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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.

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