时标上两类带有p-Laplacian算子的多点边值问题解的存在性

发布时间：2018-03-25 20:25

本文选题：时标　切入点：p-Laplacian算子　出处：《延边大学》2016年硕士论文

【摘要】：1988年Stefan-Hilger提出了时标上的动力方程理论。在时标理论没有出现之前,对于一些连续变化的现象或者连续的变化过程可以用微分方程去刻画；对于某些离散的现象或者变化过程,则用差分方程去刻画。但对于一些既包括连续状态又包括离散状态的数学模型却无从下手,时标理论提供给我们一种新的方法,去研究实际生活中许多没有规律的现象,所以时标理论的出现,引起了众多学者的关注和研究。时标上的相关理论发展速度很快,并且在不断的趋于成熟。时标上边值问题解的存在性就是众多学者关注的问题之一。众多学者的证明方法也多种多样,例如：锥上不动点定理、单调迭代方法、上下解方法、Leggett-Williams不动点定理,以及Leray-Schauder非线性选择定理等。并且已经得出了很多有价值的结论。通过阅读大量的参考文献可以发现,到目前为止大部分的作者所研究的都是一阶动力学方程或者二阶动力学方程的解的存在性问题,而对于时标上的三阶多点边值问题解的存在性的研究却不多,自然对带有P-Laplacian算子的三阶多点边值问题解的存在性进行研究的文章就更少了。本文主要是利用前人用过的定理去研究了时标上两类三阶的带有p-Laplacian算子的多点边值问题正解的存在性。对三阶问题的研究表面上增加了求方程解难度,实际上对时标上高阶边值问题的研究有助于我们更好地去解决生活中许多问题,也变得更加有实用价值。对于第一个三阶边值问题,我利用两种办法证明了其存在多重正解的充分条件。第一个方法是Leggett-Williams不动点定理；第二个方法是一锥上不动点定理。对于第二个三阶多点边值问题,我利用Leray-Schauder非线性选择定理,得出了其至少存在一个正解的充分条件。全文共分为五章,第一章是引言,简单的叙述了时标理论的历史背景、研究价值和我自己所做的一些主要工作；第二章是预备知识,该部分详细列出了论文证明过程中涉及的所有时标理论上的定义和引理；第三章主要研究了如下三阶的带有p-Laplacian算子的多点边值问题首先,利用时标上的相关理论及性质解出该边值问题的解的表达式；其次,要对这个方程建立一个合适的Banach空间和适当的锥,在锥上定义一个算子Q,然后利用Leggett-Williams不动点定理和一个锥上的不动点定理分别给出方程存在正解的充分条件；最后,我给出两个实际例子说明该部分的成果。第四章主要研究了带有p-Laplacian算子的边值问题首先,利用时标上相关理论及其性质解出该问题的解的表达式；其次,要对该方程建立一个合适的Banach空间和适当的锥,在锥上定义一个算子R；然后利用Leray-schauder非线性选择定理,得出了该边值问题至少存在一个正解的充分条件。最后一个章节是结束语以及参考文献。
[Abstract]:In 1988, Stefan-Hilger put forward the theory of dynamic equation on time scale. Before the theory of time scale appeared, some phenomena of continuous change or continuous process of change can be described by differential equation, and for some discrete phenomena or processes of change, But for some mathematical models which include both continuous state and discrete state, time scale theory provides us with a new method to study many irregular phenomena in real life. Therefore, the emergence of time scale theory has attracted the attention and research of many scholars. The related theories on the time scale have developed rapidly. The existence of solutions to the boundary value problems over time scales is one of the problems that many scholars have paid close attention to. There are a variety of methods to prove these problems, such as fixed point theorems on cones, monotone iterative methods, etc. The methods of upper and lower solutions are Leggett-Williams fixed point theorem, Leray-Schauder nonlinear selection theorem, etc. And many valuable conclusions have been drawn. Up to now, most of the authors have studied the existence of solutions for first-order or second-order dynamical equations, but there are few studies on the existence of solutions for third-order multi-point boundary value problems on time scales. Naturally, there are few studies on the existence of solutions for third order multipoint boundary value problems with P-Laplacian operators. In this paper, we mainly use the theorems used by our predecessors to study the multipoint boundary values of two classes of third order p-Laplacian operators on time scales. The existence of positive solution of the problem. The study of the third order problem increases the difficulty of solving the equation on the surface. In fact, the study of higher-order boundary value problems on time scales helps us to solve many problems in life better and become more practical. For the first third-order boundary value problem, I have proved the sufficient conditions for the existence of multiple positive solutions by using two methods. The first method is the Leggett-Williams fixed point theorem; the second method is the fixed point theorem on a cone. For the second third order multipoint boundary value problem, I use the Leray-Schauder nonlinear selection theorem. A sufficient condition for the existence of at least one positive solution is obtained. The paper is divided into five chapters. The first chapter is an introduction, which briefly describes the historical background of the theory of time scale, the research value and some main works I have done; the second chapter is the preparatory knowledge. In this part, the theoretical definitions and Lemma of all the time scales involved in the process of proving are listed in detail. In chapter 3, the following third order multipoint boundary value problems with p-Laplacian operator are studied. The expression of the solution of the boundary value problem is obtained by using the relevant theories and properties on the time scale. Secondly, an appropriate Banach space and a proper cone are established for the equation. In this paper, we define an operator Q on a cone, then by using Leggett-Williams fixed point theorem and fixed point theorem on a cone, we give the sufficient conditions for the existence of positive solutions for the equation. I give two practical examples to illustrate the results of this part. In chapter four, we mainly study the boundary value problem with p-Laplacian operator. First, we solve the solution of the problem by using the theory of correlation on time scale and its properties. In order to establish an appropriate Banach space and a proper cone for the equation, an operator R is defined on the cone, and then the Leray-schauder nonlinear selection theorem is used. A sufficient condition for the existence of at least one positive solution for the boundary value problem is obtained. The last chapter is the conclusion and references.
【学位授予单位】：延边大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175.8

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