分数阶微分包含边值问题解的存在性
发布时间：2018-09-09 09:39
【摘要】：作为非线性分析理论的重要内容之一,微分包含与许多数学分支,例如最优控制、最优化理论等都有着密不可分的关系。分数阶微分包含是整数阶微分包含的推广,也是分数阶微分方程的推广,它不仅具有整数阶微分包含的不确定性,同时分数阶微分方程可以看成分数阶微分包含的某种特殊情况。分数阶微分包含基于对系统过程有一定了解但不完全确定而建立起来的动力系统,用于揭示不确定动力系统以及不连续动力系统未来规律的工具,因而具有更丰富的理论研究意义和应用价值。近年来,随着分数阶微分方程理论的发展及其在各个领域的广泛应用,越来越多的学者致力于研究分数阶微分包含的相关内容。本文主要研究了不同边值条件下分数阶微分包含解的存在性,其中包括积分边值条件、Sturm-Liouville边值条件、可分离与不可分离边值条件、三点边值条件以及含有参数的边值条件等多种不同类型,同时还研究了q-差分包含、分数阶微分包含耦合系统、混杂型分数阶微分包含等多种分数阶微分包含的形式,涉及解的存在性和可控性,得到了一些富有创新性的结果。第一章主要介绍了分数阶微分包含的研究背景、发展现状以及在理论与实际中的应用,给出了分数阶微积分和集值映射的基本定义、相关引理和本文所运用的主要方法,最后简单介绍本文的主要研究内容。第二章研究了含参数的带有Sturm-Liouville边值条件和积分边值条件的分数阶微分包含。本章主要是通过构造适当的Banach空间,利用Leary-Schauder型非线性抉择不动点定理及其相应推论、对于上半连续的集值映射的锥拉伸压缩不动点定理以及对于具有可分解值的下半连续的集值映射的压缩原理,根据参数不同的取值范围,得出了几个新的解的存在性和可控性的结果。第三章研究了两类分数阶q-差分包含边值问题。第一节利用分数阶q-微积分和集值映射的基本概念和理论,以及压缩型非线性抉择定理,得出了边值问题解的存在性;第二节通过标准的不动点定理,给出了具有可分离边值条件和不可分离边值条件的分数阶q-差分包含解的存在性。第四章研究了一类带有耦合边值条件的混杂型分数阶微分方程和微分包含耦合系统解的存在性问题。本章主要利用Leary-Schauder非线性抉择定理得出了混杂型分数阶微分方程耦合系统解的存在性;通过定义截断算子,利用Bohnenblust-Karlin不动点定理,给出了混杂型分数阶微分包含耦合系统解存在的充分条件,同时给出了混杂型分数阶微分包含耦合系统解与上下解的关系。第五章研究了分数阶微分包含边值问题在物理和生物系统中的实际应用。第一节研究的是一类Langevin分数阶微分包含三点边值问题,通过集值映射的可溶性不动点定理,得出了问题解的存在性;第二节研究了一类生物分室模型系统,根据Leary-Schauder非线性抉择定理以及Leary-Schauder度理论,给出了系统解的存在性结果;第三节研究了一类时间分数阶导数微分包含边值问题,根据端点理论以及非线性抉择定理,给出了问题解的存在性。第六章全文的总结与展望。本章将总结全文的主要工作和创新点,同时对该领域未来的发展进行展望。
[Abstract]:As an important part of nonlinear analysis theory, differential inclusion is closely related to many mathematical branches, such as optimal control and optimization theory. Fractional differential inclusion is a generalization of integer differential inclusion and fractional differential equation. It has not only the uncertainty of integer differential inclusion, but also the uncertainty of integer differential inclusion. The fractional differential inclusion is a dynamic system based on a certain understanding of the process of the system but not fully deterministic. It is a tool for revealing the future laws of uncertain dynamical systems and discontinuous dynamical systems, and therefore has a richer theory. In recent years, with the development of the theory of fractional differential equations and its wide application in various fields, more and more scholars have devoted themselves to the study of fractional differential inclusions. Sturm-Liouville boundary conditions, separable and non-separable boundary conditions, three-point boundary conditions and parametric boundary conditions, and many different types of fractional differential inclusions, including q-difference inclusions, fractional differential inclusion coupled systems, hybrid fractional differential inclusions and so on, are studied, involving the existence of solutions. In the first chapter, the research background, development status and application in theory and practice of fractional differential inclusion are introduced. The basic definitions of fractional calculus and set-valued mapping, related lemmas and main methods used in this paper are given. Finally, the main contents of this paper are briefly introduced. In Chapter 2, we study fractional differential inclusions with Sturm-Liouville boundary value conditions and integral boundary value conditions with parameters. In this chapter, we use Leary-Schauder type nonlinear alternative fixed point theorem and its corresponding corollaries to cone stretching and compression for semi-continuous set-valued mappings by constructing appropriate Banach spaces. Fixed point theorem and compression principle for lower semi-continuous set-valued mappings with decomposable values are given. Several new results on the existence and controllability of solutions are obtained according to the range of parameters. Chapter 3 studies two kinds of fractional q-difference inclusion boundary value problems. Section 1 uses the basis of fractional q-calculus and set-valued mappings. In the second section, by means of the standard fixed point theorem, the existence of solutions for fractional q-difference inclusions with separable boundary conditions and non-separable boundary conditions is given. In the fourth chapter, a class of hybrid type with coupled boundary conditions is studied. In this chapter, the existence of solutions for coupled systems of fractional differential equations and differential inclusions is obtained by using Leary-Schauder nonlinear choice theorem, and the existence of solutions for hybrid fractional differential equations is given by defining truncation operators and using Bohnenblust-Karlin fixed point theorem. In the fifth chapter, we study the practical applications of fractional differential inclusion boundary value problems in physical and biological systems. In the first section, we study a class of Langevin fractional differential inclusion three-point boundary value problems through set values. In the second section, we study a class of biological compartment model system, and give the existence result of the system solution according to Leary-Schauder nonlinear choice theorem and Leary-Schauder degree theory. In the third section, we study a class of time fractional derivative differential inclusion boundary value problem. Endpoint theory and nonlinear choice theorem give the existence of solutions to the problem. Chapter 6 summarizes and prospects the full text. This chapter will summarize the main work and innovations of the full text, while the future development of the field is prospected.
【学位授予单位】：济南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
本文编号：2232008
[Abstract]:As an important part of nonlinear analysis theory, differential inclusion is closely related to many mathematical branches, such as optimal control and optimization theory. Fractional differential inclusion is a generalization of integer differential inclusion and fractional differential equation. It has not only the uncertainty of integer differential inclusion, but also the uncertainty of integer differential inclusion. The fractional differential inclusion is a dynamic system based on a certain understanding of the process of the system but not fully deterministic. It is a tool for revealing the future laws of uncertain dynamical systems and discontinuous dynamical systems, and therefore has a richer theory. In recent years, with the development of the theory of fractional differential equations and its wide application in various fields, more and more scholars have devoted themselves to the study of fractional differential inclusions. Sturm-Liouville boundary conditions, separable and non-separable boundary conditions, three-point boundary conditions and parametric boundary conditions, and many different types of fractional differential inclusions, including q-difference inclusions, fractional differential inclusion coupled systems, hybrid fractional differential inclusions and so on, are studied, involving the existence of solutions. In the first chapter, the research background, development status and application in theory and practice of fractional differential inclusion are introduced. The basic definitions of fractional calculus and set-valued mapping, related lemmas and main methods used in this paper are given. Finally, the main contents of this paper are briefly introduced. In Chapter 2, we study fractional differential inclusions with Sturm-Liouville boundary value conditions and integral boundary value conditions with parameters. In this chapter, we use Leary-Schauder type nonlinear alternative fixed point theorem and its corresponding corollaries to cone stretching and compression for semi-continuous set-valued mappings by constructing appropriate Banach spaces. Fixed point theorem and compression principle for lower semi-continuous set-valued mappings with decomposable values are given. Several new results on the existence and controllability of solutions are obtained according to the range of parameters. Chapter 3 studies two kinds of fractional q-difference inclusion boundary value problems. Section 1 uses the basis of fractional q-calculus and set-valued mappings. In the second section, by means of the standard fixed point theorem, the existence of solutions for fractional q-difference inclusions with separable boundary conditions and non-separable boundary conditions is given. In the fourth chapter, a class of hybrid type with coupled boundary conditions is studied. In this chapter, the existence of solutions for coupled systems of fractional differential equations and differential inclusions is obtained by using Leary-Schauder nonlinear choice theorem, and the existence of solutions for hybrid fractional differential equations is given by defining truncation operators and using Bohnenblust-Karlin fixed point theorem. In the fifth chapter, we study the practical applications of fractional differential inclusion boundary value problems in physical and biological systems. In the first section, we study a class of Langevin fractional differential inclusion three-point boundary value problems through set values. In the second section, we study a class of biological compartment model system, and give the existence result of the system solution according to Leary-Schauder nonlinear choice theorem and Leary-Schauder degree theory. In the third section, we study a class of time fractional derivative differential inclusion boundary value problem. Endpoint theory and nonlinear choice theorem give the existence of solutions to the problem. Chapter 6 summarizes and prospects the full text. This chapter will summarize the main work and innovations of the full text, while the future development of the field is prospected.
【学位授予单位】：济南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8
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