Periodic Solutions For Sublinear Duffing Equations With Bounded Impulsive Effects

With regard to the study of periodic solutions for impulsive equations, most results were gotten that the existence of periodic solutions. In this article, we study for the existence of infinitely periodic solutions (subharmonic solutions) for sublinear Duffing equation with bounded impulsive effects.The main tool is the Poincare-Birkhoff twist theorem. It is more difficult than the superlinear Duffing equations in angle estimating The reason lies in its very weak relatively twisting, slowly twisting problem, its internal borders would be find extremely difficult if we apply the Poincare-Birkhoff twist theorem to determine.For the Slowly twisting problem, main difficulties we consider is the pulse inter-fere with estimating Poincare mapping. In order to solve this problem, we change the sublinear-Duffing equation in to a new plane Hamilton system. Then, under the bounded impulsive effects, we give some properties of differential equation. As we all known, When the solution moves around the origin, it still arrives the origin. So, we should change the Jump mapping that is itself out of the outer and Identity map in of the inner. Between the inner and the outer, changed jump mapping is smooth, we can find a increasedly function which can control the inner boundary.Last, the outer boundary can be found by dubliner. By using Pioncare-Birkhoff theorem in the twist annulus, we can obtain the existence of the fixed points for the Pioncare map which corresponding to the subharmonic solutions for the new Hamilton system. The rotation estimations for the subharmonic solutions imply that these solutions just the solutions of the old equation.