Some critical point theorems and their applications to periodic solution for second order Hamiltonian systems

Journal of Differential Equations

Volumn 248, Issue 4, 2010, Pages 660-692

  Tang, Chun Lei ^a   Wu, Xing Ping ^a  

^a SOUTHWEST UNIVERSITY   (China)

Author keywords

Critical point; Periodic solution; Perturbation argument; Reduction method; Second order Hamiltonian systems; The least action principle

Indexed keywords

EID: 73549093365     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.11.007     Document Type: Article

References (57)

1. Amann H. Saddle points and multiple solutions of differential equations. Math. Z. 169 2 (1979) 127-166
2. Ambrosetti A., and Rabinowitz P.H. Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973) 349-381
3. Aubin J.-P., and Ekeland I. Applied Nonlinear Analysis. Pure Appl. Math. (N. Y.) (1984), A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York
4. Bates P.W., and Ekeland I. A saddle-point theorem. Differential Equations, Proc. Eighth Fall Conf., Oklahoma State Univ., Stillwater, Okla., 1979 (1980), Academic Press, New York 123-126
5. Benci V. Some critical point theorems and applications. Comm. Pure Appl. Math. 33 2 (1980) 147-172
6. Benci V., and Rabinowitz P.H. Critical point theorems for indefinite functionals. Invent. Math. 52 3 (1979) 241-273
7. Berger M.S., and Schechter M. On the solvability of semilinear gradient operator equations. Adv. Math. 25 2 (1977) 97-132
8. Boukhrisse H., and Moussaoui M. Critical point theorems. Int. J. Math. Math. Sci. 29 7 (2002) 429-438
9. Castro A., and Lazer A.C. Applications of a max-min principle. Rev. Colombiana Mat. 10 4 (1976) 141-149
10. Chang K.C. Infinite-Dimensional Morse Theory and Multiple Solution Problems. Progr. Nonlinear Differential Equations Appl. vol. 6 (1993), Birkhäuser Boston, Inc., Boston, MA
11. Chang K.C. On the periodic nonlinearity and the multiplicity of solutions. Nonlinear Anal. 13 5 (1989) 527-537
12. Chang K.C. Critical Point Theory and Its Applications. Mod. Math. Ser. (1986), Shanghai Kexue Jishu Chubanshe, Shanghai (in Chinese)
13. Diestel J. Geometry of Banach Spaces-Selected Topics. Lecture Notes in Math. vol. 485 (1975), Springer-Verlag, Berlin
14. Ding Y. Variational Methods for Strongly Indefinite Problems. Interdiscip. Math. Sci. vol. 7 (2007), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ
15. Fei G. On periodic solutions of superquadratic Hamiltonian systems. Electron. J. Differential Equations 8 (2002) 12 pp
16. Han Z. Solvability of elliptic boundary value problems without standard Landesman-Lazer condition. Acta Math. Sin. (Engl. Ser.) 16 2 (2000) 349-360
17. Jabri Y., and Moussaoui M. A critical point theorem without compactness and applications. Nonlinear Anal. 32 3 (1998) 363-380
18. Jiang Q., and Tang C.-L. Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 328 1 (2007) 380-389
19. Lazer A.C., Landesman E.M., and Meyers D.R. On saddle point problems in the calculus of variations, the Ritz algorithm, and monotone convergence. J. Math. Anal. Appl. 52 3 (1975) 594-614
20. Li S.J., and Willem M. Applications of local linking to critical point theory. J. Math. Anal. Appl. 189 1 (1995) 6-32
21. Liu J.Q. A generalized saddle point theorem. J. Differential Equations 82 2 (1989) 372-385
22. Long Y.M. Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials. Nonlinear Anal. 24 12 (1995) 1665-1671
23. Luan S.X., and Mao A.M. Periodic solutions of nonautonomous second order Hamiltonian systems. Acta Math. Sin. (Engl. Ser.) 21 4 (2005) 685-690
24. Manasevich R.F. A nonvariational version of a max-min principle. Nonlinear Anal. 7 6 (1983) 565-570
25. Manasevich R.F. A min max theorem. J. Math. Anal. Appl. 90 1 (1982) 64-71
26. Mawhin J. Semicoercive monotone variational problems. Acad. Roy. Belg. Bull. Cl. Sci. (5) 73 3-4 (1987) 118-130
27. Mawhin J., and Willem M. Critical Point Theory and Hamiltonian Systems. Appl. Math. Sci. vol. 74 (1989), Springer-Verlag, New York
28. n action. Trans. Amer. Math. Soc. 310 1 (1988) 303-311
29. Rabinowitz P.H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math. vol. 65 (1986), American Mathematical Society, Providence, RI Published for the Conference Board of the Mathematical Sciences, Washington, DC
30. Rabinowitz P.H. Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31 2 (1978) 157-184
31. Rabinowitz P.H. On subharmonic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 33 5 (1980) 609-633
32. Schechter M. Periodic non-autonomous second-order dynamical systems. J. Differential Equations 223 2 (2006) 290-302
33. Schechter M. Linking Methods in Critical Point Theory (1999), Birkhäuser Boston, Inc., Boston, MA
34. Shapiro V.L. Saddle points and Ritz-Galerkin approximations. SIAM J. Math. Anal. 11 3 (1980) 577-595
35. Struwe M. Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (1990), Springer-Verlag, Berlin
36. Tang C.-L. An existence theorem of solutions of semilinear equations in reflexive Banach spaces and its applications. Acad. Roy. Belg. Bull. Cl. Sci. (6) 4 7-12 (1993) 317-330
37. Tang C.-L. Periodic solutions of non-autonomous second order systems with γ-quasisubadditive potential. J. Math. Anal. Appl. 189 3 (1995) 671-675
38. Tang C.-L. Periodic solutions of non-autonomous second order systems. J. Math. Anal. Appl. 202 2 (1996) 465-469
39. Tang C.-L. Periodic solutions for nonautonomous second order systems with sublinear nonlinearity. Proc. Amer. Math. Soc. 126 11 (1998) 3263-3270
40. Tang C.-L. Existence and multiplicity of periodic solutions for nonautonomous second order systems. Nonlinear Anal. 32 3 (1998) 299-304
41. Tang C.-L., and Wu X.-P. A note on periodic solutions of nonautonomous second-order systems. Proc. Amer. Math. Soc. 132 5 (2004) 1295-1303
42. Tang C.-L., and Wu X.-P. Notes on periodic solutions of subquadratic second order systems. J. Math. Anal. Appl. 285 1 (2003) 8-16
43. Tang C.-L., and Wu X.-P. Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems. J. Math. Anal. Appl. 275 2 (2002) 870-882
44. Tang C.-L., and Wu X.-P. Periodic solutions for second order systems with not uniformly coercive potential. J. Math. Anal. Appl. 259 2 (2001) 386-397
45. Tao Z.-L., and Tang C.-L. Periodic and subharmonic solutions of second-order Hamiltonian systems. J. Math. Anal. Appl. 293 2 (2004) 435-445
46. Tao Z.-L., Yan S.-A., and Wu S.-L. Periodic solutions for a class of superquadratic Hamiltonian systems. J. Math. Anal. Appl. 331 1 (2007) 152-158
47. Wu X. Saddle point characterization and multiplicity of periodic solutions of non-autonomous second-order systems. Nonlinear Anal. 58 7-8 (2004) 899-907
48. Willem M. Minimax Theorems. Progr. Nonlinear Differential Equations Appl. vol. 24 (1996), Birkhäuser Boston, Inc., Boston, MA
49. Willem M. Oscillations forcées de systemes hamiltoniens. Publ. Sémin. Analyse Non Linéaire (1981), Univ. Besancon
50. Willem M. Periodic oscillations of odd second order Hamiltonian systems. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (6) 3 2 (1984) 293-304
51. Wu X.-P. Periodic solutions for nonautonomous second-order systems with bounded nonlinearity. J. Math. Anal. Appl. 230 1 (1999) 135-141
52. Wu X.-P., and Tang C.-L. Periodic solutions of a class of non-autonomous second-order systems. J. Math. Anal. Appl. 236 2 (1999) 227-235
53. Wu X.-P., and Tang C.-L. Periodic solutions of nonautonomous second-order Hamiltonian systems with even-typed potentials. Nonlinear Anal. 55 6 (2003) 759-769
54. Yosida K. Functional Analysis. sixth edition (1980), Springer-Verlag, Berlin
55. Zhao F., and Wu X. Saddle point reduction method for some non-autonomous second order systems. J. Math. Anal. Appl. 291 2 (2004) 653-665
56. Zhao F., and Wu X. Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity. Nonlinear Anal. 60 2 (2005) 325-335
57. Zou W., and Schechter M. Critical Point Theory and Its Applications (2006), Springer-Verlag, New York