含时滞导数项的二阶微分方程的正周期解  

Positive periodic solutions for the second-order differential equations with delayed derivative terms

作　　者：朱俐玫 ZHU Limei(College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Chin)

机构地区：[1]西北师范大学数学与统计学院,兰州730070

出　　处：《黑龙江大学自然科学学报》2018年第2期164-170,共7页Journal of Natural Science of Heilongjiang University

基　　金：国家自然科学基金资助项目(11261053;16611071);甘肃省自然科学基金资助项目(1208RJZA129)

摘　　要：利用锥映射不动点指数理论,研究含时滞导数项的二阶微分方程u″(t)+a(t)u(t)=f(t,u(t-τ1),u'(t-τ2))正ω-周期解的存在性。讨论该方程对应的线性微分方程u″(t)+a(t)u(t)=h(t)的周期问题,运用正算子扰动的方法,建立该线性方程周期解的正性及正周期解的强正性估计和C1-估计:u(t)≥σ‖u‖c,|u'(τ)|≤C1|u(t)|;以Banach空间E=C1ω(R)为工作空间,定义凸锥:K={u∈C1ω(R)|u(t)≥σ‖u‖C,|u'(τ)|≤C1|u(t)|,t,τ∈R}。将所研究方程的正ω-周期解问题转化为一个锥K上的算子A:K→K的不动点问题,应用锥上的不动点指数理论讨论算子A的非平凡不动点的存在性。Using the fixed point index theory in coins, the existence of positive to -periodic solutions for the second-order differential equations with delayed derivative terms u″（t）＋a（t）u（t）=f（t,u（t-τ1）,u＇（t-τ2）） is considered. The periodic solution of the linear second-order differential equation u″（t） ＋ a （t）u （t） = h （t） is first discussed. Applying the perturbation theorem of positive operator, it is estab- lished the positivity of periodic solutions and the strongly positive estimate and C1 -estimate of positive pe- riodic solutions of this linear differential equation： u（t）≥σ‖u‖c,｜u＇（τ）｜≤C1｜u（t）｜. Next, a working space E = Cω1（R） is chosen and a setK={u∈C1ω（R）｜u（t）≥σ‖u‖C,｜u＇（τ）｜≤C1｜u（t）｜,t,τ∈R} is defined. Then, the positive to -periodic solutions are transformed the fixed point of A： K→ K in coins. Finally, by applying the fixed point index theory, the existence of nontrivial fixed point of A is discussed.

关 键 词：二阶微分方程 正周期解 锥 不动点指数 

分 类 号：O175.15[理学—数学]