Periodic solutions for one dimensional wave equation with bounded nonlinearity

This paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients. The constant coefficient model corresponds to the classical wave equation, while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For finding the periodic solutions of variable coefficient wave equation, it is usually required that the coefficient u(x) satisfies ess infηu(x)>0 with ηu(x)=12u″u−14(u′u)2, which actually excludes the classical constant coefficient model. For the case ηu(x)=0, it is indicated to remain an open problem by Barbu and Pavel (1997) [6]. In this work, for the periods having the form T=2p−1q (p,q are positive integers) and some types of boundary value conditions, we find some fundamental properties for the wave operator with either constant or variable coefficients. Based on these properties, we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded. Such nonlinearity may cross multiple eigenvalues of the corresponding wave operator. In particular, we do not require the condition ess infηu(x)>0.