The Periodic Problem Of Second-order Differential Equations With Singularities

In this paper, we are concerned with the periodic problem of second-order dif-ferential equations rely on the method of upper and lower functions, also sufficient conditions are obtained.It includes three parts:The first section contains the background and history of differential equations, also the main work of this paper is given. In the second and third parts, we discuss the periodic solution of the second order differential equations with mixed singularities using upper and lower solutions, and obtain some common sufficient conditions under the periodic boundary condition to guarantee the existence of positive solution. Also we built concrete x0and w, then a sufficient condition is given for the equations. The reverse upper and lower functions also can guarantee the existence of positive solu-tion. As a result, we extend the conclusion of related literature [Journal of Differential Equations,2010,248,111-126].