On the Conley index in Hilbert spaces in the absence of uniqueness (Q2773309)

The ordinary differential equation NEWLINE\[NEWLINE\dot x= f(x):=Lx+K(x)\tag{1}NEWLINE\]NEWLINE on an infinite-dimensional Hilbert space \(E\) is considered. Here \(L\) is a bounded linear strongly indefinite operator on \(E\), \(K(x)\) is a completely continuous but not necessarily locally Lipschitzian map. For every isolating neighborhood \(N\) relative to (1) the Conley-type index \(h(f,N)\) of \(N\) is defined. It is shown that all properties of the \(LS\)-Conley index theory [\textit{K. Geba, M. Izydorek} and \textit{A. Pruszko}, Stud. Math. 134, No. 3, 217-233 (1999; Zbl 0927.58004)] hold in this more general setting. It is proved that the index depends only on the isolated invariant set in question and not on the choice of its isolating neighborhood. The introduced extended \(LS\)-Conley index is applicable to strongly indefinite variational problems \(\nabla\Phi(x)=0\), where \(\Phi:E\rightarrow R\) is merely a \(C^1\)-function.