Conley index of nontrivial invariant sets in a Hilbert space

Abstract

We study flows defined in a Hilbert space by potential completely continuous fields Id-K(·), where K(·) is an operator close to a homogeneous one. The Conley index of the set of fixed points and separatrices joining them (a nontrivial invariant set) is defined for such flows. By using this index, we prove that the equation K(x) = x has infinitely many solutions of arbitrarily large norm provided that the potential φ: ∇φ(·) = K(·) is coercive and has an even leading part. As a corollary, we justify the stability of an arbitrary finite number of solutions under small perturbations of the field. We show that the Conley index differs from the classical rotation theory of vector fields when proving existence theorems.