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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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[4] | Moudafi, A., Viscosity approximation methods for fixed-point problems, J. math. anal. appl., 241, 46-55, (2000) · Zbl 0957.47039 |

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[8] | Takahashi, W., Nonlinear functional analysis, (2000), Yokohama Publishers Yokohama |

[9] | Takahashi, W., Convex analysis and approximation of fixed points, (2000), Yokohama Publishers Yokohama, (in Japanese) |

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[11] | Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 58, 486-491, (1992) · Zbl 0797.47036 |

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