分数阶微分包含及变分不等式解的存在性

发布时间：2018-03-22 01:39

本文选题：微分包含　切入点：分数阶导数　出处：《广西民族大学》2017年硕士论文　论文类型：学位论文

【摘要】：分数阶微分包含来源于经济,最优控制,随机分析等某些问题的数学建模中,在工程,物理和经济学的各个领域的许多现象中得到广泛的应用.而变分不等式作为非线性规划问题中重要的分支,在弹性学,结构分析,经济学,数学规划,工程力学等方面都具有重要的应用.近年来,许多学者对分数阶微分包含和变分不等式进行了深入的研究,获得了显著的发展.本篇论文将应用分数阶微积分,集值映射和不动点定理知识研究高阶分数阶微分包含解的存在性,分数阶变分不等式解的存在性.全文共分为七章.第一章,简要地介绍了问题的研究背景,国内外研究发展现状和未来发展趋势,以及本文研究的主要内容.第二章,介绍了函数空间,分数阶微积分,集值映射,不动点定理及微分包含和变分不等式的一些基本性质,以及本文用到的相关引理等预备知识.第三章,通过运用积导合成和Banach压缩映射原理,研究了一类高阶脉冲分数阶微分方程解的存在性问题.我们得到了两个结果.第一个结果在合适的假设条件下,利用数学归纳法得到了高阶分数阶微分方程解的存在性;第二个结果利用了Banach空间中的压缩映射原理得到了高阶脉冲分数阶微分方程解的存在唯一性问题.第四章,通过给出适当的假设条件研究了一类带有积分边值条件的Caputo型高阶分数阶微分包含方程解的存在性.讨论了当集值映射是凸的情况下,利用集值映射的不动点定理得到了解的存在性.第五章,研究了一类高阶脉冲微分包含解的存在性,在第四章的基础上,我们将高阶分数阶微分包含推广到高阶分数阶脉冲微分包含,在给出适当的假设条件下,结合脉冲的相关性质,利用集值映射的不动点定理得到其解的存在性问题.第六章,研究了有限维空间中一类分数阶脉冲变分不等式解的存在性.在前人研究的基础上,给出适当的假设条件,通过验证条件的等价性,在集值映射存在可测选择的基础上,利用非线性选择不动点定理和Filippov引理得到了其解的存在性.第七章,总结目前的研究工作,并提出未来的研究设想.
[Abstract]:Fractional differential inclusions are derived from the mathematical modeling of some problems, such as economy, optimal control, stochastic analysis, etc., in engineering, Variational inequalities, as important branches of nonlinear programming problems, are widely used in many fields of physics and economics, such as elasticity, structural analysis, economics, mathematical programming, Engineering mechanics has important applications. In recent years, many scholars have made great progress in the study of fractional differential inclusions and variational inequalities. The existence of Solutions of higher order Fractional differential inclusions and the existence of Solutions of Fractional variational inequalities are studied in the knowledge of set-valued mappings and fixed point theorems. In chapter 2, some basic properties of function space, fractional calculus, set-valued mapping, fixed point theorem, differential inclusions and variational inequalities are introduced. In the third chapter, by using the principle of product derivative synthesis and Banach contraction mapping, In this paper, we study the existence of solutions for a class of fractional differential equations with higher order impulses. We obtain two results. In the first case, we obtain the existence of solutions of higher order fractional differential equations by mathematical induction under suitable assumptions. In the second result, the existence and uniqueness of solutions for higher order impulsive fractional differential equations are obtained by using the contraction mapping principle in Banach spaces. In this paper, the existence of solutions for a class of Caputo type fractional differential inclusions with integral boundary conditions is studied by giving appropriate assumptions. When set-valued mappings are convex, the existence of solutions is discussed. The existence of solutions is obtained by using fixed point theorem of set-valued mappings. In chapter 5, the existence of solutions for a class of higher order impulsive differential inclusions is studied. We generalize higher-order fractional differential inclusions to higher-order fractional impulsive differential inclusions. Under appropriate assumptions, combining the properties of impulses, we obtain the existence of solutions by using fixed point theorems of set-valued mappings. The existence of solutions for a class of fractional order impulsive variational inequalities in finite dimensional space is studied. On the basis of previous studies, appropriate assumptions are given. By verifying the equivalence of the conditions, the existence of measurable options for set-valued mappings is obtained. The existence of the solution is obtained by using the fixed point theorem of nonlinear selection and Filippov Lemma. Chapter 7 summarizes the current research work and puts forward the future research ideas.
【学位授予单位】：广西民族大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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