Sujit Patel,
- Assistant Professor Department of Mathematics, Seva Sadan College, Burhanpur , Madhya Pradesh India
Abstract
Fixed points in non-expansive mappings are elements that remain unchanged under the action of the mapping. Non-expansive mappings preserve or contract distances in a metric space. The rigorous proof of the Banach fixed-point theorem is presented at the outset of the paper, highlighting its fundamental significance. According to this theorem, every contraction mapping (a particular kind of non-expansive mapping that strictly reduces distances) has a single fixed point in a full metric space. Wide-ranging ramifications and applications of this result can be found in many areas of mathematics, such as optimization, dynamical systems, and differential equations.
This paper not only provides a rigorous proof of this theorem but also explores its diverse applications across various mathematical domains. The study explores the Non expansive Mapping Theorem, which goes beyond the Banach fixed-point theorem. By tackling non-expansive mappings, which may not always shorten distances but nevertheless maintain or shorten them, this theorem widens its application. Within specific kinds of non-expansive mappings in entire metric spaces, the Non expansive Mapping Theorem affirms the existence and uniqueness of fixed points. Considering the greater range of mappings and possible applications that this extension encompasses, it is imperative. The applicability and versatility of these theorems are demonstrated by the paper’s methodical exploration of their applications across many mathematical disciplines. These theorems provide valuable tools for problem solving in a variety of mathematics and related fields by defining the circumstances under which fixed points exist and are unique. Furthermore, it introduces the Non expansive Mapping Theorem, which extends our understanding by establishing the existence and uniqueness of fixed points within specific categories of non expansive mappings residing in complete metric spaces.
Keywords: Fixed Point, Complete metric space ,Non expansive mapping, contraction mapping, Nonlinear Operator Theory
[This article belongs to Research & Reviews: Discrete Mathematical Structures(rrdms)]
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Research & Reviews: Discrete Mathematical Structures
| Volume | 11 |
| Issue | 01 |
| Received | June 25, 2024 |
| Accepted | June 30, 2024 |
| Published | July 11, 2024 |