Abstract
In the present study, a novel version of the contraction theory on a relational partial metric space associated with a binary relation is exhibited. In this process, we observe that numerous relations can be used in many well-known fixed point theorems earlier introduced by multiple authors. We solve an elliptic boundary value problem and an integral equation to make the significance of the main conclusion evident. Exploring a nonlinear system, governed by nonlinear equations, underscores the importance of comprehending fixed points - pivotal states where the dynamics of a system remain unaltered.