## Multiplicity of periodic solutions for a class of second-order perturbed Hamiltonian systems.(English)Zbl 1477.34066

The authors consider the following second-order Hamiltonian systems with non-autonomous perturbed term
$\left\{\begin{array}{l} \ddot{u}(t)+\nabla F(u) = \nabla_u G(t,u), ~~t \in \mathbb R,\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,~~ T>0, \end{array}\right.$ where $$F(u)=-K(u)+W(u),~K,~W \in C^2(\mathbb R^N,\mathbb R),~G \in C^2(\mathbb R \times \mathbb R^N,\mathbb R)$$ with $$G(t+T,u)=G(t,u)$$, $$K$$ and $$W$$ are even functions.
It is assumed that $$G(t,u)$$ has no parity in $$u$$ and is subquadratic at infinity, $$W$$ is superquadratic at infinity and $$K(u)/|u|^2$$ is bounded by two constants. By using Bolle’s perturbation method, the authors prove the existence of infinitely many periodic solutions. This extends some results in the current literature.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 34D10 Perturbations of ordinary differential equations 37C60 Nonautonomous smooth dynamical systems
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