Abstract

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 mapping with {kn}⊂[1,∞) such that ∑n=1∞(kn−1)<∞ and F(T) is nonempty, where F(T) denotes the fixed points set of T. Let {αn}, {αn'}, and {αn''} be real sequences in (0,1) and ε≤αn,αn',αn''≤1−ε for all n∈ℕ and some ε>0. Starting from arbitrary x1∈K, define the sequence {xn} by x1∈K, zn=P(αn''T(PT)n−1xn+(1−αn'')xn), yn=P(αn'T(PT)n−1zn+(1−αn')xn), xn+1=P(αnT(PT)n−1yn+(1−αn)xn). (i) If the dual E* of E has the Kadec-Klee property, then { xn} converges weakly to a fixed point p∈F(T); (ii) if T satisfies condition (A), then {xn} converges strongly to a fixed point p∈F(T).