General Convergence Analysis for Three-step Projection Methods and Applications to Variational Problems

First a general model for a three-step projection method is introduced, and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let H be a real Hilbert space and K be a nonempty closed convex subset of H. For arbitrarily chosen initial points x0,y0,z0 ∈ K, compute sequences xn,y-n,zn such thatxn+1 = (1 - αn - rn)xn + αkPK[yn - ρTyn] + rnun,yn = (1 - βn - δn)xn + βnPK[zn - ηTzn] + δnvn,zn = (1 - an - λn)xn + akPK[xn - γTxn] + λnωn.For η,ρ,γ 0 are constants,{αn},{βn},{an},{rn},{δn},{λn} - [0,1], {un},{vn},{ωn} are sequences in K, and 0 ≤ αn + rn ≤ 1,0 ≤ βn + δn ≤ 1,0 ≤ an + λn ≤ 1,-n ≥ 0, where T : K → H is a nonlinear mapping onto K. At last three-step models are applied to some variational inequality problems.