On the existence of nonlocal integral manifolds of ordinary differential equations (Q1968647)

The author studies the existence of nonlocal integral manifolds of stationary points for the autonomous ordinary differential equation \[ \dot x=f(x),\quad x\in D \subset {\mathbb{R}}^n, \quad f\in\text{Lip}(D). \tag{1} \] The usual approach to the construction of global integral manifolds is the following: a global integral manifold is constructed as the graph of a function for system (1) either with a small enough nonlinearity \(f\) or with \(f\) suitably modified outside some neighborhood of a stationary point by using the cut-off function. Because of the use of the cut-off function the manifold is really constructed in a small and often unknown neighborhood of a stationary point. In the paper, the author constructs an integral manifold \(M\) being the graph of a Lipschitz continuous function \(s\) defined on a set \(W_0\). The main differences in comparison with other results of this kind are: 1. The set \(W_0\) is explicitly known. 2. System (1) has to be defined only in an arbitrary small neighborhood \(D\) of the explicitly defined set \(D_0\). 3. Under the assumptions of the theorems, which are required only in an arbitrary small neighborhood \(\widetilde{D}_0\subseteq D\) of \(D_0\), \(D_0\) is the smallest set in which the existence of \(M\) can be guaranteed. 4. The vector field \(f\) is modified only outside of \(D_0\) in \(\widetilde{D}_0\) and only in a manner that the modified system satisfies an outflowing condition. Some applications of the results obtained are given.