On a two-step algorithm for hierarchical fixed point problems and variational inequalities. (English) Zbl 1180.47040

The paper is concerned with the variational inequality problem of finding \(x^* \in \text{Fix}(T)\) with \(\langle (I-S)x^*, x-x^* \rangle \geq 0\) for all \(x\in \text{Fix}(T)\), where \(T,S: C\to C\) are nonexpansive mappings such that Fix\((T)\), the set of fixed points set of \(T\), is nonempty, and \(C\) is a closed convex subset of a Hilbert space \(H\). Let \(f: C \to C\) be a contraction. The authors study convergence properties of the iterative process \(x_{n+1}=\alpha_n f(x_n)+(1-\alpha_n) T y_n\), \(y_n=\beta_n S x_n+(1-\beta_n) x_n\), where \(\alpha_n,\beta_n \in [0,1]\).


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