Generalized system for relaxed cocoercive variational inequalities and projection methods. (English) Zbl 1056.49017

Summary: Let \(K\) be a nonempty closed convex subset of a real Hilbert space \(H\). The approximate solvability of a system of nonlinear variational inequality problems, based on the convergence of projection methods, is discussed as follows: find an element \((x^*,y^*)\in K\times K\) such that \[ \bigl\langle\rho T(y^*,x^*)+x^*-y^*,x-x^*\bigr \rangle\geq 0,\quad \forall x\in K\text{ and }\rho>0, \]
\[ \bigl\langle\eta T(x^*,y^*)+y^*-x^*,x-y^*\bigr\rangle\geq 0,\quad \forall x\in K\text{ and }\eta>0, \] where \(T:K\times K\to H\) is a nonlinear mapping on \(K\times K\).


49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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