Even homoclinic orbits for a class of damped vibration systems

In this paper, we study the existence of even homoclinic solutions for a damped vibration system ẍ(t)+q(t)ẋ(t)+V′(t,x(t))=0,where q is a continuously differentiable function and V ∈C1(R×RN,R), V(t,x)=−K(t,x)+W(t,x). We use a new kind of superquadratic condition instead of the global Ambrosetti-Rabinowitz condition. For the proof we use the Mountain Pass Theorem.