Strong Convergence of a Selection of Ishikawa-Reich-Sabach-type Algorithm

Abstract

We establish the strong convergence of a selection of an Ishikawa-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points $F(T)$ of a multi-valued (or single-valued) 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 contribution to the study on the computability and applicability of algorithms

for approximating the solutions of equilibrium problems for bifunctions involving the construction of the sequence $\{K_n\}_{n=1}^\infty$ of closed convex subsets of $H$ from an arbitrary $x_0\in H$ and the sequence $\{x_n\}_{n=1}^\infty$ of the metric projections of $x_0$ into $K_n$. The results obtained are contributions to the resolution of the controversy over the computability and applicability of such algorithms in the contemporary literature.