A note on the Poincaré-Birkhoff fixed point theorem and periodic solutions of planar systems

Nonlinear Analysis, Theory, Methods and Applications

Volumn 29, Issue 3, 1997, Pages 291-311

  Rebelo, Carlota ^a,b  

^a SISSA   (Italy)

^b Ctro Mathematica Aplicacoes F   (Portugal)

Author keywords

Periodic solutions of planar systems; Poincar Birkhoff fixed point theorem

Indexed keywords

DIFFERENTIAL EQUATIONS; FUNCTIONS; LAGRANGE MULTIPLIERS; PROBLEM SOLVING;

HOMEOMORPHISM; PERIODIC SOLUTIONS; PLANAR SYSTEMS; POINCARE-BIRKHOFF FIXED POINT THEOREM;

THEOREM PROVING;

EID: 0031209701     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0362-546X(96)00065-X     Document Type: Article

References (37)

1. POINCARÉ, H., Sur un théorème de geométrie. Rend. Circ. Mat. Palermo. 1912, 33, 375-407.
2. MORSE, M., George David Birkhoff and his mathematical work. Bull. Am. Math. Soc., 1946, 52, 357-391.
3. BIRKHOFF, G. D., Proof of Poincaré's geometric theorem. Trans. Am. Math. Soc., 1913, 14, 14-22 (Collected Mathematical Papers, Vol. 1, pp. 673-681. Dover, New York, 1968).
4. BIRKHOFF, G. D., Proof of Poincaré's geometric theorem. Trans. Am. Math. Soc., 1913, 14, 14-22 (Collected Mathematical Papers, Vol. 1, pp. 673-681. Dover, New York, 1968).
5. BIRKHOFF, G. D., Dynamical systems with two degrees of freedom. Trans. Am. Math. Soc., 1917, 18, 199-300 (Collected Mathematical Papers, Vol. 2, pp. 1-102. Dover, New York, 1968).
6. BIRKHOFF, G. D., Dynamical systems with two degrees of freedom. Trans. Am. Math. Soc., 1917, 18, 199-300 (Collected Mathematical Papers, Vol. 2, pp. 1-102. Dover, New York, 1968).
7. BIRKHOFF, G. D., An extension of Poincaré's last geometric theorem. Acta Math., 1925, 47, 297-311 (Collected Mathematical Papers, Vol. 2, pp. 252-266. Dover, New York, 1968).
8. BIRKHOFF, G. D., An extension of Poincaré's last geometric theorem. Acta Math., 1925, 47, 297-311 (Collected Mathematical Papers, Vol. 2, pp. 252-266. Dover, New York, 1968).
9. CARTER, P., An improvement of the Poincaré-Birkhoff fixed point theorem. Trans. Am. Math. Soc., 1982, 269, 285-299.
10. GUILLOU, L., Théorème de translation plane de Brouwer et génèralisations du théorème de Poincaré-Birkhoff. Topology, 1994, 33, 333-351.
11. BROWN, M. & NEUMANN, W. D., Proof of the Poincaré-Birkhoff fixed point theorem. Michigan Math. J., 1977, 24, 21-31.
12. JACOBOWITZ, H., Periodic solutions of x″ + f (t, x) = 0 via the Poincaré-Birkhoff theorem. J. Diff. Eqns, 1976, 20, 37-52; Corrigendum, the existence of the second fixed point: a correction to "Periodic solutions of x″ + f (t, x) = 0 via the Poincaré-Birkhoff theorem". J. Diff. Eqns, 1977, 25, 148-149.
13. JACOBOWITZ, H., Periodic solutions of x″ + f (t, x) = 0 via the Poincaré-Birkhoff theorem. J. Diff. Eqns, 1976, 20, 37-52; Corrigendum, the existence of the second fixed point: a correction to "Periodic solutions of x″ + f (t, x) = 0 via the Poincaré-Birkhoff theorem". J. Diff. Eqns, 1977, 25, 148-149.
14. BIRKHOFF, G. D., Une généralisation à n dimensions du dernier théorème de géometrie de Poincaré. Comptes rendus des séances de l'Académie des Sciences, 1931, 192, 196-198.
15. HARTMAN, P., On boundary value problems for superlinear second order differential equations. J. Diff. Eqns, 1977, 26, 37-53.
16. BIRKHOFF, G. D., Dynamical systems. Am. Math. Soc., Colloquium Publ. Vol. IX, New York, 1927.
17. BUTLER, G. J., The Poincaré-Birkhoff "twist" theorem and periodic solutions of second-order nonlinear differential equations. In Proceedings of the 8th Fall Conference on Differential Equations, Stillwater, OK, 1979, ed. S. Ahmad, M. Keener and A. C. Lazer. Academic Press, New York, 1980, pp. 135-147.
18. MAWHIN, J., Oscillatory properties of solutions and nonlinear differential equations with periodic boundary conditions. Rocky Mountain J. Math., 1995, 25, 7-37.
19. DING, W.-Y., A generalization of the Poincaré-Birkhoff theorem. Proc. Am. Math. Soc., 1983, 88, 341-346.
20. FRANKS, J., Generalizations of the Poincaré-Birkhoff theorem. Ann. Math., 1988, 128, 139-151.
21. DEL PINO, M., MANÁSEVICH, R. & MURUA, A., On the number of 2π periodic solutions for u″ + g(u) = s(1 + h(t)) using the Poincaré-Birkhoff theorem. J. Diff. Eqns, 1992, 95, 240-258.
22. DING, T., An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance. Proc. Am. Math. Soc., 1982, 86, 47-54.
23. DING, T. & ZANOLIN, F., Periodic solutions of Duffing's equations with superquadratic potential. J. Diff. Eqns, 1992, 97, 328-378.
24. DING, W.-Y., Fixed points of twist mappings and periodic solutions of ordinary differential equations. Acta Math. Sin., 1982, 25, 227-235 (Chinese).
25. MOSER, J., Ordinary differential equations, periodic orbits. Lectures given at the Courant Institute.
26. 3 = p(t). Proc. Cambridge Phil. Soc., 1965, 61, 157-164.
27. MASSEY, W., Algebraic Topology: An Introduction. Springer, Berlin, 1984.
28. MOISE, E., Geometric Topology in Dimensions 2 and 3. Springer, New York, 1977.
29. OXTOBY, J. & ULAM, S., Measure preserving homeomorphisms and metrical transitivity. Ann. Math., 1941, 42, 874-920.
30. NEUMANN, W. D., Generalizations of the Poincaré Birkhoff fixed point theorem. Bull. Austral. Math. Soc., 1977, 17, 375-389.
31. ZEIDLER, E., Nonlinear Functional Analysis and its Applications, Vol. 1. Springer, New York, 1986.
32. DING, T. & ZANOLIN, F., Subharmonic solutions of second order nonlinear equations: a time-map approach. Nonlinear Analysis, 1993, 20, 509-532.
33. NUSSBAUM, R. D., The Fixed Point Index and some Applications. Les Presses de l'Université de Montréal, 1985.
34. DING, T. & ZANOLIN, F., Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type. In Proc. of the 1st World Congress of Nonlinear Analysts, Tampa, 1992, ed. V. Lakshmikantham. de Gruyter, Berlin, 1996, pp. 395-406.
35. REBELO, C. & ZANOLIN, F., Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities. Trans. Am. Math. Soc., 1996, 348, 2349-2389.
36. SCHWARTZ, I. B. & ERNEUX, T., Subharmonic hysteresis and periodic doubling bifurcations for a periodically driven laser. SIAM J. Appl. Math., 1994, 54, 1083-1100.
37. FONDA, A. & RAMOS, M., Large-amplitude subharmonic oscillations for scalar second-order differential equations with asymmetric nonlinearities. J. Diff. Eqns, 1994, 109, 354-372.