Absztrakt: | [For the entire collection see Zbl 0607.00010.] \par
The paper
is concerned with the generalization of the invariance principle
[see {\it J. P. La Salle}: Dynamical Systems, An International
Symposium, Vol. I, 211-222 (1976; Zbl 0356.34047)] to nonautonomous
systems. One particular application is to the nonlinear differential
equation $\ddot x+a(t)\dot x+b(t)f(x)=0,$ $x\in R$, where f:R$\to
R$ is continuous. Under the following conditions, it is shown
that $\lim\sb{t\to 0}\dot x(t)=0$ for every solution: (i) $\lim\sb{\vert
x\vert \to \infty}f(x)=\infty,$ (ii) $2a(t)b(t)+\dot b(t)\ge
0,$ (t$\ge 0)$, (iii) $\lim\sb{t\to \infty}[2\int\sp{t}\sb{0}[a(s)/b(s)]ds-
1/b(t)]=\infty,$ (iv) $\lim\sb{t\to \infty}[a(t)/b(t)]=\infty.$}
\RV{P.Smith |