Multiple Steps Projection Algorithms

In this paper,we introduce some theories and developments of variational inequal-ity,and we establish k-step iterative projection algorithms.Let H be a real Hilbert space and K be a non-empty closed convex subset of H.For arbitrarily initial points x1,0,x2,0,x3,0,…,xk,0∈K,we calculate sequences{x1,n},{x2,n},{x3,n},…,{xk,n} as follows: Where T:H→H is a nonlinear mapping,ρ1,ρ2,ρ3,…,ρk are contants,{α1,n},{α2,n}, {α3,n},…,{αk,n},{β1,n},{β2,n},{β3,n},…,{βk,n}(?)[0,1],and{u1,n},{u2,n},{u3,n},…,{uk,n} are bounded sequences in K,and 0≤α1,n+β1,n≤1,0≤α2,n+β2,n≤1, 0≤α3,n+β3,n≤1,….0≤αk,n+βk,n≤1,(?)n≥0.The k-step projection method can be applied to some variational inequalities,and it is a generaliation of two-step method([9])and three-step method([5]).For any initial points,the k-step projection method is convergence.