Iterative Algorithms to Fixed Point of Nonexpansive Mapping
Yao Ru · 2007
Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly Gateaux differentiable norm and T be a nonexpansive self-mapping of C with F(T)≠0: 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 = tu + (1-t)T for arbitrary u←C. Let {αn}, {βn} and {γn} be three real sequences in [0,1] which satisfies the following conditions: (i)αn+βn+γn = 1; (ii) limn→∞αn = 0 and ; (iii) 0 liminfn→∞βn≤limsupn→∞βn 1. For arbitrary x0∈C, let the sequence {xn} be defined by xn+1 =αnu +βnxn +γnTxn. Then, {xn} converges strongly to a fixed point of T.