On periodic solutions to second-order Duffing type equations

Sufficient and necessary conditions are found for the existence of a positive periodic solution to the Duffing type equation u ′ ′ = p ( t ) u + q ( t , u ) u . The results obtained are compared with facts well known for the autonomous Duffing equation y ′ ′ + a y − b y 3 = 0 . Uniqueness of solutions and possible generalizations are discussed, as well.