A general iterative method for an infinite family of nonexpansive mappings

Abstract

Let H be a real Hilbert space. Consider the iterative sequence x n + 1 = α n γ f ( x n ) + β n x n + ( ( 1 − β n ) I − α n A ) W n x n , where γ > 0 is some constant, f : H → H is a given contractive mapping, A is a strongly positive bounded linear operator on H and W n is the W -mapping generated by an infinite countable family of nonexpansive mappings T 1 , T 2 , … , T n , … and λ 1 , λ 2 , … , λ n , … such that the common fixed points set F ≔ ⋂ n = 1 ∞ Fix ( T n ) ≠ 0̸ . Under very mild conditions on the parameters, we prove that { x n } converges strongly to p ∈ F where p is the unique solution in F of the following variational inequality: 〈 ( A − γ f ) p , p − x ∗ 〉 ≤ 0 for all x ∗ ∈ F , which is the optimality condition for the minimization problem min x ∈ F 1 2 〈 A x , x 〉 − h ( x ) .