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Cím:On partial asymptotic stability and instability. III.
Szerző:Hatvani, L.
Forrás:Acta Sci. Math. 49, 157-167 (1985).
Nyelv:English
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 
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