Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems

Abstract

Abstract In this paper, we show the hierarchical convergence of the following implicit double-net algorithm: x s , t = s [ t f ( x s , t ) + ( 1 - t ) ( x s , t - μ A x s , t ) ] + ( 1 - s ) 1 λ s ∫ 0 λ s T ( v ) x s , t d ν , ∀ s , t ∈ ( 0 , 1 ) , where f is a ρ-contraction on a real Hilbert space H, A : H → H is an α-inverse strongly monotone mapping and S = {T(s)} s ≥ 0: H → H is a nonexpansive semi-group with the common fixed points set Fix(S) ≠ ∅, where Fix(S) denotes the set of fixed points of the mapping S, and, for each fixed t ∈ (0, 1), the net {x s, t } converges in norm as s → 0 to a common fixed point x t ∈ Fix(S) of {T(s)} s ≥ 0and, as t → 0, the net {x t } converges in norm to the solution x* of the following variational inequality: x * ∈ F i x ( S ) ; 〈 A x * , x - x * 〉 ≥ 0 , ∀ x ∈ F i x ( S ) . MSC(2000): 49J40; 47J20; 47H09; 65J15.