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Projection methods, algorithms, and a new system of nonlinear variational inequalities. (English) Zbl 0995.47042
Let $$H$$ be a real Hilbert space and $$T: K \to H$$ a strongly monotone and Lipschitz continuous mapping from a closed convex subset $$K \subset H$$ into $$H$$. The paper is concerned with the problem of finding elements $$x^*, y^* \in K$$ such that $$x^*=P_K [y^*-\rho T(y^*)], y^*=P_K [x^*-\gamma T(x^*)]$$, where $$\rho, \gamma >0$$ and $$P_K$$ is the projection of $$H$$ onto $$K$$. To solve the problem the author proposes and studies the following iterative algorithm: $$x^{k+1}=(1-a^k)x^k+ a^k P_K [y^k-\rho T(y^k)], y^k=P_K [x^k-\gamma T(x^k)]$$, where $$0 \leq a^k <1$$ and $$\sum_{k=0}^{\infty} a^k =\infty$$. The strong convergence of $$\{ x^k\}$$ to $$x^*$$ is established provided that $$\rho$$ and $$\gamma$$ are sufficiently small.

##### MSC:
 47J20 Variational and other types of inequalities involving nonlinear operators (general) 47J25 Iterative procedures involving nonlinear operators 49J40 Variational inequalities
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