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.
|Title of host publication||Springer Optimization and Its Applications|
|Publisher||Springer International Publishing|
|Number of pages||29|
|State||Published - 2011|
|Name||Springer Optimization and Its Applications|
Bibliographical noteFunding 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.
© 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