We study the periodic boundary value problem associated with the second order nonlinear differential equation u'' + c u' + (a^+(t) - μ a^-(t)) g(u) = 0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill's equation. We prove the existence of 2^m-1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G. J. Butler in 1976. The proof is based on Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.
