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Approximation of solutions of variational inequalities for monotone mappings. (English) Zbl 1060.49006
Summary (extended): Let \(K\) be a closed convex subset of a real Hilbert space \(H\) and let \(P_K\) be the metric projection of \(H\) onto \(K\). A mapping \(A\) of \(K\) into \(H\) is called monotone if for all \(u,v\in K\), \(\langle Au-Av,u-v\rangle\geq 0\). The variational inequality problem is to find a \(u_0\in K\) such that \[ \langle Au_0,u-u_0\rangle\geq 0 \] for all \(u\in K\). The set of solutions of the variational inequality problem is denoted by \(\text{VI}(K,A)\). For \(\alpha>0\), a mapping \(A\) of \(K\) into \(H\) is called \(\alpha\)-inverse-strongly-monotone if \(A\) satisfies \[ \langle Au-Av,u-v\rangle\geq\alpha\|Au-Av\|^2 \] for all \(u,v\in K\).
We deal with an iterative process: \(x_0=x\in K\) and \(x_{n+1}=P_K(\alpha_n x_n+ (1-\alpha_n)P_K(x_n-\lambda_nAX_n))\) for every \(n=0,1,2,\dots,\) where \(\lambda_n>0\), \(-1<\alpha_n<1\), and \(A\) is an \(\alpha\)-inverse-strongly-monotone mapping and then we show that the sequence \(\{x_n\}\) converges weakly to an element of \(\text{VI}(K,A)\). We also prove a strong convergence theorem for \(\alpha\)-inverse-strongly-monotone mappings by applying the hybrid method in mathematical programming.

49J40 Variational inequalities