In this paper, we introduce a new class of Lipschitzian maps and prove some weak and strong convergence results for explicit iterative process using a more satisfactory definition of self mappings. Our results approximate common fixed point of a total asymptotically quasi-$I$-nonexpansive mapping $T$ and a total asymptotically quasi-nonexpansive mapping $I$, defined on a nonempty closed convex subset of a Banach space.