分?jǐn)?shù)階微分包含邊值問題解的存在性
發(fā)布時間：2018-09-09 09:39
【摘要】：作為非線性分析理論的重要內(nèi)容之一,微分包含與許多數(shù)學(xué)分支,例如最優(yōu)控制、最優(yōu)化理論等都有著密不可分的關(guān)系。分?jǐn)?shù)階微分包含是整數(shù)階微分包含的推廣,也是分?jǐn)?shù)階微分方程的推廣,它不僅具有整數(shù)階微分包含的不確定性,同時分?jǐn)?shù)階微分方程可以看成分?jǐn)?shù)階微分包含的某種特殊情況。分?jǐn)?shù)階微分包含基于對系統(tǒng)過程有一定了解但不完全確定而建立起來的動力系統(tǒng),用于揭示不確定動力系統(tǒng)以及不連續(xù)動力系統(tǒng)未來規(guī)律的工具,因而具有更豐富的理論研究意義和應(yīng)用價值。近年來,隨著分?jǐn)?shù)階微分方程理論的發(fā)展及其在各個領(lǐng)域的廣泛應(yīng)用,越來越多的學(xué)者致力于研究分?jǐn)?shù)階微分包含的相關(guān)內(nèi)容。本文主要研究了不同邊值條件下分?jǐn)?shù)階微分包含解的存在性,其中包括積分邊值條件、Sturm-Liouville邊值條件、可分離與不可分離邊值條件、三點邊值條件以及含有參數(shù)的邊值條件等多種不同類型,同時還研究了q-差分包含、分?jǐn)?shù)階微分包含耦合系統(tǒng)、混雜型分?jǐn)?shù)階微分包含等多種分?jǐn)?shù)階微分包含的形式,涉及解的存在性和可控性,得到了一些富有創(chuàng)新性的結(jié)果。第一章主要介紹了分?jǐn)?shù)階微分包含的研究背景、發(fā)展現(xiàn)狀以及在理論與實際中的應(yīng)用,給出了分?jǐn)?shù)階微積分和集值映射的基本定義、相關(guān)引理和本文所運用的主要方法,最后簡單介紹本文的主要研究內(nèi)容。第二章研究了含參數(shù)的帶有Sturm-Liouville邊值條件和積分邊值條件的分?jǐn)?shù)階微分包含。本章主要是通過構(gòu)造適當(dāng)?shù)腂anach空間,利用Leary-Schauder型非線性抉擇不動點定理及其相應(yīng)推論、對于上半連續(xù)的集值映射的錐拉伸壓縮不動點定理以及對于具有可分解值的下半連續(xù)的集值映射的壓縮原理,根據(jù)參數(shù)不同的取值范圍,得出了幾個新的解的存在性和可控性的結(jié)果。第三章研究了兩類分?jǐn)?shù)階q-差分包含邊值問題。第一節(jié)利用分?jǐn)?shù)階q-微積分和集值映射的基本概念和理論,以及壓縮型非線性抉擇定理,得出了邊值問題解的存在性;第二節(jié)通過標(biāo)準(zhǔn)的不動點定理,給出了具有可分離邊值條件和不可分離邊值條件的分?jǐn)?shù)階q-差分包含解的存在性。第四章研究了一類帶有耦合邊值條件的混雜型分?jǐn)?shù)階微分方程和微分包含耦合系統(tǒng)解的存在性問題。本章主要利用Leary-Schauder非線性抉擇定理得出了混雜型分?jǐn)?shù)階微分方程耦合系統(tǒng)解的存在性;通過定義截斷算子,利用Bohnenblust-Karlin不動點定理,給出了混雜型分?jǐn)?shù)階微分包含耦合系統(tǒng)解存在的充分條件,同時給出了混雜型分?jǐn)?shù)階微分包含耦合系統(tǒng)解與上下解的關(guān)系。第五章研究了分?jǐn)?shù)階微分包含邊值問題在物理和生物系統(tǒng)中的實際應(yīng)用。第一節(jié)研究的是一類Langevin分?jǐn)?shù)階微分包含三點邊值問題,通過集值映射的可溶性不動點定理,得出了問題解的存在性;第二節(jié)研究了一類生物分室模型系統(tǒng),根據(jù)Leary-Schauder非線性抉擇定理以及Leary-Schauder度理論,給出了系統(tǒng)解的存在性結(jié)果;第三節(jié)研究了一類時間分?jǐn)?shù)階導(dǎo)數(shù)微分包含邊值問題,根據(jù)端點理論以及非線性抉擇定理,給出了問題解的存在性。第六章全文的總結(jié)與展望。本章將總結(jié)全文的主要工作和創(chuàng)新點,同時對該領(lǐng)域未來的發(fā)展進(jìn)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.
【學(xué)位授予單位】：濟(jì)南大學(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：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.
【學(xué)位授予單位】：濟(jì)南大學(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 文浩;金棟平;胡海巖;;基于微分包含的繩系衛(wèi)星時間最優(yōu)釋放控制[J];力學(xué)學(xué)報;2008年01期
2 史樹中;微分包含與經(jīng)濟(jì)均衡[J];高校應(yīng)用數(shù)學(xué)學(xué)報A輯(中文版);1986年01期
，本文編號：2232008
本文鏈接：http://sikaile.net/kejilunwen/yysx/2232008.html
最近更新
教材專著
