We prove the weak and strong convergence of S iterative scheme to a common fixed point of a family of nonself asymptotically $I$-nonexpansive mappings $\{T_{i}\}_{i}^{N}$ and a family of nonself asymptotically nonexpansive mappings $\{I_{i}\}_{i}^{N}$ defined on a nonempty closed convex subset of a Banach space. Our scheme converges faster than Mann and Ishikawa iteration for contractions. Our weak convergence theorem is proved under more general setup of space as different from weak convergence theorems proved in previously.