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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 in the intermediate sense such that F = F (S) ∩ F (T ) = {x ∈ K : Sx = Tx = x} 6= ∅. Suppose {xn} is generated iteratively by x1 ∈ K, xn+1 = (1 − αn)Txn + αnSyn, yn = (1 − βn)xn + βnTxn, n ≥ 1, where {αn} and {βn} are real sequences in [a, b] for some a, b ∈ (0, 1). If S and T satisfy condition (B) or either S or T is semi-compact, then the sequence {xn} converges strongly to some q ∈ F and if E satisfying the Opial’s condition, then the sequence {xn} converges weakly to some q ∈ F .

Convergence to common fixed point for two asymptotically quasi-nonexpansive mappings in the intermediate sense in Banach spaces