##
**Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings.**
*(English)*
Zbl 1090.47055

In [Numer. Funct. Anal. Optim. 22, 767–773 (2001; Zbl 0999.47043)], H.-K. Xu and R. G. Ori introduced an implicit iteration process for a finite family of nonexpansive mapping and proved the weak convergence of this process to a common fixed point of a finite family of nonexpansive maps in a Hilbert space. They posed the question: What assumptions would have to be made on the finite family of nonexpansive mappings and/or their parameters \(\{ \alpha _{n}\} \) to guarantee the strong convergence of the sequence \(\{ x_{n}\} \) generated by their implicit iteration process? The authors prove the strong convergence of the sequence \(\{ x_{n}\} \) in a much more general uniformly convex Banach space under the condition that one member of the finite family of nonexpansive mappings is semi-compact or any one of the contractive assumptions of Proposition 3.4 of their paper holds.

Reviewer: Edward Prempeh (Kumasi)

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

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

47H10 | Fixed-point theorems |

47J05 | Equations involving nonlinear operators (general) |

### Citations:

Zbl 0999.47043
PDF
BibTeX
XML
Cite

\textit{C. E. Chidume} and \textit{N. Shahzad}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 62, No. 6, 1149--1156 (2005; Zbl 1090.47055)

Full Text:
DOI

### References:

[1] | C.E. Chidume, Iterative algorithms for nonexpansive mappings and some of their generalizations, Nonlinear Anal. and Appl., R.P. Agarwal, et al. (Eds.), Kluwer Academic Publishers, 2003, pp. 383-429. · Zbl 1057.47003 |

[2] | Kirk, W.A., On nonlinear mappings of strongly semicontractive type, J. math. anal. appl., 27, 409-412, (1969) · Zbl 0183.15103 |

[3] | Petryshyn, W.V., Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces, Trans. amer. math. soc., 182, 323-352, (1973) · Zbl 0277.47033 |

[4] | Shahzad, N., Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear anal., 61, 1031-1039, (2005) · Zbl 1089.47058 |

[5] | Tan, K.K.; Xu, H.K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. math. anal. appl., 178, 301-308, (1993) · Zbl 0895.47048 |

[6] | Xu, H.K., Inequalities in Banach spaces with applications, Nonlinear anal., 16, 1127-1138, (1991) · Zbl 0757.46033 |

[7] | Xu, H.K.; Ori, R., An implicit iterative process for nonexpansive mappings, Numer. funct. anal. optim., 22, 767-773, (2001) · Zbl 0999.47043 |

[8] | Zhou, Y.; Chang, S.S., Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. funct. anal. appl., 23, 911-921, (2002) · Zbl 1041.47048 |

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.