Periodic solutions of a class of superquadratic second order Hamiltonian systems.(English)Zbl 0974.34039

Here, based on saddle point theorem, the authors obtain a sufficient condition for the periodic solution to autonomous second-order differential equations $$\ddot x + Ax + \nabla F(x) = 0,$$ where the function $$F$$ satisfies superquadratic conditions, and $$A$$ is a symmetric matrix. Related results can be found in A. Bahri and H. Berestycki [Commun. Pure Appl. Math. 33, 403-442 (1984; Zbl 0588.34028)]and Y. Long [Trans. Am. Math. Soc. 311, 749-780 (1989; Zbl 0676.34026)]. In particular, Long’s result can be applied to the above equations to obtain a stronger conclusion.
Reviewer: Bin Liu (Beijing)

MSC:

 34C25 Periodic solutions to ordinary differential equations 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 70H05 Hamilton’s equations

Keywords:

periodic solutions; critical points

Citations:

Zbl 0588.34028; Zbl 0676.34026
Full Text:

References:

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