Strong Convergence of Averaging Iterations of Nonexpansive Nonself-Mappings in Banach Spaces

Abstract. Let E be a real uniformly convex Banach space whose norm is uni-formly Gâteaux differentiable, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E with P a sunny nonexpansive retrac-tion. Let T: C − → E be a nonexpansive nonself-mapping with F (T) = ∅. For any x, x0, y, y0 ∈ C, let {xn} and {yn} be generated by the algorithm xn+1 = αnx + (1 − αn) 1n+1 ∑n j=0(PT) jxn and yn+1 = 1 n+1 ∑n j=0 P (αny + (1 − αn)(TP)jyn) for all n ≥ 0 respectively, where {αn} ⊆ [0, 1] is a sequence satisfying certain con-ditions. Then the strong convergence of both {xn} and {yn} to an element of fixed points of T is proved. The results presented in this paper generalize, extend and improve the corresponding results of Matsushita and Kuroiwa [Strong convergence of averaging iterations of nonexpansive nonself-mappings, J. Math. Anal. Appl. 294 (2004) 206-214] and many others