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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.

##### MSC:
 49J40 Variational inequalities