An invariance principle of Lyapunov-Razumikhin type for neutral functional differential equations
The problem \({d\over dt} (Dx_ t)= f(x_ t)\), \(x_ 0=\varphi\) is studied under the assumptions that \(C= C([-r,0],\mathbb{R}^ n)\), \(f: C\to \mathbb{R}^ n\) is completely continuous, \(D: C\to\mathbb{R}^ n\) is linear, continuous, atomic at zero and stable. Uniqueness, continuous dependence and continuation of solutions of that problem are assumed. The \(\omega\)- limit set \(\Omega(\varphi)\) of \(\varphi\) is located by using a Lyapunov function. The results are based on the lemma: If \(\{x_ t(\varphi): t\geq 0\}\) is bounded and there exists a stable subset \(S\subseteq C\) such that \(\Omega(\varphi)\cap S\) is nonempty, then \(x_ t(\varphi)\to S\) as \(t\to \infty\). By a special choice of the set \(S\) sufficient conditions for the asymptotic stability and the asymptotic constancy are derived.
neutral functional differential equation
scientific article; zbMATH DE number 553870