Approximating fixed points of non‐self asymptotically nonexpansivemappings in Banach spaces
Yongfu Su, Xiaolong Qin · International Journal of Stochastic Analysis · 2006
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 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).