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**Iterative algorithms for solving the system of mixed equilibrium problems, fixed-point problems, and variational inclusions with application to minimization problem.**
*(English)*
Zbl 1235.65062

Summary: We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for a \(\beta\)-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

### MSC:

65K05 | Numerical mathematical programming methods |

49J40 | Variational inequalities |

47H10 | Fixed-point theorems |

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\textit{T. Chamnarnpan} and \textit{P. Kumam}, J. Appl. Math. 2012, Article ID 538912, 29 p. (2012; Zbl 1235.65062)

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### References:

[1] | E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1-4, pp. 123-145, 1994. · Zbl 0888.49007 |

[2] | P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117-136, 2005. · Zbl 1109.90079 |

[3] | S. D. Flåm and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29-41, 1997. · Zbl 0890.90150 |

[4] | 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 |

[5] | T. Jitpeera, U. Witthayarat, and P. Kumam, “Hybrid algorithms of common solutions of generalized mixed equilibrium problems and the common variational inequality problems with applications,” Fixed Point Theory and Applications, vol. 2011, Article ID 971479, 28 pages, 2011. · Zbl 1215.49014 |

[6] | T. Jitpeera and P. Kumam, “An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings,” Journal of Nonlinear Analysis and Optimization: Theory & Applications, vol. 1, no. 1, pp. 71-91, 2010. · Zbl 1413.47112 |

[7] | T. Jitpeera and P. Kumam, “A new hybrid algorithm for a system of equilibrium problems and variational inclusion,” Annali dell’Universita di Ferrara, vol. 57, no. 1, pp. 89-108, 2011. · Zbl 1216.47097 |

[8] | T. Jitpeera and P. Kumam, “Hybrid algorithms for minimization problems over the solutions of generalized mixed equilibrium and variational inclusion problems,” Mathematical Problems in Engineering, vol. 2011, Article ID 648617, 25 pages, 2011. · Zbl 1235.65071 |

[9] | P. Kumam, U. Hamphries, and P. Katchang, “Common solutions of generalized mixed equilibrium problems, variational inclusions and common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings,” Journal of Applied Mathematics, vol. 2011, Article ID 953903, 27 pages, 2011. · Zbl 1297.47084 |

[10] | P. Sunthrayuth and P. Kumam, “A new general iterative method for solution of a new general system of variational inclusions for nonexpansive semigroups in Banach spaces,” Journal of Applied Mathematics, vol. 2011, Article ID 187052, 29 pages, 2011. · Zbl 1221.47110 |

[11] | P. Katchang and P. Kumam, “Convergence of iterative algorithm for finding common solution of fixed points and general system of variational inequalities for two accretive operators,” Thai Journal of Mathematics, vol. 9, no. 2, pp. 319-335, 2011. · Zbl 06148520 |

[12] | W. Kumam, P. Junlouchai, and P. Kumam, “Generalized systems of variational inequalities and projection methods for inverse-strongly monotone mappings,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 976505, 24 pages, 2011. · Zbl 1290.47068 |

[13] | P. Kumam, “A relaxed extragradient approximation method of two inverse-strongly monotone mappings for a general system of variational inequalities, fixed point and equilibrium problems,” Bulletin of the Iranian Mathematical Society, vol. 36, no. 1, pp. 227-250, 2010. · Zbl 1231.47067 |

[14] | P. Kumam and C. Jaiboon, “Approximation of common solutions to system of mixed equilibrium problems, variational inequality problem, and strict pseudo-contractive mappings,” Fixed Point Theory and Applications, vol. 2011, Article ID 347204, 30 pages, 2011. · Zbl 1215.47075 |

[15] | C. Jaiboon and P. Kumam, “A general iterative method for addressing mixed equilibrium problems and optimization problems,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 73, no. 5, pp. 1180-1202, 2010. · Zbl 1205.49011 |

[16] | Y. J. Cho, I. K. Argyros, and N. Petrot, “Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2292-2301, 2010. · Zbl 1205.65185 |

[17] | Y. J. Cho, N. Petrot, and S. Suantai, “Fixed point theorems for nonexpansive mappings with applications to generalized equilibrium and system of nonlinear variational inequalities problems,” Journal of Nonlinear Analysis and Optimization, vol. 1, no. 1, pp. 45-53, 2010. · Zbl 1413.47076 |

[18] | Y. J. Cho and N. Petrot, “On the system of nonlinear mixed implicit equilibrium problems in Hilbert spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 437976, 12 pages, 2010. · Zbl 1184.49003 |

[19] | Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242-250, 2011. · Zbl 1266.90186 |

[20] | W. A. Kirk, “Fixed point theorem for mappings which do not increase distances,” The American Mathematical Monthly, vol. 72, pp. 1004-1006, 1965. · Zbl 0141.32402 |

[21] | P. Hartman and G. Stampacchia, “On some non-linear elliptic differential-functional equations,” Acta Mathematica, vol. 115, pp. 271-310, 1966. · Zbl 0142.38102 |

[22] | J.-C. Yao and O. Chadli, “Pseudomonotone complementarity problems and variational inequalities,” in Handbook of Generalized Convexity and Monotonicity, J. P. Crouzeix, N. Haddjissas, and S. Schaible, Eds., vol. 76, pp. 501-558, Springer, New York, NY, USA, 2005. · Zbl 1106.49020 |

[23] | Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2011, article 79, 2011. · Zbl 1270.47066 |

[24] | Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” European Journal of Operational Research. In Press. · Zbl 1280.90097 |

[25] | Y. Yao, Y.-C. Liou, and C.-P. Chen, “Algorithms construction for nonexpansive mappings and inverse-strongly monotone mappings,” Taiwanese Journal of Mathematics, vol. 15, no. 5, pp. 1979-1998, 2011. |

[26] | Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical and Computer Modelling, vol. 55, pp. 1506-1515, 2012. · Zbl 1275.47130 |

[27] | Y. Yao, Y.-C. Liou, S. M. Kang, and Y. Yu, “Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 17, pp. 6024-6034, 2011. · Zbl 1393.47042 |

[28] | Y. Yao, Y.-C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization. In Press. · Zbl 1260.47085 |

[29] | 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 |

[30] | Y. Hao, “Some results of variational inclusion problems and fixed point problems with applications,” Applied Mathematics and Mechanics, vol. 30, no. 12, pp. 1589-1596, 2009. · Zbl 1182.47052 |

[31] | M. Liu, S. S. Chang, and P. Zuo, “An algorithm for finding a common solution for a system of mixed equilibrium problem, quasivariational inclusion problem, and fixed point problem of nonexpansive semigroup,” Journal of Inequalities and Applications, vol. 2010, Article ID 895907, 23 pages, 2010. · Zbl 1206.47078 |

[32] | S.-s. Zhang, J. H. W. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variational inclusion and fixed point problems,” Applied Mathematics and Mechanics, vol. 29, no. 5, pp. 571-581, 2008. · Zbl 1196.47047 |

[33] | R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 46-55, 2000. · Zbl 0222.47017 |

[34] | B. Lemaire, “Which fixed point does the iteration method select?” in Recent Advances in Optimization, vol. 452 of Lecture Note in Economics and Mathematical Systems, pp. 154-167, Springer, Berlin, Germany, 1997. · Zbl 0882.65042 |

[35] | 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 |

[36] | H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 61, no. 3, pp. 341-350, 2005. · Zbl 1093.47058 |

[37] | Y. Su, M. Shang, and X. Qin, “An iterative method of solution for equilibrium and optimization problems,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 69, no. 8, pp. 2709-2719, 2008. · Zbl 1170.47047 |

[38] | H. He, S. Liu, and Y. J. Cho, “An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4128-4139, 2011. · Zbl 1368.47060 |

[39] | T. Jitpeera and P. Kumam, “A general iterative algorithm for generalized mixed equilibrium problems and variational inclusions approach to variational inequalities,” International Journal of Mathematics and Mathematical Sciences, Article ID 619813, 25 pages, 2011. · Zbl 1215.49013 |

[40] | H. Brézis, “Opérateur maximaux monotones,” in Mathematics Studies, vol. 5, North-Holland, Amsterdam, The Netherlands, 1973. |

[41] | Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591-597, 1967. · Zbl 0179.19902 |

[42] | 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 |

[43] | F. E. Browder, Ed., Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, vol. 18 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 1976. · Zbl 0327.47022 |

[44] | J.-W. Peng, Y.-C. Liou, and J.-C. Yao, “An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions,” Fixed Point Theory and Applications, vol. 2009, Article ID 794178, 21 pages, 2009. · Zbl 1163.91463 |

[45] | C. Klin-Eam and S. Suantai, “A new approximation method for solving variational inequalities and fixed points of nonexpansive mappings,” Journal of Inequalities and Applications, vol. 2009, Article ID 520301, 16 pages, 2009. · Zbl 1185.47077 |

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