General convergence analysis for two-step projection methods and applications to variational problems

First a general model for two-step projection methods 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∈K, compute sequences {xk} and {yk} such that xk+1=(1−ak)xk+akPK[yk−ρT(yk)]for ρ>0yk=(1−bk)xk+bkPK[xk−ηT(xk)]for η>0, where T:K→H is a nonlinear mapping on K,PK is the projection of H onto K, and 0≤ak,bk≤1. The two-step model is applied to some variational inequality problems.