A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings

Abstract EN

Let X be a real uniformly convex Banach space and C a closed convex nonempty subset of X.

Let {Ti}i=1r be 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(1)=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.