Fixed point theorems for noncommutative functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:401

Article

Abduvalieva, Gulnara¹  Kaliuzhnyi-Verbovetskyi, Dmitry S.¹ 

[1] Drexel Univ, Dept Math, Philadelphia, PA 19104 USA

关键词: Fixed point theorem;    Noncommutative function;    Contractive mapping;    Principle of nested noncommutative sets;    ODEs in noncommutative spaces;   

DOI  :  10.1016/j.jmaa.2012.12.038

来源: Elsevier

【 摘 要 】

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. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free