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 F T / ∅. Let {xn} be generated by a new composite iterative scheme: yn λnf xn 1−λn Txn, xn 1 1−βn yn βnTyn, n ≥ 0 . It is proved that {xn} converges strongly to a point in F T , which is a solution of certain variational inequality provided that the sequence {λn} ⊂ 0, 1 satisfies limn→∞λn 0 and ∑∞ n 1λn ∞, {βn} ⊂ 0, a for some 0 < a < 1 and the sequence {xn} is asymptotically regular.