Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions

Let $f:\mathbb{R}\times \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ be a continuous function and let $h:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous and strictly positive function. A sufficient condition such that the equation $\dot{x}=f\left( t,x\right) $ admits solutions $x:\mathbb{R}\rightarro...

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Bibliographic Details
Main Author: C. Avramescu
Format: Article
Language:English
Published: University of Szeged 2003-07-01
Series:Electronic Journal of Qualitative Theory of Differential Equations
Online Access:http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=159
Description
Summary:Let $f:\mathbb{R}\times \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ be a continuous function and let $h:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous and strictly positive function. A sufficient condition such that the equation $\dot{x}=f\left( t,x\right) $ admits solutions $x:\mathbb{R}\rightarrow \mathbb{R}^{N}$ satisfying the inequality $\left| x\left( t\right) \right| \leq k\cdot h\left( t\right) ,$ $t\in \mathbb{R},$ $k>0$, where $\left| \cdot \right| $ is the euclidean norm in $\mathbb{R}^{N},$ is given. The proof of this result is based on the use of a special function of Lyapunov type, which is often called guiding function. In the particular case $h\equiv 1$, one obtains known results regarding the existence of bounded solutions.
ISSN:1417-3875