Strong convergence and control condition of modifiedHalpern iterations in Banach spaces
Yonghong Yao, Rudong Chen, Haiyun Zhou · International Journal of Mathematics and Mathematical Sciences · 2006
Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly Gâteaux differentiable norm. Let T ∈ ΓC and f ∈ ΠC. Assume that {xt} converges strongly to a fixed point z of T as t → 0, where xt is the unique element of C which satisfies xt = tf(xt) + (1 − t)Txt. Let {αn} and {βn} be two real sequences in (0, 1) which satisfy the following conditions: (C1)limn→∞αn = 0; (C2)(C6)0 < liminfn→∞βn ≤ limsupn→∞βn < 1. For arbitrary x0 ∈ C, let the sequence {xn} be defined iteratively by yn = αnf(xn) + (1 − αn)Txn, n ≥ 0, xn+1 = βnxn + (1 − βn)yn, n ≥ 0. Then {xn} converges strongly to a fixed point of T.