ON PERIODIC SOLUTIONS FOR A CLASS OF SUPER-QUADRATIC DAMPED VIBRATION PROBLEMS

In this paper, we study the existence of nontrivial periodic solutions for the following damped vibration equation u(t) + q(t)u (t) + Bu (t) + 1/2 q(t)Bu(t) − L(t)u(t) + ∇W(t, u(t)) = 0, where q : R −→ R is a continuous, periodic function with mean value zero , B is an antisymmetric N ×N constant matrix, L(t) is a continuous, periodic and symmetric N × N matrix-valued function and W ∈ C1(R×RN,R) is periodic in the first variable. We use a new kind of superquadratic condition instead of the global Ambrosetti-Rabinowitz superquadratic condidition. By applying a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we obtain a nontrivial periodic solution for the above system.