具p-Laplacian算子分数阶微分方程边值问题解的存在性研究

发布时间：2018-01-13 15:26

本文关键词：具p-Laplacian算子分数阶微分方程边值问题解的存在性研究　出处：《安徽大学》2015年硕士论文　论文类型：学位论文

【摘要】：非线性泛函分析是现代数学中一个既有深刻理论意义又有广泛应用价值的研究方向,它以数学和自然科学各个领域中出现的非线性问题为背景,建立了处理非线性问题的若干一般性理论和方法。它的研究成果可以广泛应用于各种非线性微分方程、积分方程和其他各种类型的方程,以及计算数学、控制理论、最优化理论、动力系统、经济数学等许多领域。目前非线性泛函分析主要内容包括拓扑度理论、临界点理论、半序方法、解析方法和单调型映射理论等。近年来,非线性分数阶微分方程边值问题是微分方程中的一个重要课题。它在扩散和运输理论、混沌与湍流、粘弹性力学及非牛顿流体力学等诸多领域都有广泛的应用。已引起国内外学者的高度重视,并取得一系列的成果,成为国际热点研究之一。本文主要利用非线性泛函分析上的不动点指数理论、Leggett-Williams不动点理论、重合度方法研究了带p-Laplacian算子分数阶微分方程边值问题解的存在性、多解性。在较弱的条件下获得了一些新的较深刻的结果。本文的组织结构如下：第一章绪论主要介绍本文所研究问题的历史背景和基本概念和引理。第二章利用不动点指数定理研究了具p-Laplacian分数阶奇异微分方程积分边值问题多个正解的存在性,得到了这类问题至少存在两个或2n个正解。第三章构造了一个格林函数,利用逼近的方法和不动点定理研究了Riemann-Stieltjes积分边界条件的p-Laplacian分数阶奇异耦合系统,得出了该类问题至少存在三个正解。第四章我们利用Mawhin重合度的方法研究了多点边界条件的p-Laplacian分数阶耦合系统共振边值问题解的存在性。第五章我们讨论了一类带p-Laplacian分数阶微分系统无穷边值问题解的存在性,使原有在有限区间上研究的结果得到进一步完善,并给出了一个实例。
[Abstract]:Nonlinear functional analysis is not only a profound theoretical significance and wide application value in the research direction in modern mathematics, nonlinear problems in mathematics and natural science fields in the background, set up to handle nonlinear problems some general theories and methods. The research results can be widely used in all kinds of nonlinear differential equations the integral equation, and various other types of equations, and the computational mathematics, control theory, optimization theory, power system, many areas of economic mathematics. The nonlinear functional analysis, the main content includes the topological degree theory, critical point theory, partial order method, analytical method and monotone mapping theory and so on. In recent years, nonlinear fractional differential equation boundary value problem is an important subject in differential equations. It is in the diffusion and transport theory, chaos and turbulence, viscoelastic mechanics and non Newtonian fluid Mechanical and other fields are widely used. Has attracted great attention of scholars at home and abroad, and made a series of achievements, become one of the international hot research. This paper uses nonlinear functional analysis and the fixed point index theory, Leggett-Williams fixed point theory, coincidence degree method to study the existence of solutions to the problem with fractional p-Laplacian differential equations of boundary value equations. Get some profound new results under weaker conditions. This paper is organized as follows: the first chapter mainly introduces the research questions in the historical background and the basic concepts and lemmas. On the issue of the existence of multiple positive solutions of integral boundary value with p-Laplacian fractional order singular differential equations by using the fixed point index theorem in chapter second, has been the problem has at least two or 2n positive solutions. In the third chapter, we construct a Green function, use P-Laplacian fractional coupling system of Riemann-Stieltjes singular integral boundary condition approximation method and fixed point theorem, the problem has at least three positive solutions. The existence of solutions for the fourth chapter p-Laplacian fractional coupling resonance of the system we use the method of Mawhin coincidence degree of multi point boundary value fifth. Chapter we discuss the existence of solution to the problem with p-Laplacian fractional order differential systems with infinite boundary value, so that the original research results on the finite interval has been further improved, and an example is given.

【学位授予单位】：安徽大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.8

【共引文献】