Let D be a nonempty closed convex subset of a real Banach space E which is both uniformly convex and q-uniformly smooth. Let T: D \rightarrow D be a uniformly L-Lipschitzian, asymptotically nonexpansive-type and asymptotically pseudocontractive mapping with a sequence kn⊂ [1, ∞),\limn→ ∞ kn =1. Assume that the set F(T ) of fixed points of T is nonempty. Then F(T ) is a sunny nonexpansive retract of D. If U is the sunny nonexpansive retraction of D onto F(T ), w is any given point of D and {an} is a real sequence in (0,1] satisfying some restrictions, then the sequence {xn} in D defined by
xn=anw+(1-an){1}/{n+1}∑nj=0[(1-aj)I+ajT j]xn, ∀ n ≥ 0
converges strongly to Uw. No boundedness assumption is made on the set D.
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