The split feasibility problem with multiple output sets in Hilbert spaces
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The split feasibility problem with multiple output sets in Hilbert spaces Simeon Reich1 · Minh Tuyen Truong2 · Thi Ngoc Ha Mai 3 Received: 17 September 2019 / Accepted: 13 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the split feasibility problem with multiple output sets in Hilbert spaces. In order to solve this problem, we propose two new algorithms. We establish a weak convergence theorem for the first one and a strong convergence theorem for the second. Keywords Hilbert space · Metric projection · Nonexpansive mapping · Split feasibility problem
1 Introduction Let C and Q be nonempty, closed and convex subsets of real Hilbert spaces H1 and H2 , respectively. Let T : H1 −→ H2 be a bounded linear operator and let T ∗ : H2 −→ H1 be its adjoint. The split convex feasibility problem (SCFP) is formulated as follows: Find an element x ∗ ∈ C such that T x ∗ ∈ Q.
(1.1)
The SCFP was first introduced by Censor and Elfving [5] for modeling certain inverse problems. It plays an important role in medical image reconstruction and in signal processing (see [2,3]). Since then, several iterative algorithms for solving (1.1) have
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Minh Tuyen Truong [email protected] Simeon Reich [email protected] Thi Ngoc Ha Mai [email protected]
1
Department of Mathematics, The Technion – Israel Institute of Technology, 32000 Haifa, Israel
2
Department of Mathematics and Informatics, Thai Nguyen University of Sciences, Thai Nguyen, Vietnam
3
Thai Nguyen University of Agriculture and Forestry, Thai Nguyen, Vietnam
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S. Reich et al.
been presented and analyzed. See, for example, [1–3,6,7,9,18–20,24,26,27,29] and references therein. Some generalizations of the SCFP have also been studied. We mention, for instance, the multiple-set SFP (MSSFP) (see [6,14]), the split common fixed point problem (SCFPP) (see [8,15]), the split variational inequality problem (SVIP) (see [7]) and the split common null point problem (SCNPP) (see [4,21–23]). A well-known method for solving the SCFP is Byrne’s C Q algorithm (see [2]). As has already been mentioned by Byrne, a special case of this method was introduced by Landweber [12]. The C Q algorithm has been extended by several authors in order to solve the multiple-set split convex feasibility problem. See, for example, the papers by Censor and Segal [8], Censor et al. [6], Masad and Reich [14], and by Xu [26,27]. In 2019, Reich and Tuyen [17] first introduced and studied the following generalized split feasibility problem (GSFP). Let Hi , i = 1, 2, . . . , N , be real Hilbert spaces and let Ci , i = 1, 2, . . . , N , be closed and convex subsets of Hi , respectively. Let Ai : Hi −→ Hi+1 , i = 1, 2, . . . , N − 1, be bounded linear operators such that −1 −1 A−1 = ∅. := C1 ∩ A−1 1 (C 2 ) ∩ . . . ∩ A1 2 . . . A N −1 (C N ) Given Hi , Ci and Ai as above, the GSFP is to find an element x † ∈ ,
(1.2)
that is, x † ∈ C1 , A1 x † ∈ C2 . . ., A N −1 A N −2 . . . A1 x † ∈ C N . Reich and Tuyen proved a strong convergence theorem for a
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