Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media.The variable coefficient characterizes the inhomogeneity of media and its presence usually leads to the destruction of the compactness of the inverse of the linear wave operator with periodic-Dirichlet boundary conditions on its range.In the pioneering work of Barbu and Pavel(1997),they gave the existence and regularity of the periodic solution for Lipschitz,nonresonant and monotone nonlinearity under the assumptionηu>0(see Section 2 for its definition)on the coefficient u(x)and left the caseηu=0 as an open problem.In this paper,by developing the invariant subspace method and using the complete reduction technique and Leray-Schauder theory,we obtain the existence of periodic solutions for such a problem when the nonlinear term satisfies the asymptotic nonresonance conditions.Our result needs neither requirements on the coefficient except the natural positivity assumption(i.e.,u(x)>0)nor the monotonicity assumption on the nonlinearity.In particular,when the nonlinear term is an odd function and satisfies the global nonresonance conditions,there is only one(trivial)solution to this problem in the invariant subspace.