Strong and Weak Convergence Theorems for Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings in Banach Spaces
Wei-Qi Deng, Lin Wang, Yi‐Juan Chen · 2012
Suppose that K is a nonempty closed convex subset of a uniformly convex and smooth Banach space E with P as a sunny nonexpan-sive retraction. Let T1, T2: K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with two se-quences {k(i)n} ⊂ [1,∞) satisfying ∑∞n=1(k(i)n − 1) < ∞ (i = 1, 2) and F (T1) ∩ F (T2) = {x ∈ K: T1x = T2x = x} = ∅, respectively. For any given x1 ∈ K, suppose that {xn} is a sequence generated iteratively by xn+1 = αn1xn + βn1(PT1)nyn + γn1(PT2)nyn, yn = αn2xn + βn2(PT1)nzn + γn2(PT2)nzn, zn = αn3xn + βn3(PT1)nxn + γn3(PT2)nxn, where {αni}, {βni}, and {γni} (i = 1, 2, 3) are sequences in [a, 1 − a] for some a ∈ (0, 1), satisfying αni + βni + γni = 1 (i = 1, 2, 3). Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of T1 and T2 are obtained.