A new accelerated self-adaptive stepsize algorithm with excellent stability for split common fixed point problems
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A new accelerated self-adaptive stepsize algorithm with excellent stability for split common fixed point problems Zheng Zhou1
· Bing Tan1 · Songxiao Li1
Received: 4 April 2020 / Revised: 19 June 2020 / Accepted: 23 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In the framework of Hilbert spaces, we study the solutions of split common fixed point problems. A new accelerated self-adaptive stepsize algorithm with excellent stability is proposed under the effects of inertial techniques and Meir–Keeler contraction mappings. The strong convergence theorems are obtained without prior knowledge of operator norms. Finally, in applications, our main results in this paper are applied to signal recovery problems. Keywords Self-adaptive stepsize · Meir–Keeler contraction · Inertial technique · Signal recovery Mathematics Subject Classification 47H10 · 47J25 · 65K10 · 65Y10
1 Introduction Based on the idea of the split feasibility problem (for short, SFP), Censor and Segal (2009) introduced the split common fixed point problem (for short, SCFPP) in 2009 as follows. Let H1 and H2 be Hilbert spaces, T : H1 → H1 and S : H2 → H2 be nonlinear mappings, F(T ) and F(S) denote the fixed point sets of T and S, respectively, and A : H1 → H2 be a bounded linear operator. The split common fixed point problem is to find: x ∗ ∈ F(T ) and Ax ∗ ∈ F(S).
(1.1)
Under certain conditions, when T = PC and S = PQ (PC and PQ are metric projections from H1 to its nonempty closed convex subset C and H2 to its nonempty convex closed subset Q,
Communicated by Baisheng Yan.
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Zheng Zhou [email protected] Bing Tan [email protected] Songxiao Li [email protected]
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Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China 0123456789().: V,-vol
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respectively). The problem (1.1) can be considered as the split feasibility problems, which is to find a point x ∗ ∈ C and Ax ∗ ∈ Q. This problem was introduced by Censor and Elfving (1994). Naturally, x ∗ is a solution of the split common fixed point problem if and only if x ∗ is a solution of the equation x ∗ = T (I − β A∗ (I − S)A)x ∗ , where A∗ is an adjoint operator of A. Furthermore, Censor and Segal (2009) proposed an algorithm to approximate a solution of the problem (1.1) in finite-dimensional Euclidean spaces by the recursive procedure: xn+1 = T (I − β AT (I − S)A)xn , where T and S are directed operators, AT is the matrix transposition of A, M is the largest eigenvalue of matrix AT A and β ∈ (0, 2/M). By the proposed algorithm, they obtained that the iterative sequence {x n } converges to a solution of the problem (1.1). In addition, the split feasibility problem and the split common fixed point problem have been widely studied in many mathematical problems, such as variational inequality problems, equilibrium problems, monotone inclusion problems, etc. (see Censor et al. 2012; Cho and Kang 2012; Chang et al. 2018; Majee and Nahak 2018; Sh
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