new sufficient conditions for global solutions

Utilized auxiliary functions to analyze solution behavior

Extended Ważewski principle to nonlinear nonautonomous systems

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.