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We provide sufficient conditions for the existence of periodic solutions of the second order Hamiltonian system , where ε is a small parameter, x ∈ R and V (t, x) is 2π-periodic in t. Moreover we provide two applications. This is a preprint of: "Periodic solutions of second order Hamiltonian systems", Jaume Llibre,doi:10.1080/14689367.2013.781133 fatcat:pkxceomx45azzclsongjjzwpda