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.
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References
Birkhoff G.D.: An extension of Poincaré’s last geometric theorem. Acta Math. 47, 297–311 (1926)
Birkhoff G. D.: Proof of Poincaré’s geometric theorem. Trans. Amer. Math. Soc. 14, 14–22 (1913)
Brown M., Neumann W. D.: Proof of the Poincaré-Birkhoff fixed point theorem. Michigan Math. J. 24, 21–31 (1977)
Carter P. H.: An improvement of the Poincaré-Birkhoff fixed point theorem. Trans. Amer. Math. Soc. 269, 285–299 (1982)
Dalbono F., Rebelo C.: Poincaré-Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar Hamiltonian system. Rend. Semin. Mat. Univ. Politec. Torino 60, 233–263 (2002)
T. Ding, Approaches to the qualitative theory of ordinary differential equations. Peking University Series in Mathematics 3, World Scientific, Hackensack, NJ, 2007.
Ding W.-Y.: A generalization of the Poincaré-Birkhoff theorem. Trans. Amer. Math. Soc. 88, 341–346 (1983)
J. Franks, Erratum to “Generalizations of Poincaré-Birkhoff theorem”. Ann. of Math. (2) 164 (2006), 1097–1098.
J. Franks, Generalizations of the Poincaré-Birkhoff theorem. Ann. of Math. (2) 128 (1988), 139–151.
Guillou L.: A simple proof of P. Carter’s theorem. Proc. Amer. Math. Soc. 125, 1555–1559 (1997)
B. Kerékjarto, The plane translation theorem of Brouwer and the last geometric theorem of Poincaré. Acta Sci. Math. (Szeged) 4 (1928-29), 86–102.
A. Margheri, C. Rebelo and F. Zanolin, Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems. J. Differential Equations 183 (2002), 342–367.
A. Pascoletti and F. Zanolin, A topological approach to bend-twist maps with applications. Int. J. Differ. Equ. 2011 (2011), Art. ID 612041, 20 p.
H. Poincaré, Sur un thèorème de geomètrie. Rend. Circ. Mat. Palermo (2) 33 (1912), 375–407.
H. E. Winkelnkemper, A generalizations of the Poincaré-Birkhoff theorem. Proc. Amer. Math. Soc. 102 (1988), 1028–1030.
L. Yong and L. Zhenghua, A constructive proof of the Poincaré-Birkhoff theorem. Trans. Amer. Math. Soc. 347 (1995), 2111–2126.
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Kirillov, A., Starkov, V. Some extensions of the Poincaré–Birkhoff theorem. J. Fixed Point Theory Appl. 13, 611–625 (2013). https://doi.org/10.1007/s11784-013-0127-2
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DOI: https://doi.org/10.1007/s11784-013-0127-2