On Unique and Nonunique Fixed Points and Fixed Circles in    M   v   b   -Metric Space and Application to Cantilever Beam Problem

Meena Joshi,Anita Tomar,Hossam A Nabwey,Reny George,Huseyin Isik

doi:10.1155/2021/6681044

Abstract

We introduce M v b -metric to generalize and improve M v -metric and unify numerous existing distance notions. Further, we define topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology on M v b -metric space and to create an environment for the survival of a unique fixed point. Also, we introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map and establish fixed circle and fixed disc theorems. Further, we verify all these results by illustrative examples to demonstrate the authenticity of the postulates. Towards the end, we solve a fourth order differential equation arising in the bending of an elastic beam.

Highlights

A Greek mathematician Euclid of Alexandria (323-283 BC) was the first to communicate the notion of distance
We demonstrate that the collection of open balls, which forms a basis on Mbv-metric space, generates a T 0 topology on it
We introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map in an Mbv-metric space and establish fixed Journal of Function Spaces circle and greatest fixed disc theorems

Summary

Introduction

A Greek mathematician Euclid of Alexandria (323-283 BC) was the first to communicate the notion of distance. We introduce topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology and to create an environment for the survival of a unique fixed point in an Mbv-metric space. With the help of examples and remarks, we demonstrate that an Mbv-metric space unifies and combines numerous distance conceptions and marks supremacy over all those spaces wherein the continuity of a map is required for the survival of a fixed point. We introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map in an Mbv-metric space and establish fixed. We solve fourth order differential equation arising in the Cantilever beam problem

Preliminaries

Main Results

Application to Cantilever Beam

Conclusion