中文摘要：

电子束聚焦理论应用广泛,为了控制电子束的运动轨迹,使其有效地聚焦目标,对电子束聚焦系统数学模型的正周期解的存在性进行研究。利用锥上的Krasnoselekii不动点定理得到了该模型至少存在一个周期解的充分条件和解的存在范围,并进行了实例验证。结果表明：对该模型的系数取值范围的界定是合理的,定理的证明方法是正确的。并且,推广和补充了已有的结果。

英文摘要：

Electron beam focusing system theories are widely applied. In order to control the moving trace of electron beam and make it focus object effectivelly, the existence of positive periodic solution is studied based on the electron beam focusing system model. The sufficient condition and the region of the existence of at least a periodic solution for the model is found by using Krasnoselekii fixed point theorem in cone, then an example is given. The results show that the limited ranges of coefficient are reasonable and the method of theorem is correct. The previous results is extended and supplemented.