Modified $\mathrm{\Re }$-rational contractions and fixed point results with applications to boundary value problems

This manuscript introduces modified $\mathrm{\Re }$-rational contractions for single self-maps, leveraging $\omega$-distance in a relational-theoretic metric space. This novel approach establishes the existence and uniqueness of a fixed point for self-maps, specifically by applying the locally $\mathrm{\Theta }$-transitivity property. We support our theoretical advancements with compelling examples and demonstrate their practical significance by solving a fourth-order boundary value problem related to transverse oscillation in a homogeneous bar and a first-order periodic boundary value problem.