A new iterative method for common fixed points of a finite family of nonexpansive mappings

Abstract

Let X be a real uniformly convex Banach space and C a closed convex nonempty subset of X. Let {Ti}i=1rbe a finite family of nonexpansive self-mappings of C. For a given x1∈ C, let {xn} and {xn(i)}, i = 1,2,.., r, be sequences defined xn(0)= xn, xn(i)= an1(1)T1xn(0)+ (1 - an1(1)xn(0), xn(2)= an2(2)T2xn(1)+ an1(2)T1xn+ (1 - an2(2)- an1(2))xn,..;, xn+1= xn(r)- anr(r)Trxn(r-1) + an(r-1)(r)Tr-1xn(r-2)+..; + an1(r)T1xn+ (1 - an(r)(r)- an(r-1)(r)-..; - an1(r)xn, n ≥ 1, where ani(j)∈ (0, 1) for all j ∈ {1, 2,..,r}, n ∈ N and i = 1, 2,..;, j. In this paper, weak and strong convergence theorems of the sequence {xn} to a common fixed point of a finite family of nonexpansive mappings Ti(i = 1, 2,..;, r) are established under some certain control conditions.