Damped vibration problems with sign-changing nonlinearities: infinitely many periodic solutions

IN×N is theN×N identity matrix, q(t) ∈ L(R;R) is T-periodic and satisfies ∫ T  q(t)dt = , A(t) = [aij(t)] is aT-periodic symmetricN×N matrix-valued functionwith aij ∈ L∞(R;R) (∀i, j = , , . . . ,N ), B = [bij] is an antisymmetric N × N constant matrix, u = u(t) ∈ C(R,RN ), H(t,u) ∈ C(R × RN ,R) is T-periodic and Hu(t,u) denotes its gradient with respect to the u variable. In fact, there are only a few results [–] of (.). In [], the authors studied a special case (B = , zero matrix) and obtained the existence and multiplicity of periodic solutions. Recently, Chen [] obtained infinitely many periodic solutions for (.) with H being asymptotically quadratic as |u| →∞. But the authors [, , ] obtained infinitely many periodic