Results via Partial-$b$ Metric and Solution of a Pair of Elliptic Boundary Value Problem

We give a  method to establish a fixed point via partial $b$-metric  for multivalued mappings.  Since the geometry of multivalued fixed points perform a significant role in numerous real-world problems and is fascinating and innovative, we introduce the notions of fixed circle and fixed disc to frame  hypotheses to establish fixed circle/ disc theorems in a  space that permits non-zero self-distance with a coefficient more significant  than one.  Stimulated by the reality that the fixed point theorem is the frequently used technique for solving boundary value problems, we solve a pair of elliptic boundary value problems.  Our developments cannot be concluded from the current outcomes in related metric spaces. Examples are worked out to substantiate the validity of the hypothesis of our results.