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On computability and applicability of Mann-Reich-Sabach-type algorithms for approximating the solutions of equilibrium problems in Hilbert spaces. (English) Zbl 1470.47061

Summary: We establish the existence of a strong convergent selection of a modified Mann-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points \(F(T)\) of a multivalued (or single-valued) \(k \)-strictly pseudocontractive-type mapping \(T\) and the set of solutions \(E P(F)\) of an equilibrium problem for a bifunction \(F\) in a real Hilbert space \(H\). This work is a continuation of the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of a sequence \(\{K_n \}_{n = 1}^{\infty}\) of closed convex subsets of \(H\) from an arbitrary \(x_0 \in H\) and a sequence \(\{x_n \}_{n = 1}^{\infty}\) of the metric projections of \(x_0\) into \(K_n\). The obtained result is a partial resolution of the controversy over the computability of such algorithms in the contemporary literature.

MSC:

47J25 Iterative procedures involving nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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