| Absztrakt: | The author considers partial asymptotic stability for the differential
equation $$ (E)\quad x'=X(x,t)\quad (t\in {\bbfR}\sb+,\quad x\in
{\bbfR}\sp k). $$ The vector $x\in {\bbfR}\sp k$ is first decomposed
as $x=(y,z)$ where $y\in {\bbfR}\sp m$, $z\in {\bbfR}\sp n$,
$1\le m\le k$, $n=k-m$. Now let $G\sb y(H)=\{(y,z)\in {\bbfR}\sp
m\times {\bbfR}\sp n:\vert y\vert <H\}$ and $\Gamma\sb y=G\sb
y(H)\times {\bbfR}\sb+$. By assuming the existence of a Lyapunov
function V that can be composed as the sum of two continuous,
locally Lipschitzian functions $V\sb i: \Gamma\sb y(H)\to {\bbfR}$
such that $V(x,t)=V\sb 1(x,t)+V\sb 2(x,t)\ge 0,$ $V\sb 1(x,t)\ge
0$, and V is nondecrescent in some extended sense, the author
shows every solution x(t) of (E) satisfies $\lim\sb{t\to \infty}V\sb
1(x(t),t)=0,\quad \lim\sb{t\to \infty}V\sb 2(x(t),t)<\infty.$
Moreover, if $V\sb 1$ is positive definite with respect to y,
then the y- component of a solution x(t) will tend to zero as
$t\to \infty$. Application to the generalized Lienard equation
$$ x''+a(t)g(x,x')x'+b(t)f(x)=0 $$ and the Lagrangian equation
$$ \frac{d}{dt}\frac{\partial T}{\partial \dot q}-\frac{\partial
T}{\partial q}=-g\frac{2\partial P\sp*}{\partial q}+Q\quad (q,\dot
q\in {\bbfR}\sp r) $$ are given. This extends the author's earlier
work on partial asymptotic stability for autonomous and limiting
equations [Acta Sci. Math. 45, 219-231 (1983; Zbl 0524.34057)
and ibid. 46, 143-156 (1983; Zbl 0581.34039)].}
\RV{P.K.Wong |