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

TL;DR

This paper develops new sufficient conditions for the existence of global solutions in nonlinear nonautonomous systems using the Wazewski topological principle, with applications to Lagrangian and almost periodic solutions.

Contribution

It introduces a novel approach combining auxiliary functions W and V to establish global existence criteria for nonautonomous systems.

Findings

Established new conditions for global solutions

Applied method to Lagrangian systems

Derived criteria for almost periodic solutions

Abstract

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.