非线性奇异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.
非线性整数阶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实际背景,一直被许多研究者所关注,并取得丰富的研究成果.近几年,分数阶微分方程在扩散和运输理论、混沌与湍流、粘弹性力学及非牛顿流体力学等诸多领域得以广泛应用.已经引起国内外数学及自然科学界的高度重视,取得了一系列研究成果,成为国际热点研究方向之一
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、Krasnosel-skii不动点定理、单调迭代方法等研究了整数阶和分数阶上几类微分方程奇异和半正边值问题(组)解得存在性、多解性等情况,同时我们研究了非线性二阶脉冲微分方程正解的存在性和多解性.通过深入的研究,在较弱的条件下获得了一些新的深刻有趣的结果.这些结果大都已经发表在国外重要的学术期刊上,如荷兰的《Commun. Non-linear Sci. Numer. Simul.》(SCI)、美国的《Abstr. Appl. Anal.》(SCI)、《Discrete Dyn. Nat. Soc.》(SCI)、《Adv. Difference Equ.》(SCI)、马来西亚的《Bull. Malays. Math. Sci. Soc.》(SCI)和匈牙利的《Electron. J. Qual. Theory Differ. Equ.》等.
本文共分六章.第一章简要介绍了非线性泛函分析的历史背景与一些基本概念和定理.第二章研究两类整数阶微分方程组非局部边值问题正解的存在性.§2.2考察了一类带有耦合边值的二阶微分方程组对称正解的存在性、多解性.§2.3我们讨论了带有积分边界条件的p-Laplacian四阶微分系统正解的存在性结果.第三章我们讨论了两类整数阶半正边值问题正解的存在性.§3.2建立一类奇异积分边值问题一个正解和两个正解的存在性.§3.3我们得到了非线性项关于两个变元均奇异的积分边值问题正解的存在性结果.第四章讨论了一类二阶脉冲微分方程积分边值问题一个正解和三个正解的存在性结果.第五章我们研究了两类高阶分数微分方程正解的存在性、多解性.§5.2我们建立了一类带有参数项的m点边值问题正解的存在唯一性定理,并讨论了正解的某些性质.§5.3我们得到了一类分数阶奇异微分方程正解的唯一性、存在性及多解性结果.第六章,我们把注意力放在两类半正分数阶方程组的研究上.§6.2,我们探讨了具有变号非线性项的奇异分数阶积分边值问题正解的存在性.§6.3研究了带有耦合边界条件的奇异分数阶方程组正解的存在性定理.
Nonlinear functional analysis is a research field of mathematics with profoundtheories and extensive applications. It constructs many general theories and methodsto deal with nonlinear problems on the basis of the study of the nonlinear problemswhich appeared in mathematics and the natural sciences. Its rich theories and ad-vanced methods are widely used in studies of solving many kinds of nonlinear difer-ential equations, nonlinear integral equations and some other types of equations, andhandling many nonlinear problems in computational mathematics, cybernetics, opti-mized theory, dynamic system, economical mathematics, etc. At present, nonlinearfunctional analysis mainly covers topology degree theory, critical point theory, partialorder method, analysis method, monotone mapping theory, and so on.
The boundary value problems of nonlinear integer diferential equations are im-portant subjects in the theory of diferential equations. Owing to the importance inboth theory and application, boundary value problems for integer diferential equa-tions have attracted many researchers’ attention, and a large number of results havebeen obtained. In recent years, fractional diferential equations have been widely usedin difusion and transport theory, chaos and turbulence, viscoelastic mechanics, non-newtonian fluid mechanics etc. As one of the hottest issues in the international researchfield, it has received highly attention of the domestic and foreign mathematics and nat-ural science field.
The theory and method of nonlinear functional analysis has been employed inthe present paper, such as cone theory, fixed point theory, fixed point index theory,Krasnoselskii fixed point theorem and monotone iterative technique, to investigate theexistence, multiplicity of positive solutions to several kinds of (singular) boundary valueproblems of nonlinear integer (fractional) diferential equations (system). Besides, theexistence of solutions and multiple solutions for nonlinear second order impulsive difer-ential equation has been investigated. Having studied thoroughly, some new interestingresults under weaker conditions have been obtained, most of which have been publishedin Commun. Nonlinear Sci. Numer. Simul.(SCI), Abstr. Appl. Anal.(SCI),Discrete Dyn. Nat. Soc.(SCI), Adv. Diference Equ.(SCI), Bull. Malays.Math. Sci. Soc.(SCI) and Electron. J. Qual. Theory Difer. Equ.(SCI), ect.
The dissertation is divided into six chapters. In Chapter I, the background ofnonlinear functional analysis and some basic concepts and theorems have been intro-duced. In Chapter II, the existence results for positive solutions are derived to twokinds of nonlocal boundary value problem of integer diferential systems. In§2.2, theexistence and multiplicity of symmetric positive solutions for a class of singular second-order diferential system with coupled boundary condition is established. In§2.3, theexistence results for positive solutions are considered as a class of singular p-Laplacianfourth-order diferential systems with integral boundary conditions. In Chapter III, theexistence of positive solutions for two kinds of semipositone boundary value problemsof singular integer diferential equation has been studied.§3.2, the existence of at leastone or two positive solutions for a class of singular integral boundary value problemhas been discussed. In§3.3, the existence results for positive solutions to a class ofintegral boundary value problem have been obtained, where the nonlinear term areboth singular at two variables. In Chapter IV, the existence of at least one, or threepositive solutions for a class of singular second order impulsive diferential equationwith integral boundary condition has been dealt with. In Chapter V, the existence andmultiplicity results for positive solutions have been studied to two kinds of higher orderfractional diferential equations. In§5.2, we established the uniqueness of positive so-lution and the dependence of solution on the parameter to a kind of m-point boundaryvalue problem, and the properties of positive solution are given. In§5.3, we obtain theuniqueness, the existence and multiplicity of positive solutions for singular fractionaldiferential equation. In Chapter VI, the study of two kinds of semipositone fractionaldiferential systems has been focused on. In§6.2, the existence of positive solutionsfor singular fractional diferential system has been discussed in a sign-changing nonlin-ear term and integral boundary condition. In§6.3, the existence of positive solutionsto a singular fractional diferential system with coupled boundary condition has beeninvestigated.
非线性整数阶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实际背景,一直被许多研究者所关注,并取得丰富的研究成果.近几年,分数阶微分方程在扩散和运输理论、混沌与湍流、粘弹性力学及非牛顿流体力学等诸多领域得以广泛应用.已经引起国内外数学及自然科学界的高度重视,取得了一系列研究成果,成为国际热点研究方向之一
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、Krasnosel-skii不动点定理、单调迭代方法等研究了整数阶和分数阶上几类微分方程奇异和半正边值问题(组)解得存在性、多解性等情况,同时我们研究了非线性二阶脉冲微分方程正解的存在性和多解性.通过深入的研究,在较弱的条件下获得了一些新的深刻有趣的结果.这些结果大都已经发表在国外重要的学术期刊上,如荷兰的《Commun. Non-linear Sci. Numer. Simul.》(SCI)、美国的《Abstr. Appl. Anal.》(SCI)、《Discrete Dyn. Nat. Soc.》(SCI)、《Adv. Difference Equ.》(SCI)、马来西亚的《Bull. Malays. Math. Sci. Soc.》(SCI)和匈牙利的《Electron. J. Qual. Theory Differ. Equ.》等.
本文共分六章.第一章简要介绍了非线性泛函分析的历史背景与一些基本概念和定理.第二章研究两类整数阶微分方程组非局部边值问题正解的存在性.§2.2考察了一类带有耦合边值的二阶微分方程组对称正解的存在性、多解性.§2.3我们讨论了带有积分边界条件的p-Laplacian四阶微分系统正解的存在性结果.第三章我们讨论了两类整数阶半正边值问题正解的存在性.§3.2建立一类奇异积分边值问题一个正解和两个正解的存在性.§3.3我们得到了非线性项关于两个变元均奇异的积分边值问题正解的存在性结果.第四章讨论了一类二阶脉冲微分方程积分边值问题一个正解和三个正解的存在性结果.第五章我们研究了两类高阶分数微分方程正解的存在性、多解性.§5.2我们建立了一类带有参数项的m点边值问题正解的存在唯一性定理,并讨论了正解的某些性质.§5.3我们得到了一类分数阶奇异微分方程正解的唯一性、存在性及多解性结果.第六章,我们把注意力放在两类半正分数阶方程组的研究上.§6.2,我们探讨了具有变号非线性项的奇异分数阶积分边值问题正解的存在性.§6.3研究了带有耦合边界条件的奇异分数阶方程组正解的存在性定理.
Nonlinear functional analysis is a research field of mathematics with profoundtheories and extensive applications. It constructs many general theories and methodsto deal with nonlinear problems on the basis of the study of the nonlinear problemswhich appeared in mathematics and the natural sciences. Its rich theories and ad-vanced methods are widely used in studies of solving many kinds of nonlinear difer-ential equations, nonlinear integral equations and some other types of equations, andhandling many nonlinear problems in computational mathematics, cybernetics, opti-mized theory, dynamic system, economical mathematics, etc. At present, nonlinearfunctional analysis mainly covers topology degree theory, critical point theory, partialorder method, analysis method, monotone mapping theory, and so on.
The boundary value problems of nonlinear integer diferential equations are im-portant subjects in the theory of diferential equations. Owing to the importance inboth theory and application, boundary value problems for integer diferential equa-tions have attracted many researchers’ attention, and a large number of results havebeen obtained. In recent years, fractional diferential equations have been widely usedin difusion and transport theory, chaos and turbulence, viscoelastic mechanics, non-newtonian fluid mechanics etc. As one of the hottest issues in the international researchfield, it has received highly attention of the domestic and foreign mathematics and nat-ural science field.
The theory and method of nonlinear functional analysis has been employed inthe present paper, such as cone theory, fixed point theory, fixed point index theory,Krasnoselskii fixed point theorem and monotone iterative technique, to investigate theexistence, multiplicity of positive solutions to several kinds of (singular) boundary valueproblems of nonlinear integer (fractional) diferential equations (system). Besides, theexistence of solutions and multiple solutions for nonlinear second order impulsive difer-ential equation has been investigated. Having studied thoroughly, some new interestingresults under weaker conditions have been obtained, most of which have been publishedin Commun. Nonlinear Sci. Numer. Simul.(SCI), Abstr. Appl. Anal.(SCI),Discrete Dyn. Nat. Soc.(SCI), Adv. Diference Equ.(SCI), Bull. Malays.Math. Sci. Soc.(SCI) and Electron. J. Qual. Theory Difer. Equ.(SCI), ect.
The dissertation is divided into six chapters. In Chapter I, the background ofnonlinear functional analysis and some basic concepts and theorems have been intro-duced. In Chapter II, the existence results for positive solutions are derived to twokinds of nonlocal boundary value problem of integer diferential systems. In§2.2, theexistence and multiplicity of symmetric positive solutions for a class of singular second-order diferential system with coupled boundary condition is established. In§2.3, theexistence results for positive solutions are considered as a class of singular p-Laplacianfourth-order diferential systems with integral boundary conditions. In Chapter III, theexistence of positive solutions for two kinds of semipositone boundary value problemsof singular integer diferential equation has been studied.§3.2, the existence of at leastone or two positive solutions for a class of singular integral boundary value problemhas been discussed. In§3.3, the existence results for positive solutions to a class ofintegral boundary value problem have been obtained, where the nonlinear term areboth singular at two variables. In Chapter IV, the existence of at least one, or threepositive solutions for a class of singular second order impulsive diferential equationwith integral boundary condition has been dealt with. In Chapter V, the existence andmultiplicity results for positive solutions have been studied to two kinds of higher orderfractional diferential equations. In§5.2, we established the uniqueness of positive so-lution and the dependence of solution on the parameter to a kind of m-point boundaryvalue problem, and the properties of positive solution are given. In§5.3, we obtain theuniqueness, the existence and multiplicity of positive solutions for singular fractionaldiferential equation. In Chapter VI, the study of two kinds of semipositone fractionaldiferential systems has been focused on. In§6.2, the existence of positive solutionsfor singular fractional diferential system has been discussed in a sign-changing nonlin-ear term and integral boundary condition. In§6.3, the existence of positive solutionsto a singular fractional diferential system with coupled boundary condition has beeninvestigated.
引文
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