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

@article{Izydorek2001ACC,
  title={A Cohomological Conley Index in Hilbert Spaces and Applications to Strongly Indefinite Problems},
  author={Marek Izydorek},
  journal={Journal of Differential Equations},
  year={2001},
  volume={170},
  pages={22-50}
}
  • M. Izydorek
  • Published 10 February 2001
  • Mathematics
  • Journal of Differential Equations
Abstract 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. 

On homotopy Conley index for multivalued flows in Hilbert spaces

An approximation approach is applied to obtain a homotopy version of the Conley type index in Hilbert spaces considered by the first author and W. Kryszewski. The definition given in the paper is

Morse cohomology in a Hilbert space via the Conley index

The main theorem of this paper states thatMorse cohomology groups in a Hilbert space are isomorphic to the cohomological Conley index. It is also shown that calculating the cohomological Conley index

E-cohomological Conley index

In this thesis we continue with developing the E-cohomological Conley index which was introduced by A.Abbondandolo. In particular, we generalize the index to non-gradient flows, we show that it an

LECTURES ON THE MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

After reviewing some classical results about hyperbolic dynamics in a Banach setting, we describe the Morse complex for gradient-like flows on an infinite-dimensional Banach manifold M, under the

THE LS-INDEX: A SURVEY

Let H be a Hilbert space. Consider a vector field f defined on H of the form f (x) = L(x) + K(x), where L is a strongly indefinite bounded linear operator and K is a completely continuous

Conley Type Index and Hamiltonian Inclusions

This paper is based mainly on the joint paper with W. Kryszewski [Dzedzej, Z., Kryszewski, W.: Conley type index applied to Hamiltonian inclusions. J. Math. Anal. Appl. 347 (2008), 96–112.], where

On the Fredholm Lagrangian Grassmannian, spectral flow and ODEs in Hilbert spaces

References

SHOWING 1-10 OF 31 REFERENCES

A Morse equation in Conley's index theory for semiflows on metric spaces

Abstract Given a compact (two-sided) flow, an isolated invariant set S and a Morse-decomposition (M1, …, Mn) of S, there is a generalized Morse equation, proved by Conley and Zehnder, which relates

On the homotopy index for infinite-dimensional semiflows

ABsRAc'r. In this paper we consider semiflows whose solution operator is eventually a conditional a-contraction. Such semiflows include solutions of retarded and neutral functional differential

On strongly indefinite functionals with applications

Recently, in their remarkable paper Critical point theory for indefinite functionals, V. Benci and P. Rabinowitz gave a direct approach-avoiding finitedimensional approximationsto the existence

Solutions of Asymptotically Linear Operator Equations via Morse Theory.

Abstract : In this paper, we use the classical Morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations. As special

MASLOV-TYPE INDEX, DEGENERATE CRITICAL POINTS, AND ASYMPTOTICALLY LINEAR HAMILTONIAN SYSTEMS

This paper introduces a Maslov-type index theory for paths in the symplectic groups, especially for the degenerate paths via rotational perturbation method, therefore gives a full classification of

Topological Methods for Variational Problems With Symmetries

Category, genus and critical point theory with symmetries.- Category and genus of infinite-dimensional representation spheres.- The length of G-spaces.- The length of representation spheres.- The

Periodic solutions of asymptotically linear Hamiltonian systems

We prove existence and multiplicity results for periodic solutions of time dependent and time independent Hamiltonian equations, which are assumed to be asymptotically linear. The periodic solutions

Nontrivial solution of a semilinear Schrodinger equation

This paper deals with strongly indefinite functionals whose gradients are Fredholm operators of index 0 and map weakly convergent sequences to weakly convergent sequences. We show bow these results

Morse‐type index theory for flows and periodic solutions for Hamiltonian Equations

An index theory for flows is presented which extends the classical Morse theory for gradient flows on compact manifolds. The theory is used to prove a Morse-type existence statement for periodic