In this paper we consider the homoclinic orbits for a class of second order Hamiltonian systems of the form
$ \ddot{q}(t)-\lambda q(t)+\nabla W(t,q(t)) = 0 $
where $ \lambda>0 $ is a parameter, $ \frac{|\nabla W(t,x)|}{|x|} $ asymptotically tends to a constant as $ |x|\rightarrow\infty $ and $ |t|\rightarrow\infty $. Via the variational method, two new theorems are proved.
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