Lipschitz isomorphism and fixed point theorem for normed groups

Abstract

This paper aims to propose normed structures for groups and to establish the Lipschitz mapping of a normed group $G$ to itself. We also investigate some conjugate and isomorphic Lipschitz mappings to determine the equivalent norm and inverse Lipschitz mappings. Specifically, in the main result, we present a fixed point theorem for self-mappings satisfying certain contraction principles on a complete normed group.

Keywords:

• Banach group
• generalized Lipschitz conditions
• fixed point theorems
• Klee property
• expansive mapping

PUBLIC INTEREST STATEMENT

In mathematical analysis, Lipschitz continuity (named after Rudolf Lipschitz) is the Picard–Lindelöf theorem’s central condition. A particular type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem (known as the contraction mapping theorem) is an essential tool that ensures the existence and uniqueness of fixed points of certain self-mappings and provides a constructive method to find those fixed points. From previous studies, it has been shown that Banach fixed-point theorem is a useful mechanism for numerous branches of mathematical analysis, general topology, functional analysis, and economics. One of the most known applications of Banach’s fixed point theorem for economists is Bellman’s functional equations.

This article mainly focused on understanding the general properties of Lipschitz self-mappings and its connection with the existence of isomorphic group-norm. We have investigated Banach fixed point theorem for normed groups to determine the existence of unique fixed points following specific conditions.

Acknowledgements

The authors are grateful to the referees and the editors for valuable comments and suggestions, which have improved the original manuscript greatly.

Conflicts of Interest

The author(s) declare(s) that there is no conflict of interest regarding the publication of this paper.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [11971493].

Notes on contributors

Yongjin Li

Dr. Yongjin Li is working as a professor at the School of Mathematics, Sun Yat-sen University. His area of specialization is Functional Analysis, Differential Equations. Under his guidance, 04 students are working for a Ph.D., and 03 post Doctors are working in his group. He has published 150 Research papers, 06 books and completed 03 projects for the National Natural Science Foundation of China.

Muhammad Sarfraz is currently a Ph.D. Research Scholar under the supervision of Professor Dr. Yongjin Li in School of Mathematics, Sun Yat-sen University. His area of specialization is Functional Analysis and Cryptography. He has published 7 Research papers.

Fawad Ali is currently a Ph.D. Research Scholar with the School of Mathematics and Statistics Xi’an Jiaotong University, Xi’an 710049, P. R. China. He has been working in algebraic graph theory since 2017.