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Zbl 1149.47051
Qin, Xiaolong; Kang, Shin Min; Shang, Meijuan
(Qin, Xiao-long; Shang, Mei-juan)
Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces. (English)
[J] Appl. Anal. 87, No. 4, 421-430 (2008). ISSN 0003-6811; ISSN 1563-504X/e

The authors study a system of nonlinear variational inequalities: $$\align\langle sT_1(y^*,x^*)+g(x^*)-g(y^*),g(x)-g(x^*))\ge 0 &\quad\forall g(x)\in C,\ s>0,\\ \langle tT_2(x^*,y^*)+g(y^*)-g(x^*),g(x)-g(y^*))\ge 0&\quad \forall g(x)\in C,\ t>0, \endalign$$ where $T_1,T_2: H\times H\rightarrow H$ and $g: H\rightarrow H$ are three nonlinear mappings and $C$ is a nonempty, closed and convex subset of a Hilbert space $H$. Under an appropriate set of assumptions, the authors produce sequences $\{x_n\}$ and $\{y_n\}$ converging to the solutions $x^*,y^*\in H$ of the system.
[Leszek Gasiński (Kraków)]

MSC 2000:: *47J20 Inequalities involving nonlinear operators
47H04 Set-valued operators
47H06 Accretive operators, etc. (nonlinear)
47J25 Methods for solving nonlinear operator equations (general)
49J40 Variational methods including variational inequalities
65B05 Extrapolation to the limit

Keywords: variational inequality; projection method; relaxed cocoercive mapping; Hilbert space

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