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

C. Avramescu, University of Craiova, Craiova, Romania

E. J. Qualitative Theory of Diff. Equ., No. 13. (2003), pp. 1-9.

Abstract: Let $f:\R\times \R^{N}\rightarrow \R^{N}$ be a continuous function and let $h:\R\rightarrow \R$ be a continuous and strictly positive function. A sufficient condition such that the equation $\dot{x}=f\left( t,x\right) $ admits solutions $x:\R\rightarrow \R^{N}$ satisfying the inequality $\left| x\left( t\right) \right| \leq k\cdot h\left( t\right) ,$ $t\in \R,$ $k>0$, where $\left| \cdot \right| $ is the euclidean norm in $\R^{N},$ 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.

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