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
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