Solving Split Common Fixed-Point Problem of Firmly Quasi-Nonexpansive Mappings without Prior Knowledge of Operators Norms

Abstract

Very recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms for the split common fixed-point problem concerned two bounded linear operators. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. It is the purpose of this paper to introduce a viscosity iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. We prove the strong convergence of the proposed algorithms for split common fixed-point problem governed by the firmly quasi-nonexpansive operators. As a consequence, we obtain strong convergence theorems for split feasibility problem and split common null point problems of maximal monotone operators. Our results improve and extend the corresponding results announced by many others.

1. Introduction

Throughout this paper, we always assume that 𝐻 is a real Hilbert space with the inner product ⟨⋅,⋅⟩ and the norm ‖⋅‖. Let 𝐼 denote the identity operator on 𝐻. Let 𝑇∶𝐻→𝐻 be a mapping. A point 𝑥∈𝐻 is said to be a fixed point of 𝑇 provided 𝑇𝑥=𝑥. In this paper, we use 𝐹(𝑇) to denote the fixed point set of 𝑇.

Let 𝐶 and 𝑄 be nonempty closed convex subsets of real Hilbert spaces 𝐻₁ and 𝐻₂, respectively. The split feasibility problem (SFP) is to find a point as follows: 𝑥∈𝐶suchthat𝐴𝑥∈𝑄, (1) where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2].

Note that if the split feasibility problem (1) is consistent (i.e., (1) has a solution), then (1) can be formulated as a fixed point equation by using the following fact: 𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗=𝑥^∗, (2) where 𝑃_𝐶 and 𝑃_𝑄 are the (orthogonal) projections onto 𝐶 and 𝑄, respectively, 𝛾>0 is any positive constant, and 𝐴^∗ denotes the adjoint of 𝐴. That is, 𝑥^∗ solves SFP (1) if and only if 𝑥^∗ solves fixed point equation (2) (see [3] for the details). This implies that we can use fixed point algorithms (see [3–6]) to solve SFP. To solve (2), Byrne [2] proposed his CQ algorithm which generates a sequence {𝑥_𝑘} by 𝑥_𝑘+1=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑘,𝑘∈𝑁, (3) where 𝛾∈(0,2/𝜆) with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴.

Censor and Segal [7] introduced the following split common fixed-point problem (SCFP): find𝑥^∗∈𝐹(𝑈)suchthat𝐴𝑥^∗∈𝐹(𝑇), (4) where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator and 𝑈∶𝐻₁→𝐻₁ and 𝑇∶𝐻₂→𝐻₂ are two nonexpansive operators with nonempty fixed-point sets 𝐹(𝑈)=𝐶 and 𝐹(𝑇)=𝑄. SCFP is in itself at the core of the modeling of many inverse problems in various areas of mathematics and physical sciences and has been used to model significant real-world inverse problems in many areas (see [8]).

To solve (4), Censor and Segal [7] proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm: 𝑥_𝑘+1=𝑈𝑥_𝑘+𝛾𝐴^𝑡(𝑇−𝐼)𝐴𝑥_𝑘,𝑘∈𝑁, (5) where 𝛾∈(0,2/𝜆), with 𝜆 being the largest eigenvalue of the matrix 𝐴^𝑡𝐴 (𝐴^𝑡 stands for matrix transposition).

Let 𝐻₁, 𝐻₂, and 𝐻₃ be real Hilbert spaces; let 𝐴∶𝐻₁→𝐻₃ and 𝐵∶𝐻₂→𝐻₃ be two bounded linear operators; let 𝑈∶𝐻₁→𝐻₁ and 𝑇∶𝐻₂→𝐻₂ be two firmly quasi-nonexpansive operators. In [9], Moudafi introduced the following split common fixed-point problem (SCFP): find𝑥∈𝐹(𝑈),𝑦∈𝐹(𝑇),suchthat𝐴𝑥=𝐵𝑦, (6) which allows asymmetric and partial relations between the variables 𝑥 and 𝑦. The interest is to cover many situations, for instance in decomposition methods for PDE’s, in a applications in game theory, and in intensity-modulated radiation therapy (IMRT). In decision sciences, this allows to consider agents who interplay only via some components of their decision variables (see [10]). In IMRT, these amounts envisage a weak coupling between the vector of doses absorbed in all voxels and that of the radiation intensity (see [11]).

If 𝐻₂=𝐻₃ and 𝐵=𝐼, then SCFP (6) reduces to SCFP (4). For solving SCFP (6), Moudafi [9] introduced the following alternating algorithm: 𝑥_𝑘+1=𝑈𝑥_𝑘−𝛾_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑦_𝑘+1=𝑇𝑦_𝑘+𝛾_𝑘𝐵^∗𝐴𝑥_𝑘+1−𝐵𝑦_𝑘, (7) for firmly quasi-nonexpansive operators 𝑈 and 𝑇, where nondecreasing sequence 𝛾_𝑘∈(𝜀,min(1/𝜆_𝐴,1/𝜆_𝐵)−𝜀) and 𝜆_𝐴 and 𝜆_𝐵 stand for the spectral radius of 𝐴^∗𝐴 and 𝐵^∗𝐵, respectively.

Very recently, Moudafi and Al-Shemas [12] introduced the following simultaneous iterative method to solve SCFP (6): 𝑥_𝑘+1=𝑈𝑥_𝑘−𝛾_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑦_𝑘+1=𝑇𝑦_𝑘+𝛾_𝑘𝐵^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘, (8) for firmly quasi-nonexpansive operators 𝑈 and 𝑇, where 𝛾_𝑘∈(𝜀,(2/(𝜆_𝐴+𝜆_𝐵))−𝜀) and 𝜆_𝐴 and 𝜆_𝐵 stand for the spectral radius of 𝐴^∗𝐴 and 𝐵^∗𝐵, respectively.

In [13], Zhao and He introduced the following alternating mann iterative algorithms for SCFP (6) governed by quasi-nonexpansive mappings and obtained weak convergence results: 𝑢_𝑘=𝑥_𝑘−𝛾_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑥_𝑘+1=𝛼_𝑘𝑢_𝑘+1−𝛼_𝑘𝑈𝑢_𝑘,𝑣_𝑘+1=𝑦_𝑘+𝛾_𝑘𝐵^∗𝐴𝑥_𝑘+1−𝐵𝑦_𝑘,𝑦_𝑘+1=𝛽_𝑘𝑣_𝑘+1+1−𝛽_𝑘𝑇𝑣_𝑘+1. (9)

Note that, in (7), (8), and (9) mentioned above, the determination of the stepsize {𝛾_𝑘} depends on the operator (matrix) norms ‖𝐴‖ and ‖𝐵‖ (or the largest eigenvalues of 𝐴^∗𝐴 and 𝐵^∗𝐵). In order to implement the above algorithms for solving SCFP (6), one has first to compute (or, at least, estimate) operator norms of 𝐴 and 𝐵, which is in general not an easy work in practice. To overcome this difficulty, López et al [14] and Zhao and Yang [15] presented a helpful method for estimating the stepsizes which do not need prior knowledge of the operator norms for solving the split feasibility problems and multiple-set split feasibility problems, respectively. Inspired by them, in this paper, we introduce a new choice of the stepsize sequence {𝛾_𝑘} for the viscosity iterative algorithm to solve SCFP (6) governed by firmly quasi-nonexpansive operators as follows: 𝛾_𝑘∈2𝜖,𝐴𝑥_𝑘−𝐵𝑦_𝑘²𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘²+𝐵^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘²−𝜖. (10) The advantage of our choice (9) of the stepsizes lies in the fact that no prior information about the operator norms of 𝐴 and 𝐵 is required, and still convergence is guaranteed.

Some algorithms have been invented to solve SCFP (6) (see [16, 17] and references therein). In this paper, inspired and motivated by the works mentioned above, to get the strong convergence of the algorithm, we introduce the viscosity iterative algorithm without prior knowledge of operator norms for solving SCFP (6) governed by firmly quasi-nonexpansine operators. The organization of this paper is as follows. Some useful definitions and results are listed for the convergence analysis of the iterative algorithm in Section 2. In Section 3, the strong convergence theorem of the proposed viscosity iterative algorithm is obtained. At last, we provide some applications.

2. Preliminaries

In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively. We use 𝜔_𝑤(𝑥_𝑘)={𝑥∶∃𝑥_𝑘𝑗⇀𝑥} to stand for the weak 𝜔-limit set of {𝑥_𝑘} and use Γ to stand for the solution set of SCFP (6).

Definition 1. An operator 𝑇∶𝐻→𝐻 is said to be(i)nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻,(ii)quasi-nonexpansive if 𝐹(𝑇)≠∅ and if ‖𝑇𝑥−𝑞‖≤‖𝑥−𝑞‖ for all 𝑥∈𝐻 and 𝑞∈𝐹(𝑇),(iii)firmly nonexpansive if ‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²−‖(𝑥−𝑦)−(𝑇𝑥−𝑇𝑦)‖² for all 𝑥,𝑦∈𝐻,(iv)firmly quasi-nonexpansive if 𝐹(𝑇)≠∅ and if ‖𝑇𝑥−𝑞‖²≤‖𝑥−𝑞‖²−‖𝑥−𝑇𝑥‖² for all 𝑥∈𝐻 and 𝑞∈𝐹(𝑇).

Remark 2. A firmly quasi-nonexpansive operator is also called a separating operator [18], cutter operator [19], directed operators [7, 20], or class-𝔗 operator which was introduced by Bauschke and Combettes [21]. Firmly quasi-nonexpansive operators are important because they include many types of nonlinear operators arising in applied mathematics such as approximation and convex optimization. For instance, the subgradient projection 𝑇 of a continuous convex function 𝑓∶𝐻→ℝ is a firmly quasi-nonexpansive operator. Recall that the subgradient projection 𝑇 is defined by, assuming that the level set {𝑥∈𝐻∶𝑓(𝑥)≤0}≠∅, 𝑓(𝑥)𝑇𝑥∶=𝑥−𝑔(𝑥)²𝑔(𝑥)(𝑥)(𝑥),𝑓>0,𝑥,𝑓≤0, (11) where 𝑔 is a selection of the subdifferential 𝜕𝑓 (i.e., 𝑔(𝑥)∈𝜕𝑓(𝑥) for all 𝑥∈𝐻).

Particularly, projections are firmly quasi-nonexpansive operators. Recall that, given a closed convex subset 𝐶 of a Hilbert space 𝐻, the projection 𝑃_𝐶∶𝐻→𝐶 assigns each 𝑥∈𝐻 to its closest point from 𝐶 defined by 𝑃_𝐶𝑥=argmin_𝑧∈𝐶‖𝑥−𝑧‖. (12) It is well known that 𝑃_𝐶𝑥 is characterized by the inequality 𝑃_𝐶𝑥∈𝐶,𝑥−𝑃_𝐶𝑥,𝑧−𝑃_𝐶𝑥≤0,𝑧∈𝐶. (13)

Lemma 3 (see [19, 21]). The fixed point set of a firmly quasi-nonexpansive operator is closed convex.

We also need other classes of operators.

Definition 4. An operator 𝑇∶𝐻→𝐻 called demiclosed at the origin if whenever the sequence {𝑥_𝑛} converges weakly to 𝑥 and the sequence {𝑇𝑥_𝑛} converges strongly to 0, then 𝑇𝑥=0.

We remark here that a firmly quasi-nonexpansive operator 𝑇 may be not nonexpansive; even 𝑇−𝐼 is demiclosed at origin. See the following example [22].

Example 5. Let 𝐻=𝑅₁ and define a mapping by 𝑇∶𝐻→𝐻 by 𝑥𝑇𝑥∶=21sin𝑥,𝑥≠0,0,𝑥=0. (14) Then, 𝐹(𝑇)={0} and |𝑇𝑥−0|²=𝑥²41sin𝑥²≤𝑥²−𝑥𝑥−21sin𝑥²=|𝑥−0|²−|𝑥−𝑇𝑥|². (15) So, 𝑇 is firmly quasi-nonexpansive but not nonexpansive. It is easy to see that 𝑇−𝐼 is demiclosed at origin.

Definition 6. An operator 𝑇∶𝐻→𝐻 is called contraction with constant 𝜌∈[0,1) if, for any 𝑥,𝑦∈𝐻, 𝑇𝑥−𝑇𝑦≤𝜌𝑥−𝑦. (16)

In real Hilbert space, we easily get the following equality: 2𝑥,𝑦=‖𝑥‖²+𝑦²−𝑥−𝑦²=𝑥+𝑦²−‖𝑥‖²−𝑦²,∀𝑥,𝑦∈𝐻. (17)

We end this section by the following lemmas, which are important in convergence analysis for our iterative algorithm.

Lemma 7 (see [23]). Assume {𝑠_𝑘} is a sequence of nonnegative real numbers such that 𝑠_𝑘+1≤1−𝜆_𝑘𝑠_𝑘+𝜆_𝑘𝛿_𝑘𝑠,𝑘≥0,_𝑘+1≤𝑠_𝑘−𝜂_𝑘+𝜇_𝑘,𝑘≥0, (18) where {𝜆_𝑘} is a sequence in (0,1), {𝜂_𝑘} is a sequence of nonnegative real numbers, and {𝛿_𝑘} and {𝜇_𝑘} are two sequences in ℝ such that(i)Σ^∞_𝑘=1𝜆_𝑘=∞;(ii)lim_𝑘→∞𝜇_𝑘=0;(iii)lim_𝑙→∞𝜂_𝑘𝑙=0 implies limsup_𝑙→∞𝛿_𝑘𝑙≤0 for any subsequence {𝑘_𝑙}⊂{𝑘}.Then, lim_𝑘→∞𝑠_𝑘=0.

Lemma 8 (see [24, Lemma 1.3]). Let {𝛿_𝑛} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {𝛿_𝑛𝑗}_𝑗≥0 of {𝛿_𝑛} which satisfies 𝛿_𝑛𝑗<𝛿_𝑛𝑗+1 for all 𝑗≥0. Also consider the sequence of integers {𝜏(𝑛)}_𝑛≥𝑛0 defined by 𝜏(𝑛)=max𝑘≤𝑛∣𝛿_𝑘<𝛿_𝑘+1. (19) Then, {𝜏(𝑛)}_𝑛≥𝑛0 is a nondecreasing sequence verifying lim_𝑛→∞𝜏(𝑛)=∞ and, for all 𝑛≥𝑛₀, it holds that 𝛿_𝜏(𝑛)≤𝛿_𝜏(𝑛)+1 and we have 𝛿_𝑛≤𝛿_𝜏(𝑛)+1. (20)

3. Viscosity Iterative Algorithm without Prior Knowledge of Operator Norms

In this section, we introduce a viscosity iterative algorithm where the stepsizes 𝛾_𝑘 do not depend on the operator norms ‖𝐴‖ and ‖𝐵‖ and prove the strong convergence of algorithm without prior knowledge of operator norms.

Algorithm 9. Let 𝑓₁∶𝐻₁→𝐻₁ and 𝑓₂∶𝐻₂→𝐻₂ be two contractions with constants 𝜌₁, 𝜌₂∈[0,1), and 𝛼_𝑘∈[0,1]. Choose an initial guess 𝑥₀∈𝐻₁, 𝑦₀∈𝐻₂ arbitrarily. Assume that the 𝑘th iterate 𝑥_𝑘∈𝐻₁, 𝑦_𝑘∈𝐻₂ has been constructed; then, we calculate the (𝑘+1)th iterate (𝑥_𝑘+1,𝑦_𝑘+1) via the formula 𝑢_𝑘=𝑥_𝑘−𝛾_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑥_𝑘+1=𝛼_𝑘𝑓₁𝑥_𝑘+1−𝛼_𝑘𝑈𝑢_𝑘,𝑣_𝑘=𝑦_𝑘+𝛾_𝑘𝐵^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑦_𝑘+1=𝛼_𝑘𝑓₂𝑥_𝑘+1−𝛼_𝑘𝑇𝑣_𝑘. (21) The stepsize 𝛾_𝑘 is chosen in such a way that 𝛾_𝑘∈2𝜖,𝐴𝑥_𝑘−𝐵𝑦_𝑘²𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘²+𝐵^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘²−𝜖,𝑘∈Ω, (22) otherwise, 𝛾_𝑘=𝛾 (𝛾 being any nonnegative value), where the set of indexes Ω={𝑘∶𝐴𝑥_𝑘−𝐵𝑦_𝑘≠0}.

Remark 10. Note that, in (22), the choice of the stepsize 𝛾_𝑘 is independent of the norms ‖𝐴‖ and ‖𝐵‖. The value of 𝛾 does not influence the considered algorithm, but it was introduced just for the sake of clarity. Furthermore, we will see from Lemma 3 that 𝛾_𝑘 is well defined.

Lemma 11. Assume the solution set Γ of (6) is nonempty. Then, 𝛾_𝑘 defined by (22) is well defined.

Proof. Taking (𝑥,𝑦)∈Γ, that is, 𝑥∈𝐹(𝑈), 𝑦∈𝐹(𝑇), and 𝐴𝑥=𝐵𝑦, we have 𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑥_𝑘−𝑥=𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝐴𝑥_𝑘,𝐵−𝐴𝑥_∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑦−𝑦_𝑘=𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝐵𝑦−𝐵𝑦_𝑘. (23) By adding the two above equalities and by taking into account the fact that 𝐴𝑥=𝐵𝑦, we obtain 𝐴𝑥_𝑘−𝐵𝑦_𝑘²=𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑥_𝑘+𝐵−𝑥_∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑦−𝑦_𝑘≤𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘⋅𝑥_𝑘+𝐵−𝑥_∗𝐴𝑥_𝑘−𝐵𝑦_𝑘⋅𝑦−𝑦_𝑘. (24) Consequently, for 𝑘∈Ω, that is, ‖𝐴𝑥_𝑘−𝐵𝑦_𝑘‖>0, we have ‖𝐴^∗(𝐴𝑥_𝑘−𝐵𝑦_𝑘)‖≠0 or ‖𝐵^∗(𝐴𝑥_𝑘−𝐵𝑦_𝑘)‖≠0. This leads that 𝛾_𝑘 is well defined.

Theorem 12. Let 𝐻₁, 𝐻₂, and 𝐻₃ be real Hilbert spaces. Given two bounded linear operators 𝐴∶𝐻₁→𝐻₃ and 𝐵∶𝐻₂→𝐻₃, let 𝑈∶𝐻₁→𝐻₁ and 𝑇∶𝐻₂→𝐻₂ be firmly quasi-nonexpansive operators with the solution set Γ of (6) being nonempty. Let the sequence {(𝑥_𝑘,𝑦_𝑘)} be generated by Algorithm 9. Assume that the following conditions are satisfied:(1)𝜌₁,𝜌₂∈[0,1/√2);(2)lim_𝑘→∞𝛼_𝑘=0 and ∑^∞_𝑘=0𝛼_𝑘=∞.(3)𝑈−𝐼 and 𝑇−𝐼 are demiclosed at origin;
Then, sequence {(𝑥_𝑘,𝑦_𝑘)} strongly converges to a solution (𝑥^∗,𝑦^∗) of (6) which solves the variational inequality problem 𝐼−𝑓₁𝑥^∗,𝑥−𝑥^∗≥0,𝐼−𝑓₂𝑦^∗,𝑦−𝑦^∗≥0, (25)

Proof. Let (𝑥^∗,𝑦^∗)∈Γ be the solution of the variational inequality problem (25). Then, 𝑥^∗∈𝐹(𝑈), 𝑦^∗∈𝐹(𝑇), and 𝐴𝑥^∗=𝐵𝑦^∗. We have 𝑢_𝑘−𝑥^∗2=𝑥_𝑘−𝛾_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘−𝑥^∗2=𝑥_𝑘−𝑥^∗2−2𝛾_𝑘𝑥_𝑘−𝑥^∗,𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘+𝛾²_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘². (26) Using (17), we have −2𝑥_𝑘−𝑥^∗,𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘=−2𝐴𝑥_𝑘−𝐴𝑥^∗,𝐴𝑥_𝑘−𝐵𝑦_𝑘=−𝐴𝑥_𝑘−𝐴𝑥^∗2−𝐴𝑥_𝑘−𝐵𝑦_𝑘²+𝐵𝑦_𝑘−𝐴𝑥^∗2. (27) By (26) and (27), we obtain 𝑢_𝑘−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−𝛾_𝑘𝐴𝑥_𝑘−𝐴𝑥^∗2−𝛾_𝑘𝐴𝑥_𝑘−𝐵𝑦_𝑘²+𝛾_𝑘𝐵𝑦_𝑘−𝐴𝑥^∗2+𝛾²_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘². (28) Similarly, we have 𝑣_𝑘−𝑦^∗2≤𝑦_𝑘−𝑦^∗2−𝛾_𝑘𝐵𝑦_𝑘−𝐵𝑦^∗2−𝛾_𝑘𝐴𝑥_𝑘−𝐵𝑦_𝑘²+𝛾_𝑘𝐴𝑥_𝑘−𝐵𝑦^∗2+𝛾²_𝑘𝐵^∗(𝐴𝑥_𝑘−𝐵𝑦_𝑘)². (29) By adding the two last inequalities and by taking into account the fact that 𝐴𝑥^∗=𝐵𝑦^∗, we obtain 𝑢_𝑘−𝑥^∗2+𝑣_𝑘−𝑦^∗2≤𝑥_𝑘−𝑥^∗2+𝑦_𝑘−𝑦^∗2−𝛾_𝑘2𝐴𝑥_𝑘−𝐵𝑦_𝑘²−𝛾_𝑘𝐴^∗(𝐴𝑥_𝑘−𝐵𝑦_𝑘)²+𝐵^∗(𝐴𝑥_𝑘−𝐵𝑦_𝑘)². (30) With assumption on 𝛾_𝑘, we obtain 𝑢_𝑘−𝑥^∗2+𝑣_𝑘−𝑦^∗2≤𝑥_𝑘−𝑥^∗2+𝑦_𝑘−𝑦^∗2. (31) Setting 𝜌=max{𝜌₁,𝜌₁}, we have 𝜌∈[0,1/√2). By 𝑈 and 𝑇 being firmly quasi-nonexpansive operators, it follows that 𝑥_𝑘+1−𝑥^∗2≤𝛼_𝑘𝑓₁𝑥_𝑘−𝑥^∗2+1−𝛼_𝑘𝑈𝑢_𝑘−𝑥^∗2≤𝛼_𝑘𝑓₁𝑥_𝑘−𝑓₁𝑥^∗+𝑓₁𝑥^∗−𝑥^∗2+1−𝛼_𝑘𝑢_𝑘−𝑥^∗2−1−𝛼_𝑘𝑢_𝑘−𝑈𝑢_𝑘²≤2𝛼_𝑘𝑓₁𝑥_𝑘−𝑓₁𝑥^∗2+𝑓₁𝑥^∗−𝑥^∗2+1−𝛼_𝑘𝑢_𝑘−𝑥^∗2−1−𝛼_𝑘𝑢_𝑘−𝑈𝑢_𝑘²≤2𝛼_𝑘𝜌²₁𝑥_𝑘−𝑥^∗2+2𝛼_𝑘𝑓₁𝑥^∗−𝑥^∗2+1−𝛼_𝑘𝑢_𝑘−𝑥^∗2−1−𝛼_𝑘𝑢_𝑘−𝑈𝑢_𝑘²,𝑦_𝑘+1−𝑦^∗2≤2𝛼_𝑘𝜌²₂𝑦_𝑘−𝑦^∗2+2𝛼_𝑘𝑓₂𝑦^∗−𝑦^∗2+1−𝛼_𝑘𝑣_𝑘−𝑦^∗2−1−𝛼_𝑘𝑣_𝑘−𝑇(𝑣_𝑘)². (32) Adding up the last two inequalities and using (31), setting 𝑠_𝑘=‖𝑥_𝑘−𝑥^∗‖²+‖𝑦_𝑘−𝑦^∗‖², we get 𝑠_𝑘+1≤1−𝛼_𝑘1−2𝜌²𝑠_𝑘+2𝛼_𝑘𝑓₁𝑥^∗−𝑥^∗2+𝑓₂𝑦^∗−𝑦^∗2−1−𝛼_𝑘𝑢_𝑘−𝑈𝑢_𝑘²+𝑣_𝑘−𝑇𝑣_𝑘², (33) which implies 𝑠_𝑘+1≤1−𝛼_𝑘1−2𝜌²𝑠_𝑘+𝛼_𝑘1−2𝜌²21−2𝜌²×𝑓₁𝑥^∗−𝑥^∗2+𝑓₂𝑦^∗−𝑦^∗2. (34) It follows from induction that 𝑠_𝑘≤max𝑠₀,21−2𝜌²𝑓₁𝑥^∗−𝑥^∗2+𝑓₂𝑦^∗−𝑦^∗2𝑘≥0, (35) which implies that {𝑥_𝑘} and {𝑦_𝑘} are bounded. It follows that {𝑢_𝑘},{𝑣_𝑘},{𝑓₁(𝑥_𝑘)} and {𝑓₂(𝑦_𝑘)} are bounded.
Note that 𝑈 is a firmly quasi-nonexpansive operator; we have 𝑥_𝑘+1−𝑥^∗2=𝛼²_𝑘𝑓₁𝑥_𝑘−𝑥^∗2+2𝛼_𝑘1−𝛼_𝑘×𝑓₁𝑥_𝑘−𝑥^∗,𝑈𝑢_𝑘−𝑥^∗+1−𝛼_𝑘²𝑈(𝑢_𝑘)−𝑥^∗2=𝛼²_𝑘𝑓₁𝑥_𝑘−𝑥^∗2+2𝛼_𝑘1−𝛼_𝑘×⟨𝑓₁𝑥_𝑘−𝑓₁𝑥^∗,𝑈𝑢_𝑘−𝑥^∗⟩+2𝛼_𝑘1−𝛼_𝑘⟨𝑓₁𝑥^∗−𝑥^∗,𝑈𝑢_𝑘−𝑥^∗⟩+1−𝛼_𝑘²𝑈(𝑢_𝑘)−𝑥^∗2≤𝛼²_𝑘𝑓₁𝑥_𝑘−𝑥^∗2+𝛼_𝑘1−𝛼_𝑘×𝑓₁𝑥_𝑘−𝑓₁𝑥^∗2+𝑈𝑢_𝑘−𝑥^∗2+2𝛼_𝑘1−𝛼_𝑘𝑓₁𝑥^∗−𝑥^∗,𝑈𝑢_𝑘−𝑥^∗+1−𝛼_𝑘²𝑈(𝑢_𝑘)−𝑥^∗2≤𝛼²_𝑘𝑓₁𝑥_𝑘−𝑥^∗2+𝛼_𝑘1−𝛼_𝑘𝜌²₁𝑥_𝑘−𝑥^∗2+𝛼_𝑘1−𝛼_𝑘𝑈𝑢_𝑘−𝑥^∗2+2𝛼_𝑘1−𝛼_𝑘𝑓₁𝑥^∗−𝑥^∗,𝑈𝑢_𝑘−𝑥^∗+1−𝛼_𝑘²𝑈𝑢_𝑘−𝑥^∗2≤1−𝛼_𝑘𝑢_𝑘−𝑥^∗2+𝛼_𝑘1−𝛼_𝑘𝜌²₁𝑥_𝑘−𝑥^∗2+𝛼_𝑘𝛼_𝑘𝑓₁(𝑥_𝑘)−𝑥^∗2+21−𝛼_𝑘×𝑓₁𝑥^∗−𝑥^∗,𝑈𝑢_𝑘−𝑥^∗. (36) Similarly, we have 𝑦_𝑘+1−𝑦^∗2≤1−𝛼_𝑘𝑣_𝑘−𝑦^∗2+𝛼_𝑘1−𝛼_𝑘𝜌²₂𝑦_𝑘−𝑦^∗2+𝛼_𝑘𝛼_𝑘𝑓₂(𝑦_𝑘)−𝑦^∗2+21−𝛼_𝑘×⟨𝑓₂𝑦^∗−𝑦^∗,𝑇𝑣_𝑘−𝑦^∗⟩. (37) So, by (31), (36), and (37), we obtain 𝑠_𝑘+1≤1−𝛼_𝑘𝑠_𝑘+𝛼_𝑘1−𝛼_𝑘𝜌²𝑠_𝑘+𝛼_𝑘𝛼_𝑘𝑓₁𝑥_𝑘−𝑥^∗2+𝑓₂𝑦_𝑘−𝑦^∗2+21−𝛼_𝑘⟨𝑓₁𝑥^∗−𝑥^∗,𝑈𝑢_𝑘−𝑥^∗⟩+⟨𝑓₂𝑦^∗−𝑦^∗,𝑇𝑣_𝑘−𝑦^∗⟩=1−𝜆_𝑘𝑠_𝑘+𝜆_𝑘𝛿_𝑘, (38) where 𝜆_𝑘=𝛼_𝑘1−1−𝛼_𝑘𝜌²,𝛿_𝑘=21−𝛼_𝑘𝑓₁𝑥^∗−𝑥^∗,𝑈𝑢_𝑘−𝑥^∗+𝑓₂𝑦^∗−𝑦^∗,𝑇𝑣_𝑘−𝑦^∗×1−1−𝛼_𝑘𝜌²⁻¹+𝛼_𝑘𝑓₁𝑥_𝑘−𝑥^∗2+𝑓₂𝑦_𝑘−𝑦^∗21−1−𝛼_𝑘𝜌². (39)
On the other hand, from (21), we have 𝑥_𝑘+1−𝑥^∗2≤𝛼_𝑘𝑓₁(𝑥_𝑘)−𝑥^∗2+1−𝛼_𝑘𝑈(𝑢_𝑘)−𝑥^∗2≤𝛼_𝑘𝑓₁𝑥_𝑘−𝑥^∗2+1−𝛼_𝑘𝑢_𝑘−𝑥^∗2−1−𝛼_𝑘𝑈𝑢_𝑘−𝑢_𝑘²,𝑦_𝑘+1−𝑦^∗2≤𝛼_𝑘𝑓₂𝑦_𝑘−𝑦^∗2+1−𝛼_𝑘𝑣_𝑘−𝑦^∗2−1−𝛼_𝑘𝑇(𝑣_𝑘)−𝑣_𝑘². (40) Adding up the last two inequalities and using (30), we obtain 𝑠_𝑘+1≤𝑢_𝑘−𝑥^∗2+𝑣_𝑘−𝑦^∗2+𝛼_𝑘𝑓₁𝑥_𝑘−𝑥^∗2+𝑓₂𝑦_𝑘−𝑦^∗2−1−𝛼_𝑘𝑈𝑢_𝑘−𝑢_𝑘²+𝑇𝑣_𝑘−𝑣_𝑘²≤𝑠_𝑘−𝛾_𝑘2𝐴𝑥_𝑘−𝐵𝑦_𝑘²−𝛾_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘²+𝐵^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘²+𝛼_𝑘𝑓₁𝑥_𝑘−𝑥^∗2+𝑓₂𝑦_𝑘−𝑦^∗2−1−𝛼_𝑘𝑈𝑢_𝑘−𝑢_𝑘²+𝑇𝑣_𝑘−𝑣_𝑘². (41) Now, by setting 𝜇_𝑘=𝛼_𝑘(‖𝑓₁(𝑥_𝑘)−𝑥^∗‖²+‖𝑓₂(𝑦_𝑘)−𝑦^∗‖²), 𝜂_𝑘=𝛾_𝑘[2‖𝐴𝑥_𝑘−𝐵𝑦_𝑘‖²−𝛾_𝑘(‖𝐴^∗(𝐴𝑥_𝑘−𝐵𝑦_𝑘)‖²+‖𝐵^∗(𝐴𝑥_𝑘−𝐵𝑦_𝑘)‖²)], and 𝜃_𝑘=(1−𝛼_𝑘)(‖𝑈(𝑢_𝑘)−𝑢_𝑘‖²+‖𝑇(𝑣_𝑘)−𝑣_𝑘‖²), (41) can be rewritten as the following form: 𝑠_𝑘+1≤𝑠_𝑘−𝜂_𝑘+𝜇_𝑘−𝜃_𝑘≤𝑠_𝑘−𝜂_𝑘+𝜇_𝑘,𝑘≥0. (42) By the assumption on 𝛼_𝑘, we get ∑^∞_𝑘=0𝜆_𝑘=∞ and lim_𝑘→∞𝜇_𝑘=0 which thanks to the boundedness of {𝑓₁(𝑥_𝑘)} and {𝑓₂(𝑦_𝑘)}.
The rest of the proof will be divided into two parts.
Case 1. Suppose that there exists 𝑘₀ such that {𝑠_𝑘}_𝑘≥𝑘0 is nonincreasing. In this situation, {𝑠_𝑘} is convergent because it is nonnegative so that lim_𝑘→∞(𝑠_𝑘+1−𝑠_𝑘)=0; hence, in light of (33) together with 𝛼_𝑘→0 and the boundedness of {𝑠_𝑘}, we obtain lim_𝑘→∞𝑢_𝑘−𝑈𝑢_𝑘=lim_𝑘→∞𝑣_𝑘−𝑇𝑣_𝑘=0. (43)
To use Lemma 8, it suffices to verify that, for all subsequences {𝑘_𝑙}⊂{𝑘}, lim_𝑙→∞𝜂_𝑘𝑙=0 implies limsup_𝑙→∞𝛿_𝑘𝑙≤0. (44) It follows from lim_𝑘→∞𝜂_𝑘𝑙=0 that lim_𝑙→∞𝛾_𝑘𝑙2𝐴𝑥_𝑘𝑙−𝐵𝑦_𝑘𝑙²−𝛾_𝑘𝑙𝐴^∗𝐴𝑥_𝑘𝑙−𝐵𝑦_𝑘𝑙²+𝐵^∗𝐴𝑥_𝑘𝑙−𝐵𝑦_𝑘𝑙²=0, (45) which yields lim_𝑙→∞‖𝐴𝑥_𝑘𝑙−𝐵𝑦_𝑘𝑙‖=0 from the assumption on 𝛾_𝑘. So, lim_𝑙→∞𝑢_𝑘𝑙−𝑥_𝑘𝑙=lim_𝑙→∞𝛾_𝑘𝑙𝐴^∗𝐴𝑥_𝑘𝑙−𝐵𝑦_𝑘𝑙=0,lim_𝑙→∞𝑣_𝑘𝑙−𝑦_𝑘𝑙=lim_𝑙→∞𝛾_𝑘𝑙𝐵^∗𝐴𝑥_𝑘𝑙−𝐵𝑦_𝑘𝑙=0. (46) Taking (𝑥,𝑦)∈𝜔_𝑤(𝑥_𝑘𝑙,𝑦_𝑘𝑙), from (46), we have (𝑥,𝑦)∈𝜔_𝑤(𝑢_𝑘𝑙,𝑣_𝑘𝑙). Combined with the demiclosednesses of 𝑈−𝐼 and 𝑇−𝐼 at 0, (43) yields 𝑈𝑥=𝑥 and 𝑇𝑦=𝑦. So, 𝑥∈𝐹(𝑈) and 𝑦∈𝐹(𝑇). On the other hand, 𝐴𝑥−𝐵𝑦∈𝜔_𝑤(𝐴𝑥_𝑘𝑙−𝐵𝑦_𝑘𝑙) and weakly lower semicontinuity of the norm imply 𝐴𝑥−𝐵𝑦≤liminf_𝑙→∞𝐴𝑥_𝑘𝑙−𝐵𝑦_𝑘𝑙=0; (47) hence, (𝑥,𝑦)∈Γ. So, 𝜔_𝑤(𝑥_𝑘𝑙,𝑦_𝑘𝑙)⊂Γ. Since lim_𝑘→∞𝛼_𝑘(‖𝑓₁(𝑥_𝑘)−𝑥^∗‖²+‖𝑓₂(𝑦_𝑘)−𝑦^∗‖²)=0 and lim_𝑘→∞(1−(1−𝛼_𝑘)𝜌²)=1−𝜌², to get (44), we only need to verify limsup_𝑙→∞𝑓₁𝑥^∗−𝑥^∗,𝑈𝑢_𝑘𝑙−𝑥^∗+𝑓₂𝑦^∗−𝑦^∗,𝑇𝑣_𝑘𝑙−𝑦^∗≤0. (48) Indeed, from (43) and (46), we have limsup_𝑙→∞𝑓₁𝑥^∗−𝑥^∗,𝑈𝑢_𝑘𝑙−𝑥^∗+𝑓₂𝑦^∗−𝑦^∗,𝑇𝑣_𝑘𝑙−𝑦^∗=limsup_𝑙→∞𝑓₁𝑥^∗−𝑥^∗,𝑢_𝑘𝑙−𝑥^∗+𝑓₂𝑦^∗−𝑦^∗,𝑣_𝑘𝑙−𝑦^∗=limsup_𝑙→∞𝑓₁𝑥^∗−𝑥^∗,𝑥_𝑘𝑙−𝑥^∗+𝑓₂𝑦^∗−𝑦^∗,𝑦_𝑘𝑙−𝑦^∗=−liminf_𝑙→∞𝐼−𝑓₁𝑥^∗,𝑥_𝑘𝑙−𝑥^∗+𝐼−𝑓₂𝑦^∗,𝑦_𝑘𝑙−𝑦^∗. (49) We can take subsequence {(𝑥_𝑘𝑙𝑗,𝑦_𝑘𝑙𝑗)} of {(𝑥_𝑘𝑙,𝑦_𝑘𝑙)} such that (𝑥_𝑘𝑙𝑗,𝑦_𝑘𝑙𝑗)⇀(𝑥,𝑦) as 𝑗→∞ and −liminf_𝑙→∞𝐼−𝑓₁𝑥^∗,𝑥_𝑘𝑙−𝑥^∗+𝐼−𝑓₂𝑦^∗,𝑦_𝑘𝑙−𝑦^∗=−lim_𝑗→∞𝐼−𝑓₁𝑥^∗,𝑥_𝑘𝑙𝑗−𝑥^∗+𝐼−𝑓₂𝑦^∗,𝑦_𝑘𝑙𝑗−𝑦^∗=−𝐼−𝑓₁𝑥^∗,𝑥−𝑥^∗+𝐼−𝑓₂𝑦^∗,𝑦−𝑦^∗. (50) Since 𝜔_𝑤(𝑥_𝑘𝑙,𝑦_𝑘𝑙)⊂Γ and (𝑥^∗,𝑦^∗) is the solution of the variational inequality problem (25), from (49) and (50), we obtain limsup_𝑙→∞𝑓₁𝑥^∗−𝑥^∗,𝑈𝑢_𝑘𝑙−𝑥^∗+𝑓₂𝑦^∗−𝑦^∗,𝑇𝑣_𝑘𝑙−𝑦^∗≤0. (51) From Lemma 8, it follows lim_𝑘→∞𝑥_𝑘−𝑥^∗2+𝑦_𝑘−𝑦^∗2=0, (52) which implies that 𝑥_𝑘→𝑥^∗ and 𝑦_𝑘→𝑦^∗.
Case 2. Suppose there exists a subsequence {𝑠_𝑘𝑗}_𝑗≥0 of {𝑠_𝑘} such that 𝑠_𝑘𝑗<𝑠_𝑘𝑗+1 for all 𝑗≥0. In this situation, we consider the sequence of indices {𝜏(𝑘)} as defined in Lemma 8. It follows that 𝑠_𝜏(𝑘)+1−𝑠_𝜏(𝑘)>0. From (42), we have 0≤𝜂_𝜏(𝑘)≤𝑠_𝜏(𝑘)−𝑠_𝜏(𝑘)+1+𝜇_𝜏(𝑘)<𝜇_𝜏(𝑘),𝑘≥0. (53) So, by lim_𝑘→∞𝜇_𝑘=0, we obtain lim_𝑘→∞𝜂_𝜏(𝑘)=0. (54) Again from (42), we get 0≤𝜃_𝜏(𝑘)≤𝜇_𝜏(𝑘)−𝜂_𝜏(𝑘); (55) hence, lim_𝑘→∞𝜃_𝜏(𝑘)=lim_𝑘→∞1−𝛼_𝜏(𝑘)×𝑈𝑢_𝜏(𝑘)−𝑢_𝜏(𝑘)²+𝑇𝑣_𝜏(𝑘)−𝑣_𝜏(𝑘)²=0. (56) In light of 𝛼_𝑘→0, we obtain lim_𝑘→∞𝑢_𝜏(𝑘)−𝑈𝑢_𝜏(𝑘)=lim_𝑘→∞𝑣_𝜏(𝑘)−𝑇𝑣_𝜏(𝑘)=0. (57) From 𝑦_𝜏(𝑘)→0, similar to Case 1, we have lim_𝑘→∞𝐴𝑥_𝜏(𝑘)−𝐵𝑦_𝜏(𝑘)=lim_𝑘→∞𝑢_𝜏(𝑘)−𝑥_𝜏(𝑘)=lim_𝑘→∞𝑣_𝜏(𝑘)−𝑦_𝜏(𝑘)=0, (58)𝜔_𝑤(𝑥_𝜏(𝑘),𝑦_𝜏(𝑘))⊂Γ, and limsup_𝑘→∞⟨𝑓₁𝑥^∗−𝑥^∗,𝑈𝑢_𝜏(𝑘)−𝑥^∗⟩+𝑓₂𝑦^∗−𝑦^∗,𝑇𝑣_𝜏(𝑘)−𝑦^∗≤0, (59) which implies limsup_𝑘→∞𝛿_𝜏(𝑘)≤0. (60) From 𝑠_𝜏(𝑘)+1−𝑠_𝜏(𝑘)>0 and (38), it follows that 𝜆_𝜏(𝑘)𝑠_𝜏(𝑘)≤𝜆_𝜏(𝑘)𝛿_𝜏(𝑘). (61) Since 𝑠_𝜏(𝑘)+1−𝑠_𝜏(𝑘)>0, again from (38), we may assume 𝜆_𝜏(𝑘)>0 for all 𝑘≥0. It follows from (60) and (61) that lim_𝑘→∞𝑠_𝜏(𝑘)=0 and hence lim_𝑘→∞𝑥_𝜏(𝑘)−𝑥^∗=lim_𝑘→∞𝑦_𝜏(𝑘)−𝑦^∗=0. (62) On the other hand, it follows that 𝑥_𝜏(𝑘)+1−𝑥_𝜏(𝑘)=𝛼_𝜏(𝑘)𝑓𝑥_𝜏(𝑘)−𝑥_𝜏(𝑘)+1−𝛼_𝜏(𝑘)𝑈𝑢_𝜏(𝑘)−𝑥_𝜏(𝑘)≤𝛼_𝜏(𝑘)𝑓𝑥_𝜏(𝑘)−𝑥_𝜏(𝑘)+1−𝛼_𝜏(𝑘)×𝑈𝑢_𝜏(𝑘)−𝑢_𝜏(𝑘)+𝑢_𝜏(𝑘)−𝑥_𝜏(𝑘), (63) which, by 𝛼_𝑘→0, (57), and (58), implies that lim_𝑘→∞𝑥_𝜏(𝑘)+1−𝑥_𝜏(𝑘)=0. (64) By (62), we obtain lim_𝑘→∞𝑥_𝜏(𝑘)+1−𝑥^∗=0. (65) Similarly, we have lim_𝑘→∞‖𝑦_𝜏(𝑘)+1−𝑦^∗‖=0; hence, lim_𝑘→∞𝑠_𝜏(𝑘)+1=lim_𝑘→∞𝑥_𝜏(𝑘)+1−𝑥^∗2+𝑦_𝜏(𝑘)+1−𝑦^∗2=0. (66) Then, recalling that 𝑠_𝑘≤𝑠_𝜏(𝑘)+1 (by Lemma 8), we get lim_𝑘→∞𝑠_𝑘=0.
So, sequence {(𝑥_𝑘,𝑦_𝑘)} strongly converges to the solution (𝑥^∗,𝑦^∗) of (6) which solves the variational inequality problem (25).

4. Another Split Problem Deduced from SCFP

We now turn our attention to providing some algorithms for solving another split problem without prior knowledge of operator norms.

4.1. Split Feasibility Problem

Taking 𝑈=𝑃_𝐶 and 𝑇=𝑃_𝑄, we have that the following viscosity iterative algorithm for split feasibility problem (SFP) under consideration is nothing but find𝑥∈𝐶,𝑦∈𝑄,suchthat𝐴𝑥=𝐵𝑦. (67)

Algorithm 13. Let 𝑥₀∈𝐻₁, 𝑦₀∈𝐻₂ be arbitrary. Consider 𝑢_𝑘=𝑥_𝑘−𝛾_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑥_𝑘+1=𝛼_𝑘𝑓₁𝑥_𝑘+1−𝛼_𝑘𝑃_𝐶𝑢_𝑘,𝑣_𝑘=𝑦_𝑘+𝛾_𝑘𝐵^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑦_𝑘+1=𝛼_𝑘𝑓₂𝑥_𝑘+1−𝛼_𝑘𝑃_𝑄𝑣_𝑘, (68) where the stepsize 𝛾_𝑘 is chosen by (22) in Algorithm 9.

In [16], Dong et al. introduced Algorithm 13 for SFP (67) without prior knowledge of operator norms. The stepsize 𝛾_𝑘 is chosen in such a way that 𝛾_𝑘∈𝜖,min𝐴𝑥_𝑘−𝐵𝑦_𝑘²𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘²,𝐴𝑥_𝑘−𝐵𝑦_𝑘²𝐵^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘²−𝜖. (69) It is easy to see that the results of this paper improve and extend the corresponding results of [16].

4.2. Split Common Null Point Problem

Given a maximal monotone operator 𝑀∶𝐻₁→2^𝐻1, it is well known that its associated resolvent mapping, 𝐽^𝑀_𝜇(𝑥)∶=(𝐼+𝜇𝑀)⁻¹, is firmly quasi-nonexpansive and 0∈𝑀(𝑥)⇔𝑥=𝐽^𝑀_𝜇(𝑥). In other words, zeroes of 𝑀 are exactly fixed-points of its resolvent mapping. By taking 𝑈=𝐽^𝑀_𝜇, 𝑇=𝐽^𝑁_𝜈, where 𝑁∶𝐻₂→2^𝐻2 is another maximal monotone operator, the problem under consideration is nothing but find𝑥^∗∈𝑀⁻¹(0),𝑦^∗∈𝑁⁻¹(0)suchthat𝐴𝑥^∗=𝐵𝑦^∗, (70) and the algorithms take the following equivalent form: ∀𝑥₀∈𝐻₁,𝑦₀∈𝐻₂,𝑢_𝑘=𝑥_𝑘−𝛾_𝑘𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑥_𝑘+1=𝛼_𝑘𝑓₁𝑥_𝑘+1−𝛼_𝑘𝐽^𝑀_𝜇𝑢_𝑘,𝑣_𝑘=𝑦_𝑘+𝛾_𝑘𝐵^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘,𝑦_𝑘+1=𝛽_𝑘𝑓₂𝑥_𝑘+1−𝛽_𝑘𝐽^𝑁_𝜈𝑣_𝑘,𝑘≥1, (71) The stepsize 𝛾_𝑘 is chosen as follows: 𝛾_𝑘∈𝜖,2𝐴𝑥_𝑘−𝐵𝑦_𝑘²×𝐴^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘²+𝐵^∗𝐴𝑥_𝑘−𝐵𝑦_𝑘²⁻¹−𝜖,𝑘∈Ω; (72) otherwise, 𝛾_𝑘=𝛾 (𝛾 being any nonnegative value), where the set of indexes Ω={𝑘∶𝐴𝑥_𝑘−𝐵𝑦_𝑘≠0}.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research was supported by Fundamental Research Funds for the Central Universities (Program no. 3122013k004).