拓扑定理及其在超线性脉冲方程中的应用  

A topological theorem and its application to the superlinear equations including impulses for the existence of periodic solutions

作　　者：王超[1] 刘期怀[2,3] 钱定边[4] 王志国[4] 

机构地区：[1]盐城师范学院数学科学学院,盐城224001 [2]桂林电子科技大学数学与计算科学学院,桂林541001 [3]复旦大学数学科学学院,上海200433 [4]苏州大学数学科学学院,苏州215006

出　　处：《中国科学：数学》2014年第9期957-968,共12页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11101299;11301106;11271277和11162004);江苏省高校自然科学基金(批准号:09KJD110005);中国博士后面上基金(批准号:2014M550210)资助项目

摘　　要：本文研究一类含脉冲的非保守超线性方程的周期振动问题.通过详细刻画跳跃映射的角度函数,并应用相平面分析方法,本文证明方程的Poincaré映射有无穷多个不动点,而这些不动点对应了方程的无穷多个2π周期解.由于本文考虑的Poincaré映射不是同胚,所以,本文重新构造并证明了一个拓扑定理用于替代经典的Poincaré-Birkhoff扭转不动点定理.In this paper,we prove the existence and multiplicity of 2π periodic solutions for a class of nonconservative super-linear equations including periodic impulses with one non-homeomorphic impulse in one period.By applying the method of the phase plane analysis,we consider the fixed points problem for the Poincare mapping of the original equation after introducing the angle variable for impulses.Note that the Poincare mapping is not homeomorphic due to the impulses,we give a new topological fixed point theorem to obtain the fixed points for the mapping instead of the Poincare-Birkhoff fixed point theorem.

关 键 词：超线性方程 脉冲 周期解 不动点 

分 类 号：O175[理学—数学]