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 forced vibrations nonhomogeneous string and propagation seismic waves in nonisotropic media. The characterizes inhomogeneity media its presence usually leads to destruction compactness inverse linear operator with periodic-Dirichlet boundary conditions on range. In pioneering work Barbu Pavel (1997), they gave existence regularity solution for Lipschitz, nonresonant monotone nonlinearity under assumption ηu > 0 (see Section 2 definition) u(x) left case = as an open problem. this paper, by developing invariant subspace method using complete reduction technique Leray-Schauder theory, we obtain such problem when nonlinear term satisfies asymptotic nonresonance conditions. Our result not only does need any requirements except natural positivity (i.e., 0) but also monotonicity nonlinearity. particular, is odd function global conditions, there one (trivial) subspace.