On Unique and Nonunique Fixed Points and Fixed Circles in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:msubsup> <a:mrow> <a:mi mathvariant="script">M</a:mi> </a:mrow> <a:mrow> <a:mi>v</a:mi> </a:mrow> <a:mrow> <a:mi>b</a:mi> </a:mrow> </a:msubsup> </a:math>-Metric Space and Application to Cantilever Beam Problem
Meena Joshi, Anita Tomar, Hossam A. Nabwey, Reny George
Abstract
We introduce <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"> <a:msubsup> <a:mrow> <a:mi mathvariant="script">M</a:mi> </a:mrow> <a:mrow> <a:mi>v</a:mi> </a:mrow> <a:mrow> <a:mi>b</a:mi> </a:mrow> </a:msubsup> </a:math> -metric to generalize and improve <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" id="M3"> <d:msub> <d:mrow> <d:mi mathvariant="script">M</d:mi> </d:mrow> <d:mrow> <d:mi>v</d:mi> </d:mrow> </d:msub> </d:math> -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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M4"> <g:msubsup> <g:mrow> <g:mi mathvariant="script">M</g:mi> </g:mrow> <g:mrow> <g:mi>v</g:mi> </g:mrow> <g:mrow> <g:mi>b</g:mi> </g:mrow> </g:msubsup> </g:math> -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.