On the Periodic Boundary Value Problem and Chaotic-like Dynamics for Nonlinear Hill’s Equations

Abstract

Abstract We present some results which show the rich and complicated structure of the solutions of the second order differential equation ẍ + 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 in [57, 58, 59], 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.