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,y∈K,ρ>0,〈ηT(x*,y*)+y*-x*,z-y*〉≥0,z∈K,η>0。文章中首先给出了希尔伯特空间H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。
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.
出处
《重庆师范大学学报（自然科学版）》
CAS
2007年第4期8-11,36,共5页
Journal of Chongqing Normal University:Natural Science
基金
国家自然科学基金(No.10471159)
关键词
松弛余强制非线性变分不等式
带误差的二步投影方法
松弛映象
余强制映象
投影方法的收敛性
relaxed cocoercive nonlinear variational inequalities
two-step projection methods with errors
relaxed mappings
cocoer-cive mappings
convergence of projection methods
