Periodic Solutions of a Nonlinear Second Order Differential Equation

let g(x) and f(x) be continuous for all x. We shall also assume, throughout, that the zeros of f(x) are isolated. To do otherwise makes the proof of the theorem awkward and, besides this nuisance, no physical significance is lost. Clearly, if f(a) = 0, then x = a is a periodic solution of (1). Excluding such trivial solutions as periodic, we shall show that (1) has a periodic solution if and only if f(x) has at least one zero at which the graph of y = f(x) crosses the x-axis from below to above. Exactly what is required is that for some real number a and some 6 > 0, (x a)f(x) > 0 for 0 <I x al < 6. If there is one nontrivial periodic solution, then there is a continuum of them. That is, for all initial conditions x = xo, = x'0 in a certain rectangle of R x R the corresponding solutions are periodic. Section 2 is devoted to a proof of this result. Equation (1) is considered by the author in [5] where the objective is to discover equations having many periodic solutions and, at the same time, many nonperiodic solutions. In addition, when written as a phase-space system of first order equations, they provide a source of interesting examples of dynamical systems. Recently Sedziwy [3, Theorem 2] has shown the existence of periodic solutions of the forced equation