  We prove the existence of positive periodic solutions for the second order
nonlinear equation $u" + a(x) g(u) = 0$, where $g(u)$ has superlinear growth at
zero and at infinity. The weight function $a(x)$ is allowed to change its sign.
Necessary and sufficient conditions for the existence of nontrivial solutions
are obtained. The proof is based on Mawhin's coincidence degree and applies
also to Neumann boundary conditions. Applications are given to the search of
positive solutions for a nonlinear PDE in annular domains and for a periodic
problem associated to a non-Hamiltonian equation.
