##
**Convergence theorem based on a new hybrid projection method for finding a common solution of generalized equilibrium and variational inequality problems in Banach spaces.**
*(English)*
Zbl 1195.47044

Summary: The purpose of this paper is to introduce a new hybrid projection method for finding a common element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of the variational inequality for an \(\alpha \)-inverse-strongly monotone mapping, and the set of solutions of the generalized equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend, and unify some well-known strong convergence results in the literature.

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

49J40 | Variational inequalities |

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

PDF
BibTeX
XML
Cite

\textit{S. Saewan} et al., Abstr. Appl. Anal. 2010, Article ID 734126, 25 p. (2010; Zbl 1195.47044)

### References:

[1] | K. Ball, E. A. Carlen, and E. H. Lieb, “Sharp uniform convexity and smoothness inequalities for trace norms,” Inventiones Mathematicae, vol. 115, no. 3, pp. 463-482, 1994. · Zbl 0803.47037 |

[2] | Y. Takahashi, K. Hashimoto, and M. Kato, “On sharp uniform convexity, smoothness, and strong type, cotype inequalities,” Journal of Nonlinear and Convex Analysis, vol. 3, no. 2, pp. 267-281, 2002. · Zbl 1030.46012 |

[3] | Y. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39-54, 1994. · Zbl 0851.47043 |

[4] | Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15-50, Marcel Dekker, New York, NY, USA, 1996. · Zbl 0883.47083 |

[5] | I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. · Zbl 0712.47043 |

[6] | S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938-945, 2002. · Zbl 1101.90083 |

[7] | W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000, Fixed Point Theory and Its Application. · Zbl 0997.47002 |

[8] | X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20-30, 2009. · Zbl 1165.65027 |

[9] | 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 |

[10] | 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 |

[11] | A. Moudafi, “Second-order differential proximal methods for equilibrium problems,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 18, 7 pages, 2003. · Zbl 1175.90413 |

[12] | S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 313-318, Marcel Dekker, New York, NY, USA, 1996. · Zbl 0943.47040 |

[13] | W. Nilsrakoo and S. Saejung, “Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 312454, 19 pages, 2008. · Zbl 1203.47061 |

[14] | Y. Su, D. Wang, and M. Shang, “Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 284613, 8 pages, 2008. · Zbl 1203.47078 |

[15] | H. Zegeye and N. Shahzad, “Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2707-2716, 2009. · Zbl 1223.47108 |

[16] | D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151-174, 2001. · Zbl 1010.47032 |

[17] | D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489-508, 2003. · Zbl 1071.47052 |

[18] | Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323-339, 1996. · Zbl 0883.47063 |

[19] | S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257-266, 2005. · Zbl 1071.47063 |

[20] | H. Iiduka, W. Takahashi, and M. Toyoda, “Approximation of solutions of variational inequalities for monotone mappings,” Panamerican Mathematical Journal, vol. 14, no. 2, pp. 49-61, 2004. · Zbl 1060.49006 |

[21] | H. Iiduka and W. Takahashi, “Weak convergence of a projection algorithm for variational inequalities in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 668-679, 2008. · Zbl 1129.49012 |

[22] | S. Matsushita and W. Takahashi, “Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2004, no. 1, pp. 37-47, 2004. · Zbl 1088.47054 |

[23] | W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 45-57, 2009. · Zbl 1170.47049 |

[24] | W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 528476, 11 pages, 2008. · Zbl 1187.47054 |

[25] | X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 6, pp. 1958-1965, 2007. · Zbl 1124.47046 |

[26] | L. Wei, Y. J. Cho, and H. Zhou, “A strong convergence theorem for common fixed points of two relatively nonexpansive mappings and its applications,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 95-103, 2009. · Zbl 1222.47125 |

[27] | P. Cholamjiak, “A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 719360, 18 pages, 2009. · Zbl 1167.65379 |

[28] | P. Kumam and K. Wattanawitoon, “Convergence theorems of a hybrid algorithm for equilibrium problems,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 386-394, 2009. · Zbl 1166.47060 |

[29] | B. Beauzamy, Introduction to Banach Spaces and Their Geometry, North-Holland, Amsterdam, The Netherlands, 2nd edition, 1995. |

[30] | H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1127-1138, 1991. · Zbl 0757.46033 |

[31] | C. Z\ualinescu, “On uniformly convex functions,” Journal of Mathematical Analysis and Applications, vol. 95, no. 2, pp. 344-374, 1983. · Zbl 0519.49010 |

[32] | 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 |

[33] | F. Kohsaka and W. Takahashi, “Strong convergence of an iterative sequence for maximal monotone operators in a Banach space,” Abstract and Applied Analysis, no. 3, pp. 239-249, 2004. · Zbl 1064.47068 |

[34] | S. Plubtieng and W. Sriprad, “An extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 591874, 16 pages, 2009. · Zbl 1186.47076 |

[35] | S. Plubtieng and W. Sriprad, “Strong and weak convergence of modified Mann iteration for new resolvents of maximal monotone operators in Banach spaces,” Abstract and Applied Analysis, vol. 2009, Article ID 795432, 20 pages, 2009. · Zbl 1186.47074 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.