Problems at resonance for equations with periodic nonlinearities

We study differential equations with periodic nonlinearities. In particular, we prove that for a small, the equation -u" - u + g(u) = h(t) + a sin(t - phi), where g is a C-1 periodic function with zero mean value and h is orthogonal to cos t and sin t, has 2pi-periodic solutions. The proof relies on an abstract result based on variational arguments; its application makes use of integral estimates by the stationary phase method. We also study the existence of multiple solutions. (C) 2003 Elsevier Ltd. All rights reserved