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

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 HH be a real Hilbert space and KK be a nonempty closed convex subset of HH. For arbitrarily chosen initial points x0,y0∈Kx0,y0∈K, compute sequences {xk}{xk} and {yk}{yk} such that View the MathML sourcexk+1=(1−ak)xk+akPK[yk−ρT(yk)]for ρ>0 Turn MathJax on View the MathML sourceyk=(1−bk)xk+bkPK[xk−ηT(xk)]for η>0, Turn MathJax on where T:K→HT:K→H is a nonlinear mapping on K,PKK,PK is the projection of HH onto KK, and 0≤ak,bk≤10≤ak,bk≤1. The two-step model is applied to some variational inequality problems.