本文選題：分?jǐn)?shù)階反應(yīng)擴(kuò)散方程　切入點(diǎn)：分?jǐn)?shù)階發(fā)展方程　出處：《曲阜師范大學(xué)》2017年博士論文

【摘要】：非線性泛函分析是現(xiàn)代數(shù)學(xué)中一個(gè)重要的數(shù)學(xué)分支,其主要內(nèi)容包括拓?fù)涠壤碚�、不�?dòng)點(diǎn)理論、半序方法等.非線性泛函分析為研究具有非線性問(wèn)題的諸多領(lǐng)域中的數(shù)學(xué)模型提供了理論基礎(chǔ)和先進(jìn)方法.在Banach空間,非線性泛函分析對(duì)非線性發(fā)展方程理論的研究具有重要應(yīng)用,已被廣泛應(yīng)用于物理、化學(xué)、金融和最優(yōu)控制等領(lǐng)域.近年來(lái),非線性發(fā)展方程初值、邊值問(wèn)題解的存在性問(wèn)題受到廣大研究者的普遍關(guān)注,并取得一系列研究成果.分?jǐn)?shù)階微積分理論由于成功應(yīng)用到分形、多孔介質(zhì)彌散、金融等領(lǐng)域而發(fā)展迅速.分?jǐn)?shù)階微分方程相比整數(shù)階微分方程能夠更好的解釋反常擴(kuò)散、粘彈性體中的應(yīng)力應(yīng)變等具有記憶和遺傳性的過(guò)程,這使得分?jǐn)?shù)階微分方程的研究也受到越來(lái)越多的關(guān)注.分?jǐn)?shù)階微分(發(fā)展)方程相比整數(shù)階微分(發(fā)展)方程的研究要困難,原因在于分?jǐn)?shù)階微分算子具有奇異和非局部的特點(diǎn).這也說(shuō)明研究分?jǐn)?shù)階發(fā)展方程在理論和實(shí)際應(yīng)用方面都具有重要意義.本文主要研究了非線性分?jǐn)?shù)階發(fā)展方程解的存在性問(wèn)題,利用半群理論(預(yù)解算子理論)、非緊性測(cè)度、不動(dòng)點(diǎn)理論等方法取得了一些新的結(jié)果.這些結(jié)果改進(jìn)并推廣了一些前人的結(jié)果.其中部分結(jié)果發(fā)表在《Appl.Math. Lett.》(SCI)和《Comput.Math. Appl.》(SCI)等國(guó)外重要的學(xué)術(shù)期刊上.本文共分五章.第一章緒論,簡(jiǎn)要介紹了分?jǐn)?shù)階微積分的發(fā)展歷史及其在相關(guān)領(lǐng)域的應(yīng)用,給出Riemann-Liouville分?jǐn)?shù)階積分算子、Riemann-Liouville分?jǐn)?shù)階微分算子和Caputo分?jǐn)?shù)階微分算子的定義,非線性泛函分析的應(yīng)用領(lǐng)域,以后各章用到的一些定義、性質(zhì)和引理以及帶非瞬時(shí)脈沖的發(fā)展方程應(yīng)用領(lǐng)域和研究現(xiàn)狀.第二章,利用預(yù)解算子理論、非緊性測(cè)度、不動(dòng)點(diǎn)定理和Banach壓縮影像原理,我們研究了一類(lèi)帶遲滯的分?jǐn)?shù)階反應(yīng)擴(kuò)散方程初邊值問(wèn)題解的存在性.在t屬于有限區(qū)間時(shí),分別討論了預(yù)解算子是緊算子和非緊算子情況下方程整體解的存在性.在t屬于無(wú)窮區(qū)間時(shí),討論了預(yù)解算子是緊算子條件下方程局部解和整體解的存在性.我們的結(jié)論改進(jìn)并完善了前人的一些結(jié)果.第三章,我們研究了一類(lèi)半線性分?jǐn)?shù)階積分微分方程局部解和整體解的存在性,利用非緊性測(cè)度和不動(dòng)點(diǎn)定理給出方程存在解的充分條件.其中本章給出了一種新的研究分?jǐn)?shù)階發(fā)展方程解的存在性的方法.最后,給出一個(gè)利用本章主要結(jié)果的應(yīng)用.另外,利用同樣的方法我們研究了一類(lèi)分?jǐn)?shù)階混合型微分方程解的存在性問(wèn)題.第四章,我們考慮了一類(lèi)帶非瞬時(shí)脈沖和遲滯的分?jǐn)?shù)階半線性積分微分方程.利用預(yù)解算子理論和不動(dòng)點(diǎn)定理,我們討論了方程解的存在性,得到一些新的結(jié)果.最后給出一個(gè)例子來(lái)說(shuō)明本章主要結(jié)果的應(yīng)用.第五章,研究了一類(lèi)帶非瞬時(shí)脈沖的分?jǐn)?shù)階半線性積分微分方程周期邊值問(wèn)題.利用預(yù)解算子理論、非緊性測(cè)度和不動(dòng)點(diǎn)定理得到方程解存在的一些新結(jié)果.最后給出一個(gè)例子來(lái)說(shuō)明本章主要結(jié)果的應(yīng)用.第六章,利用廣義Banach壓縮影像原理研究了 一類(lèi)帶遲滯和瞬時(shí)脈沖的分?jǐn)?shù)階非自治發(fā)展方程初值問(wèn)題解的存在性和唯一性,給出其解的迭代序列和誤差估計(jì)并討論了其唯一解是連續(xù)依賴(lài)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.

【學(xué)位授予單位】：曲阜師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O175.8

【參考文獻(xiàn)】

相關(guān)期刊論文 前5條

1 馬和平;劉斌;;EXACT CONTROLLABILITY AND CONTINUOUS DEPENDENCE OF FRACTIONAL NEUTRAL INTEGRO-DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY[J];Acta Mathematica Scientia(English Series);2017年01期

2 崔靜;閆理坦;;CONTROLLABILITY OF NEUTRAL STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION[J];Acta Mathematica Scientia(English Series);2017年01期

3 劉玉記;;PIECEWISE CONTINUOUS SOLUTIONS OF INITIAL VALUE PROBLEMS OF SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS[J];Acta Mathematica Scientia(English Series);2016年05期

4 李楠;王長(zhǎng)有;;NEW EXISTENCE RESULTS OF POSITIVE SOLUTION FOR A CLASS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS[J];Acta Mathematica Scientia;2013年03期

5 Dimplekumar;N. CHALISHAJAR;K. KARTHIKEYAN;;EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES[J];Acta Mathematica Scientia;2013年03期

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本文編號(hào)：1672791

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