*(English)*Zbl 1122.47056

The authors provide a result strictly related to Theorem 1 in [*A. Tada* and *W. Takahashi*, Proc. NACA (Okinawa, 2005), 609–617 (2007; Zbl 1122.47055), reviewed above]. Indeed, they consider the equilibrium problem: find $x\in C$ such that

where $C$ is a nonempty, closed and convex subset of a real Hilbert space $H$, and $F:C\times C\to \mathbb{R}$. The set of solutions is denoted by $EP\left(F\right)$.

Under the same assumptions of Theorem 1, and given in addition a contraction $f:H\to H$, they find a way to generate two sequences of points, namely $\left\{{x}_{n}\right\}$ and $\left\{{u}_{n}\right\}$, approximating in the viscosity sense the equilibria that are also the fixed points $F\left(S\right)$ of a nonexpansive map $S$, i.e., both of them converge strongly to a point $z\in EP\left(F\right)\cap F\left(S\right)$, where $z$ is the projection of $f\left(z\right)$ onto $EP\left(F\right)\cap F\left(S\right)$. As corollaries, they get results previously obtained by *R. Wittman* [Arch. Math. 58, No. 5, 486–491 (1992; Zbl 0797.47036)] and *P. L. Combettes* and *S. A. Hirstoaga* [J. Nonlinear Convex Anal. 6, No. 1, 117–136 (2005; Zbl 1109.90079)].

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47J20 | Inequalities involving nonlinear operators |

49J40 | Variational methods including variational inequalities |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |

47H09 | Mappings defined by “shrinking” properties |

65J15 | Equations with nonlinear operators (numerical methods) |

90C47 | Minimax problems |