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

AbstractIn this paper, we show the hierarchical convergence of the following implicit double-net algorithm: xs,t=s[tf(xs,t)+(1-t)(xs,t-μAxs,t)]+(1-s)1λs∫0λsT(v)xs,tdν,∀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 {xs, t} converges in norm as s → 0 to a common fixed point xt ∈ Fix(S) of {T(s)}s ≥ 0and, as t → 0, the net {xt} converges in norm to the solution x* of the following variational inequality: x*∈Fix(S);〈Ax*,x-x*〉≥0,∀x∈Fix(S).MSC(2000): 49J40; 47J20; 47H09; 65J15.