Repelling conditions for boundary sets using Liapunov-like functions. I: Flow-invariance, terminal value problem and weak persistence (Q1120052)

Noncontinuable solutions x(\(\cdot)\) of the equation (1) \(x'=f(t,x)\) satisfying the initial condition (2) \(x(t_ 0)=x_ 0\in \Omega\) \((t_ 0\in J)\) are considered. In (1), (2) f: \(J\times \Omega \to {\mathbb{R}}^ d\) is a continuous function, J is a nondegenerate real interval, \(\Omega \subset {\mathbb{R}}^ d\) is a nonempty open set and \({\mathbb{R}}^ d\) is the d- dimensional real euclidean space. Let additionally \(G,S\subset {\mathbb{R}}^ d\) be nonempty sets, with \(G\subset \Omega\) and \(S\cap G=\emptyset\). Then a purpose of the article is to find various criteria relating behaviour of the solution of (1), (2) for \(x(t_ 0)\in G\), with the set S. The following situations 1) solutions of (1) never reach S from the set G; 2) there are no solutions x(\(\cdot)\) of (1), with x(t)\(\in G\) for all \(t\in I_ x\) and such that \(\lim_{t\to t_ x^-}x(t)=u\in S;\) 3) solutions of (1), with x(t)\(\in G\) for all \(t\in I_ x\), are asymptotically far from S; 4) S is a repeller for the solutions of (1) which remain in \(G(t_ x:=Sup I_ x,I_ x\) is the right maximal interval of existence of x(\(\cdot))\), are discussed.