Oscillatory properties of solutions and nonlinear differential equations with periodic boundary conditions (Q1898295)

This is a paper to the memory of Geoffrey J. Butler and to the dedication of his colleagues in Edmonton. The author concisely describes the historical development of the work initiated by G. J. Butler on the use of the Poincaré-Birkhoff fixed point theorem for the obtention of nontrivial \(T\)-periodic solutions of second order differential equations of the form \(\ddot x+ f(t, x)= 0\), \((f(t+ T, x)\equiv f(t, x)\), \(f(t, 0)\equiv 0)\), and \(\ddot x+ g(x)= p(t)\), \((p(t+ T)\equiv p(t)\), \( g(0)= 0)\). It shows how the combination of the use of oscillatory properties of solutions with topological, symplectic and variational techniques provides sharp results about the existence and the number of periodic solutions for the above equations. Most new results related to the author's topics are briefly reported.