Parallel iterative methods for solving the generalized split common null point problem in Hilbert spaces
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Parallel iterative methods for solving the generalized split common null point problem in Hilbert spaces Simeon Reich1 · Truong Minh Tuyen2 Received: 13 October 2019 / Accepted: 10 July 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract We study the recently introduced generalized split common null point problem in Hilbert spaces. In order to solve this problem, we propose two new parallel algorithms and establish strong convergence theorems for both of them. Our schemes combine the hybrid and shrinking projection methods with the proximal point algorithm. Keywords Hilbert space · Metric projection · Monotone operator · Nonexpansive mapping · Split common null point problem Mathematics Subject Classification 47H05 · 47H09 · 49J53 · 90C25
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 [6] for modeling certain inverse problems. It plays an important role in medical image reconstruction and in signal processing (see [3,4]). Several iterative algorithms for solving (1.1) were presented and analyzed in [2– 4,7,8,11,20,22,23,29–32], and in references therein. Some generalizations of the SCFP have also been studied. We mention, for example, the multiple-set SFP (MSSFP) (see [7,15]), the split common fixed point problem (SCFPP) (see
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Truong Minh Tuyen [email protected] Simeon Reich [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 0123456789().: V,-vol
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[9,16]), the split variational inequality problem (SVIP) (see [8]) and the split common null point problem (SCNPP) (see [5,24–26]). A popular method for solving the SCFP is Byrne’s C Q algorithm (see [3]). As has already been mentioned by Byrne, a special case of this method was introduced by Landweber in [14]. 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 [9], Censor et al. [7], Masad and Reich [15], and by Xu [30,31]. It is known that the SCFP is a special case of the split common null point problem (SCNPP), which is formulated as follows. Let A : H1 −→ 2 H1 and B : H2 −→ 2 H2 be two monotone operators and let T : H1 −→ H2 be a bounded linear operator such that S = A−1 (0) ∩ T −1 (B −1 (0)) = ∅. The SCNPP is to find an element x ∗ ∈ S. The SCNPP has recently been studied by several authors. We mention, for instance, Byrne et al. [5], Dadashi [11], Takahashi and Takahashi [20], Takahashi [23], and Wang [27–29]. In 2019 Reich and Tuyen [17] first introduced and studied the generalized split common null point
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