General Method for Solving the Split Common Fixed Point Problem

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• 作者：Andrzej Cegielski (1)
  
  1. Faculty of Mathematics ; Computer Science and Econometrics ; University of Zielona G贸ra ; ul. Szafrana 4a ; 65-516 ; Zielona Gora ; Poland
  
• 关键词：Split feasibility problem ; Split common fixed point problem ; Quasi ; nonexpansive operators ; Block ; iterative procedure ; Demi ; closedness principle ; 47J25 ; 47N10 ; 65J15 ; 90C25
• 刊名：Journal of Optimization Theory and Applications
• 出版年：2015
• 出版时间：May 2015
• 年：2015
• 卷：165
• 期：2
• 页码：385-404
• 全文大小：250 KB
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  2. Byrne, C (2002) Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18: pp. 441-453 10.1088/0266-5611/18/2/310" target="_blank" title="It opens in new window">CrossRef
  3. Qu, B, Xiu, N (2005) A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21: pp. 1655-1665 10.1088/0266-5611/21/5/009" target="_blank" title="It opens in new window">CrossRef
  4. Qu, B, Xiu, N (2008) A new halfspace-relaxation projection method for the split feasibility problem. Lin. Algebra Appl. 428: pp. 1218-1229 10.1016/j.laa.2007.03.002" target="_blank" title="It opens in new window">CrossRef
  5. Xu, H.-K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010). 17 pp
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  18. Cegielski, A., Censor, Y.: Opial-type theorems and the common fixed point problem. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications, vol. 49, pp. 155鈥?83. New York, NY, USA (2011)
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  24. Cegielski, A., Zalas, R.: Properties of a class of approximately shrinking operators and their applications. Fixed Point Theory. 15, 399鈥?26 (2014)
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• 刊物主题：Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operations Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2878

文摘

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.