Singular boundary value problems via the Conley index

Tomáš Gedeon, Konstantin Mischaikow


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.


Singular boundary value problems; Conley index

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