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.


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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