Subharmonic solutions of conservative systems with nonconvex potentials

Authors:
A. Fonda and A. C. Lazer

Journal:
Proc. Amer. Math. Soc. **115** (1992), 183-190

MSC:
Primary 34C25; Secondary 34B15, 47H15, 58F22, 70K40

MathSciNet review:
1087462

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the system of second order differential equations

**[1]**Vieri Benci and Donato Fortunato,*A “Birkhoff-Lewis” type result for a class of Hamiltonian systems*, Manuscripta Math.**59**(1987), no. 4, 441–456. MR**915996**, 10.1007/BF01170847**[2]**G. D. Birkhoff and D. C. Lewis,*On the periodic motions near a given periodic motion of a dynamical system*, Ann. Mat. Pura Appl.**12**(1934), no. 1, 117–133. MR**1553217**, 10.1007/BF02413852**[3]**Frank H. Clarke and I. Ekeland,*Nonlinear oscillations and boundary value problems for Hamiltonian systems*, Arch. Rational Mech. Anal.**78**(1982), no. 4, 315–333. MR**653545**, 10.1007/BF00249584**[4]**C. Conley and E. Zehnder,*Subharmonic solutions and Morse theory*, Phys. A**124**(1984), no. 1-3, 649–657. Mathematical physics, VII (Boulder, Colo., 1983). MR**759212**, 10.1016/0378-4371(84)90282-6**[5]**T. Ding and F. Zanolin,*Periodic solutions of Duffing's equations with superquadratic potential*, preprint.**[6]**I. Ekeland and H. Hofer,*Subharmonics for convex nonautonomous Hamiltonian systems*, Comm. Pure Appl. Math.**40**(1987), no. 1, 1–36. MR**865356**, 10.1002/cpa.3160400102**[7]**A. Fonda, M. Ramos, and M. Willem,*Subharmonic solutions for second order differential equations*, Topol. Methods Nonlinear Anal.**1**(1993), no. 1, 49–66. MR**1215258****[8]**A. Fonda and M. Willem,*Subharmonic oscillations of forced pendulum-type equations*, J. Differential Equations**81**(1989), no. 2, 215–220. MR**1016079**, 10.1016/0022-0396(89)90120-4**[9]**Howard Jacobowitz,*Periodic solutions of 𝑥′′+𝑓(𝑥,𝑡)=0 via the Poincaré-Birkhoff theorem*, J. Differential Equations**20**(1976), no. 1, 37–52. MR**0393673****[10]**Jean Mawhin and Michel Willem,*Critical point theory and Hamiltonian systems*, Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. MR**982267****[11]**Ray Michalek and Gabriella Tarantello,*Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems*, J. Differential Equations**72**(1988), no. 1, 28–55. MR**929197**, 10.1016/0022-0396(88)90148-9**[12]**Paul H. Rabinowitz,*On subharmonic solutions of Hamiltonian systems*, Comm. Pure Appl. Math.**33**(1980), no. 5, 609–633. MR**586414**, 10.1002/cpa.3160330504**[13]**Paul H. Rabinowitz,*Minimax methods in critical point theory with applications to differential equations*, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR**845785****[14]**G. Tarantello,*Subharmonic solutions for Hamiltonian systems via**-pseudo index theory*, preprint.**[15]**M. Willem,*Subharmonic oscillations of convex Hamiltonian systems*, Nonlinear Anal.**9**(1985), no. 11, 1303–1311. MR**813660**, 10.1016/0362-546X(85)90037-9**[16]**Z. Yang,*The existence of subharmonic solutions for sublinear Duffing's equation*, preprint.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1992-1087462-X

Keywords:
Critical point,
Saddle Point Theorem,
Palais-Smale condition

Article copyright:
© Copyright 1992
American Mathematical Society