Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces

Abstract

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=1∞λn=∞, {βn}⊂[0,a) for some 0<a<1 and the sequence {xn} is asymptotically regular.