infinitely many fast homoclinic solutions for damped vibration systems with combined nonlinearities

This article  concerns the existence of fast homoclinic solutions for the following damped vibration systemd/dt (p (t)u˙ (t)) + q(t)p (t)u˙ (t) − l(t)u(t) + ∇w (t, u(t)) = 0,where p, l ∈ c r, rn2 are symmetric and positive definite matri- ces, q ∈ c (r, r) and w ∈ c1 (r × rn , r . applying the fountain theorem and dual fountain theorem, we prove the above system possesses two different sequences of fast homoclinic solutions when l satisfies a new coercive condition and the potential w (t, x) is combined nonlinearity.