Application of sign-constant functions to the theory of integral manifolds (Q1103092)

Consider a differential equation (*) \(X'=X(t,x)\), where \(X: \Omega\to R^ n\) \((\Omega =I\times U,I=[a,b]\), \(U\subset R^ n\)). The authors prove that a manifold \({\mathcal M}\subset \Omega\) is: (i) r-integral manifold of (*) iff there is a function \(V: \Omega\to [0,+\infty)\) with \(\ker V= {\mathcal M}\) such that for every solution \(x(0;t_ 0,x_ 0)\) of (*) the function \(V(0,x(0;t_ 0,x_ 0))\) is non increasing. (ii) Integral manifold of (*) iff there is a function \(V: I\times \Omega \to [0,+\infty)\) with \(\ker V(0,\tau,0)= {\mathcal M}\) such that for every solution \(x(0;t_ 0,x_ 0)\) of (*) the function \(V(0,\tau,x(0;t_ 0,x_ 0))\) is non increasing for \(t\geq \tau\) and non decreasing for \(t\leq \tau\).