Abstract.A module structure of the cohomology Conley index is used to define a relative cup-length. This invariant is applied then to prove a multiplicity theorem for periodic solutions to Hamiltonian systems.Mathematics Subject Classification (2010). 37B30, 37J20, 37J45, 34C25, 58E05.

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In the paper we develop the theory of a cohomological index of the Conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional Hilbert space. An application to the existence of periodic orbits to asymptotically linear Hamiltonian inclusions is presented.

An equivariant degree is defined for equivariant completely continuous multivalued vector fields with compact convex values. Then it is applied to obtain a result on existence of solutions to a second order BVP for differential inclusions carrying some symmetries. MSC:47H11, 34A60, 34B99

Abstract.In [C] and [F1] the connection matrix theory for Morse decomposition is developed in the case of continuous dynamical systems. Our purpose is to study the case of discrete time dynamical systems.The connection matrices are matrices between the homology indices of the sets in the Morse decomposition. They provide information about the structure of the Morse decomposition; in particular, they give an algebraic condition for the existence of connecting orbit set between different Morse sets.0. Introduction. One of the methods by which the Conley index theory studies isolated invariant sets is to decompose them into subinvariant sets (Morse sets) and connecting orbits between them. This structure is called a Morse decomposition of an isolated invariant set. A filtration of index pairs associated with a Morse decomposition can be used to find connections between Morse sets. The existence of such a filtration in the case of continuous dynamical systems has been proved in [CoZ], [Sal] and [F1]. In [BD] we have proved the existence of index triples and index filtrations for dynamical systems given by a homeomorphism (discrete dynamical systems).Studying the topology of the sets in the filtration provides some information on the structure of the Morse decomposition. The principal tool for this purpose is the connection matrix. In [C] and [F1, F2] the connection matrix theory for Morse decompositions is developed for flows. We wish to investigate this theory for a homeomorphism. Once an index filtration has been found, most steps of the construction of the connection matrix in the continuous case can be carried over to the discrete case. The main difference is that the homology Conley index is not simply the homology of the index pair but the Leray reduction of its homology (see [M2] for more details).