Suppose K is a nonempty closed convex subset of a real uniformly convex Banach space E. Let S, T: K → K be two asymptotically quasi-nonexpansive mappings with sequences {un} and {vn} respectively, such that Σ∞ n = 1un < ∞ Σ∞ n=1 < ∞ and F = F(S) F(T) = {x ∈ K: Sx = Tx = x} ≠ 0. Suppose {xn} is generated iteratively by x1 ∈ K, xn+1 = (1 - αn)Tnxn + αn Snyn yn = (1 - βn)xn + βnTnxn. n > 1, where {αn} and {βn} are real sequences in [δ,1 - δ] for some δ ∈ (0,1). If E also has a Frechet differentiable norm or its dual E∗ has the Kedec-Klee property, then the weak convergence of the sequence {xn} to some q ∈ F are obtained. © 2014 Allahabad Mathematical Society.