Asymptotic behavior of the potential and existence of a periodic solution for a second order differential equation

We establish the existence of a 2π-periodic solution for a second order semilinear equation in terms of the asymptotic behavior of the potential of the nonlinearity. Our condition includes the case in which the nonlinearity is asymptotically linear with slopes at infinity of the jumping nonlinearity between the positive axes and the first Fucik spectrum curve (see S. Fucik (1976 Fucik, S. 1976. Boundary value problems with jumping nonlinearities. Casopis Pest. Mat., 101: 69–87.  Google Scholar]). Boundary value problems with jumping nonlinearities. Casopis Pest. Mat., 101, 69–87.]). Our results extend those of L. Fernandez and F. Zanolin (1988 Fernandez, L. and Zanolin, F. 1988. Periodic solutions of a second order differential equation with one-sided growth restriction on the restoning term. Arch. Math., 51: 151–163.  Google Scholar]). Periodic solutions of a second order differential equation with one-sided growth restriction on the restoning term. Arch. Math., 51, 151–163.].