非线性分数阶发展方程初边值问题解的存在性研究

发布时间：2018-03-27 18:52

本文选题：分数阶反应扩散方程　切入点：分数阶发展方程　出处：《曲阜师范大学》2017年博士论文

【摘要】：非线性泛函分析是现代数学中一个重要的数学分支,其主要内容包括拓扑度理论、不动点理论、半序方法等.非线性泛函分析为研究具有非线性问题的诸多领域中的数学模型提供了理论基础和先进方法.在Banach空间,非线性泛函分析对非线性发展方程理论的研究具有重要应用,已被广泛应用于物理、化学、金融和最优控制等领域.近年来,非线性发展方程初值、边值问题解的存在性问题受到广大研究者的普遍关注,并取得一系列研究成果.分数阶微积分理论由于成功应用到分形、多孔介质弥散、金融等领域而发展迅速.分数阶微分方程相比整数阶微分方程能够更好的解释反常扩散、粘弹性体中的应力应变等具有记忆和遗传性的过程,这使得分数阶微分方程的研究也受到越来越多的关注.分数阶微分(发展)方程相比整数阶微分(发展)方程的研究要困难,原因在于分数阶微分算子具有奇异和非局部的特点.这也说明研究分数阶发展方程在理论和实际应用方面都具有重要意义.本文主要研究了非线性分数阶发展方程解的存在性问题,利用半群理论(预解算子理论)、非紧性测度、不动点理论等方法取得了一些新的结果.这些结果改进并推广了一些前人的结果.其中部分结果发表在《Appl.Math. Lett.》(SCI)和《Comput.Math. Appl.》(SCI)等国外重要的学术期刊上.本文共分五章.第一章绪论,简要介绍了分数阶微积分的发展历史及其在相关领域的应用,给出Riemann-Liouville分数阶积分算子、Riemann-Liouville分数阶微分算子和Caputo分数阶微分算子的定义,非线性泛函分析的应用领域,以后各章用到的一些定义、性质和引理以及带非瞬时脉冲的发展方程应用领域和研究现状.第二章,利用预解算子理论、非紧性测度、不动点定理和Banach压缩影像原理,我们研究了一类带迟滞的分数阶反应扩散方程初边值问题解的存在性.在t属于有限区间时,分别讨论了预解算子是紧算子和非紧算子情况下方程整体解的存在性.在t属于无穷区间时,讨论了预解算子是紧算子条件下方程局部解和整体解的存在性.我们的结论改进并完善了前人的一些结果.第三章,我们研究了一类半线性分数阶积分微分方程局部解和整体解的存在性,利用非紧性测度和不动点定理给出方程存在解的充分条件.其中本章给出了一种新的研究分数阶发展方程解的存在性的方法.最后,给出一个利用本章主要结果的应用.另外,利用同样的方法我们研究了一类分数阶混合型微分方程解的存在性问题.第四章,我们考虑了一类带非瞬时脉冲和迟滞的分数阶半线性积分微分方程.利用预解算子理论和不动点定理,我们讨论了方程解的存在性,得到一些新的结果.最后给出一个例子来说明本章主要结果的应用.第五章,研究了一类带非瞬时脉冲的分数阶半线性积分微分方程周期边值问题.利用预解算子理论、非紧性测度和不动点定理得到方程解存在的一些新结果.最后给出一个例子来说明本章主要结果的应用.第六章,利用广义Banach压缩影像原理研究了一类带迟滞和瞬时脉冲的分数阶非自治发展方程初值问题解的存在性和唯一性,给出其解的迭代序列和误差估计并讨论了其唯一解是连续依赖于初值的.
[Abstract]:Nonlinear functional analysis is an important branch of mathematics in modern mathematics, the main contents include the topological degree theory, fixed point theory, partial order method. Provide a theoretical basis and methods in many fields to study the mathematical model with nonlinear problems in nonlinear functional analysis. In Banach space, nonlinear functional analysis has important the application of the theory of nonlinear evolution equations, has been widely used in physics, chemistry, finance and optimal control and other fields. In recent years, the initial value of the nonlinear evolution equation, boundary value concern the existence of solutions of problems by the majority of researchers, and achieved a series of research results. The theory of fractional calculus due to the successful application to fractal, porous diffusion, finance and other fields and developed rapidly. Compared to the fractional differential equations of integer order differential equation can be expanded to better explain the anomalous dispersion, viscoelastic The stress and strain of the body has a memory and hereditary process, which makes the research of fractional differential equations has attracted more and more attention. The fractional differential equations (Development) compared to the integer order differential equation (Development) research to be difficult, because the fractional differential operators with singular and non local characteristics it also shows that the research of fractional evolution equations has important significance both in theory and practical application. This paper mainly studies the existence of solutions of nonlinear fractional evolution equations, using semigroup theory (resolvent operator theory), measure of noncompactness, fixed point theory and other methods to achieve some new results. The results improve and generalize some previous results. Some of the results published in the (SCI) and (SCI) and other important academic journals. This paper is divided into five chapters. The first chapter Theory, this paper briefly introduces the development history of fractional calculus and its application in related fields, given Riemann-Liouville fractional integral operator, the definition of Riemann-Liouville fractional differential operator and Caputo fractional differential operator, the application field of nonlinear functional analysis, the chapter used the definition, properties and application of lemma and equations with non instantaneous the pulse of the development and research status. The second chapter, by using the resolvent operator theory, measure of noncompactness, fixed point theorem and Banach image compression principle, we study a class of fractional reaction diffusion with hysteresis the existence of solutions for boundary value problems. In the early T equation belonging to a finite interval, discussed resolvent the operator is the existence of global solutions of compact operator and non compact operator equation. In the case of T belongs to the infinite interval, the resolvent operator is a compact operator equation under the condition of local solution and The existence of global solutions. Our results improve and improve some recent results. In the third chapter, we study the existence of global solutions and local solutions of a class of Semilinear Integro differential equations of fractional order, using the measure of noncompactness and fixed point theorem equations are sufficient conditions for existence of solutions. The solution method in this chapter we give a new study of fractional evolution equations. Finally, this chapter gives an application by the main results. In addition, we study the existence problem of a class of fractional order mixed type differential equations by using the same method. In the fourth chapter, we consider a class of Semilinear fractional integral differential equation with non instantaneous pulse and delay. By using the resolvent operator theory and fixed point theorem, we discuss the existence of solutions of the equation, some new results are obtained. Finally an example is given to illustrate the main results of this chapter The application. The fifth chapter is to study a class of periodic fractional order semilinear Integro differential equation of non instantaneous pulse boundary value problem. By using the resolvent operator theory, measure of noncompactness and fixed point theorem to obtain some new results on existence equations. Finally gives an example of application to the Akimoto Akiko. In the sixth chapter, by using the generalized Banach of non existence and uniqueness of the autonomous development of equations solutions with hysteresis and instantaneous pulse fractional image compression principle, gives the solution of iterative sequence and error estimation and discusses the uniqueness of solution is continuously dependent on the initial value.

【学位授予单位】：曲阜师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.8

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