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**A new iterative method for finding common solutions of a system of equilibrium problems, fixed-point problems, and variational inequalities.**
*(English)*
Zbl 1206.47080

Summary: We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove a strong convergence theorem. The results in this paper unify and generalize some well-known results in the literature.

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

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

### Keywords:

extragradient method; viscosity approximation method; nonexpansive mappings; relaxed cocoercive mapping; Hilbert space; strong a convergence
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\textit{J.-W. Peng} et al., Abstr. Appl. Anal. 2010, Article ID 428293, 27 p. (2010; Zbl 1206.47080)

### References:

[1] | P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, pp. 117-136, 2005. · Zbl 1109.90079 |

[2] | J.-W. Peng and J.-C. Yao, “A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. 6001-6010, 2009. · Zbl 1178.47047 |

[3] | V. Colao, G. L. Acedo, and G. Marino, “An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2708-2715, 2009. · Zbl 1175.47058 |

[4] | S. Saeidi, “Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4195-4208, 2009. · Zbl 1225.47110 |

[5] | S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506-515, 2007. · Zbl 1122.47056 |

[6] | M. Shang, Y. Su, and X. Qin, “A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2007, Article ID 95412, 9 pages, 2007. · Zbl 1158.47317 |

[7] | S. Plubtieng and R. Punpaeng, “A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 455-469, 2007. · Zbl 1127.47053 |

[8] | X. Qin, M. Shang, and Y. Su, “A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3897-3909, 2008. · Zbl 1170.47044 |

[9] | X. Qin, M. Shang, and Y. Su, “Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems,” Mathematical and Computer Modelling, vol. 48, no. 7-8, pp. 1033-1046, 2008. · Zbl 1187.65058 |

[10] | E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” Mathematics Students, vol. 63, pp. 123-145, 1994. · Zbl 0888.49007 |

[11] | A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359-370, 2007. · Zbl 1147.47052 |

[12] | S. Plubtieng and R. Punpaeng, “A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 548-558, 2008. · Zbl 1154.47053 |

[13] | V. Colao, G. Marino, and H.-K. Xu, “An iterative method for finding common solutions of equilibrium and fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 340-352, 2008. · Zbl 1141.47040 |

[14] | J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1401-1432, 2008. · Zbl 1185.47079 |

[15] | J.-W. Peng and J.-C. Yao, “Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping,” Journal of Global Optimization, vol. 46, no. 3, pp. 331-345, 2010. · Zbl 1208.90171 |

[16] | J.-W. Peng and J.-C. Yao, “A modified CQ method for equilibrium problems, fixed points and variational inequality,” Fixed Point Theory, vol. 9, no. 2, pp. 515-531, 2008. · Zbl 1172.47051 |

[17] | S.-S. Chang, H. W. Joseph Lee, and C. K. Chan, “A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3307-3319, 2009. · Zbl 1198.47082 |

[18] | L.-C. Ceng, S. Al-Homidan, Q. H. Ansari, and J.-C. Yao, “An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 967-974, 2009. · Zbl 1167.47307 |

[19] | A. N. Iusem and W. Sosa, “Iterative algorithms for equilibrium problems,” Optimization, vol. 52, no. 3, pp. 301-316, 2003. · Zbl 1176.90640 |

[20] | T. T. V. Nguyen, J. J. Strodiot, and V. H. Nguyen, “A bundle method for solving equilibrium problems,” Mathematical Programming, vol. 116, no. 1-2, pp. 529-552, 2009. · Zbl 1155.49006 |

[21] | H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659-678, 2003. · Zbl 1043.90063 |

[22] | G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43-52, 2006. · Zbl 1095.47038 |

[23] | A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46-55, 2000. · Zbl 0957.47039 |

[24] | X. Qin, M. Shang, and H. Zhou, “Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 242-253, 2008. · Zbl 1147.65048 |

[25] | Z. Opial, “Weak convergence of the sequence of successive approximation for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 561-597, 1967. · Zbl 0179.19902 |

[26] | R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75-88, 1970. · Zbl 0222.47017 |

[27] | H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240-256, 2002. · Zbl 1013.47032 |

[28] | W. Takahashi and K. Shimoji, “Convergence theorems for nonexpansive mappings and feasibility problems,” Mathematical and Computer Modelling, vol. 32, no. 11-13, pp. 1463-1471, 2000. · Zbl 0971.47040 |

[29] | K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387-404, 2001. · Zbl 0993.47037 |

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