Projection methods, algorithms, and a new system of nonlinear variational inequalities.

*(English)*Zbl 0995.47042Let \(H\) 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.

Reviewer: Mikhail Yu.Kokurin (Yoshkar-Ola)

##### MSC:

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

47J25 | Iterative procedures involving nonlinear operators |

49J40 | Variational inequalities |

##### Keywords:

system of nonlinear variational inequalities; system of complementarity problems; iterative algorithms; projection methods; strongly monotone and Lipschitz continuous mapping
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\textit{R. U. Verma}, Comput. Math. Appl. 41, No. 7--8, 1025--1031 (2001; Zbl 0995.47042)

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