Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems

We are interested in nonnegative and nonpositive solutions of the boundary value problem u " = f (t, u), u(0) = u(1), u'(0) = u'(1), where f fulfils t he Caratheodory conditions on [0, 1] x 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 boundar y value problems for nonlinear Duffing equations whose right-hand sides hav e a repulsive or attractive singularity at the origin. We extend or general ize existence results by A. C. Lazer and S. Solimini and other authors. Mor eover, we get some multiplicity results and in the case of a repulsive sing ularity we also admit a weak singularity, in constrast to the previous pape rs on this subject. Our proofs are based on the method of lower and upper f unctions and topological degree arguments and the results are tested on exa mples. (C) 2001 Academic Press.