Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings

Abstract

Let C be a nonempty closed convex subset of a real strictly convex and reflexive Banach space E which has a uniformly Gâteaux differentiable norm. Let f : C → C be a given contractive mapping and { T n } n = 1 ∞ : C → C be an infinite family of nonexpansive mappings such that the common fixed point sets F : = ⋂ n = 1 ∞ F ( T n ) ≠ ∅ . Let { α n } and { β n } be two real sequences in [0, 1]. For given x 0 ∈ C arbitrarily, let the sequence { x n } be generated iteratively by x n + 1 = α n f ( x n ) + β n x n + ( 1 - α n - β n ) W n x n , where W n is the W-mapping generated by the mappings T n , T n - 1 , … , T 1 and ξ n , ξ n - 1 , … , ξ 1 . Suppose the iterative parameters { α n } and { β n } satisfy the following control conditions: (C1) lim n → ∞ α n = 0 ; (C2) ∑ n = 0 ∞ α n = ∞ ; (B5) limsup n → ∞ β n < 1 . Then the sequence { x n } converges strongly to p ∈ F where p is the unique solution in F to the following variational inequality: 〈 ( I - f ) p , j ( p - x ∗ ) 〉 ⩽ 0 for all x ∗ ∈ F .