On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations

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.