##
**Strong convergence theorems for modifying Halpern iterations for quasi-\(\phi\)-asymptotically nonexpansive multivalued mapping in Banach spaces with applications.**
*(English)*
Zbl 1263.47083

Summary: An iterative sequence for quasi-\(\phi\)-asymptotically nonexpansive multivalued mapping for modifying Halpern’s iterations is introduced. Under suitable conditions, some strong convergence theorems are proved. The results presented in the paper improve and extend the corresponding results in the work by S.-S. Chang et al. [Appl. Math. Comput. 217, No. 18, 7520–7530 (2011; Zbl 1221.65132)].

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

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

47H04 | Set-valued operators |

### Keywords:

iterative algorithm; quasi-\(\phi\)-asymptotically nonexpansive multivalued mapping; modified Halpern iteration; strong convergence### Citations:

Zbl 1221.65132
PDF
BibTeX
XML
Cite

\textit{L. Yi}, J. Appl. Math. 2012, Article ID 912545, 11 p. (2012; Zbl 1263.47083)

Full Text:
DOI

### References:

[1] | 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 |

[2] | 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, A. G. Kartosator, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15-50, Marcel Dekker, New York, NY, USA, 1996. · Zbl 0883.47083 |

[3] | S. S. Chang, C. K. Chan, and H. W. J. Lee, “Modified block iterative algorithm for quasi-\varphi -asymptotically nonexpansive mappings and equilibrium problem in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 18, pp. 7520-7530, 2011. · Zbl 1221.65132 |

[4] | W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506-510, 1953. · Zbl 0050.11603 |

[5] | A. Genel and J. Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics, vol. 22, no. 1, pp. 81-86, 1975. · Zbl 0314.47031 |

[6] | B. Halpern, “Fixed points of nonexpansive maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957-961, 1967. · Zbl 0177.19101 |

[7] | K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372-379, 2003. · Zbl 1035.47048 |

[8] | S. Y. 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 |

[9] | S. Matsushita and W. Takahashi, “An iterative algorithm for relatively nonexpansive mappings by hybrid method and applications,” in Proceedings of the 3rd International Conference on Nonlinear Analysis and Convex Analysis, pp. 305-313, 2004. · Zbl 1086.47055 |

[10] | S.-y. 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 |

[11] | X. Qin, Y. J. Cho, S. M. Kang, and H. Zhou, “Convergence of a modified Halpern-type iteration algorithm for quasi-\varphi -nonexpansive mappings,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 22, no. 7, pp. 1051-1055, 2009. · Zbl 1179.65061 |

[12] | Z. Wang, Y. Su, D. Wang, and Y. Dong, “A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2364-2371, 2011. · Zbl 1213.65082 |

[13] | 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 |

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.