Existence of positive periodic solutions for the second order and fourth order nonlinear ordinary differential system

In this paper, by using the fixed-point index theory and product of topology degree theory, the existence results on positive solutions is established to the second order and fourth order nonlinear ordinary differential system with periodic boundary value. Our result is essential and do not require the nonlinear terms are superlinear or sublinear at ends (zero or infinty) and the system must have the same order.