Abstract
We study some monotonicity and iteration inequality of the Maslov-type index i −1 of linear Hamiltonian systems. As an application we prove the existence of symmetric periodic solutions with prescribed minimal period for first order nonlinear autonomous Hamiltonian systems which are semipositive, even, and superquadratic at zero and infinity. This result gives a positive answer to Rabinowitz’s minimal period conjecture in this case without strictly convex assumption. We also give a different proof of the existence of symmetric periodic solutions with prescribed minimal period for classical Hamiltonian systems which are semipositive, even, and superquadratic at zero and infinity which was proved by Fei, Kim and Wang in 2001.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosetti A, Coti Zelati V. Solutions with minimal period for Hamiltonian systems in a potential well. Ann Inst H Poincaré Anal non Linéaire, 1987, 4: 242–275
Ambrosetti A, Mancini G. Solutions of minimal period for a class of convex Hamiltonian systems. Math Ann, 1981, 255: 405–421
An T, Long Y. Index theories of second order Hamiltonian systems. Nonlinear Anal, 1998, 27: 585–592
Cappell S E, Lee R, Miller E Y. On the Maslov-type index. Comm Pure Appl Math, 1994, 47: 121–186
Chang K C, Liu J Q, Liu M. Nontrivial period solutions for strong resonance Hamiltonian systems. Ann Inst H Poincaré Anal non Linéaire, 1997, 14: 103–117
Dong D, Long Y. The iteration theory of the Maslov-type index theory with applications to nonlinear Hamiltonian systems. Trans Amer Math Soc, 1997, 349: 2619–2661
Ekeland I. Convexity Methods in Hamiltonian Mechanics. Berlin: Springer-Verlag, 1990
Ekeland I, Hofer E. Periodic solutions with percribed period for convex autonomous Hamiltonian systems. Invent Math, 1985, 81: 155–188
Fei G, Qiu Q. Minimal period solutions of nonlinear Hamiltonian systems. Nonlinear Anal, 1996, 27: 821–839
Fei G, Qiu Q. Periodic solutuions of asymptotically linear Hamiltonian systems. Chinese Ann Math Ser B, 1997, 18: 359–372
Fei G, Kim S-K, Wang T. Minimal Period Estimates of Period Solutions for Superquadratic Hamiltonian Syetems. J Math Anal Appl, 1999, 238: 216–233
Fei G, Kim S-K, Wang T. Solutions of minimal period for even classical Hamiltonian systems. Nonlinear Anal, 2001, 43: 363–375
Girardi M, Matzeu M. Some results on solutions of minimal period to superquadratic Hamiltonian equations. Nonlinear Anal, 1983, 7: 475–482
Girardi M M, Matzeu M. Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem. Nonlinear Anal TMA, 1986, 10: 371–382
Girardi M, Matzeu M. Periodic solutions of convex Hamiltonian systems with a quadratic growth at the origin and superquadratic at infinity. Ann Math Pure Appl, 1987, 147: 21–72
Girardi M, Matzeu M. Dual Morse index estimates for periodic solutions of Hamiltonian systems in some nonconvex superquadratic case. Nonlinear Anal TMA, 1991, 17: 481–497
Ghuussoub N. Location, multiplicity and Morse indices of min-max critical points. J Reine Angew Math, 1991, 417: 27–76
Lazer A, Solomini S. Nontrivial solution of operator equations and Morse indices of critical points of min-max type. Nonlinear Anal, 1988, 12: 761–775
Liu C, Long Y. An optimal increasing estimate for iterated Maslov-type indices. Chinese Sci Bull, 1997, 42: 2275–2277
Liu C, Long Y. Iteration inequalities of the Maslov-type index theory with applications. J Differ Equ, 2000, 165: 355–376
Liu C. A note on the monotonicity of Maslov-type index of Linear Hamiltonian systems with applications. Proc Royal Soc Edinburg Ser A, 2005, 135: 1263–1277
Long Y. The minimal period problem of classical Hamiltonian systems with even potentials. Ann I H Poincaré Anal non Linéaire, 1993, 10: 605–626
Long Y. The minimal period problem of period solutions for autonomous superquadratic second Hamiltonian systems. J Differ Equ, 1994, 111: 147–174
Long Y. On the minimal period for periodic solution problem of nonlinear Hamiltonian systems. Chinese Ann Math Ser B, 1997, 18: 481–484
Long Y. Bott formula of the Maslov-type index theory. Pacific J Math, 1999, 187: 113–149
Long Y. Index Theory for Symplectic Paths with Applications. Basel: Birkhäuser, 2002
Long Y, Zhang D, Zhu C. Multiple brake orbits in bounded convex symmetric domains. Adv Math, 2006, 203: 568–635
Rabinowitz P H. Periodic solution of Hamiltonian systems. Comm Pure Appl Math, 1978, 31: 157–184
Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. In: CBMS Regional Conf Ser in Math, vol. 65. Providence, RI: Amer Math Soc, 1986
Solimini S. Morse indes estimates in min-max theorems. Manuscripta Math, 1989, 63: 421–453
Szulkin A. Cohomology and Morse theory for strongly indefinite functions. Math Z, 1992, 209: 375–418
Xiao Y. Periodic solutions with prescribed minimal period for second order Hamiltonian systems with even potentials. Acta Math Sin English Ser, 2010, 26: 825–830
Zhang D. P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurface in ℝ2n. Discrete Contin Dyn Syst, 2013, 33: 947–964
Zhu C, Long Y. Maslov index theory for symplectic paths and spectral flow(I). Chinese Ann Math Ser B, 1999, 208: 413–424
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, D. Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems. Sci. China Math. 57, 81–96 (2014). https://doi.org/10.1007/s11425-013-4598-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-013-4598-9