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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.

47J20Inequalities involving nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
49J40Variational methods including variational inequalities
Full Text: DOI
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