Morse decompositions in the absence of uniqueness

Maria C. Carbinatto, Krzysztof P. Rybakowski

DOI: http://dx.doi.org/10.12775/TMNA.2001.032

Abstract


In this paper we define attractors and Morse
decompositions in an abstract framework of curves in a metric
space. We establish some basic properties of these concepts
including their stability under perturbations. This extends
results known for flows and semiflows on metric spaces to large
classes of ordinary or partial differential equations with
possibly nonunique solutions of the Cauchy problem. As an
application, we first prove a Morse equation in the context of a
Conley index theory
which was recently defined in
[M. Izydorek and K. P. Rybakowski, < i> On the Conley index in Hilbert spaces in the
absence of uniqueness< /i> , Fund. Math.] for problems without uniqueness,
and then apply this
equation to give an elementary proof of two multiplicity results for strongly
indefinite elliptic systems previously obtained in
[S. Angenent and R. van der Vorst, < i> A superquadratic indefinite elliptic system and
its Morse–Conley–Floer homology< /i> , Math. Z. < b> 231< /b> (1999), 203–248] using Morse-Floer
homology.

Keywords


Attractor-repeller pairs; Morse decompositions; Conley index; strongly indefinite elliptic systems

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