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

In this paper, we show the hierarchical convergence of the following implicit double-net algorithm: x(s,t) = s[tf(x(s,t))+(1-t)(x(s,t)-mu Ax(s,t))]+(1-s)1/lambda(s) integral(lambda s)(0) T(nu)x(s,t)d nu, for all s,t epsilon (0,1), where f is a rho-contraction on a real Hilbert space H, A : H -> H is an alpha-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) not equal a..., where Fix(S) denotes the set of fixed points of the mapping S, and, for each fixed t epsilon (0, 1), the net {x(s,t)} converges in norm as s -> 0 to a common fixed point x(t) epsilon Fix(S) of {T(s)}(s >= 0)and, as t -> 0, the net {x(t)} converges in norm to the solution x* of the following variational inequality: {x* epsilon Fix(S); (Ax*, x-x*)>= 0,for all x epsilon Fix(S).