Singular boundary value problems via the Conley index
The authors consider a singular system of the form \[ \varepsilon^2 D u'' + f(t,u,\varepsilon u')=0 \] with \(u(t) \in {\mathbb R}^n\) and \(D\) a diagonal matrix, together with boundary conditions which generalize linear boundary conditions of Sturm-Liouville type. For the main result it is assumed that in case \(\varepsilon =0\) there exist two hyperbolic saddle points, and hypotheses are made on the corresponding stable/unstable manifolds. Thus, an existence result is given for \(\varepsilon\) sufficiently small. The proof is performed in the framework of Conley index theory. An interesting feature of this paper is that Conley index theory is used for establishing solutions to a boundary value problem and not (what is more standard) for detecting the existence of invariant sets.
scientific article