In this paper we present conditions ensuring the existence and localization of lower and upper functions of the periodic boundary value problem u '' + k u = f(t, u), u(0) = u(2 π), u 0 (0) = u 0 (2π), k ∈ R, k ≠ 0. These functions are constructed as solutions of some related generalized linear problems and can be nonsmooth in general.