Banach空间中四类脉冲微分方程多点边值问题的正解
发布时间：2018-09-03 13:39
【摘要】：微分方程边值问题是微分方程理论中常见的一种基本问题,脉冲微分方程边值问题又是微分方程边值问题的一个重要分支,具有很高的应用价值,脉冲微分方程是研究一个过程突然发生变化的基本工具,能够充分体现瞬时突变现象对系统的影响,能更加真实的地描述自然界状态,脉冲系统在现代科学领域中是广泛存在的,它的理论在经济学、社会科学、生物学、物理学、工程学等有着广泛的运用,因此,对脉冲微分方程的研究早己引起了国内外同行的广泛关注.本学位论文讨论了四类脉冲微分方程多点边值问题正解的存在性,利用锥拉伸锥压缩不动点定理和Leggett-Williams不动点定理得出了四类脉冲微分方程多点边值问题正解存在性的充分条件,全文具体内容如下.第一章为绪论,主要介绍了脉冲微分方程边值问题研宄的相关背景和基本情况,以及给出了文中用到的定义和定理.第二章考虑了非线性项带有一阶导数的二阶脉冲积分微分方程m点边值问题(?)运用锥上的不动点定理得出该边值问题至少存在一个正解.第三章考虑带p-Laplacian算子的二阶脉冲微分方程m点边值问题(?)同样是运用锥拉伸锥压缩不动点定理得出该问题至少存在一个正解.第四章考虑带p-Laplacian算子积分边界条件下的四阶脉冲微分方程多点边值问题(?)采用将四阶微分方程边值问题转化为与之等价的两个二阶微分方程边值问题的方法,然后运用锥拉伸锥压缩不动点定理得出该问题至少存在一个正解的两个充分条件,并在文章最后给出了应用举例.第五章利用Leggett-Williams不动点定理,考虑了一类带p-Laplacian算子积分边界条件下的四阶脉冲微分方程边值问题(?)同样是采用将四阶边值问题转化为等价的两个二阶边值问题的方法,然后运用Leggett-Williams不动点定理得出该边值问题至少存在三个正解的充分条件.第六章为本论文的结束语,总结了本文的主要工作,并对进一步可以研究的问题做了设想.
[Abstract]:Boundary value problem of differential equation is a basic problem in the theory of differential equation. Boundary value problem of impulsive differential equation is an important branch of boundary value problem of differential equation. Impulsive differential equation is a basic tool to study the sudden change of a process. It can fully reflect the influence of the transient sudden change on the system, and can describe the state of nature more realistically. Pulse system is widely used in the field of modern science. Its theory is widely used in economics, social science, biology, physics, engineering and so on. The study of impulsive differential equations has attracted wide attention from domestic and foreign counterparts. In this paper, we discuss the existence of positive solutions of multipoint boundary value problems for four kinds of impulsive differential equations. By using the fixed point theorem of conical stretching cone contraction and Leggett-Williams 's fixed point theorem, we obtain sufficient conditions for the existence of positive solutions of multipoint boundary value problems for four classes of impulsive differential equations. The full text is as follows. The first chapter is the introduction, which mainly introduces the background and basic situation of the boundary value problems of impulsive differential equations, and gives the definitions and theorems used in this paper. In chapter 2, we consider the m-point boundary value problem of second order impulsive integrodifferential equations with first derivative. By using the fixed point theorem on the cone, it is obtained that there is at least one positive solution to the boundary value problem. In chapter 3, the m-point boundary value problem of second order impulsive differential equation with p-Laplacian operator is considered. The fixed point theorem of cone-stretching cone compression is also used to obtain that there is at least one positive solution to the problem. In chapter 4, the multipoint boundary value problem of fourth order impulsive differential equation with p-Laplacian operator integral boundary condition is considered. In this paper, the boundary value problem of fourth order differential equation is transformed into two boundary value problems of second order differential equation, and then two sufficient conditions for the existence of at least one positive solution are obtained by using the fixed point theorem of cone stretching cone contraction. At the end of the article, an application example is given. In chapter 5, by using Leggett-Williams fixed point theorem, the boundary value problem of fourth order impulsive differential equation with p-Laplacian operator integral boundary condition is considered. The same method is used to transform the fourth order boundary value problem into two equivalent second order boundary value problems. Then the sufficient conditions for the existence of at least three positive solutions for the boundary value problem are obtained by using the Leggett-Williams fixed point theorem. The sixth chapter is the conclusion of this paper, summarizes the main work of this paper, and makes a tentative plan for further research.
【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
本文编号：2220131
[Abstract]:Boundary value problem of differential equation is a basic problem in the theory of differential equation. Boundary value problem of impulsive differential equation is an important branch of boundary value problem of differential equation. Impulsive differential equation is a basic tool to study the sudden change of a process. It can fully reflect the influence of the transient sudden change on the system, and can describe the state of nature more realistically. Pulse system is widely used in the field of modern science. Its theory is widely used in economics, social science, biology, physics, engineering and so on. The study of impulsive differential equations has attracted wide attention from domestic and foreign counterparts. In this paper, we discuss the existence of positive solutions of multipoint boundary value problems for four kinds of impulsive differential equations. By using the fixed point theorem of conical stretching cone contraction and Leggett-Williams 's fixed point theorem, we obtain sufficient conditions for the existence of positive solutions of multipoint boundary value problems for four classes of impulsive differential equations. The full text is as follows. The first chapter is the introduction, which mainly introduces the background and basic situation of the boundary value problems of impulsive differential equations, and gives the definitions and theorems used in this paper. In chapter 2, we consider the m-point boundary value problem of second order impulsive integrodifferential equations with first derivative. By using the fixed point theorem on the cone, it is obtained that there is at least one positive solution to the boundary value problem. In chapter 3, the m-point boundary value problem of second order impulsive differential equation with p-Laplacian operator is considered. The fixed point theorem of cone-stretching cone compression is also used to obtain that there is at least one positive solution to the problem. In chapter 4, the multipoint boundary value problem of fourth order impulsive differential equation with p-Laplacian operator integral boundary condition is considered. In this paper, the boundary value problem of fourth order differential equation is transformed into two boundary value problems of second order differential equation, and then two sufficient conditions for the existence of at least one positive solution are obtained by using the fixed point theorem of cone stretching cone contraction. At the end of the article, an application example is given. In chapter 5, by using Leggett-Williams fixed point theorem, the boundary value problem of fourth order impulsive differential equation with p-Laplacian operator integral boundary condition is considered. The same method is used to transform the fourth order boundary value problem into two equivalent second order boundary value problems. Then the sufficient conditions for the existence of at least three positive solutions for the boundary value problem are obtained by using the Leggett-Williams fixed point theorem. The sixth chapter is the conclusion of this paper, summarizes the main work of this paper, and makes a tentative plan for further research.
【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
【参考文献】
相关期刊论文 前2条
1 李宝麟;樊瑞宁;;Banach空间二阶脉冲积分-微分方程三点边值问题[J];甘肃科学学报;2009年04期
2 张学梅;葛渭高;;一类带p-Laplace算子的奇异脉冲特征值问题[J];北京理工大学学报;2008年12期
，本文编号：2220131
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