Conley index of nontrivial invariant sets in a Hilbert space (Q843661)

The author investigates the flows defined in a Hilbert space \(H\) by potential completely continuous vector fields \(I-K(\cdot)\), with \(K(\cdot)\) close to a homogeneous one. It is supposed neither isolation of critical points of functional, nor monotonicity by cone of its gradient, while the oddness of the operator \(K(\cdot)\) is changed by the oddness of its main part (vectors \(K(x)\) and \(K(-x)\) are not co-directed, when both are nonzero). Instead of this the condition of nearity of the operator \(K(\cdot)\) to homogeneous one is supposed. For such flows Conley index is determined for the set of equilibria and joining them separatricies as nontrivial invariant set. Conley index possesses basic properties of compact index and allow prove the existence theorems for solutions of the equation \(K(x)=x\) under the condition that the potential \(\varphi: \nabla\varphi(\cdot)= K(\cdot)\) is coercive and has the even main part. From here the stability of a finite set of solutions to small perturbations follows. The distinction of the Conley index from the classical rotation theory is shown.