## Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions.(English)Zbl 1044.34008

The paper deals with the nonautonomous ordinary differential equation $\dot x=f(t,x), \;x\in\mathbb R^N,$ where $$f:\mathbb R\times\mathbb R^N\to\mathbb R^N$$ is a continuous function. For a given continuous and strictly positive function $$h:\mathbb R\to\mathbb R$$, the author states and proves a sufficient condition for the existence of a solution $$x(t)$$ satisfying the inequality $$| x(t)| \leq kh(t)$$, $$k>0$$, for any $$t\in \mathbb R$$. The question studied in the paper is an analogue of the global Lyapunov stability problem for an autonomous ordinary differerential equation. A way to prove the global stability is to try to construct a {Lyapunov function} – a function decreasing along the phase curves. The proof of the main result in the paper is based on the method of guiding functions, which are nonautonomous analogues of Lyapunov functions.

### MSC:

 34D05 Asymptotic properties of solutions to ordinary differential equations 34D23 Global stability of solutions to ordinary differential equations

### Keywords:

differential inequalities; guiding functions
Full Text: