The authors prove a weak maximum principle for the solutions of the telegraph equation \[ u_{tt}- u_{xx}+ cu_t+\lambda u= f(t,x)\qquad (c>0) \] which are \(2\pi\)-periodic with respect to \(x\) and bounded over \(\mathbb{R}\) with respect to \(t\). Then the maximum principle is generalized to the case where \(f\) is replaced by an element in a suitable class of measures. Using this tool, the authors extend the method of upper and lower solutions for nonlinear perturbations of this equation under some monotonicity conditions. The method of upper and lower solutions is then applied to the forced dissipative sine-Gordon equation \(u_{tt}- u_{xx}+ cu_t+ a\sin u= p(t,x)\).