## 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 $$K\times K$$.

### MSC:

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

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