We propose a one-step iterative scheme to approximate common fixed points of multi-valued non-expansive mappings in a convex metric space ${\rm Y}$, under certain mild boundary conditions. We present a numerical example to verify our convergence result obtained herein. We also establish some convergence results for multi-valued asymptotically non-expansive mappings in this context, which either improve or generalize results obtained for multi-valued non-expansive mappings on ${\rm Y}$. Furthermore, we demonstrate how these findings can be applied to solve a system of two equations defined by multi-valued asymptotically non-expansive mappings on a convex metric space.