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Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces. (English) Zbl 1149.47051
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.

47J20Inequalities involving nonlinear operators
47H04Set-valued operators
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47J25Iterative procedures (nonlinear operator equations)
49J40Variational methods including variational inequalities
65B05Extrapolation to the limit, deferred corrections
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