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On the Conley index in Hilbert spaces — a multivalued case. (English) Zbl 1114.37013

Jachymski, Jacek (ed.) et al., Fixed point theory and its applications. Proceedings of the international conference, Bȩdlewo, Poland, August 1–5, 2005. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 77, 61-68 (2007).
Introduction: K. Gȩba, M. Izydorek and A. Pruszko [Stud. Math. 134, No. 3, 217–233 (1999; Zbl 0927.58004)] constructed an invariant which is a version of Conley index for flows determined by compact perturbations of special linear operators in an infinite-dimensional Hilbert space. Their method of construction is very similar to the definition of the Leray-Schauder degree. This invariant has been then applied to obtain existence and multiplicity results in variational problems with strongly indefinite functionals. On the other hand, M. Mrozek [J. Differ. Equations 84, No. 1, 15–51 (1990; Zbl 0703.34019)] considered a cohomological index for multivalued flows on compact spaces. The aim of this paper is to give an infinite-dimensional version of Mrozek’s index by use of the above method. Nontriviality of the obtained invariant gives existence results of invariant sets for inclusions in Hilbert spaces. We can expect also further applications to differential inclusions coming from e.g. non-smooth analysis.
One has to mention also the other definitions of Conley index for multivalued flows by M. Kunze [Non-smooth dynamical systems, Lecture Notes in Mathematics, 1744, Berlin: Springer (2000; Zbl 0965.34026)], and [Differ. Integral Equ. 13, No. 4–6, 479–502 (2000; Zbl 0968.37006)] and G. Gabor [Set-Valued Anal. 13, No. 2, 125–149 (2005; Zbl 1096.54016)] consisting in approximation of the generators of the flow by more smooth one (locally Lipschitz). Other applications and an equivariant version will be a subject of further research. We have limited our attention to the simplest case of flows. A local version of all considerations here is natural.
For the entire collection see [Zbl 1112.47302].

MSC:

37B30 Index theory for dynamical systems, Morse-Conley indices
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34A60 Ordinary differential inclusions
47H11 Degree theory for nonlinear operators
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