Split Common Fixed Point Problem for Quasi-Pseudo-Contractive Mapping in Hilbert Spaces
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Split Common Fixed Point Problem for Quasi-Pseudo-Contractive Mapping in Hilbert Spaces Shih-sen Chang1 · Lin Wang2 · Y. H. Zhao2 · G. Wang2 · Z. L. Ma3 Received: 3 September 2019 / Revised: 26 July 2020 / Accepted: 16 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, the split common fixed point problem for quasi-pseudo-contractive mappings is studied in Hilbert spaces. By using the hybrid projection method, a new algorithm and some strong convergence theorems are established under suitable assumptions. Our results not only improve and generalize some recent results but also give an affirmative answer to an open question. Keywords Split common fixed point problem · Quasi-pseudo-contractive mapping · Demicontractive operator · Quasi-nonexpansive mapping · Directed operator · Firmly nonexpansive mapping · Strong convergence Mathematics Subject Classification 47J25 · 47J20 · 49N45 · 65J15
Communicated by Rosihan M. Ali.
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Shih-sen Chang [email protected] Lin Wang [email protected] Y. H. Zhao [email protected] G. Wang [email protected] Z. L. Ma [email protected]
1
Center for General Education, China Medical University, Taichung 40402, Taiwan
2
Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
3
College of Public Foundation, Yunnan Open University, Kunming, Yunnan 650223, China
123
S.-s. Chang et al.
1 Introduction Let H1 and H2 be two real Hilbert spaces and S : H1 → H1 and T : H2 → H2 be two nonlinear operators. Denote the fixed point sets of S and T by Fi x(S) and Fi x(T ), respectively. Let A : H1 → H2 be a bounded linear operator with adjoint A∗ . The “so-called” split common fixed point problem is to find a point x ∗ ∈ H1 such that x ∗ ∈ Fi x(S) and Ax ∗ ∈ Fi x(T ).
(1.1)
As well known, the split common fixed point problem (1.1) is a generalization of the split feasibility problem arising from signal processing and image restoration ([1–8]). Note that solving (1.1) can be translated to solve the following fixed point equation: x ∗ = S(x ∗ − τ A∗ ((I − T )Ax ∗ ), τ > 0.
(1.2)
In order to solve Eq. (1.2), Censor and Segal [9] proposed an algorithm for directed operators. Since then, there has been growing interest in the split common fixed point problem ([8,10–15]). In particular, in 2017, Wang [16] introduced the following new iterative algorithm for the split common fixed point problem for firmly nonexpansive mappings. Algorithm 1.1 Choose an arbitrary initial guess x0 ∈ H1 . Step 1 Given xn , compute the next iteration via the formula: xn+1 = xn − ρn [xn − Sxn + A∗ (I − T )Axn ], n ≥ 0.
(1.3)
Step 2 If the following equality holds ||xn+1 − Sxn+1 + A∗ (I − T )Axn+1 || = 0,
(1.4)
then stop; otherwise, go to step 1. Subsequently, he proved the following result. Theorem 1.2 (Wang [16]). Assume the following conditions are satisfied: (1) A is a bounded linear operator; (2) the solution set of problem (1.1), denoted by , is nonempty; (3) both S and T are firmly nonexpansive operators. ∞ 2 If the
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