In this paper, we introduce the concepts of $\mathcal{R}$-completeness and weak $\mathcal{R^{\neq}}$-preserving mapping and employ them to prove some fixed point results for multi-valued mappings satisfying an implicit relation in rectangular $b$-metric spaces. We also deduce a fixed point result for the same in rectangular $b$-metric spaces endowed with graph $\mathcal{G}$. Furthermore, we adopt some examples to exhibit the utility of our definitions and results.