几类集值微分方程解的收敛性分析

发布时间：2018-01-25 22:29

本文关键词： 集值微分方程上下解方法拟线性化方法平方收敛高阶收敛　出处：《河北大学》2017年硕士论文　论文类型：学位论文

【摘要】：近年来,定义在半线性度量空间上的集值微分方程的研究引起了国内外学者的广泛关注.集值微分方程作为常微分方程的推广之一,在物理学、生物学和工程学中有许多应用.然而由于集值微分方程自身的复杂性,其定性与稳定性问题的相关结果还不多.本文主要利用拟线性化方法,讨论几种类型的集值微分方程解的收敛性问题.全文分为以下三部分内容:第一部分在右端函数满足较弱条件下,利用广义拟线性化方法,获得函数项为三项和的集值微分方程初值问题解的平方收敛性.第二部分在右端函数满足一定的条件下,证明比较定理,利用拟线性化方法,讨论具有最大项的集值微分方程周期边值问题解的平方收敛性.第三部分在右端函数满足一定条下,通过引入超凸超凹的定义,利用拟线性化方法,研究集值微分方程初值问题解的高阶收敛性;以及在右端函数分别满足超凸或超凹的条件下,通过引入耦合上下解的定义,证明函数项为两项和的集值微分方程初值问题的高阶收敛性.
[Abstract]:In recent years, the study of set-valued differential equations defined on semilinear metric spaces has attracted wide attention of scholars at home and abroad. As one of the generalizations of ordinary differential equations, set-valued differential equations are widely used in physics. There are many applications in biology and engineering. However, due to the complexity of set-valued differential equations, there are few results related to their qualitative and stability problems. This paper discusses the convergence of solutions of several types of set-valued differential equations. This paper is divided into the following three parts: in the first part, the generalized quasi-linearization method is used to satisfy the weak condition of the right end function. The quadratic convergence of the solution of the initial value problem of set-valued differential equations with a trisomponent term is obtained. In the second part, the comparison theorem is proved and the quasi-linearization method is used under the condition that the function at the right end satisfies certain conditions. The quadratic convergence of solutions of periodic boundary value problems for set-valued differential equations with the largest term is discussed. In the third part, by introducing the definition of hyperconvex hyperconcave, the quasi-linearization method is used. The higher order convergence of solutions for initial value problems of set-valued differential equations is studied. By introducing the definition of coupled upper and lower solutions, the higher order convergence of the initial value problem of set-valued differential equations with binomial terms is proved under the condition that the right end function satisfies hyperconvex or hyperconcave, respectively.
【学位授予单位】：河北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

【参考文献】