This paper deals with a nonlinear periodic boundary-value problem of the following form \[ u_{tt} - u_{xx} = F[u,u_{t},u_{x}], \tag{1} \] \[ u(0,t) = u( \pi , t)=0, \tag{2} \] \[ u(x, t+T) = u(x,t), (x,t) \in {\mathbb{R}}^{2}, \tag{3} \] where \( F[u,u_{t},u_{x}] \) is a nonlinear operator mapping a smooth function into continuous scalar one. The author proves a theorem on sufficient conditions for existence and uniqueness of smooth solution to the problem (1)-(3).