Weak Almost-Convergence Theorem without Opial’s Condition

Let E be Banach space with propertyŽU, m, m 1, ., R, m N, and a uniformly Gateaux differentiable norm; J: E E* a duality mapping; D a nonempty closed convex bounded subset of E; and T: D D a uniformly L-Lipschitzian asymptotically hemicontractive mapping with L NŽE.1 2 where NŽE. is the normal structure coefficient of E satisfying the condition x Tny 2 ² x Tny, JŽx y.:for all x, y D, n N 04. Under the above condi- tions, the convergence of JŽxn .4 for the sequence xn4 of the modified Ishikawa iteration process is established and then it is used to prove weak convergence of the process. The modified Ishikawa iteration process is defined as follows: For D a convex subset of a Banach space X and T a mapping D into itself, the sequence xn4 n 0in D is defined by x0 D, xn 1 Ž1 n.xn 1 Partially supported by Department of Science and Technology, New Delhi, India, 1997. 636 0022-247X 01 $35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved. WEAK ALMOST-CONVERGENCE THEOREM 637 nTŽŽ1 n. xn nTxn., n 0, where n4 and n4 satisfy 0 n, n 1 for all n and Ý n 0 n .