Let be a real Banach space, a closed convex nonempty subset of , and asymptotically quasi-nonexpansive mappings with sequences (resp.)
satisfying as , and . Let
be a sequence in . Define a sequence by , , , , , . Let . Necessary and sufficient conditions for a strong convergence of the sequence to a common fixed point of the family are proved. Under some appropriate conditions, strong and weak convergence theorems are also proved.