Singular boundary value problems via the Conley index

Abstract

We use Conley index theory to solve the singular boundary value problem $\varepsilon^2D u_{xx} + f(u,\varepsilon u_x,x) = 0$ on an interval $[-1,1]$, where $u \in \mathbb R^n$ and $D$ is a diagonal matrix, with separated boundary conditions. Since we use topological methods the assumptions we need are weaker then the standard set of assumptions. The Conley index theory is used here not for detection of an invariant set, but for tracking certain cohomological information, which guarantees existence of a solution to the boundary value problem.