几类微分方程边值问题解的研究

发布时间：2018-05-04 13:48

  本文选题：边值问题 + 压缩映射原理　； 参考：《山东科技大学》2017年硕士论文

【摘要】：非线性微分方程边值问题是微分方程理论中一个颇为重要的研究分支,在建筑工程、航天工程、生物工程、物理学等领域都有着广泛的应用。因此,通过研究微分方程的边值问题将会对于解决这些领域中常见的难题提供帮助。而且在实际的生活和科研中,许多问题所构成的数学模型通常都是微分方程。可见,研究非线性微分方程的边值问题显得格外的重要。第一章主要介绍微分方程边值问题研究的历史及发展现状,叙述了本文的研究目的和意义,并说明对微分方程边值问题解的研究不仅具有重要的理论意义,而且具有重要的实际意义。第二章讨论了奇异边值问题解的唯一性,其中h(t)在t=1处奇异。本章的新颖之处在于Lipschitz系数与相应线性算子的第一特征值有关,我们通过u0-范数以及压缩映射原理来给出上述边值问题的唯一解。第三章讨论了分数阶微分方程边值问题至少存在一个正解,其中2p≤3是一个实数.本章通过运用相应线性算子的第一特征值及其相关性质,以及不动点指数等理论得到其至少存在一个正解的证明.第四章应用单调迭代法以及上下解方法研究四阶两点边值问题解的存在性,其中f:[0,1]×R→R为连续函数。本章的新颖之处在于,对此边值问题利用线性算子的性质建立一个比较结果,从而研究其极值解的存在性。
[Abstract]:Boundary value problem of nonlinear differential equation is an important branch of differential equation theory. It is widely used in architectural engineering, aerospace engineering, biological engineering, physics and so on. Therefore, studying the boundary value problems of differential equations will be helpful to solve the common problems in these fields. Moreover, in real life and scientific research, the mathematical models of many problems are usually differential equations. Therefore, it is very important to study the boundary value problem of nonlinear differential equation. The first chapter mainly introduces the history and development of the boundary value problem of differential equation, describes the purpose and significance of this paper, and shows that the study of the solution of the boundary value problem of differential equation is not only of great theoretical significance. And has important practical significance. In chapter 2, we discuss the uniqueness of the solution of the singular boundary value problem, where ht) is singular at t ~ (1). The novelty of this chapter is that the Lipschitz coefficient is related to the first eigenvalue of the corresponding linear operator. We obtain the unique solution of the above boundary value problem by using u0-norm and contraction mapping principle. In chapter 3, we discuss the existence of at least one positive solution for the boundary value problem of fractional differential equations, where 2p 鈮，

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