三阶微分方程周期边值问题的正解  

Positive Solutions to Periodic Boundary Value Problems of Third-Order Differential Equations

作　　者：郑欣 

机构地区：[1]兰州理工大学,甘肃 兰州

出　　处：《理论数学》2021年第6期1181-1201,共21页Pure Mathematics

摘　　要：微分方程正解的存在性在现实生活中有着重要的意义。对于一般的三阶线性周期边值问题,运用微分方程基本理论,求解出该齐次线性微分方程周期边值问题的特征函数,并用Cardano公式进行转换,将具体问题分成四种情况,求解出在不同情况下特征根的具体形式,再联立其周期边界条件得到该齐次线性周期边值问题的唯一解,并在满足一定的条件下使得该唯一解大于零。最后运用锥上的不动点指数理论,求解出下述三阶微分方程周期边值问题 正解的存在性,其中f∈C([0,2π]×[0,+∞)), a,b,c∈R。The existence of positive solutions of differential equations is of great significance in real life. For general third-order linear periodic boundary value problems, the basic theory of differential equations is used to solve the characteristic function of the homogeneouslinear differential equa-tion periodic boundary value problem, and the Cardano formula is used to transform the specific problem into four cases. The solution is: in different situations, the specific forms of the charac-teristic roots are combined with their periodic boundary conditions to obtain the unique solution of the homogeneous linear periodic boundary value problem, and it is concluded that the unique solution is greater than zero under certain conditions. Finally, using the fixed point index theory on the cone, solve the existence of positive solutions to the periodic boundary value problem of the following third-order differential equations where and satisfy f∈C([0,2π]×[0,+∞)), a,b,c∈R.

关 键 词：微分方程 周期边值问题 格林函数 正解 不动点指数理论 

分 类 号：O17[理学—数学]