### Singular boundary value problems via the Conley index

DOI: http://dx.doi.org/10.12775/TMNA.2006.030

#### 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.

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.

#### Keywords

Singular boundary value problems; Conley index

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