*(English)*Zbl 1215.49020

Summary: We devote this paper to solving the Variational Inequality (VI) of finding ${x}^{*}$ with property ${x}^{*}\in \text{Fix}\left(T\right)$ such that $\langle (A-\gamma f){x}^{*},x-{x}^{*}\rangle \ge 0$ for all $x\in \text{Fix}\left(T\right)$. Note that this hierarchical problem is associated with some convex programming problems. For solving the above VI, we suggest two algorithms:

Implicit Algorithm: ${x}_{t}=TPc[I-t(A-\gamma f)]{x}_{t}$ for all $t\in (0,1)$ and

Explicit Algorithm: ${x}_{n+1}={\beta}_{n}{x}_{n}+(1-{\beta}_{n})TPc[1-{\alpha}_{n}(A-\gamma f)]{x}_{n}$ for all $n\ge 0$ .

It is shown that these two algorithms converge strongly to the unique solution of the above VI. As special cases, we prove that the proposed algorithms strongly converge to the minimum norm fixed point of $T$.

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