This article introduces a new concept of compatible contraction in metric spaces. By utilizing graph properties, namely being orbitally edge-preserving, we establish the existence of fixed points under these conditions. Furthermore, under specific assumptions on the mapping, the uniqueness of the fixed point is guaranteed. We provide illustrative examples to clarify the theoretical developments. To demonstrate the practical utility of this framework, we apply it to specific classes of integral and differential equations, specifically focusing on nonlinear fourth-order boundary value problems. We show that these problems satisfy the proposed contraction criteria, ensuring their solution via iterative methods. Numerical experiments on various fourth-order boundary value problems validate the effectiveness of our approach.
Poochinapan et al. (2026) studied this question.