Research Article Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces

Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E, f: C → C a contractive mapping or a weakly contractive mapping, and T: C → C nonexpansive mapping with the fixed point set FT / ∅. Let {xn} be generated by a new composite iterative scheme: yn λnfxn1−λnTxn, xn1 1−βnynβnTyn, n ≥ 0. It is proved that {xn} converges strongly to a point in FT, which is a solution of certain variational inequality provided that the sequence {λn} ⊂ 0, 1 satisfies limn→∞λn 0 and∑∞ n1λn ∞, {βn} ⊂ 0, a for some 0 < a < 1 and the sequence {xn} is asymptotically regular. Copyright q 2008 Jong Soo Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1