Nonlinear second-order periodic boundary value problems

Abstract In this paper we study a periodic boundary value problem (PBVP, for brevity) of second order. It is well known that if β and α are upper and lower solutions, respectively, with α ⩽ β, then there exists a solution u such that α ⩽ u ⩽ β. When β ⩽ α nothing was known. The main purpose of this paper is to consider this last case. Also, we study the structure of the set of solutions of the PBVP in the sector [α, β] when α ⩽ β, and we apply the results to the case β ⩽ α.

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