A Note on Approximating Fixed Points of Nonexpansive Mappings by the Ishikawa Iteration Process

Let X be a uniformly convex Banach space that satisfies Opial’s condition or whose norm is Fr´echet differentiable, let C be a bounded closed convex subset of X, and let T: CªC be a nonexpansive mapping. It is shown that for any initial data x0gC, the Ishikawa iterates xn4, defined by xnq1stnT snTxnq 1y sn.xn.q 1ytn.xn, nG0, with the restrictions that limn sn is less than 1, and for any subsequence nk4`ks0 of n4`ns0 , `ks0 tnk 1ytnk.diverges, converge to a fixed point of T weakly. Thus, such a result complements Theorem 1 of Tan and Xu J. Math. Anal. Appl. 178 1993., 301]308. and generalizes, to a certain extent, Theorem 2 of Reich J. Math. Anal. Appl. 67 1979., 274]276..