摘要本文主要研究不动点理论及其应用，特别是在高等数学数列极限、积分方程和微分方程中的应用。不动点理论的出现是整个数学史上的一个重要突破，它具有广泛的实际应用背景，数学中的许多重要定理，例如微分方程存在性定理和隐函数存在性定理，利用不动点定理可以给出更加易于理解的证明。本篇论文将以不动点定理（即是压缩映射定理）开始，然后给出不动点定理在数列极限、证明常微分方程和积分方程解的存在性和唯一性中的应用，最后介绍不动点理论在线性方程组方面的实际使用方法。而涉及到这些方面的应用过程中，尤为关键的是构造压缩算子，把问题转化为求的不动点问题基于不动点在数学、物理等领域的广泛应用，我们对不动点理论及其应用的研究具有很重要的理论和实际应用价值，对我们求解微分方程、积分方程及研究解的性质有着重要的指导作用。82315

毕业论文关键词：Banach不动点定理；完备度量空间；Banach空间；压缩映射原理。

Abstract

In this paper, we mainly study the fixed point theory and its applications, especially the applications of sequence limit、integral equations and differential equations in higher mathematics。 The emergence of the Fixed Point Theorem is an important breakthrough in the history of the whole mathematics, it includes many branches of mathematics and has broadly practical application。 Many important theorems in mathematics, such as the existence theorem of solutions of differential equations, the implicit function theorem and so on can be verified by the fixed point theorems more clearly。 First, in this paper we introduce the Banach Fixed Point Theorem, i。e。, the contraction mapping principle。 Second, we give the application of the fixed point theory in sequence limit、the existence and uniqueness of the solutions of ordinary differential equations and integral equations。 Finally, we generalize the application of the fixed point theory to the nonlinear systems。 Moreover, involving the application of these aspects, the key is to construct contraction operator L and put the problems into looking the fixed points of 。Based on the widely use of the fixed point theory in the field of mathematics、physics and so on , our research on the fixed point theory and its applications has an important theoretical and practical value。 Which has an important guiding role to solve the differential and integral equations and study the properties of the solutions。

Keywords: Banach Fixed Point Theorem; Complete Metric Space; Banach space; Compression Mapping。

目  录

第一章 绪论 1

1。1导论 1

1。1。1 选题背景 1

1。1。2 选题意义 2

1。1。3 主要研究内容 3

1。2 研究现状 3

1。3 本章小结 4

第二章 不动点定理简介 5

2。1 不动点定理的有关概念 5

2。2 不动点定理的拓广使用和近年发展 6

第三章 不动点定理的应用 10

3。1 求数列的通项公式 11

3。2 数列的有界性 13

3。3 数列的单调性及其收敛性 15

3。3。1 关于数列的单调性和收敛性的重要结论 15