几类脉冲微分方程边值问题多个正解的存在性

发布时间：2018-03-16 10:38

本文选题：脉冲微分方程　切入点：滞后型二阶脉冲微分方程　出处：《北京信息科技大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文通过运用Leggett-Williams不动点定理,不动点指数理论,特征值理论,变换技巧和H?lder不等式系统地研究了包括二阶、四阶和n阶在内的脉冲微分方程边值问题多个正解的存在性,正解对参数的连续依赖性以及正解存在的最优区间。根据研究内容和研究方法,全文共分为五章。第一章绪论,介绍脉冲微分方程边值问题的研究背景与意义,并根据国内和国外的研究现状提出了本文所研究的主要内容,最后给出本文所需要的一些基本概念和定理。第二章讨论了一类带积分边界条件的滞后型二阶脉冲微分方程边值问题。首先给出了对应的齐次边值问题的Green函数的表达式,并研究了其性质。然后利用Leggett-Williams不动点定理和H?lder不等式得到了边值问题至少存在三个正解的结果。最后给出了一个相应的实例以说明我们的结论。第三章研究了一类四阶脉冲微分方程边值问题多个正解的存在性以及对参数的依赖性。文章通过使用两个变换和不动点定理,确立脉冲梁方程的正解存在性,多解性和正解对参数的依赖性。值得一提的是,我们不仅给出了解的范数估计形式,还讨论了解对参数的依懒性并在最后通过一个实例验证了主要结果的正确性。第四章考察了一类n阶超前型特征值问题正解的存在性。文章通过使用变换技巧,H?lder不等式以及特征值理论确立了参数l的最优区间,并且在这个区间上,我们证明了这个具超前变元的n阶脉冲微分方程存在正解。第五章对全篇文章进行总结并展望了今后的研究工作。
[Abstract]:In this paper, Leggett-Williams fixed point theorem, fixed point exponent theory, eigenvalue theory, transformation technique and H? Lder inequality systematically studies the existence of several positive solutions of boundary value problems for impulsive differential equations, including second-order, fourth-order and n-order boundary value problems, the continuous dependence of positive solutions on parameters and the optimal interval of existence of positive solutions. The first chapter introduces the background and significance of the research on boundary value problems of impulsive differential equations, and puts forward the main contents of this paper according to the current research situation both at home and abroad. Finally, some basic concepts and theorems needed in this paper are given. In chapter 2, the boundary value problems of second order impulsive differential equations with integral boundary conditions are discussed. First, the expression of the Green function of the corresponding homogeneous boundary value problem is given. Then by using Leggett-Williams fixed point theorem and H? Lder inequality has obtained that there are at least three positive solutions for boundary value problems. Finally, a corresponding example is given to illustrate our conclusion. In chapter 3, we study the existence of multiple positive solutions for a class of fourth order impulsive differential equation boundary value problems. In this paper, by using two transformations and fixed point theorem, The existence of positive solutions, the multiplicity of solutions and the dependence of positive solutions on parameters of impulsive beam equations are established. Finally, an example is given to verify the correctness of the main results. Chapter 4th investigates the existence of positive solutions for a class of n-order advanced eigenvalue problems. Lder inequality and eigenvalue theory establish the optimal interval of parameter l, and on this interval, We prove the existence of positive solutions for this n-order impulsive differential equation with advanced arguments. Chapter 5th summarizes the whole paper and looks forward to the future research work.
【学位授予单位】：北京信息科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8

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