Opial-type theorems and the common fixed point problem

Andrzej Cegielski, Yair Censor

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


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 languageEnglish
Title of host publicationSpringer Optimization and Its Applications
PublisherSpringer International Publishing
Number of pages29
StatePublished - 2011
Externally publishedYes

Publication series

NameSpringer Optimization and Its Applications
ISSN (Print)1931-6828
ISSN (Electronic)1931-6836

Bibliographical note

Publisher Copyright:
© Springer Science+Business Media, LLC 2011.


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

ASJC Scopus subject areas

  • Control and Optimization


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