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Zbl 1056.49017
Verma, R.U.
Generalized system for relaxed cocoercive variational inequalities and projection methods.
(English)
[J] J. Optimization Theory Appl. 121, No. 1, 203-210 (2004). ISSN 0022-3239; ISSN 1573-2878/e

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 $$\bigl\langle\rho T(y^*,x^*)+x^*-y^*,x-x^*\bigr \rangle\ge 0,\quad \forall x\in K\text{ and }\rho>0,$$ $$\bigl\langle\eta T(x^*,y^*)+y^*-x^*,x-y^*\bigr\rangle\ge 0,\quad \forall x\in K\text{ and }\eta>0,$$ where $T:K\times K\to H$ is a nonlinear mapping on $K\times K$.
MSC 2000:
*49J40 Variational methods including variational inequalities
47J20 Inequalities involving nonlinear operators

Keywords: relaxed cocoercive nonlinear variational inequalities; projection methods; relaxed cocoercive mappings; convergence of projection methods

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