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 system d/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.