三类脉冲微分方程边值问题正解的存在性
发布时间：2018-08-23 14:15
【摘要】：在科学研究的应用中,微分方程作为一种重要工具在物理,化学,生物中扮演着重要的角色,其中脉冲微分方程是微分方程的一个重要分支.本文主要讨论三类脉冲微分方程正解的存在性,其中有两类脉冲微分方程带有时滞现象.脉冲现象和时滞现象在现代科技各领域的实际问题中是普遍存在的.相比没有脉冲和时滞的微分方程而言,脉冲微分方程和时滞微分方程能更真实地反映事物的发展过程,其最突出的特点是能够充分考虑到瞬时突变和时间延滞现象对状态的影响,能够更精确、更深刻地反映事物的变化规律.本文主要分为四章,具体内容安排如下:第一章,概述脉冲微分方程的历史背景和研究现状.第二章,研究二阶脉冲微分方程周期边值问题正解的存在性,本章主要通过假设f和IK是递增的条件下,通过Leggett-Williams不动点定理,证明所给系统至少存在三个正周期解.第三章,研究一类带有时滞的一阶脉冲微分方程,主要通过转化技术把脉冲微分方程转化为非脉冲微分方程,再运用不动点定理去建立时滞脉冲微分方程正周期解的存在性.在一定的条件下证明所给系统至少存在两个正周期解,使得研究结果得到进一步完善.第四章,研究一类二阶时滞脉冲微分方程边值问题,二阶时滞脉冲微分方程往往在求解过程中由于其脉冲性而较为繁琐,本章借助转化技术将二阶时滞脉冲微分方程转化为二阶时滞微分方程,并通过锥拉伸与锥压缩不动点定理,研究所给系统正解的存在性.
[Abstract]:In the application of scientific research, differential equations play an important role in physics, chemistry and biology as an important tool, among which impulsive differential equations are an important branch of differential equations. In this paper, we discuss the existence of positive solutions for three kinds of impulsive differential equations, in which two kinds of impulsive differential equations have delay phenomena. Pulse phenomenon and delay phenomenon are common in various fields of modern science and technology. Compared with differential equations without impulses and delays, impulsive differential equations and delay differential equations can more truly reflect the development of things. Its most outstanding feature is that it can fully consider the influence of transient abrupt change and time delay on the state, and can more accurately and profoundly reflect the changing law of things. This paper is divided into four chapters. The main contents are as follows: chapter 1, the historical background and research status of impulsive differential equations. In chapter 2, we study the existence of positive solutions for periodic boundary value problems of second order impulsive differential equations. In this chapter, we prove that there are at least three positive periodic solutions for the system under the assumption that f and IK are increasing, and by Leggett-Williams fixed point theorem. In chapter 3, a class of first order impulsive differential equations with time delay is studied. The impulsive differential equations are transformed into nonimpulsive differential equations by transformation technique, and the existence of positive periodic solutions of delay impulsive differential equations is established by using fixed point theorem. Under certain conditions, it is proved that there are at least two positive periodic solutions for the given system, which further improves the research results. In chapter 4, the boundary value problems of a class of second order delay impulsive differential equations are studied. In this chapter, the second order delay impulsive differential equation is transformed into the second order delay differential equation by means of transformation technique, and the existence of positive solutions of the system is studied by using the fixed point theorem of cone stretching and cone contraction.
【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：2199317
[Abstract]:In the application of scientific research, differential equations play an important role in physics, chemistry and biology as an important tool, among which impulsive differential equations are an important branch of differential equations. In this paper, we discuss the existence of positive solutions for three kinds of impulsive differential equations, in which two kinds of impulsive differential equations have delay phenomena. Pulse phenomenon and delay phenomenon are common in various fields of modern science and technology. Compared with differential equations without impulses and delays, impulsive differential equations and delay differential equations can more truly reflect the development of things. Its most outstanding feature is that it can fully consider the influence of transient abrupt change and time delay on the state, and can more accurately and profoundly reflect the changing law of things. This paper is divided into four chapters. The main contents are as follows: chapter 1, the historical background and research status of impulsive differential equations. In chapter 2, we study the existence of positive solutions for periodic boundary value problems of second order impulsive differential equations. In this chapter, we prove that there are at least three positive periodic solutions for the system under the assumption that f and IK are increasing, and by Leggett-Williams fixed point theorem. In chapter 3, a class of first order impulsive differential equations with time delay is studied. The impulsive differential equations are transformed into nonimpulsive differential equations by transformation technique, and the existence of positive periodic solutions of delay impulsive differential equations is established by using fixed point theorem. Under certain conditions, it is proved that there are at least two positive periodic solutions for the given system, which further improves the research results. In chapter 4, the boundary value problems of a class of second order delay impulsive differential equations are studied. In this chapter, the second order delay impulsive differential equation is transformed into the second order delay differential equation by means of transformation technique, and the existence of positive solutions of the system is studied by using the fixed point theorem of cone stretching and cone contraction.
【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1条
1 ;Periodic boundary value problem for the first order functional differential equations with impulses[J];Applied Mathematics:A Journal of Chinese Universities(Series B);2009年01期
，本文编号：2199317
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