A Note on Periodic Solutions of Second Order Differential Equations without Damping

where, in addition to satisfying conditions which insure existence and uniqueness of solutions, g(x) and p(t) satisfy the following conditions: (i) there exist positive constants r and k such that for all t, p(t) = p(t+r) and \p(t)\ k. We show that for r sufficiently small, Equation (1) has a solution x(t) of period r. Our method of proof consists of showing that a certain mapping of the phase plane of Equation (1) has a fixed point, not by use of the Brouwer theorem, but by a somewhat more general result concerning the index of the bounding curve of a simply-connected region relative to a vector field induced by the mapping; cf. [1, p. 337]. We consider Equation (1) in terms of the system: