Nonlinear vibrations of a nonhomogeneous string

We consider a semilinear equation for the forced vibrations of a finite string with $x$-dependent coefficients under Dirichlet boundary conditions. The existence of the time-periodic solution in nonresonant case is proved. It isn't require the Lipschitz condition. The proof uses the method of monotonic operators and principe Leray–Schauder of the fixed point.