几类脉冲耦合系统边值问题弱解的存在性研究

发布时间：2018-03-28 12:45

本文选题：脉冲效应　切入点：微分耦合系统　出处：《中国矿业大学》2017年硕士论文

【摘要】：微分方程边值问题是数学基础理论和应用研究中的一个重要分支.随着研究的逐渐深入,脉冲微分方程边值问题引起了众多学者的关注,该类方程广泛地应用在航天、工程力学、材料以及控制等众多学科领域当中.此外,分数阶微分方程边值问题也是目前数学工作者关注的热点问题之一.本文研究了几类带脉冲项的耦合系统边值问题弱解的存在性,全文一共分为四章,具体内容安排如下:在第一章绪论中,介绍了本课题相关工作的研究背景和研究意义,概括了相关问题的研究现状,最后阐述了本文的研究内容.在第二章中,介绍了山路引理,并且利用山路引理研究了一类整数阶带p-Laplacian算子的脉冲耦合系统的弱解的存在性问题.在第三章中,运用山路引理,研究了一类分数阶脉冲耦合系统弱解的存在性与多解性问题.在第四章中,对本文工作进行了总结,并且对后续问题进行了展望.本文的创新点主要体现在下述两个方面:首先,p-Laplacian算子在p = 2时为二阶线性算子,研究的问题退化为二阶线性耦合系统的问题,因此本文较前人的工作更具一般性,推广了一类微分方程系统边值问题的研究结果.其次,分数阶脉冲微分系统的研究目前相对较少,本文在前人工作基础上,研究了一类分数阶脉冲耦合系统弱解的存在性问题,因此,本文在某种程度上推广和丰富了分数阶微分方程边值问题研究和脉冲耦合系统的已有研究成果。
[Abstract]:Boundary value problems of differential equations is an important branch of mathematical theory and application research. With the deepening of the investigation, the problem has aroused the concern of many scholars of impulsive differential equation boundary value, the equations are widely used in aerospace, mechanical engineering, material control and many other disciplines. In addition, fractional differential boundary value problem is currently one of the hot issues of mathematics educators pay close attention to. The existence of the weak solutions of coupling system is studied in this paper several kinds of impulses of the boundary value, this paper is divided into four chapters, the main contents are as follows: in the first chapter, introduces the related work of the research background and research significance and summarizes the current research status of related problems, finally elaborated the research content of this paper. In the second chapter, introduces the mountain pass lemma, and the mountain pass lemma is studied for a class of integer order with p-Laplac The existence of weak solutions of pulse coupling system of the Ian operator. In the third chapter, using the mountain pass lemma is studied for a class of fractional impulsive coupling system of the existence of weak solutions and multiple solutions. In the fourth chapter, the work of this thesis is summarized and the follow-up issues are discussed in this paper. The innovation is mainly reflected in the following two aspects: first, p-Laplacian operator at P = 2 for two order linear operator, the research problem is reduced to two order linear coupling system, this paper compared with the previous work is more general, the promotion of a system of differential equations with boundary value problems of the research results. Secondly, study the fractional order impulsive differential system is relatively small, on the basis of previous work, study a class of fractional impulsive coupling system existence problem, weak solution so this to some extent promote and enrich the fractional order The research on boundary value problems of differential equations and the existing research results of the pulse coupling system.

【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8

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