PreprintArticleVersion 1Preserved in Portico This version is not peer-reviewed
The Nonlinear Analysis Related to Schauder Fixed Points, Best Approximation, Birkhoff-Kellogg Problems and Principle of Leray-Schauder Alternatives in p-Vector Spaces
Version 1
: Received: 20 April 2022 / Approved: 21 April 2022 / Online: 21 April 2022 (10:22:55 CEST)
How to cite:
Yuan, G.X. The Nonlinear Analysis Related to Schauder Fixed Points, Best Approximation, Birkhoff-Kellogg Problems and Principle of Leray-Schauder Alternatives in p-Vector Spaces. Preprints2022, 2022040203. https://doi.org/10.20944/preprints202204.0203.v1
Yuan, G.X. The Nonlinear Analysis Related to Schauder Fixed Points, Best Approximation, Birkhoff-Kellogg Problems and Principle of Leray-Schauder Alternatives in p-Vector Spaces. Preprints 2022, 2022040203. https://doi.org/10.20944/preprints202204.0203.v1
Yuan, G.X. The Nonlinear Analysis Related to Schauder Fixed Points, Best Approximation, Birkhoff-Kellogg Problems and Principle of Leray-Schauder Alternatives in p-Vector Spaces. Preprints2022, 2022040203. https://doi.org/10.20944/preprints202204.0203.v1
APA Style
Yuan, G.X. (2022). The Nonlinear Analysis Related to Schauder Fixed Points, Best Approximation, Birkhoff-Kellogg Problems and Principle of Leray-Schauder Alternatives in p-Vector Spaces. Preprints. https://doi.org/10.20944/preprints202204.0203.v1
Chicago/Turabian Style
Yuan, G.X. 2022 "The Nonlinear Analysis Related to Schauder Fixed Points, Best Approximation, Birkhoff-Kellogg Problems and Principle of Leray-Schauder Alternatives in p-Vector Spaces" Preprints. https://doi.org/10.20944/preprints202204.0203.v1
Abstract
It is known that the class of p-normed spaces (0 < p ≤ 1) is an important generalization of usual normed spaces with rich topological and geometrical structure, but the most of tools and general principles in nature with nonlinearity have not been developed yet, thus the main goal of this paper is to develop tools for nonlinear analysis under the framework of p-vector spaces. In particular, we first develop the general fixed point theorems which provide solutions to answer Schauder conjecture since 1930’s in the affirmative for p-vector spaces when p = 1 (which is just general topological vector spaces); then the one best approximation result for upper semi-continuous mappings is given, which is used as a powerful tool to establish fixed points for non-self set-valued mappings with either inward or outward set conditions; and finally we establish comprehensive existence results of solutions for Birkhoff-Kellogg Problems, and the general principle of nonlinear alternative by including Leray-Schauder alternative and related results. The results given in this paper not only include the corresponding results in the existing literature as special cases, but also expected to be useful tools for the study of nonlinear problems arising from theory to practice.
Keywords
nonlinear analysis; best approximation; Birkhoff-Kellogg theorem; schauder conjecture; fixed point theorem; nonlinear alternative; Leray-Schauder alternative; p-convex; p-inward and p-outward set; p-vector space
Subject
Computer Science and Mathematics, Analysis
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Commenter:
The commenter has declared there is no conflict of interests.
The new title is: Nonlinear analysis by applying best approximation method in p-vector spaces
(by George Xianzhi Yuan)
https://fixedpointtheoryandapplications.springeropen.com/articles/10.1186/s13663-022-00730-x