学术
广义松弛余强制变分不等式体系及二步投影方法
彭再云[1] 唐平[2]
[1]重庆交通大学理学院,重庆400067 [2]重庆文理学院数学与计算机科学系,重庆402160
摘 要:
设H为希尔伯特空间，〈＊，＊〉，‖＊‖分别表示希尔伯特空间H中的内积和范数。K为H中的闭凸子集，T：K×K→H为K×K上的任一映象。本文将重点讨论下面一类非线性变分体系（SNVI）问题：求x^＊，y^＊∈K使得（ρT（y^＊，x^＊）＋x^＊-y^＊，y-x^＊）≥0，任意y∈K，ρ〉0，（ηT（x^＊,y^＊）＋y^＊-x^＊,z-y^＊）≥0，任意z∈K，η〉0。文章中首先给出了希尔伯特空间日中一类带误差的二步投影方法，然后借助于投影方法的收敛性证明了由该算法生成的迭代序列强收敛于此类广义松弛余强制变分不等式体系（SNVI）问题的精确解。文中结果主要推广了Verma和S．S．Chang等的主要结论。[著者文摘]
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文章出处:
《重庆师范大学学报：自然科学版》-2007年24卷4期,36 -8-11,36页
栏目信息:
分 类 号:
文献标识码:
A
文章编号:
1672-6693（2007）04-0008-04
Generalized System of Relaxed Cocoercive Variational Inequalities and Projection Methods in Hilbert Space
PENG Zai-yun, TANG Ping （ 1. College of Science, Chongqing JiaoTong University, Chongqing 400067 ; 2. Dept. of Mathematics and Computer Science, Chongqing College of Liberal Arts and Science, Chongqing 402160）
Abstract:
Let H be a real Hilbert space with the inner product,〈＊,＊〉 and norm ‖＊‖. Let T ： K × K→H be any mapping on K × K, and let K be a closed convex subset of H. We consider a system of nonlinear variational inequality（SNVI） problem as follows： to find x^＊,y^＊∈K suchthat （ρT（y^＊,x^＊）＋x^＊-y^＊,y-x^＊）≥0,arbitary y∈K,ρ〉0,（ηT（x^＊,y^＊）＋y^＊-x^＊,z-y^＊）≥0. arbitary x∈K,η〉0.Based on the convergence of projection methods, the approximate solvability of generalized system of relaxed cocoercive nonlinear variational inequality problems and two-step projection methods with errors in the setting of Hilbert space are considered. Let T ： K × K → H a relaxed （γ,r）-cocoercive and μ-Lipschitz continuous in the first variable. Suppose that （x^＊ ,y^＊ ） ∈K × K is a solution to （SNVI） problem （ 1 ） , （2） and that {xn} , {yn} are sequences generated by Algorithm 1. If {un} ,{vn} are bounded sequences in K, {αn},{βn} ,{dn} , {en} are four sequences in [ 0,1 ] satisfying the following conditions： 1 ） β→1, en→0 （ n→∞） ;2 ）∑n=0^∞αn=∞ ,）∑n=0^∞ dn〈∞;3）0〈p,η〈2（r-γμ^2）/μ^2. Then the sequences {xn} and {yn} converge strong to x^＊ and y^＊ are solved respectively. The results presented in the paper improve the main results in Verma, Chang and the references therein.[著者文摘]
Key words:
relaxed cocoercive nonlinear variational inequalities ; two-step projection methods with errors ; relaxed mappings ; cocoer-cive mappings; convergence of projection methods
基金资助:
国家自然科学基金（No.10471159）.致谢：感谢张石生教授的指导与帮助!
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