A continuation lemma and its applications to periodic solutions of Rayleigh differential equations with subquadratic potential conditions

In this paper, we study the existence of periodic solutions of Rayleigh equationsx″+f(t,x′)+g(x)=e(t),x″+f(t,x′)+g(x)=e(t), where f, g, e are continuous functions and f is T-periodic with the first variable, e is T-periodic. By developing a continuation lemma for Rayleigh equations, we prove that the given equation has at least one T  -periodic solution provided that f(t,y)f(t,y) is sublinear with respect to the variable y   and G(x)(=∫0xg(u)du) satisfies some subquadratic conditions.