Hyperbolic characteristics on star-shaped hypersurfaces

Liu, Chun-Gen; Long, Yiming

Annales de l'I.H.P. Analyse non linéaire, Tome 16 (1999) no. 6, pp. 725-746.

[1] A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian systems, Acta Mathematica, Vol. 152, 1984, pp. 143-197. | MR | Zbl

[2] H. Berestycki, J.M. Lasry, G. Mancini and B. Rof, Existence of multiple periodic orbits on starshaped Hamiltonian systems, Comm. pure Appl. Math., Vol. 38, 1985, pp. 253-289. | MR | Zbl

[3] V. Brousseau, Espaces de Krein et index des systémes hamiltoniens, Ann. Inst. H. Poincaré, Anal. non linéaire, Vol. 7, 1990, pp. 525-560. | Numdam | MR | Zbl

[4] K.C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhäuser, Boston, 1993. | MR | Zbl

[5] C. Conley and E. Zehnder, Maslov-type index theory for flows and periodic solutions for Hamiltonian equations, Commun. Pure Appl. Math., Vol. 37, 1984, pp. 207-253. | MR | Zbl

[6] G. Dell'Antonio, Variational calculus and stability of periodic solutions of a class of Hamiltonian systems, SISSA Ref. (185/92/FM (Oct. 1992)). | MR

[7] G. Dell'Antonio, B. D'Onofrio and I. Ekeland, Les systém hamiltoniens convexes et pairs ne sont pas ergodiques en general, C. R. Acad. Sci. Paris, t. 315, Series I, 1992, pp. 1413-1415. | MR | Zbl

[8] D. Dong and Y. Long, The Iteration Formula of the Maslov-type Index Theory with Applications to Nonlinear Hamiltonian Systems, Trans. Amer. Math. Soc., Vol. 349, 1997, pp. 2619-2661. | MR | Zbl

[9] I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Physics, Vol. 113, 1987, pp. 419-467. | MR | Zbl

[10] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, Berlin, 1990. | MR | Zbl

[11] I. Ekeland, An index theory for periodic solutions of convex Hamiltonian systems, In Nonlinear Funct. Anal. and its Appl. Proc. Symposia in pure Math., Vol. 45-1, 1986, pp. 395-423. | MR | Zbl

[12] I. Ekeland, Une thórie de Morse pour les systèms hamiltoniens convexes, Ann. Inst. Henri poincaté. Anal. Non Linéair, Vol. 1, 1984, pp. 19-78. | Numdam | MR | Zbl

[13] G. Fei and Q. Qiu, Periodic solutions of asymptotically linear Hamiltonian systems, Preprint, 1996, Chinese Ann. of Math. (To appear). | MR | Zbl

[14] N. Ghoussoub, Location, multiplicity and Morse indices of minimax critical points, J. Reine Angew Math., Vol. 417, 1991, pp. 27-76. | MR | Zbl

[15] C. Liu, Monotonicity of the Maslov-type index and the ω-index theory, Acta of Nankai University (Chinese). to appear.

[16] C. Liu and Y. Long, An optimal increasing estimate of the Maslov-type indices for iterations, Chinese Sci. Bull, Vol. 42, 1997, pp. 2275-2277 (Chinese edition), Vol. 43, 1998, pp. 1063-1066 (English edition). | MR | Zbl

[17] Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Science in China (Scientia Sinica), Vol. Series A.33, 1990, pp. 1409-1419. | MR | Zbl

[18] Y. Long, A Maslov-type theory and asymptotically linear Hamiltonian systems, In Dyna. Syst. and Rel. Topics. K. Shiraaiwa ed. World Sci., 1991, pp. 333-341. | MR

[19] Y. Long, The Index Theory of Hamiltonian Systems with Applications, (In Chinese) Science Press, Beijing, 1993.

[20] Y. Long, Bott formula of the Maslov-type index theory, Nankai Inst. of Math. Nankai Univ. Preprint, 1995, Revised 1996, 1997, Pacific J. Math., Vol. 187, 1999, pp. 113-149. | MR | Zbl

[21] Y. Long, Hyperbolic closed characteristics on compact convex smooth hypersurfaces in R2n, Nankai Inst. of Math. Preprint, 1996, Revised 1997, J. Diff. Equa., Vol. 150, 1998, pp. 227-249. | MR | Zbl

[22] Y. Long, A Maslov-type index theory for symplectic paths, Top. Meth. Nonl. Anal., Vol. 10, 1997, pp. 47-78. | MR | Zbl

[23] Y. Long and E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, Stoc. Proc. Phys. and Geom., S. Albeverio et al. ed. World Sci., 1990, pp. 528-563. | MR

[24] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., Vol. 31, 1978, pp. 157-184. | MR | Zbl

[25] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conf. Ser. in Math. Amer. Math. Soc., Vol. 65, 1986. | MR | Zbl

[26] C. Viterbo, A Proof of the Weinstein conjecture in R2n, Ann. IHP. Analyse nonlinéaire, Vol. 4, 1987, pp. 337-357. | Numdam | MR | Zbl

[27] C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math. Soc., Vol. 311, 1989, pp. 621-655. | MR | Zbl

[28] T. Wang and G. Fei, Subharmonicsfor superquadratic Hamiltonian systems via the iteration method of the Maslov-type index theory, Preprint, 1996.

[29] A. Weinstein, On the Hypotheses of Rabinowitz' Periodic Orbit Theorems, J. Diff. Equa., Vol. 33, 1979, pp. 353-358. | MR | Zbl

[30] J.A. Yorke, periods of periodic solutions and the Lipschitz contant, Proc. Amer. Math. Soc., Vol. 22, 1969, pp. 509-512. | MR | Zbl