Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations

AbstractConsider the second order scalar ordinary differential equation x″(t) + ƒ(t, x(t)) = 0 (′ = ddt), where ƒ(t, x) is ω-periodic in t and ƒ(t, 0) = 0 for all t. The usual existence results for periodic solutions employing degree theory or other fixed point arguments are generally unhelpful in this case, since the periodic solution that they predict may well be the trivial solution.Jacobowitz has recently succeeded in applying the Poincaré-Birkhoff “twist” theorem to demonstrate that this equation has infinitely many (nontrivial) periodic solutions when ƒ satisfies a suitable “strong nonlinearity” condition with respect to x. Essential to his method of proof, however, is the condition xƒ(t, x) > 0 for all t (x = ≠ 0). In this note we show how this hypothesis may be relaxed, by modifying a technique used by the author when considering the problem of the global existence of solutions which occurs with the removal of the sign condition