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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
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