Morse decompositions in the absence of uniqeness. (Q696167)

Let \(X\) be a metric space and let \(\mathcal C\) be the space of continuous maps \(X\to \mathbb R\). For a given family \(\mathcal T\subset \mathcal C\), the notions of \(\mathcal T\)-invariance, \(\mathcal T\)-attractor, \(\mathcal T\)-repeller, \(\mathcal T\)-limit sets, \(\mathcal T\)-Morse decompositions, etc. are defined and their properties are proved. A motivation for the introduction of those notions comes from differential equations without the uniqueness property: for such an equation \(\mathcal T\) can be equal to the set of its full solutions. In this case, results on the Conley index and the Morse inequalities are presented. They are stated also in the context of infinite-dimensional equations, following ideas of \textit{M. Izydorek} and \textit{K. P. Rybakowski} [Fundam. Math. 171, 31--52 (2002; Zbl 0994.58006)]. As an application, a proof of a theorem on a strongly indefinite elliptic system from the paper of \textit{S. Angenent} and \textit{R. van der Vorst} [Math. Z. 231, 203--248 (1999; Zbl 0939.58015)], is given.