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Cím:Stability theorems for nonautonomous functional differential equations by Lyapunov functionals.
Szerző:Burton, Theodore; Hatvani, L\'aszl\'o
Forrás:T\^ohoku Math. J., II. Ser. 41, No.1, 65-104 (1989).
Absztrakt:We consider a system of functional differential equations (1) $x'(t)=F(t,x\sb t)$ having finite delay h and satisfying $F(t,0)=0$. The object is to give conditions on Lyapunov functionals to ensure stability without asking that F(t,$\phi)$ be bounded for $\phi$ bounded. We begin by surveying seven classical examples in which we note that an $L\sp 2$- norm of the solution x(t) frequently appears in the derivative of the Lyapunov functional, but that investigators ignore it in favor of a pointwise norm on x(t). We show that the $L\sp 2$-norm can be much more useful than the pointwise norm. \par A measurable function $\eta$ : $R\to R$ is said to be integrally positive with parameter $\delta >0$ ($\eta\in IP(\delta))$ if whenever $\{t\sb i\}$ and $\{\delta\sb i\}$ satisfy $t\sb i+\delta\sb i<t\sb{i+1}$, $\delta\sb i\ge \delta$, then $\sum\sp{\infty}\sb{i=1}\int\sp{t\sb i+\delta\sb i}\sb{t\sb i}\eta (t)dt=\infty.$ Let $W\sb i$ denote a nonnegative increasing function with $W\sb i(0)=0$ and let $\Vert\vert \phi \Vert\vert$ denote the $L\sp 2$-norm of $\phi$. The following is a typical result. Theorem. Let $\eta \in IP(h\sb 1)$ with $0<h\sb 1<h$ and suppose there is a continuous functional V(t,$\phi)$ satisfying $(i)\quad W\sb 1(\vert \phi (0)\vert)\le V(t,\phi)\le W\sb 2(\vert \phi (0)\vert +W\sb 3(\Vert\vert \phi \Vert\vert)$ and $(ii)\quad V'\sb{(1)}(t,x\sb t)\le -\eta (t)W\sb 4(\Vert\vert x\sb t\Vert\vert.$ Then $x=0$ is asymptotically stable. It is interesting to note that this result fails for ODE's.} \RV{T.Burton 
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