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Wang, Yachao; Guo, Tiexin

The fixed point theorems and invariant approximations for random nonexpansive mappings in random normed modules. (English) Zbl 07852378

J. Math. Anal. Appl. 531, No. 1, Part 2, Article ID 127796, 11 p. (2024).
By making full use of the theory of random sequential compactness in random normed modules, in this paper the authors established a noncompact Dotson fixed point theorem. Furthermore, they obtained an existence result for best approximations in random normed modules, which generalizes the classical result of Smoluk. In addition, they also got an existence result for invariant approximations in random normed modules. The \(\sigma\)-stability of both the sets and mappings involved in the random setting plays a prominent part in the proofs of the main results of this paper.
The theory of random normed modules has been put forward by Prof. Tiexin Guo several years ago, who creatively defined a new random framework of spaces that encompasses the classical ones. The main results in this paper are new and interesting, which enriches the theory of random functional analysis, and shed some new light on the study of fixed point results in the random case.
Reviewer: Chuanxi Zhu (Nanchang)

MSC:

47H10 Fixed-point theorems
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)

Keywords:

random normed modules; random sequential compactness; random nonexpansive mappings; Dotson fixed point theorem; invariant approximations
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References:

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