Nonautonomous nonlinear ODEs: Nonresonance conditions and rotation numbers

We deal with the T-periodic problem associated with a nonlinear scalar differential equation like x″+f(t,x)=0, where, for x→0 and |x|→+∞ the nonlinearity f is assumed behave linearly, with a time-dependent coefficient. We prove that the Landesman–Lazer conditions at zero and at infinity possess a rotational effect on “small” and “large” (in the phase-plane) solutions of (1). As a consequence, we are able to generalize previous multiplicity results in the resonant case - i.e., when the linearizations at zero and/or at infinity are resonant, through the use of the Poincaré–Birkhoff fixed point theorem.