Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

Source abstract

In this paper, we study the convergence of the sequence defined by $$x 0 ∈ C , x n + 1 = α n x + ( 1 − α n ) T x n , n = 0 , 1 , 2 , … , x_0\in C , \;\; x_{n+1} = \alpha _{n}x + (1-\alpha _{n})Tx_n , \;\; n=0,1,2, \ldots ,$$ where 0 ≤ α n ≤ 1 0 \leq \alpha _n \leq 1 and T T is a nonexpansive mapping from a closed convex subset of a Banach space into itself.