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General Method for Solving the Split Common Fixed Point Problem

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  • Andrzej Cegielski

    (University of Zielona Góra)

Abstract

The split common fixed point problem (also called the multiple-sets split feasibility problem) is to find a common fixed point of a finite family of operators in one real Hilbert space, whose image under a bounded linear transformation is a common fixed point of another family of operators in the image space. In the literature one can find many methods for solving this problem as well as for its special case, called the split feasibility problem. We propose a general method for solving both problems. The method is based on a block-iterative procedure, in which we apply quasi-nonexpansive operators satisfying the demi-closedness principle and having a common fixed point. We prove the weak convergence of sequences generated by this method and show that the convergence for methods known from the literature follows from our general result.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:165:y:2015:i:2:d:10.1007_s10957-014-0662-z
    DOI: 10.1007/s10957-014-0662-z
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Jinzuo Chen & Mihai Postolache & Li-Jun Zhu, 2019. "Iterative Algorithms for Split Common Fixed Point Problem Involved in Pseudo-Contractive Operators without Lipschitz Assumption," Mathematics, MDPI, vol. 7(9), pages 1-13, August.
    2. Jason Xu & Eric C. Chi & Meng Yang & Kenneth Lange, 2018. "A majorization–minimization algorithm for split feasibility problems," Computational Optimization and Applications, Springer, vol. 71(3), pages 795-828, December.
    3. Ismat Beg & Mujahid Abbas & Muhammad Waseem Asghar, 2023. "Approximation of the Solution of Split Equality Fixed Point Problem for Family of Multivalued Demicontractive Operators with Application," Mathematics, MDPI, vol. 11(4), pages 1-16, February.
    4. Dianlu Tian & Lining Jiang & Luoyi Shi, 2019. "Gradient Methods with Selection Technique for the Multiple-Sets Split Equality Problem," Mathematics, MDPI, vol. 7(10), pages 1-10, October.
    5. Fenghui Wang, 2022. "The Split Feasibility Problem with Multiple Output Sets for Demicontractive Mappings," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 837-853, December.

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