Strong Convergence of Composite Iterative Methods for Nonexpansive Mappings

Abstract

Abstract. Let E be a reflexive Banach space with a weakly sequentially continuous duality mapping, C be a nonempty closed convex subset of E, f : C → C a contractive mapping (or a weakly contractive mapping), and T : C → C a nonexpansive mapping with the fixed point set F (T ) = ∅. Let {xn} be generated by a new composite iterative scheme: yn = λnf (xn)+(1−λn)T xn, x n+1 = (1−βn)yn +βnT yn, (n ≥ 0). It is proved that {xn} converges strongly to a point in F (T ), which is a solution of certain variational inequality provided the sequence {λn} ⊂ (0, 1) satisfies limn→∞ λn = 0 and 2 ∞ n=0 λn = ∞, {βn} ⊂ [0, a) for some 0 < a < 1 and the sequence {xn} is asymptotically regular.