An iterative method for solving proximal split feasibility problems and fixed point problems
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(2019) 38:177
An iterative method for solving proximal split feasibility problems and fixed point problems Wongvisarut Khuangsatung1 · Pachara Jailoka2 · Suthep Suantai2 Received: 4 February 2019 / Revised: 12 July 2019 / Accepted: 24 September 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract The purpose of this research is to introduce a regularized algorithm based on the viscosity method for solving the proximal split feasibility problem and the fixed point problem in Hilbert spaces. A strong convergence result of our proposed algorithm for finding a common solution of the proximal split feasibility problem and the fixed point problem for nonexpansive mappings is established. We also apply our main result to the split feasibility problem, and the fixed point problem of nonexpansive semigroups, respectively. Finally, we give numerical examples for supporting our main result. Keywords Fixed point problems · Proximal split feasibility problems · Nonexpansive mappings Mathematics Subject Classification 47H09 · 47H10
1 Introduction Throughout this article, let H1 and H2 be two real Hilbert spaces. Let f : H1 → R ∪ {+∞} and g : H2 → R ∪ {+∞} be two proper and lower semicontinuous convex functions and A : H1 → H2 be a bounded linear operator.
Communicated by Ernesto G. Birgin.
B
Suthep Suantai [email protected] Wongvisarut Khuangsatung [email protected] Pachara Jailoka [email protected]
1
Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Thanyaburi, Pathumthani 12110, Thailand
2
Data Science Research Center, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
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In this paper, we focus our attention on the following proximal split feasibility problem (PSFP): find a minimizer x ∗ of f , such that Ax ∗ minimizes g, namely x ∗ ∈ argmin f such that Ax ∗ ∈ argmin g,
(1.1)
where argmin f := {x¯ ∈ H1 : f (x) ¯ ≤ f (x) for all x ∈ H1 } and argmin g := { y¯ ∈ H2 : g( y¯ ) ≤ g(y) for all y ∈ H2 }. We assume that the problem (1.1) has a nonempty solution set := argmin f ∩ A−1 (argmin g). Censor and Elfving (1994) introduced the split feasibility problem (in short, SFP). The split feasibility problem (SFP) has been used for many applications in various fields of science and technology, such as in signal processing and image reconstruction, and especially applied in medical fields such as intensity-modulated radiation therapy (IMRT) (for details, see Censor et al. (2006) and the references therein). Let C and Q be nonempty, closed, and convex subsets of H1 and H2 , respectively, and then, the SFP is to find a point: x ∈ C such that Ax ∈ Q,
(1.2)
where A : H1 → H2 is a bounded linear operator. For solving the problem (1.2), Byrne (2002) introduced a popular algorithm which is called the C Q algorithm as follows: xn+1 = PC (xn − μn A∗ (I − PQ )Axn ), ∀n ≥ 1, where PC and PQ denote the metr
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