Periodic solutions of some second-order differential equations with desultorily sublinear nonlinearities

In this paper, we study the existence of periodic solutions of the Rayleigh equations x″+f(x′)+g(x)=e(t).x″+f(x′)+g(x)=e(t). The nonlinear term gg satisfies the following conditions: lim infx→+∞g(x)x⩽0, and lim infx→+∞,x∈Saxg′(x)g(x)>−∞, where aa is a non-negative constant, Sa={x∈R+:g(x)>a} with supSa=+∞supSa=+∞ and ff satisfies the sublinear condition. Using the Leray–Schauder continuation theorem, we give sufficient conditions for the existence of periodic solutions of the given equations.