Suppose K is a nonempty closed convex subset of a real Banach space E . Let S , T : K → K be two asymptotically quasi-nonexpansive maps with sequences { u n } , { v n } ⊂ [ 0 , ∞ ) such that ∑ n = 1 ∞ u n < ∞ and ∑ n = 1 ∞ v n < ∞ , and F = F ( S ) ∩ F ( T ) : = { x ∈ K : S x = T x = x } ≠ ∅ . Suppose { x n } is generated iteratively by x 1 ∈ K , x n + 1 = ( 1 − α n ) x n + α n S n [ ( 1 − β n ) x n + β n T n x n ] , n ≥ 1 where { α n } and { β n } are real sequences in [ 0 , 1 ] . It is proved that (a) { x n } converges strongly to some x ∗ ∈ F if and only if liminf n → ∞ d ( x n , F ) = 0 ; (b) if X is uniformly convex and if either T or S is compact, then { x n } converges strongly to some x ∗ ∈ F . Furthermore, if X is uniformly convex, either T or S is compact and { x n } is generated by x 1 ∈ K , x n + 1 = α n x n + β n S n [ α ′ n x n + β ′ n T n x n + γ ′ n z ′ n ] + γ n z n , n ≥ 1 , where { z n } , { z ′ n } are bounded, { α n } , { β n } , { γ n } , { α ′ n } , { β ′ n } , { γ ′ n } are real sequences in [ 0 , 1 ] such that α n + β n + γ n = 1 = α ′ n + β ′ n + γ ′ n and { γ n } , { γ ′ n } are summable; it is established that the sequence { x n } (with error member terms) converges strongly to some x ∗ ∈ F .