On the Conley index in Hilbert spaces in the absence of uniqueness
Volume 171 / 2002
Abstract
Consider the ordinary differential equation $$\dot x=Lx+K(x)\tag 1 $$ on an infinite-dimensional Hilbert space $E$, where $L$ is a bounded linear operator on $E$ which is assumed to be strongly indefinite and $K : E\to E$ is a completely continuous but not necessarily locally Lipschitzian map. Given any isolating neighborhood $N$ relative to equation (1) we define a Conley-type index of $N$. This index is based on Galerkin approximation of equation (1) by finite-dimensional ODEs and extends to the non-Lipschitzian case the ${\cal L}{\cal S}$-Conley index theory introduced in [9]. This extended ${\cal L}{\cal S}$-Conley index allows applications to strongly indefinite variational problems $\nabla {\mit \Phi }(x)=0$ where ${\mit \Phi } : E\to {\mathbb R}$ is merely a $C^1$-function.
