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)


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
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