Journal of Fixed Point Theory and Applications

, Volume 13, Issue 2, pp 611–625

Some extensions of the Poincaré–Birkhoff theorem


DOI: 10.1007/s11784-013-0127-2

Cite this article as:
Kirillov, A. & Starkov, V. J. Fixed Point Theory Appl. (2013) 13: 611. doi:10.1007/s11784-013-0127-2


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.

Mathematics Subject Classification

Primary 54H25Secondary 37E40


Poincaré–Birkhoff theoremfixed pointcontinuous map

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Institute of Applied Mathematical ResearchKarelian Research Centre of the Russian Academy of SciencesPetrozavodskRussia
  2. 2.Department of MathematicsPetrozavodsk State UniversityPetrozavodskRussia