We are interested in nonnegative and nonpositive solutions of the boundary value problem uoe=f (t, u), u(0)=u(1), uOE(0)=uOE (1), where f fulfils the Carathéodory conditions on [0, 1] × R. We generalize the results reached by M. N. Nkashama, J. Santanilla and L. Sanchez and present estimates for solutions. In addition, we apply our existence theorems to periodic boundary value problems for nonlinear Duffing equations whose right-hand sides have a repulsive or attractive singularity at the origin. We extend or generalize existence results by A. C. Lazer and S. Solimini and other authors. Moreover, we get some multiplicity results and in the case of a repulsive singularity we also admit a weak singularity, in constrast to the previous papers on this subject. Our proofs are based on the method of lower and upper functions and topological degree arguments and the results are tested on examples.

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