三步投影方法的广义收敛及其在变分不等式中的应用

摘要

在这篇文章中,首先介绍了带有误差估计的三步投影法的广义模型,其次将其应用到解决一组在Hilbert空间中的非线性变分不等式的近似解.令H是实值Hilbert空间,K是H中的非空闭凸集.对任意选定的起始点x0,y0,z0∈K,计算序列{xn},{yn}and{zn},使得xn+1=(1-an-dn)xn+an PK[zn-ρT(zn)]+dnun forρ＞0yn =(1-bn-en)xn +bnPK[xn-ηT(xn)]+envn forη＞0zn =(1-cn -fn)xn +cnPK[yn -λT(yn)]+ fnwn forλ＞0其中T:K→H是K上的非线性映射,PK是H到K的投影且o≤an,bn,cn,dn,en,fn≤1,{un},{vn},{wn}是K中的有界序列.三步投影模型应用到许多变分不等式问题.%In this paper, first we introduce a general model with error estimate for three-step projection methods 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 }, {yn} and {zn } such thatxn+1 =(1-an-dn)xn +anPK[zn-ρT(zn)]+dnun for ρ＞0yn =(1-bn-en)xn +bnPK[xn-ηT(xn)]+envn forη＞0zn =(1-cn -fn)xn +cnPK[yn -λT(yn)]+ fnwn forλ＞0where T:K→H is a nonlinear mapping on K, PK is the projection of H onto K, and o ≤ an, bn, cn, dn,en, fn ≤ 1, {un }, {vn }, {wn } are bounded sequences of K. The three-step model is applied to some variational inequality problems.