Banach空间中单调算子零点的粘性逼近方法
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摘要
在一致光滑严格凸的具有对偶空间的自反Banach空间中,针对单调算子的零点逼近,提出了新的粘性迭代算法.通过定义两个单调算子确定的联合预解式,分析了该联合预解式的基本性质并推导联合预解式与常规预解式之间的关系;随后利用联合预解式构造了强单调算子与单调算子公共零点的粘性逼近序列,并在一定条件下证明了该迭代序列的强收敛性,获得了强收敛定理,同时证明了该收敛点正是某类变分不等式问题的唯一解,推广和统一了部分非线性算子类文献的结论.
The viscosity iterative algorithm for the zero of monotone operators in convex closed Banach spaces with dual space is studied in this paper. In the reflexive real Banach spaces,the resolvent technique is discussed to obtain the viscosity approximation method for the zeros of monotone operators. The strong convergence theorems are proved under some appropriate conditions,at the same time,the approximated zero is exactly the solution of some variation inequality.The results have extended and unified some similar results of nonlinear operators.
The viscosity iterative algorithm for the zero of monotone operators in convex closed Banach spaces with dual space is studied in this paper. In the reflexive real Banach spaces,the resolvent technique is discussed to obtain the viscosity approximation method for the zeros of monotone operators. The strong convergence theorems are proved under some appropriate conditions,at the same time,the approximated zero is exactly the solution of some variation inequality.The results have extended and unified some similar results of nonlinear operators.
引文
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[6]唐艳,闻道君.非扩张映射不动点的粘性逼近方法[J].重庆工商大学学报(自然科学版),2009,26(5):420-423TANG Y,WEN D J.Viscosity Approximation Method for the Fixed Point of Non-expansive Mappings[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2009,26(5):420-423
[7]徐灵.一种带有投影校正步的部分并行分离算法[J].重庆工商大学学报(自然科学版),2016,33(6):31-35XU L.A Partial Parallel Separation Method with Projection Correction Step[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2016,33(6):31-35
[8]AOYAMA K,TOYODA M.Approximation of Zeros of Accetive Operators in Banach spapces[J].Israel Journal of Mathematics,2017,220:803-816
[9]CHANG S S.JOSEPH H W.CHAN C K.On Reich's Strong Convergence Theorem for Asymptotically Nonexpansive Mappings in Banach Spaces[J].Nonlinear Analysis,Theory,Methods and Applications,2017,66:2364-2374
[2]ROCKAFELLAR R T.Monotone Operators and the Proximal Point Algorithm[J].SIAM Journal of Control and Optimization,1976(14):877-898
[3]GULER O.On the Convergence of the Proximal Point Algorithm for Convex Optimization[J].SIAM J Control Optim,1991,29:403-419
[4]XU H K.A Regularization Method for the Proximal Point Algorithm[J].J Glob Optim,2006,36:115-125
[5]YAO Y H,NASEER S.Strong Convergence of a Proximal Point Algorithm with General Errors[J].Optim Lett,2012(6):621-628
[6]唐艳,闻道君.非扩张映射不动点的粘性逼近方法[J].重庆工商大学学报(自然科学版),2009,26(5):420-423TANG Y,WEN D J.Viscosity Approximation Method for the Fixed Point of Non-expansive Mappings[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2009,26(5):420-423
[7]徐灵.一种带有投影校正步的部分并行分离算法[J].重庆工商大学学报(自然科学版),2016,33(6):31-35XU L.A Partial Parallel Separation Method with Projection Correction Step[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2016,33(6):31-35
[8]AOYAMA K,TOYODA M.Approximation of Zeros of Accetive Operators in Banach spapces[J].Israel Journal of Mathematics,2017,220:803-816
[9]CHANG S S.JOSEPH H W.CHAN C K.On Reich's Strong Convergence Theorem for Asymptotically Nonexpansive Mappings in Banach Spaces[J].Nonlinear Analysis,Theory,Methods and Applications,2017,66:2364-2374