Fixed point theorems for noncommutative functions
Fixed point theorems for noncommutative functions
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非交换函数的不动点定理
DOI：
10.1016/j.jmaa.2012.12.038
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发表时间：
2012
影响因子：
1.3
通讯作者：
Dmitry S. Kaliuzhnyi
中科院分区：
文献类型：
--
作者：
G. Abduvalieva;Dmitry S. Kaliuzhnyi
我们建立了一个不动点定理，适用于所有大小的方阵映射，它与矩阵大小和矩阵的直接和有关。如果这样的映射也尊重矩阵相似性，即是一个非交换函数，则结论更强。作为一个特例，我们证明了相应的压缩映射定理，该定理可以看作是Banach不动点定理的一个新版本。然后应用这一结果证明了非交换空间中微分方程初值问题解的存在唯一性。作为本文发展思想的副产品，我们建立了嵌套闭集原理的非交换版本。
We establish a fixed point theorem for mappings of square matrices of all sizes which respect the matrix sizes and direct sums of matrices. The conclusions are stronger if such a mapping also respects matrix similarities, i.e., is a noncommutative function. As a special case, we prove the corresponding contractive mapping theorem which can be viewed as a new version of the Banach Fixed Point Theorem. This result is then applied to prove the existence and uniqueness of a solution of the initial value problem for ODEs in noncommutative spaces. As a by-product of the ideas developed in this paper, we establish a noncommutative version of the principle of nested closed sets.