Periodic solutions for a coupled system of wave equations with x-dependent coefficients

This paper is concerned with the periodic solutions for a coupled system of wave equations with x-dependent coefficients. Such a model arises naturally when two waves propagate simultaneously in the nonisotrpic media. In this paper, for the periods having the form T = 2 a − 1 b ( a , b are positive integers ) $T=\frac{2a-1}{b}\left(a,b \text{are\,positive\,integers}\right)$ and some types of boundary conditions, we obtain the existence of the time periodic solutions and analyze the asymptotic behaviors as the coupled parameter goes to zero, when the nonlinearities are superlinear and monotone, by using the variational method. In particular, the condition ess inf η ϱ (x) > 0 is not required.