一类微分方程零解全局弱吸引和全局吸引的充要条件

Necessary and Sufficient Conditions for the Global Weak Attractivity and Global Attractivity of a Class of Nonlinear Differential Equations

考虑二阶微分方程x=ω（y）-F（x）,y=-g（x）q（y） （E） 零解的全局弱吸引和全局吸引性，说明了Filippov条件（A2）不能排除最大椭圆扇形S^＊的存在性，也不能排除δS^＊作为其外侧邻域轨线正向极限集的可能．全面回答了文献[8]末提出的问题；得到了方程（E）满足或不满足Filippov条件时零解全局弱吸引和全局吸引的一系列充分必要条件，同时也得到了零解全局渐近稳定的一些新条件．

This paper deals with the global weak attractivity and global attractivity for the nonlinear differential equations x=ω（y）-F（x）,y=-g（x）q（y） （E） It is shown that Filippov condition （A2） cannot exclude the existence of the maximum elliptic sector S^＊ and it cannot exclude the possibility of δS^＊ as the ω-limit set of the orbits departing from the exterior of S^＊. The problem proposed by Jiang Jifa in Nonlinear Analysis, 28（5）, 855-870（1997） is answered by a negative answer. A series of necessary and sufficient conditions for the global weak attractivity and global attractivity are established for both the cases that Filippov condition holds and Filippov condition fails. Some new conditions for the global asymptotic stability are also obtained.