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.
The paper deals with the boundary value problem u '' + k u = g(u) + e(t), u(0) = u(2π), u ' (0) = u ' (2π), where k ∈ R, g : (0, ∞) → R is continuous, e ∈ L[0, 2π] and lim x→0+ ∫1 x g(s) ds = ∞. In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.