Fixed‐Point Result for Generalized Enriched Contractions With Applications in Cantilever Beam Problem and Homotopy Theory

In this study, we introduce a novel class of mappings called orthogonal extended interpolative enriched Ćirić–Reich–Rus type ‐contractions and establish fixed‐point results within the framework of orthogonal complete convex extended b ‐metric spaces. The unique fixed point is approximated using a Krasnoselskii‐type iterative method. To demonstrate the practical significance of our findings, we present several illustrative examples. Furthermore, recognizing that certain nonlinear systems can be reformulated as integral equations, we validate the applicability of our main results by proving the existence and uniqueness of solution to a Cantilever beam problem. In addition, we extend our analysis to homotopy theory, establishing the existence of a unique solution to related problems.