A Cohomological Conley Index in Hilbert Spaces and Applications to Strongly Indefinite Problems

A cohomological Conley index is defined for flows on infinite dimensional real Hilbert spaces generated by vector fields of the form f: H→H, f(x)=Lx+K(x), where L: H→H is a bounded linear operator satisfying certain technical assumptions and K is a completely continuous perturbation. Generalized Morse inequalities for Morse decompositions of isolated invariant sets are proved. Simple examples are presented to show how the theory can be applied to strongly indefinite problems.