## Abstract

The well-known Opial theorem says that an orbit of a nonexpansive and asymptotically regular operator T having a fixed point and defined on a Hilbert space converges weakly to a fixed point of T. In this paper, we consider recurrences generated by a sequence of quasi-nonexpansive operators having a common fixed point or by a sequence of extrapolations of an operator satisfying Opial’s demiclosedness principle and having a fixed point. We give sufficient conditions for the weak convergence of sequences defined by these recurrences to a fixed point of an operator which is closely related to the sequence of operators. These results generalize in a natural way the classical Opial theorem. We give applications of these generalizations to the common fixed point problem.

Original language | English |
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Title of host publication | Springer Optimization and Its Applications |

Publisher | Springer International Publishing |

Pages | 155-183 |

Number of pages | 29 |

DOIs | |

State | Published - 2011 |

Externally published | Yes |

### Publication series

Name | Springer Optimization and Its Applications |
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Volume | 49 |

ISSN (Print) | 1931-6828 |

ISSN (Electronic) | 1931-6836 |

### Bibliographical note

Funding Information:We thank two anonymous referees for their constructive comments. This work was partially supported by Award Number R01HL070472 from the National Heart, Lung and Blood Institute and by United States-Israel Binational Science Foundation (BSF) grant No. 2009012.

Publisher Copyright:

© Springer Science+Business Media, LLC 2011.

## Keywords

- Common fixed point
- Cutter operators
- Dos Santos method
- Opial theorem
- Quasi-nonexpansive operators

## ASJC Scopus subject areas

- Control and Optimization