PERIODIC SOLUTIONS FOR A CLASS OF SECOND ORDER DAMPED VIBRATION PROBLEMS

In this paper, we study the following damped vibration problems{u(t) +q(t) _u(t) +B_u(t) +12q(t)Bu(t)−L(t)u(t) +∇W(t, u(t)) = 0u(0)−u(T) = _u(0)−_u(T) = 0whereT >0,q:R−→Ris a continuous,T−periodic function with∫T0q(t)dt= 0,Q(t) =∫t0q(s)ds,Bis an antisymmetricN×Nconstant matrix,L(t) is a continuousT−periodic and symmetricN×Nmatrix-valued function andW:R×RN−→Ris a continuously di erentiable function which isT−periodic in the rst variable. Using another type of superquadratic conditions instead of theglobal Ambrosetti-Rabinowitz condition, we prove the existence of solutions for the above systemby applying a variant of Local Linking Theorem. Our result extends some recent results in theliterature.