Positive subharmonic solutions to nonlinear ODEs with indefinite weight

We prove that the superlinear indefinite equation [Formula: see text] where [Formula: see text] and [Formula: see text] is a [Formula: see text]-periodic sign-changing function satisfying the (sharp) mean value condition [Formula: see text], has positive subharmonic solutions of order [Formula: see text] for any large integer [Formula: see text], thus providing a further contribution to a problem raised by Butler in its pioneering paper [Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations 22 (1976) 467–477]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincaré–Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it).