Strong and weak convergence theorems for asymptotically nonexpansive mappings

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {k_n}_n≥1 ⊂[1, ∞), limk_n = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {x_n}_n≥1 is generated iteratively by x₁ ∈ K, x_n+1 = P((1-α_n x_n+α_nT(PT)^n-1x_n), n≥1, where {α_n}_n≥1 ⊂ (0, 1) is such that ∈ < 1 - α_n < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑_n≥1 (k_n² - 1) < ∞ and T is completely continuous, strong convergence of {x_n} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {x_n} to some x* ∈ F(T) is obtained.