Iterative approximation of fixed points of nonexpansive mappings

Let K be a nonempty closed convex subset of a real Banach space E   which has a uniformly Gâteaux differentiable norm and T:K→K be a nonexpansive mapping with F(T):={x∈K:Tx=x}≠∅F(T):={x∈K:Tx=x}≠∅. For a fixed δ∈(0,1)δ∈(0,1), define S:K→K by Sx:=(1−δ)x+δTxSx:=(1−δ)x+δTx, ∀x∈K∀x∈K. Assume that {zt}{zt} converges strongly to a fixed point z of T   as t→0t→0, where ztzt is the unique element of K   which satisfies zt=tu+(1−t)Tztzt=tu+(1−t)Tzt for arbitrary u∈Ku∈K. Let {αn}{αn} be a real sequence in (0,1)(0,1) which satisfies the following conditions: C1:limαn=0; C2:∑αn=∞. For arbitrary x0∈Kx0∈K, let the sequence {xn}{xn} be defined iteratively byxn+1=αnu+(1−αn)Sxn.xn+1=αnu+(1−αn)Sxn. Then, {xn}{xn} converges strongly to a fixed point of T.