Orbits with minimal period for a class of autonomous second order one-dimensional Hamiltonian systems

Abstract

Abstract We show, under an iterative condition which is similar to but stronger than that of Ambrosetti and Rabinowitz and by using a variational method, the existence of a T-periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential z ¨ + V ′ ⁢ ( z ) = 0 , z ∈ ℝ , \ddot{z}+V^{\prime}(z)=0,\quad z\in\mathbb{R}, for any T > 0 {T>0} . Moreover, such a solution has T / k {T/k} as a minimal period for some integer 1 ≤ k ≤ 3 {1\leq k\leq 3} .