Morse decompositions in the absence of uniqueness

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, On the Conley index in Hilbert spaces in the absence of uniqueness , 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, A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology , Math. Z. 231 (1999), 203–248] using Morse-Floer homology