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.
Highlights
IntroductionIn 1994 Censor and Elfving [1] introduced a notion of the split feasibility problem (SFP), which is to find an element of a closed convex subset of the Euclidean space
In 1994 Censor and Elfving [1] introduced a notion of the split feasibility problem (SFP), which is to find an element of a closed convex subset of the Euclidean spaceCommunicated by Alfredo N
In this paper we study the behavior of a general method for solving the split common fixed point problem (SCFPP); we prove the weak convergence of sequences generated by this method and show that the convergence of several known methods follows from the results presented in the paper
Summary
In 1994 Censor and Elfving [1] introduced a notion of the split feasibility problem (SFP), which is to find an element of a closed convex subset of the Euclidean space. In this paper we study the behavior of a general method for solving the SCFPP; we prove the weak convergence of sequences generated by this method and show that the convergence of several known methods follows from the results presented in the paper. we present various methods known from the literature for solving the split common fixed point problem. we show that the class of strongly quasi-nonexpansive operators having a common fixed point and satisfying the demi-closedness principle is closed under convex combination and composition. The main result of the paper is contained, where we show in particular that a block-iterative CQ-type method with intermittent control in which the metric projections are replaced by strongly quasi-nonexpansive operators satisfying the demi-closedness principle generates sequences converging to a solution of a consistent SCFPP.
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