On Computability and Applicability of Mann-Reich-Sabach-Type Algorithms for Approximating the Solutions of Equilibrium Problems in Hilbert Spaces

Abstract

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 $EP(F)$ 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 $\{K_n{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ of closed convex subsets of $H$ from an arbitrary $x_{\mathrm{0}}\in H$ and a sequence $\{x_n{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ of the metric projections of $x_{\mathrm{0}}$ into $K_n$. The obtained result is a partial resolution of the controversy over the computability of such algorithms in the contemporary literature.