We present some results which show the rich and complicated structure of the
solutions of the second order differential equation x''+ w(t)g(x) = 0 when the
weight w(t) changes sign and g is sufficiently far from the linear case. New
applications, motivated by recent studies on the superlinear Hill’s equation, are then proposed for some asymptotically linear equations and for
some sublinear equations with a sign-indefinite weight. Our results are based on a
fixed point theorem for maps which satisfy a stretching condition along the paths
on two-dimensional cells.
