On proving existence and smoothness of invariant manifolds in singular perturbation problem (Q1327684)

The author considers the autonomous singularly perturbed system (*) \(du/dt= f(u,v,\varepsilon)\), \(\varepsilon dv/dt= g(u,v,\varepsilon)\) under the following assumptions (i) \(f\), \(g\) are \(C^ r\)-bounded maps on \(\mathbb{R}^ m\times \mathbb{R}^ n\times \mathbb{R}\), (ii) There is a \(C^ r\)- bounded map \(h\) such that \(g(u, h(u),0) \equiv 0\), (iii) \(g_ v (u,h(u), 0)\) is uniformly hyperbolic with respect to \(u\). Under these conditions it is well-known that (*) has an invariant manifold \(v=h(u,\varepsilon)\in C^{r-1}\). The author provides a proof of this theorem which is simpler and shorter than other ones and has the advantage that it can be used to derive a corresponding result for maps and semiflows.