Banach空間中四類脈沖微分方程多點(diǎn)邊值問題的正解
發(fā)布時間：2018-09-03 13:39
【摘要】：微分方程邊值問題是微分方程理論中常見的一種基本問題,脈沖微分方程邊值問題又是微分方程邊值問題的一個重要分支,具有很高的應(yīng)用價值,脈沖微分方程是研究一個過程突然發(fā)生變化的基本工具,能夠充分體現(xiàn)瞬時突變現(xiàn)象對系統(tǒng)的影響,能更加真實(shí)的地描述自然界狀態(tài),脈沖系統(tǒng)在現(xiàn)代科學(xué)領(lǐng)域中是廣泛存在的,它的理論在經(jīng)濟(jì)學(xué)、社會科學(xué)、生物學(xué)、物理學(xué)、工程學(xué)等有著廣泛的運(yùn)用,因此,對脈沖微分方程的研究早己引起了國內(nèi)外同行的廣泛關(guān)注.本學(xué)位論文討論了四類脈沖微分方程多點(diǎn)邊值問題正解的存在性,利用錐拉伸錐壓縮不動點(diǎn)定理和Leggett-Williams不動點(diǎn)定理得出了四類脈沖微分方程多點(diǎn)邊值問題正解存在性的充分條件,全文具體內(nèi)容如下.第一章為緒論,主要介紹了脈沖微分方程邊值問題研宄的相關(guān)背景和基本情況,以及給出了文中用到的定義和定理.第二章考慮了非線性項(xiàng)帶有一階導(dǎo)數(shù)的二階脈沖積分微分方程m點(diǎn)邊值問題(?)運(yùn)用錐上的不動點(diǎn)定理得出該邊值問題至少存在一個正解.第三章考慮帶p-Laplacian算子的二階脈沖微分方程m點(diǎn)邊值問題(?)同樣是運(yùn)用錐拉伸錐壓縮不動點(diǎn)定理得出該問題至少存在一個正解.第四章考慮帶p-Laplacian算子積分邊界條件下的四階脈沖微分方程多點(diǎn)邊值問題(?)采用將四階微分方程邊值問題轉(zhuǎn)化為與之等價的兩個二階微分方程邊值問題的方法,然后運(yùn)用錐拉伸錐壓縮不動點(diǎn)定理得出該問題至少存在一個正解的兩個充分條件,并在文章最后給出了應(yīng)用舉例.第五章利用Leggett-Williams不動點(diǎn)定理,考慮了一類帶p-Laplacian算子積分邊界條件下的四階脈沖微分方程邊值問題(?)同樣是采用將四階邊值問題轉(zhuǎn)化為等價的兩個二階邊值問題的方法,然后運(yùn)用Leggett-Williams不動點(diǎn)定理得出該邊值問題至少存在三個正解的充分條件.第六章為本論文的結(jié)束語,總結(jié)了本文的主要工作,并對進(jìn)一步可以研究的問題做了設(shè)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.
【學(xué)位授予單位】：廣西師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.8
本文編號：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.
【學(xué)位授予單位】：廣西師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.8
【參考文獻(xiàn)】
相關(guān)期刊論文 前2條
1 李寶麟;樊瑞寧;;Banach空間二階脈沖積分-微分方程三點(diǎn)邊值問題[J];甘肅科學(xué)學(xué)報(bào);2009年04期
2 張學(xué)梅;葛渭高;;一類帶p-Laplace算子的奇異脈沖特征值問題[J];北京理工大學(xué)學(xué)報(bào);2008年12期
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