两类非线性脉冲微分方程边值问题的正解
发布时间：2018-08-16 11:36
【摘要】：二十世纪五十年代,非线性泛函分析已初步形成完整的理论体系,作为其重要组成部分,非线性微分积分问题受到了国内外数学界乃至整个自然科学界的高度重视,因为它的"能良好解释众多自然现象的功效性".从发展上看,非线性微分-积分问题来源于应用数学和物理学的多个方面,在应用数学物理学和工程学等应用学科上有着极为重要的应用价值,研究此类问题的意义就在于此.随着物理学、航空航天技术、生物技术等分支领域中实际问题的不断出现,非线性泛函分析已成为解决这些非线性问题的重要理论工具,其中非线性微分积分方程解的存在性与多重性已成为重要的研究课题之一,它能清晰地刻画在物理、化学、经济等应用学科中出现的各种非线性问题.近些年来,由于边值问题在化学工程,热弹力学,人口动态,热传导,等离子物理中的各种应用,带有积分边界条件的边值问题更是得到广泛的关注.微分方程边值问题的存在性研究一般是将微分方程结合边界条件,转化为积分方程,寻求积分方程的不动点.本文研究了两类Banach空间上的带积分边值条件的非线性微分方程,丰富了微分方程边值问题正解的研究理论.本文的结构安排如下:第一章,主要介绍了积分边值微分方程的研究背景和研究现状,并且给出了一些关于积分边值问题的基本定义、基本性质以及一些重要的不动点定理.第二章,研究了如下具有单调同态和积分边值条件的三阶非线性脉冲微分-积分方程在实Banach空间中正解的存在性.利用不动点指数定理和Guo-Krasnosel'skill不动点定理给出了一些积分边值问题正解的存在性的一些充分条件,并用实例验证了相应的结果的合理性.第三章,通过构造格林函数,研究了如下p-Laplace微积分方程的积分边值问题在实Banach空间中正解的存在性：其中，，φ_p是p-Laplace算子，即φ_p(u)=|u|~(p-2)u(p1)，φ_p~(-1)(u)= φ_q(u)(1/p+1/q = 1).利用不动点指数定理和Guo-Krasnosel'skill不动点定理给出了一些积分边值问题正解的存在性的一些充分条件，并用实例验证了相应的结果的合理性。
[Abstract]:In the 1950s, nonlinear functional analysis had formed a complete theoretical system. As an important part of it, nonlinear differential and integral problems were highly valued by the mathematics field at home and abroad and even the whole natural science field. Because of its "good explanation of the efficacy of many natural phenomena." From the point of view of development, nonlinear differential-integral problems come from many aspects of applied mathematics and physics, and have extremely important application value in applied subjects such as applied mathematics, physics and engineering, so the significance of studying such problems lies in this. With the emergence of practical problems in the fields of physics, aerospace technology and biotechnology, nonlinear functional analysis has become an important theoretical tool to solve these nonlinear problems. Among them, the existence and multiplicity of solutions of nonlinear differential integral equations have become one of the important research topics, which can clearly describe all kinds of nonlinear problems in physics, chemistry, economics and other applied disciplines. In recent years, boundary value problems with integral boundary conditions have been paid more and more attention due to their applications in chemical engineering, thermoelastic mechanics, population dynamics, heat conduction and plasma physics. In general, the existence of boundary value problems for differential equations is transformed into integral equations by combining boundary conditions, and the fixed points of integral equations are found. In this paper, we study two kinds of nonlinear differential equations with integral boundary value conditions on Banach spaces, which enrich the research theory of positive solutions for boundary value problems of differential equations. The structure of this paper is arranged as follows: in the first chapter, the research background and present situation of integro-boundary value differential equations are introduced, and some basic definitions, basic properties and some important fixed point theorems are given. In chapter 2, we study the existence of positive solutions for the third order nonlinear impulsive differential-integral equations with monotone homomorphism and integral boundary value conditions in real Banach spaces. By using the fixed point exponent theorem and Guo-Krasnosel'skill fixed point theorem, some sufficient conditions for the existence of positive solutions of some integral boundary value problems are given, and the rationality of the corresponding results is verified by an example. In chapter 3, by constructing Green's function, we study the existence of positive solutions of integral boundary value problems for p-Laplace calculus equations in real Banach spaces: where 蠁 _ p _ p is a p-Laplace operator, that is, 蠁 _ P _ p (u) = u ~ (p-2) u (p _ (1), 蠁 p ~ (-1) (u) = 蠁 Q (u) (1 / p ~ (1 / Q = 1). By using the fixed point exponent theorem and Guo-Krasnosel'skill fixed point theorem, some sufficient conditions for the existence of positive solutions of some integral boundary value problems are given, and the rationality of the corresponding results is verified by an example.
【学位授予单位】：昆明理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
本文编号：2185865
[Abstract]:In the 1950s, nonlinear functional analysis had formed a complete theoretical system. As an important part of it, nonlinear differential and integral problems were highly valued by the mathematics field at home and abroad and even the whole natural science field. Because of its "good explanation of the efficacy of many natural phenomena." From the point of view of development, nonlinear differential-integral problems come from many aspects of applied mathematics and physics, and have extremely important application value in applied subjects such as applied mathematics, physics and engineering, so the significance of studying such problems lies in this. With the emergence of practical problems in the fields of physics, aerospace technology and biotechnology, nonlinear functional analysis has become an important theoretical tool to solve these nonlinear problems. Among them, the existence and multiplicity of solutions of nonlinear differential integral equations have become one of the important research topics, which can clearly describe all kinds of nonlinear problems in physics, chemistry, economics and other applied disciplines. In recent years, boundary value problems with integral boundary conditions have been paid more and more attention due to their applications in chemical engineering, thermoelastic mechanics, population dynamics, heat conduction and plasma physics. In general, the existence of boundary value problems for differential equations is transformed into integral equations by combining boundary conditions, and the fixed points of integral equations are found. In this paper, we study two kinds of nonlinear differential equations with integral boundary value conditions on Banach spaces, which enrich the research theory of positive solutions for boundary value problems of differential equations. The structure of this paper is arranged as follows: in the first chapter, the research background and present situation of integro-boundary value differential equations are introduced, and some basic definitions, basic properties and some important fixed point theorems are given. In chapter 2, we study the existence of positive solutions for the third order nonlinear impulsive differential-integral equations with monotone homomorphism and integral boundary value conditions in real Banach spaces. By using the fixed point exponent theorem and Guo-Krasnosel'skill fixed point theorem, some sufficient conditions for the existence of positive solutions of some integral boundary value problems are given, and the rationality of the corresponding results is verified by an example. In chapter 3, by constructing Green's function, we study the existence of positive solutions of integral boundary value problems for p-Laplace calculus equations in real Banach spaces: where 蠁 _ p _ p is a p-Laplace operator, that is, 蠁 _ P _ p (u) = u ~ (p-2) u (p _ (1), 蠁 p ~ (-1) (u) = 蠁 Q (u) (1 / p ~ (1 / Q = 1). By using the fixed point exponent theorem and Guo-Krasnosel'skill fixed point theorem, some sufficient conditions for the existence of positive solutions of some integral boundary value problems are given, and the rationality of the corresponding results is verified by an example.
【学位授予单位】：昆明理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
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1 徐玉梅;Banach空间中二阶非线性脉冲微分-积分方程初值问题的整体解[J];数学物理学报;2005年01期
相关硕士学位论文 前1条
1 龚平;几类非线性分数阶微分方程边值问题正解的研究[D];昆明理工大学;2015年
本文编号：2185865
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