Strong Convergence of the Cq Method for Fixed Point Iteration Processes

(3) { zn = βnxn + (1− βn)Txn, xn+1 = αnxn + (1− αn)Tzn, n ≥ 0, where the initial guess x0 is taken arbitrarily and {αn}n=0 and {βn} are sequences in the interval [0, 1]. In general not much has been known regarding the convergence of the iteration processes (1)-(3) unless the underlying space X has nice properties which we briefly mention here. The iteration process (1) has been proved to be strongly convergent in both Hilbert spaces [7, 12, 24] and uniformly smooth Banach spaces [17, 20, 25] whenever the sequence {tn} satisfies the conditions: (i) tn → 0; (ii) ∑∞ n=0 tn = ∞; and (iii) either ∑∞ n=0 |tn − tn+1| < ∞ or limn→∞(tn/tn+1) = 1. Due to the restriction of condition (ii), the process (1) is widely believed to have slow convergence though the rate of convergence has not been determined. Moreover, Halpern [7] proves that the conditions (i) and (ii) are indeed necessary in the sense that if the process (1) is strongly convergent for all closed convex subsets C of a Hilbert space H and all nonexpansive mappings T on C, then the sequence {tn} must satisfy the conditions (i) and (ii). Thus to improve the rate of convergence of the process (1), one