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 (PT 1) n x n + γ n (PT 2) n x n , for all n ≥ 1, where {α n }, {β n }, and {γ n } are three real sequences in [ ,1 − ] for some > 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.

We present a new algorithm for solving the two-set split common fixed point problem with total quasi-asymptotically pseudocontractive operators and consider the case of quasi-pseudocontractive operators. Under some appropriate conditions, we prove that the proposed algorithms have strong convergence. The results presented in this paper improve and extend the previous algorithms and results of Censor and Segal (2009), Moudafi (2011 and 2010), Mohammed (2013), Yang et al. (2011), Chang et al. (2012), and others.