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In this paper, we introduce a new class of Lipschitzian maps and prove the weak and strong convergence of the explicit iterative process to a 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.

On weak and strong convergence of an explicit iteration process for a total asymptotically quasi-I-nonexpansive mapping in Banach space