Periodic solutions of equations with oscillating nonlinearities

In this paper, we consider the 2π-periodic problem for the equation x″ + n2x = f(x) − b(t), where n is a positive integer, b(t) is continuous and 2π-periodic, and f(x) is bounded and continuous. We give a new formulation for the Lazer-Leach conditions for the existence of 2π-periodic solutions, and new sufficient conditions for the existence of unbounded sequences of such solutions. © 2000 Elsevier Science Ltd. All rights reserved.