本文選題：時標(biāo)　切入點：p-Laplacian算子　出處：《延邊大學(xué)》2016年碩士論文

【摘要】：1988年Stefan-Hilger提出了時標(biāo)上的動力方程理論。在時標(biāo)理論沒有出現(xiàn)之前,對于一些連續(xù)變化的現(xiàn)象或者連續(xù)的變化過程可以用微分方程去刻畫；對于某些離散的現(xiàn)象或者變化過程,則用差分方程去刻畫。但對于一些既包括連續(xù)狀態(tài)又包括離散狀態(tài)的數(shù)學(xué)模型卻無從下手,時標(biāo)理論提供給我們一種新的方法,去研究實際生活中許多沒有規(guī)律的現(xiàn)象,所以時標(biāo)理論的出現(xiàn),引起了眾多學(xué)者的關(guān)注和研究。時標(biāo)上的相關(guān)理論發(fā)展速度很快,并且在不斷的趨于成熟。時標(biāo)上邊值問題解的存在性就是眾多學(xué)者關(guān)注的問題之一。眾多學(xué)者的證明方法也多種多樣,例如：錐上不動點定理、單調(diào)迭代方法、上下解方法、Leggett-Williams不動點定理,以及Leray-Schauder非線性選擇定理等。并且已經(jīng)得出了很多有價值的結(jié)論。通過閱讀大量的參考文獻(xiàn)可以發(fā)現(xiàn),到目前為止大部分的作者所研究的都是一階動力學(xué)方程或者二階動力學(xué)方程的解的存在性問題,而對于時標(biāo)上的三階多點邊值問題解的存在性的研究卻不多,自然對帶有P-Laplacian算子的三階多點邊值問題解的存在性進(jìn)行研究的文章就更少了。本文主要是利用前人用過的定理去研究了時標(biāo)上兩類三階的帶有p-Laplacian算子的多點邊值問題正解的存在性。對三階問題的研究表面上增加了求方程解難度,實際上對時標(biāo)上高階邊值問題的研究有助于我們更好地去解決生活中許多問題,也變得更加有實用價值。對于第一個三階邊值問題,我利用兩種辦法證明了其存在多重正解的充分條件。第一個方法是Leggett-Williams不動點定理；第二個方法是一錐上不動點定理。對于第二個三階多點邊值問題,我利用Leray-Schauder非線性選擇定理,得出了其至少存在一個正解的充分條件。全文共分為五章,第一章是引言,簡單的敘述了時標(biāo)理論的歷史背景、研究價值和我自己所做的一些主要工作；第二章是預(yù)備知識,該部分詳細(xì)列出了論文證明過程中涉及的所有時標(biāo)理論上的定義和引理；第三章主要研究了如下三階的帶有p-Laplacian算子的多點邊值問題首先,利用時標(biāo)上的相關(guān)理論及性質(zhì)解出該邊值問題的解的表達(dá)式；其次,要對這個方程建立一個合適的Banach空間和適當(dāng)?shù)腻F,在錐上定義一個算子Q,然后利用Leggett-Williams不動點定理和一個錐上的不動點定理分別給出方程存在正解的充分條件；最后,我給出兩個實際例子說明該部分的成果。第四章主要研究了帶有p-Laplacian算子的邊值問題首先,利用時標(biāo)上相關(guān)理論及其性質(zhì)解出該問題的解的表達(dá)式；其次,要對該方程建立一個合適的Banach空間和適當(dāng)?shù)腻F,在錐上定義一個算子R；然后利用Leray-schauder非線性選擇定理,得出了該邊值問題至少存在一個正解的充分條件。最后一個章節(jié)是結(jié)束語以及參考文獻(xiàn)。
[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.
【學(xué)位授予單位】：延邊大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：O175.8

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