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

Abstract

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.