非线性微分方程边值问题的正解及其应用
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摘要
非线性泛函分析是现代数学中一个既有深刻理论意义,又有广泛应用价值的研究方向,它以数学和自然科学各个领域中出现的非线性问题为背景,建立处理许多非线性问题的若干一般性理论和方法.它的研究成果可以广泛地应用于各种非线性微分方程、积分方程和其他各种类型的方程以及计算数学、控制理论、最优化理论、动力系统、经济数学等许多领域.目前非线性泛函分析主要内容包括拓扑度理论、临界点理论、半序方法、解析方法和单调型映射理论等.由于非线性问题已经引起国内外数学界和自然科学界的高度重视,对非线性泛函分析及其应用的研究,无疑具有重要的理论意义和应用价值.
非线性微分方程边值问题是微分方程理论中的一个重要课题,由于其重要的理论价值和物理背景,一直被许多研究者所关注,并取得了丰富的研究成果.在泛函分析理论和实际问题的推动下,非线性微分方程边值问题的研究发展非常迅速.特别是近年来随着非线性泛函分析理论的发展和新的非线性问题的出现,非线性微分方程边值问题形成了许多新的研究方向,取得了一系列研究成果,成为一个研究热点.
本文主要利用非线性泛函分析的锥理论、不动点理论、不动点指数理论、Kras-nosel'skii不动点定理、全局连续定理、单调迭代方法和上下解方法等研究了几类非线性微分方程(奇异)边值问题(方程组)正解的存在性、多解性、解对参数的依赖性和解的单调性等情况,这中间包括一些周期边值问题、高阶高奇性问题、非局部问题、半正问题、脉冲边值问题和含p-Laplacian算子的微分系统等.通过深入的研究,我们得到了一些新的深刻而有趣的成果.
全文分为六章.第一章,我们对非线性泛函分析的历史发展和一些基本概念和定理作简要的介绍.第二章我们得到了两类含有参数的二阶周期边值问题正解的存在性、非存在性和多解性,并对解的唯一性和解对参数的依赖性做了研究.第三章我们讨论了三类奇异高阶非局部边值问题正解的存在性和多解性.§3.2我们建立了一类n-阶m-点奇异边值问题正解的存在性结果,其中非线性项含有未知函数的导数且允许奇异；§3.3我们利用线性算子的第一特征值研究了带有积分边界条件的n-阶奇异边值问题,其中积分边界条件由带有广义测度的Riemann-Stieltjes积分给出；§3.4在Banach空间中得到了奇异n-阶非局部边值问题的多个正解.第四章,我们把注意力放在两类高阶微分方程边值问题单调正解的研究上.§4.1我们得到了半正右聚焦边值问题的单调正解,其非线性项可以下无界；§4.2我们建立了高阶积分边值问题多个单调正解的存在性定理.第五章讨论了一类含有积分边界条件的非线性脉冲微分方程Sturm-Liouville边值问题的正解.第六章,综合利用锥上的不动点指数理论和上下解方法,研究了一类含两个参数的p-Laplacian算子系统奇异边值问题正解的存在性、非存在性和多解性,得到了一条由参数决定的连续曲线,它决定了解的分布情况.
Nonlinear functional analysis is a research field of mathematics which has pro-found theories and extensive applications. It takes the nonlinear problems appearing in mathematics and the natural sciences as background to establish some general the-ories and methods to handle nonlinear problems. Its rich theory and advanced method have provided the effective theory tool for solving many kinds of nonlinear differential equations, nonlinear integral equations and some other types of equations, and han-dling many nonlinear problems in computational mathematics, cybernetics, optimized theory, dynamic system, economical mathematics and so on. At present, the contents of nonlinear functional analysis mainly have topology degree theory, critical point the-ory, partial order method, analysis method, monotone mapping theory and so on. In recent years nonlinear problems have received highly attention of the domestic and foreign mathematics and natural science field, so the research on nonlinear functional analysis and its applications is very important in both theory and applications.
The boundary value problems of nonlinear differential equations are important subjects in the theory of differential equations. Owing to the important in both theory and in applications, boundary value problems for ordinary diffrential equations have been attracted many researchers, and a large number of results have been obtained. Under the impetus of functional analysis and practical problems, the development of the research on boundary value problems for nonlinear differential equations is rapid.
The present paper employs the nonlinear functional analysis theory and method, such as cone theory, fixed point theory, fixed point index theory, Krasnosel'skii fixed point theorem, Global Continuation Theorem, monotone iterative technique and the method of lower and upper solutions, to investigate the existence, multiplicity, depen-denceon a parameter and monotony for positive solutions to several kinds of (singular) boundary value problems of nonlinear differential equations (system), including some periodic boundary value problems, high singularity of higher order differential equa-tions, nonlocal problems, semipositone problems, impulsive boundary value problems and singular boundary value problems with p-Laplacian operator systems. By deep study, we obtain some new interesting results.
The thesis is divided into six chapters. In Chapter I, we mainly introduce the background of nonlinear functional analysis and some basic concepts and theorems. In Chapter II, the existence, multiplicity and nonexistence results for positive solutions are derived to second order periodic boundary value problems, and the uniqueness of solutions and the dependence of solutions on the parameter are also studied. In Chapter III, we discuss three kinds of singular higher order nonlocal boundary value problems. In§3.2, the existence of positive solutions is established for nth-order m-point singular boundary value problem depends on higher derivatives of unknown function. In§3.3, using the first eigenvalue corresponding to the relevant linear operator, we study the nonlinear nth-order singular boundary value problem with nonlocal condition which is given by Riemann-Stieltjes integral with a signed measure.§3.4 deals with multiple positive solutions to nth-order singular nonlocal boundary value problem in Banach space. Chapter IV focuses on the study of monotone positive solution for higher order boundary value problems.§4.1 establishes the existence of monotone positive solution for semipositone right focal boundary value problem with a sign-changing nonlinear term which may be unbounded from below. In§4.2, the existence of multiple monotone positive solutions for higher order integral boundary value problems is established. Chapter V deals with the positive solutions to a class of nonlinear impulsive Sturm-Liouville problem with integral boundary conditions. In Chapter VI, the existence, nonexistence and multiplicity of positive solutions for singular boundary value problems with p-Laplacian operator systems involving two parameters are investigated. By the fixed point index theory and the upper-lower solutions method, a continuous curve which resolves the distribution of positive solutions is derived.
非线性微分方程边值问题是微分方程理论中的一个重要课题,由于其重要的理论价值和物理背景,一直被许多研究者所关注,并取得了丰富的研究成果.在泛函分析理论和实际问题的推动下,非线性微分方程边值问题的研究发展非常迅速.特别是近年来随着非线性泛函分析理论的发展和新的非线性问题的出现,非线性微分方程边值问题形成了许多新的研究方向,取得了一系列研究成果,成为一个研究热点.
本文主要利用非线性泛函分析的锥理论、不动点理论、不动点指数理论、Kras-nosel'skii不动点定理、全局连续定理、单调迭代方法和上下解方法等研究了几类非线性微分方程(奇异)边值问题(方程组)正解的存在性、多解性、解对参数的依赖性和解的单调性等情况,这中间包括一些周期边值问题、高阶高奇性问题、非局部问题、半正问题、脉冲边值问题和含p-Laplacian算子的微分系统等.通过深入的研究,我们得到了一些新的深刻而有趣的成果.
全文分为六章.第一章,我们对非线性泛函分析的历史发展和一些基本概念和定理作简要的介绍.第二章我们得到了两类含有参数的二阶周期边值问题正解的存在性、非存在性和多解性,并对解的唯一性和解对参数的依赖性做了研究.第三章我们讨论了三类奇异高阶非局部边值问题正解的存在性和多解性.§3.2我们建立了一类n-阶m-点奇异边值问题正解的存在性结果,其中非线性项含有未知函数的导数且允许奇异；§3.3我们利用线性算子的第一特征值研究了带有积分边界条件的n-阶奇异边值问题,其中积分边界条件由带有广义测度的Riemann-Stieltjes积分给出；§3.4在Banach空间中得到了奇异n-阶非局部边值问题的多个正解.第四章,我们把注意力放在两类高阶微分方程边值问题单调正解的研究上.§4.1我们得到了半正右聚焦边值问题的单调正解,其非线性项可以下无界；§4.2我们建立了高阶积分边值问题多个单调正解的存在性定理.第五章讨论了一类含有积分边界条件的非线性脉冲微分方程Sturm-Liouville边值问题的正解.第六章,综合利用锥上的不动点指数理论和上下解方法,研究了一类含两个参数的p-Laplacian算子系统奇异边值问题正解的存在性、非存在性和多解性,得到了一条由参数决定的连续曲线,它决定了解的分布情况.
Nonlinear functional analysis is a research field of mathematics which has pro-found theories and extensive applications. It takes the nonlinear problems appearing in mathematics and the natural sciences as background to establish some general the-ories and methods to handle nonlinear problems. Its rich theory and advanced method have provided the effective theory tool for solving many kinds of nonlinear differential equations, nonlinear integral equations and some other types of equations, and han-dling many nonlinear problems in computational mathematics, cybernetics, optimized theory, dynamic system, economical mathematics and so on. At present, the contents of nonlinear functional analysis mainly have topology degree theory, critical point the-ory, partial order method, analysis method, monotone mapping theory and so on. In recent years nonlinear problems have received highly attention of the domestic and foreign mathematics and natural science field, so the research on nonlinear functional analysis and its applications is very important in both theory and applications.
The boundary value problems of nonlinear differential equations are important subjects in the theory of differential equations. Owing to the important in both theory and in applications, boundary value problems for ordinary diffrential equations have been attracted many researchers, and a large number of results have been obtained. Under the impetus of functional analysis and practical problems, the development of the research on boundary value problems for nonlinear differential equations is rapid.
The present paper employs the nonlinear functional analysis theory and method, such as cone theory, fixed point theory, fixed point index theory, Krasnosel'skii fixed point theorem, Global Continuation Theorem, monotone iterative technique and the method of lower and upper solutions, to investigate the existence, multiplicity, depen-denceon a parameter and monotony for positive solutions to several kinds of (singular) boundary value problems of nonlinear differential equations (system), including some periodic boundary value problems, high singularity of higher order differential equa-tions, nonlocal problems, semipositone problems, impulsive boundary value problems and singular boundary value problems with p-Laplacian operator systems. By deep study, we obtain some new interesting results.
The thesis is divided into six chapters. In Chapter I, we mainly introduce the background of nonlinear functional analysis and some basic concepts and theorems. In Chapter II, the existence, multiplicity and nonexistence results for positive solutions are derived to second order periodic boundary value problems, and the uniqueness of solutions and the dependence of solutions on the parameter are also studied. In Chapter III, we discuss three kinds of singular higher order nonlocal boundary value problems. In§3.2, the existence of positive solutions is established for nth-order m-point singular boundary value problem depends on higher derivatives of unknown function. In§3.3, using the first eigenvalue corresponding to the relevant linear operator, we study the nonlinear nth-order singular boundary value problem with nonlocal condition which is given by Riemann-Stieltjes integral with a signed measure.§3.4 deals with multiple positive solutions to nth-order singular nonlocal boundary value problem in Banach space. Chapter IV focuses on the study of monotone positive solution for higher order boundary value problems.§4.1 establishes the existence of monotone positive solution for semipositone right focal boundary value problem with a sign-changing nonlinear term which may be unbounded from below. In§4.2, the existence of multiple monotone positive solutions for higher order integral boundary value problems is established. Chapter V deals with the positive solutions to a class of nonlinear impulsive Sturm-Liouville problem with integral boundary conditions. In Chapter VI, the existence, nonexistence and multiplicity of positive solutions for singular boundary value problems with p-Laplacian operator systems involving two parameters are investigated. By the fixed point index theory and the upper-lower solutions method, a continuous curve which resolves the distribution of positive solutions is derived.
引文
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