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**Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.**
*(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
\[
F(x,y)\geq 0\quad \forall y\in C,
\]
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 \(EP(F)\).

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 \(\{x_n\}\) and \(\{u_n\}\), approximating in the viscosity sense the equilibria that are also the fixed points \(F(S)\) of a nonexpansive map \(S\), i.e., both of them converge strongly to a point \(z\in EP(F)\cap F(S)\), where \(z\) is the projection of \(f(z)\) onto \(EP(F)\cap F(S)\). 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)].

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 \(\{x_n\}\) and \(\{u_n\}\), approximating in the viscosity sense the equilibria that are also the fixed points \(F(S)\) of a nonexpansive map \(S\), i.e., both of them converge strongly to a point \(z\in EP(F)\cap F(S)\), where \(z\) is the projection of \(f(z)\) onto \(EP(F)\cap F(S)\). 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)].

Reviewer: Rita Pini (Milano)

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

49J40 | Variational inequalities |

47H10 | Fixed-point theorems |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

65J15 | Numerical solutions to equations with nonlinear operators |

90C47 | Minimax problems in mathematical programming |

### Keywords:

viscosity approximation method; equilibrium problem; fixed point; nonexpansive mapping; strong convergence
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\textit{S. Takahashi} and \textit{W. Takahashi}, J. Math. Anal. Appl. 331, No. 1, 506--515 (2007; Zbl 1122.47056)

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### References:

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[2] | Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, 117-136 (2005) · Zbl 1109.90079 |

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