  We deal with the problem of existence of periodic solutions for the scalar
differential equation x" + f (t, x) = 0 when the asymmetric nonlinearity
satisfies a one-sided superlinear growth at infinity. The nonlinearity is asked
to be next to resonance and a Landesman-Lazer type of condition will be
introduced in order to obtain a positive answer. Moreover we provide also the
corresponding result for equations with a singularity and asymptotically linear
growth at infinity, showing a further application to radially symmetric
systems.
