Suppose that K is a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T 1 , T 2 : K → E be two weakly inward and asymptotically nonexpansive mappings with respect to P with sequences { K n } , { l n } ⊂ [ 1 , ∞ ) ,   lim n → ∞ k n = 1 ,   lim n → ∞ l n = 1 , F ( T 1 ) ∩ F ( T 2 ) = { x ∈ K : T 1 x = T 2 x = x } ≠ ∅ , respectively. Suppose that { x n } is a sequence in K generated iteratively by x 1 ∈ K , x n + 1 = α n x n + β n ( P T 1 ) n x n + γ n ( P T 2 ) n x n , for all n ≥ 1 , where { α n } , { β n } , and { γ n } are three real sequences in [ ε , 1 − ε ] for some ε > 0 which satisfy condition α n + β n + γ n = 1 . Then, we have the following. (1) If one of T 1 and T 2 is completely continuous or demicompact and ∑ n = 1 ∞ ( k n − 1 ) < ∞ , ∑ n = 1 ∞ ( l n − 1 ) < ∞ , then the strong convergence of { x n } to some q ∈ F ( T 1 ) ∩ F ( T 2 ) is established. (2) If E is a real uniformly convex Banach space satisfying Opial's condition or whose norm is Fréchet differentiable, then the weak convergence of { x n } to some q ∈ F ( T 1 ) ∩ F ( T 2 ) is proved.