Time-periodic solutions to a nonlinear wave equation with periodic or anti-periodic boundary conditions

This paper is concerned with the existence of time-periodic solutions to the nonlinear wave equation with x-dependent coefficients u(x)y(tt) - (u(x)y(x))(x)+au(x)(y)+vertical bar y vertical bar(p-2) y=f(x, t) on (0, pi) x R under the periodic or anti-periodic boundary conditions y(0, t) = +/- y(pi, t), y(x)(0, t) = +/- y(x)(pi, t) and the time-periodic conditions y(x, t+T) = y(x, t), y(t)(x, t+T) = y(t)(x, t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. A main concept is the notion 'weak solution' to be given in (sic)2. For T = 2 pi/k(k is an element of R), we establish the existence of time-periodic solutions in the weak sense by investigating some important properties of the wave operator with x - dependent coefficients.