Generalized $\mathcal{R}$ - contractions and fixed point theorems with an application to boundary value problem governing transverse oscillation in homogeneous bar

We introduce the notion of generalized $\mathcal{R}$ - contractions for single self-maps via w - distance in relational-theoretic metric space. We utilized this notion to find the existence and uniqueness of a fixed point for self-map using the locally Ψ-transitivity property. In order to validate our findings, some examples with graphical representation are also presented. At last, we incorporate this work with an application to find the existence of solution to a forth order boundary value problem governing transverse oscillation in a homogeneous bar.