, Volume 13, Issue 2, pp 611-625

Some extensions of the Poincaré–Birkhoff theorem

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Abstract

We represent several results on the existence of fixed points of the arbitrary topological annulus maps. The celebrated boundary twist condition of the Poincaré–Birkhoff theorem is replaced by its essentially weakest analogue for two points in the annulus. We do not use area-preserving and homeomorphic maps. We consider continuous maps satisfying some modification of T. Ding’s bend condition and a special monotonicity condition. We also reject the often used 2π-periodicity angle displacement condition. Besides, we obtain the description of the fixed points set structure for continuously differentiable maps.