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Opial-Type Theorems and the Common Fixed Point Problem

In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Author

Listed:
  • Andrzej Cegielski

    (University of Zielona Góra)

  • Yair Censor

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.

Suggested Citation

  • Andrzej Cegielski & Yair Censor, 2011. "Opial-Type Theorems and the Common Fixed Point Problem," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 155-183, Springer.
    • Handle: RePEc:spr:spochp:978-1-4419-9569-8_9
    DOI: 10.1007/978-1-4419-9569-8_9
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    Cited by:

    1. Andrzej Cegielski, 2015. "General Method for Solving the Split Common Fixed Point Problem," Journal of Optimization Theory and Applications, Springer, vol. 165(2), pages 385-404, May.
    2. Yair Censor & Daniel Reem & Maroun Zaknoon, 2022. "A generalized block-iterative projection method for the common fixed point problem induced by cutters," Journal of Global Optimization, Springer, vol. 84(4), pages 967-987, December.

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