On unique and nonunique fixed points and fixed circles in \(\mathcal{M}_v^b\)-metric space and application to cantilever beam problem (Q2024582)

Summary: We introduce \(\mathcal{M}_v^b\)-metric to generalize and improve \(\mathcal{M}_v\)-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 \(\mathcal{M}_v^b\)-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.