Convergence of the Ishikawa Iteration Process for Nonexpansive Mappings

Let D be a subset of a normed space X and T: DªX be a nonexpansive mapping. Given a sequence xn4 in D and two real sequences tn4 and sn4 satisfying i. 0FtnFt-1 and `ns1 tns`, ii. 0FsnF1 and `ns1 sn-`, iii. xnq1stnT snTxnq 1ysn.xn.q 1ytn.xn, ns1, 2, 3, . . . , we prove that if xn4 is bounded, then limnª`5Txnyxn5s0. The conditions on D,X, and T are shown which guarantee the weak and strong convergence of the Ishikawa iteration process to a fixed point of T.