压缩映射的构造及Banach不动点定理的应用

Construction of compression mapping and application of Banach fixed point theorem

Banach不动点定理是泛函分析中最常用、最简单的存在性定理之一,也是数学分析中许多定理结果的特殊情形。因其应用广泛,倍受学者们青睐,关于该定理的应用性文章也层出不穷。然而,应用Banach不动点定理的关键是合理的定义压缩映射。基于此,笔者给出了3种不同条件下构造压缩映射的方法:即利用区间长度的比例构造压缩映射、利用线性方程组形的定义形式构造压缩映射和利用Lipschitz条件构造压缩映射,并对所构造的压缩映射进行了证明。同时,针对每种情况,举例说明了该种构造方法在应用Banach不动点定理解决问题中的作用。

Banach fixed point theorem is one of the most common and simple existence theorems in functional analysis,is also the special case of many theorems of mathematical analysis.Because of its wide range of applications,a number of scholars favor it.Articles about the application of which are endless.However,the key of using Banach fixed point theorem is a reasonable definition compression mapping.Based on this,the paper gives three tectonic compression mapping methods under different conditions： that is,using the ratio of the interval length,using the linear equations definition form and the Lipschitz condition,and then,proves them to be compression mappings.In the same time,in each case,an example was given to show that how to use the method to solve problems.