Conley index in Hilbert spaces versus the generalized topological degree

The equivariant spectral flow and bifurcation of periodic solutions of Hamiltonian systems

“…The aim of this section is to construct a G = Z 2 -invariant family of functionals f such that the Hessians L are a loop in F S(H) G having a non-vanishing G-equivariant spectral flow. Thus by Theorem 3.1 there is a bifurcation of critical points that could not have been found by any invariant that only depends on the endpoints L 0 , L 1 of the path such as [5], [9] and [14]. Moreover, our example also has the feature that sf(L) = 0 and consequently Theorem 1.3 fails as well.…”

“…are not isomorphic as G-representations, then there is a bifurcation of critical points for f . By (9), we also reobtain the main theorem of Fitzpatrick, Pejsachowicz and Recht's work [11] that we stated in the introduction in Theorem 1.3.…”

The existence of isolating blocks for multivalued semiflows

“…The importance of this topology definition can be measured by the large amount of studies that followed the above references. Just to cite a few of them, this concept was used in applications (see for instance, [18,15]), it was also defined for flows on Hilbert spaces (see [10,4,11]), for non-autonomous semiflows on Banach spaces [12] and also for multivalued semiflows (see [8,16]).…”

Nonlocal quasilinear variations of the Chafee-Infante problem

“…The importance of this topology definition can be measured by the large amount of studies that followed the above references. Just to cite a few of them, this concept was used in applications (see for instance, (RYBAKOWSKI, 1987;MISCHAIKOW, 1995)), it was also defined for flows on Hilbert spaces (see (G ĘBA; IZY-DOREK; PRUSZKO, 1999;BŁaSZCZYK;GOŁ ĘBIEWSKA;RYBICKI, 2017;IZYDOREK et al, 2017)), for non-autonomous semiflows on Banach spaces (JäNIG, 2019) and also for multivalued semiflows (see (DZEDZEJ;GABOR, 2011;MROZEK, 1990)).…”