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 equations 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 ) is sublinear with respect to the variable y and G ( x ) ( = ∫ 0 x g ( u ) d u ) satisfies some subquadratic conditions.