Periodic solutions for second order periodic differential equations under scalable control

Second order periodic differential equations such as the Hill's equation are used to model damped oscillators, vibrating elliptical drumheads, rotating electric dipoles, etc. and existence of periodic solutions are important. In this paper we study second order equations subject to 'adjustable' feedback control functions with delays. Based on the fixed point theorem for an ordered Banach space, the existence, multiplicity, and the nonexistence of periodic solutions for our equation are obtained. The uniqueness of periodic solutions and the dependence of periodic solutions on the adjustable scale factor are also studied.