An Implicit Algorithm for the Split Fixed Point and Convex Feasibility Problems

Abstract

We consider an implicit algorithm for the split fixed point and convex feasibility problems. Strong convergence theorem is obtained.

1. Introduction

Due to their broad applicability in many areas, especially in signal processing (e.g., phase retrieval) and image restoration, the split feasibility problems continue to receive great attention; see, for example, [1–6]. The present paper is devoted to this topic. Now we recall that the split feasibility problem originally introduced by Censor and Elfving [7] is to find 𝑥^† such that 𝑥^†∈ℂ,𝔸𝑥^†∈ℚ, (1) where ℂ and ℚ are two closed convex subsets of two Hilbert spaces ℍ₁ and ℍ₂, respectively, and 𝔸∶ℍ₁→ℍ₂ is a bounded linear operator. A special case of (1) is when ℚ={𝑏} is singleton, and then (1) is reduced to the convexly constrained linear inverse problem 𝑥^†∈ℂ,𝔸𝑥^†=𝑏, (2) which has received considerable attention. We can use projected Landweber algorithm to solve (2). The projected Landweber algorithm generates a sequence {𝑥^𝑘} in such a way that 𝑥^𝑘+1=proj_ℂ𝑥^𝑘+𝛾𝔸^∗𝑏−𝔸𝑥^𝑘, (3) where proj_ℂ denotes the nearest point projection from ℍ₁ onto ℂ, 𝛾>0 is a parameter such that 0<𝛾<2/‖𝔸‖², and 𝔸^∗ is the transpose of 𝔸. When the system (2) is reduced to the unconstrained linear system 𝔸𝑥^∗=𝑏, (4) then the projected Landweber algorithm is turned to the Landweber algorithm 𝑥^𝑘+1=𝑥^𝑘+𝛾𝔸^∗𝑏−𝔸𝑥^𝑘. (5) Note that (1) is equivalent to the fixed point equation 𝑥^†=proj_ℂ𝐼−𝜂𝔸^∗𝐼−proj_ℚ𝔸𝑥^†. (6) Using this relation, we can suggest the following iterative algorithm: 𝑥^𝑘+1=proj_ℂ𝑥^𝑘−𝜁𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘,𝑘∈ℕ, (7) which is refereed as ℂℚ algorithm and was devised by Byrne [8]. ℂℚ algorithm has been extensively studied; see, for instance, [9–11].

The ℂℚ algorithm (7) is proved to converge weakly but fails to converge in norm in general infinite-dimensional Hilbert spaces ℍ₁ and ℍ₂. Tikhonov's regularization method can solve this problem. First, we define a convex function 𝑓 by 𝑓(𝑥)=12𝔸𝑥−proj_ℚ𝔸𝑥² (8) with its gradient ∇𝑓(𝑥)=𝔸^∗𝐼−proj_ℚ𝔸𝑥 (9) and consider the minimization problem min_𝑥∈ℂ𝑓(𝑥). (10) It is known that 𝑥^∗∈ℂ solves (1) if and only if 𝑓(𝑥^∗). We know that (10) is ill-posed. So regularization is needed. We consider Tikhonov's regularization: min_𝑥∈ℂ𝑓_𝛼(𝑥)1∶=2𝐼−proj_ℚ𝔸𝑥²+12𝛼‖𝑥‖², (11) where 𝛼>0 is the regularization parameter. The gradient ∇𝑓_𝛼 of 𝑓_𝛼 is given by ∇𝑓_𝛼(𝑥)=∇𝑓(𝑥)+𝛼𝑥=𝔸^∗𝐼−proj_ℚ𝔸𝑥+𝛼𝑥. (12) Define a Picard iterates 𝑥_𝛼^𝑘+1=proj_ℂ𝐼−𝛿𝔸^∗𝐼−proj_ℚ𝔸+𝛼𝐼𝑥^𝑘_𝛼,𝑘∈ℕ. (13) Xu [12] proves that if (1) is solvable, then as 𝑘→∞, 𝑥^𝑘_𝛼→𝑥_𝛼 and consequently the strong lim_𝛼→0𝑥_𝛼 exists and is the minimum-norm solution of (1). Note that (13) is a double-step iteration. Xu [12] introduced a single step regularized method: 𝑥^𝑘+1=proj_ℂ𝐼−𝛿_𝑘∇𝑓_𝜉𝑘𝑥^𝑘=proj_ℂ1−𝜉_𝑘𝛿_𝑘𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘,𝑘∈ℕ. (14) It is shown that the sequence {𝑥^𝑘} generated by (14) converges to the solution of (1) provided that the parameters {𝜉_𝑘}⊂(0,1) and {𝛿_𝑘}⊂(0,𝜉_𝑛/(‖𝐴‖²+𝜉_𝑛)) satisfy lim_𝑘→∞𝜉_𝑘=0,^∞_𝑘=1𝜉_𝑘𝛿_𝑘=∞,lim_𝑘→∞𝛿_𝑘+1−𝛿_𝑘+𝛿_𝑘𝜉_𝑘+1−𝜉_𝑘𝜉_𝑘+1𝛿_𝑘+1²=0. (15)

Inspired by (14), Ceng et al. [3] introduced the following relaxed extragradient method: 𝑦^𝑘=proj_ℂ𝑥^𝑘−𝜉_𝑘∇𝑓𝑥_𝑘+𝜃_𝑘𝑥^𝑘,𝑥^𝑘+1=𝛽_𝑘𝑥^𝑘+𝛾_𝑘𝑦^𝑘+𝛿_𝑘proj_ℂ×𝑥^𝑘−𝜉_𝑘∇𝑓𝑦^𝑘+𝜃_𝑘𝑦^𝑘,𝑘∈ℕ, (16) where the sequences {𝜃_𝑘}⊂(0,1), {𝛽_𝑘}⊂(0,1), {𝛾_𝑘}⊂(0,1), {𝛿_𝑘}⊂(0,1), and {𝜉_𝑘}⊂(0,𝜃_𝑘/(‖𝐴‖²+𝜃_𝑘)²) satisfy the conditions lim_𝑘→∞𝜃_𝑘=lim_𝑘→∞𝜉_𝑘=lim_𝑘→∞𝜉_𝑘+1−𝜉_𝑘+𝜉_𝑘𝜃_𝑘+1−𝜃_𝑘𝜃³_𝑘+1𝜉²_𝑘+1𝛿_𝑘+1=0,_∞𝑘=1𝜃²_𝑘𝜉_𝑘𝛿_𝑘=∞,2𝛿_𝑘‖𝐴‖²+𝜃_𝑘≤𝛾_𝑘𝜉_𝑘 (17) for all 𝑘∈ℕ. Ceng et al. proved that the sequence {𝑥^𝑘} generated by (16) converges to the solution of (1) which is the the minimum-norm element. Recently, Ceng et al. [13] further introduced another regularization for the split feasibility problem and the fixed point problem: 𝑦^𝑘=proj_ℂ𝑥^𝑘−𝛾_𝑘∇𝑓𝑥_𝑘+𝜃_𝑘𝑥^𝑘,𝑥^𝑘+1=𝜗_𝑘𝑥^𝑘+1−𝜗_𝑘×𝑈proj_ℂ𝑥^𝑘−𝛾_𝑘∇𝑓𝑦^𝑘+𝜃_𝑘𝑦^𝑘,𝑘∈ℕ. (18) Ceng et al. proved that algorithm (18) has weak convergence.

Motivated by the above works, in this paper, our main purpose is to introduce an implicit algorithm for solving the split fixed point and convex feasibility problems. We show that the implicit algorithm converges strongly to the solution of the split fixed point and convex feasibility problems.

2. Preliminaries

Let ℍ be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. Let ℂ be a nonempty closed convex subset of ℍ.

Definition 1. A mapping 𝕌∶ℂ→ℂ is called nonexpansive if 𝕌𝑝−𝕌𝑞≤𝑝−𝑞 (19) for all 𝑝,𝑞∈ℂ.

We will use Fix(𝕌) to denote the set of fixed points of 𝕌; that is, Fix(𝕌)={𝑥^‡∈𝐶∶𝑥^‡=𝕌𝑥^‡}.

Definition 2. A mapping 𝒞∶ℂ→ℂ is called contractive if 𝒞𝑝−𝒞𝑞≤𝜅𝑝−𝑞 (20) for all 𝑝,𝑞∈ℂ and for some constant 𝜅∈(0,1). In this case, we call 𝒞 a 𝜅-contraction.

Definition 3. A linear bounded operator 𝔹∶ℍ→ℍ is called strongly positive if there exists a constant 𝛾>0 such that 𝔹𝑥^‡,𝑥^‡≥𝛾𝑥^‡2 (21) for all 𝑥^‡∈ℍ.

Definition 4. We call that proj_ℂ∶ℍ→ℂ is the metric projection if for each 𝑥^‡∈ℍ𝑥^‡−proj_ℂ𝑥^‡=inf𝑥^‡−𝑥∶𝑥∈ℂ. (22)

It is well known that the metric projection proj_ℂ∶ℍ→ℂ is characterized by 𝑥^‡−proj_ℂ𝑥^‡,𝑥−proj_ℂ𝑥^‡≤0 (23) for all 𝑥^‡∈ℍ, 𝑥∈ℂ. From this, we can deduce that proj_ℂ is firmly nonexpansive; that is, proj_ℂ𝑥^‡−proj_ℂ(𝑥)²≤𝑥^‡−𝑥,proj_ℂ𝑥^‡−proj_ℂ(𝑥) (24) for all 𝑥^‡,𝑥∈ℍ. Hence proj_ℂ is nonexpansive.

Lemma 5 (see [14]). Let ℂ be a closed convex subset of a real Hilbert space ℍ, and let 𝕌∶ℂ→ℂ be a nonexpansive mapping. Then, the mapping 𝐼−𝕌 is demiclosed. That is, if {𝑥^𝑘} is a sequence in 𝐶 such that 𝑥^𝑘→𝑥^§ weakly and (𝐼−𝕌)𝑥^𝑘→𝑦 strongly, then (𝐼−𝕌)𝑥^§=𝑦.

3. Main Result

In this section, we first introduce our algorithm for solving this problem and consequently we give convergence analysis.

Let ℍ₁ and ℍ₂ be two Hilbert spaces and ℂ⊂ℍ₁ and ℚ⊂ℍ₂ two nonempty closed convex sets. Let 𝔸∶ℍ₁→ℍ₂ be a bounded linear operator with its adjoint 𝔸^∗. Let 𝔹 be a strongly positive bounded linear operator on ℍ₁ with coefficient 𝛾>0. Let 𝒞∶ℍ₁→ℍ₁ be a 𝜅-contraction. Let 𝕍∶ℚ→ℚ and 𝕌∶ℂ→ℂ be two nonexpansive mappings.

In the sequel, our objective is to Find𝑥^†∈ℂ∩Fix(𝕌)suchthat𝔸𝑥^†∈ℚ∩Fix(𝕍). (25) We use Ω to denote the solution set of (25); that is, Ω=𝑥^†∣𝑥^†∈ℂ∩Fix(𝕌),𝔸𝑥^†∈ℚ∩Fix(𝕍). (26)

Now, we introduce the following implicit algorithm.

Algorithm 6. Define an implicit algorithm {𝑥_𝑡} as follows: 𝑥_𝑡=𝑡𝜁𝒞𝑥_𝑡+(𝐼−𝑡𝔹)𝕌proj_ℂ×𝑥_𝑡−𝛿𝔸^∗𝐼−𝕍proj_ℚ𝔸𝑥_𝑡,𝑡∈(0,1), (27) where 𝜁∈(0,𝛾/𝜅) and 𝛿∈(0,1/‖𝐴‖²) are two constants.

Remark 7. {𝑥_𝑡} is well-defined. Define a mapping ℝ∶ℂ→ℂ as ℝ𝑥=proj_ℂ𝑥−𝛿𝔸^∗𝐼−𝕍proj_ℚ𝔸𝑥,∀𝑥∈ℂ. (28) Then, we have ℝ𝑥−ℝ𝑦²=proj_ℂ𝑥−𝛿𝔸^∗𝐼−𝕍proj_ℚ𝔸𝑥−proj_ℂ𝑦−𝛿𝔸^∗𝐼−𝕍proj_ℚ𝔸𝑦²≤𝑥−𝛿𝔸^∗𝐼−𝕍proj_ℚ−𝔸𝑥𝑦−𝛿𝔸^∗𝐼−𝕍proj_ℚ𝔸𝑦²=𝑥−𝑦+𝛿𝔸^∗×𝕍proj_ℚ𝔸𝑥−𝔸𝑥−𝕍proj_ℚ𝔸𝑦−𝔸𝑦²=𝑥−𝑦²+2𝛿𝑥−𝑦,𝔸^∗𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦+𝛿²𝔸^∗𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦²=𝑥−𝑦²+2𝛿𝔸𝑥−𝑦,𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦+𝛿²𝔸^∗𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦²=𝑥−𝑦²+2𝛿𝕍proj_ℚ𝔸𝑥−𝕍proj_ℚ𝔸𝑦,𝕍proj_ℚ𝔸𝑥−𝔸𝑥−𝕍proj_ℚ𝔸𝑦−𝔸𝑦−2𝛿𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦²+𝛿²𝔸^∗𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦²=𝑥−𝑦²+𝛿𝕍proj_ℚ𝔸𝑥−𝕍proj_ℚ𝔸𝑦²+𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦²−𝔸𝑥−𝔸𝑦²−2𝛿𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦²+𝛿²𝔸^∗𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦²≤𝑥−𝑦²−𝛿𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦²+𝛿²‖𝔸‖²𝕍proj_ℚ−𝔸𝑥−𝔸𝑥𝕍proj_ℚ𝔸𝑦−𝔸𝑦²≤𝑥−𝑦². (29) This indicates that 𝑅 is nonexpansive. Consequently, for fixed 𝑡∈(0,1), we have that the mapping 𝑡𝜁𝒞+(𝐼−𝑡𝔹)𝕌𝑅 is contractive due to the facts that 𝒞 is a 𝜅-contraction and 𝕌 is nonexpansive. Therefore, {𝑥_𝑡} is well-defined.

Next, we prove the convergence of (27).

Theorem 8. Suppose that Ω≠∅. Then the net {𝑥_𝑡} generated by algorithm (25) converges strongly to 𝑝^†=proj_Ω(𝜁𝒞+𝐼−𝔹)𝑝^† which solves the following variational inequality: (𝜁𝒞−𝔹)𝑥,𝑦−𝑥≤0,∀𝑦∈Ω. (30)

Proof. Set 𝑟_𝑡=proj_ℚ𝔸𝑥_𝑡, 𝑣_𝑡=𝑥_𝑡−𝛿𝔸^∗(𝐼−𝕍proj_ℚ)𝔸𝑥_𝑡, and 𝑢_𝑡=proj_ℂ(𝑥_𝑡−𝛿𝔸^∗(𝐼−𝕍proj_ℚ)𝔸𝑥_𝑡) for all 𝑡∈(0,1). Then 𝑢_𝑡=proj_ℂ𝑣_𝑡. It is clear that the solution of (30) is unique. Let 𝑝^†=proj_Ω(𝜁𝒞+𝐼−𝔹)𝑝^†. Then, we have 𝑝^†∈ℂ∩Fix(𝕌) and 𝔸𝑝^†∈ℚ∩Fix(𝕍). First, we easily deduce the following three inequalities: 𝑟_𝑡−𝔸𝑝^†=proj_ℚ𝔸𝑥_𝑡−𝔸𝑝^†≤𝔸𝑥_𝑡−𝔸𝑝^†𝑢_𝑡,−𝑝^†=proj_ℂ𝑣_𝑡−𝑝^†≤𝑣_𝑡−𝑝^†,𝕍𝑟_𝑡−𝔸𝑝^†2≤𝑟_𝑡−𝔸𝑝^†2≤𝔸𝑥_𝑡−𝔸𝑝^†2−𝑟_𝑡−𝔸𝑥_𝑡². (31)(32)(33) From (25), we have 𝑥_𝑡−𝑝^†=𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†+(𝐼−𝑡𝔹)𝕌𝑢_𝑡−𝑝^†≤𝑡𝜁𝒞𝑥_𝑡−𝒞𝑝^†+𝑡𝜁𝒞𝑝^†−𝔹𝑝^†+1−𝑡𝛾𝑢_𝑡−𝑝^†. (34) Note that 𝑣_𝑡−𝑝^†2=𝑥_𝑡−𝑝^†+𝛿𝔸^∗𝕍𝑟_𝑡−𝔸𝑥_𝑡²=𝑥_𝑡−𝑝^†2+𝛿²𝔸^∗(𝕍𝑟_𝑡−𝔸𝑥_𝑡)²+2𝛿𝑥_𝑡−𝑝^†,𝔸^∗𝕍𝑟_𝑡−𝔸𝑥_𝑡. (35) Since 𝐴 is a linear operator and 𝔸^∗ is the adjoint of 𝔸, we get 𝑥_𝑡−𝑝^†,𝔸^∗𝕍𝑟_𝑡−𝔸𝑥_𝑡=𝔸𝑥_𝑡−𝑝^†,𝕍𝑟_𝑡−𝔸𝑥_𝑡=𝔸𝑥_𝑡−𝔸𝑝^†+𝕍𝑟_𝑡−𝔸𝑥_𝑡−𝕍𝑟_𝑡−𝔸𝑥_𝑡,𝕍𝑟_𝑡−𝔸𝑥_𝑡=𝕍𝑟_𝑡−𝔸𝑝^†,𝕍𝑟_𝑡−𝔸𝑥_𝑡−𝕍𝑟_𝑡−𝔸𝑥_𝑡². (36) At the same time, we know ⟨𝕍𝑟_𝑡−𝔸𝑝^†,𝕍𝑟_𝑡−𝔸𝑥_𝑡⟩=12𝕍𝑟_𝑡−𝔸𝑝^†2+𝕍𝑟_𝑡−𝔸𝑥_𝑡²−𝔸𝑥_𝑡−𝔸𝑝^†2. (37) By (33), (36), and (37), we get 𝑥_𝑡−𝑝^†,𝔸^∗𝕍𝑟_𝑡−𝔸𝑥_𝑡=12𝕍𝑟_𝑡−𝔸𝑝^†2+𝕍𝑟_𝑡−𝔸𝑥_𝑡²−𝔸𝑥_𝑡−𝔸𝑝^†2−𝕍𝑟_𝑡−𝔸𝑥_𝑡²≤12𝔸𝑥_𝑡−𝔸𝑝^†2−𝑟_𝑡−𝔸𝑥_𝑡²+𝕍𝑟_𝑡−𝔸𝑥_𝑡²−𝔸𝑥_𝑡−𝔸𝑝^†2−𝕍𝑟_𝑡−𝔸𝑥_𝑡²1=−2𝑟_𝑡−𝔸𝑥_𝑡²−12𝕍𝑟_𝑡−𝔸𝑥_𝑡². (38) Substituting (38) into (35) to deduce 𝑣_𝑡−𝑝^†2≤𝑥_𝑡−𝑝^†2+𝛿²‖𝔸‖²𝕍𝑟_𝑡−𝔸𝑥_𝑡²−1+2𝛿2𝑟_𝑡−𝔸𝑥_𝑡²−12𝕍𝑟_𝑡−𝔸𝑥_𝑡²=𝑥_𝑡−𝑝^†2+𝛿²‖𝔸‖²−𝛿𝕍𝑟_{𝑡−𝔸𝑥𝑡}²−𝛿𝑟_𝑡−𝔸𝑥_𝑡²≤𝑥_𝑡−𝑝^†2. (39) It follows that 𝑣_𝑡−𝑝^†≤𝑥_𝑡−𝑝^†. (40) Thus, from (34), we get 𝑥_𝑡−𝑝^†≤𝑡𝜁𝜅𝑥_𝑡−𝑝^†+𝑡𝜁𝒞𝑝^†−𝔹𝑝^†+1−𝑡𝛾𝑥_𝑡−𝑝^†=1−𝛾−𝜁𝜅𝑡𝑥_𝑡−𝑝^†+𝑡𝜁𝒞𝑝^†−𝐵𝑝^†. (41) So, 𝑥_𝑡−𝑝^†≤𝜁𝒞𝑝^†−𝐵𝑝^†.𝛾−𝜁𝜅 (42) The boundedness of the net {𝑥_𝑡} yields.
Since 𝑥_𝑡−𝕌𝑢_𝑡=𝑡(𝜁𝒞(𝑥_𝑡)−𝔹𝕌𝑢_𝑡), we obtain lim_𝑡→0𝑥_𝑡−𝕌𝑢_𝑡=0. (43) Using the firmly nonexpansive necessity of proj_ℂ, we have 𝑢_𝑡−𝑝^†2=proj_ℂ𝑣_𝑡−𝑝^†2≤𝑣_𝑡−𝑝^†2−proj_ℂ𝑣_𝑡−𝑣_𝑡²=𝑣_𝑡−𝑝^†2−𝑢_𝑡−𝑣_𝑡². (44) From (34), we derive that 𝑥_𝑡−𝑝^†2=𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†+(𝐼−𝑡𝔹)𝕌𝑢_𝑡−𝑝^†2≤(𝐼−𝑡𝔹)𝕌𝑢_𝑡−𝑝^†2+2𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†,𝑥_𝑡−𝑝^†≤‖𝐼−𝑡𝔹‖𝕌𝑢_𝑡−𝑝^†2+2𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†𝑥_𝑡−𝑝^†≤1−𝑡𝛾²𝑢_𝑡−𝑝^†2+2𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†𝑥_𝑡−𝑝^†. (45) This together with (44) implies that 𝑥_𝑡−𝑝^†2≤𝑣_𝑡−𝑝^†2−𝑢_𝑡−𝑣_𝑡²+2𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†𝑥_𝑡−𝑝^†≤𝑥_𝑡−𝑝^†2−𝑢_𝑡−𝑣_𝑡²+2𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†𝑥_𝑡−𝑝^†. (46) It follows that 𝑢_𝑡−𝑣_𝑡²≤2𝑡𝜁𝒞𝑥_𝑡−𝐵𝑝^†𝑥_𝑡−𝑝^†. (47) Hence, lim_𝑡→0𝑢_𝑡−𝑣_𝑡=0. (48) Returning to (45) and using (39), we have 𝑥_𝑡−𝑝^†2≤1−𝑡𝛾²𝑢_𝑡−𝑝^†2+2𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†𝑥_𝑡−𝑝^†≤𝑣_𝑡−𝑝^†2+2𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†𝑥_𝑡−𝑝^†≤𝑥_𝑡−𝑝^†2+𝛿²‖𝔸‖²−𝛿𝕍𝑟_𝑡−𝔸𝑥_𝑡²−𝛿𝑟_𝑡−𝔸𝑥_𝑡²+2𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†𝑥_𝑡−𝑝^†. (49) Thus, 𝛿−𝛿²‖𝔸‖²𝕍𝑟_𝑡−𝔸𝑥_𝑡²+𝛿𝑟_𝑡−𝔸𝑥_𝑡²≤2𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†𝑥_𝑡−𝑝^†, (50) which implies that lim_𝑡→0𝕍𝑟_𝑡−𝔸𝑥_𝑡=lim_𝑡→0𝑟_𝑡−𝔸𝑥_𝑡=0. (51) So, lim_𝑡→0𝕍𝑟_𝑡−𝑟_𝑡=0. (52) Note that 𝑣_𝑡−𝑥_𝑡=𝛿𝔸^∗𝕍proj_ℚ−𝐼𝔸𝑥_𝑡≤𝛿‖𝔸‖𝕍𝑟_𝑡−𝔸𝑥_𝑡. (53) It follows from (51) that lim_𝑡→0𝑥_𝑡−𝑣_𝑡=0. (54) From (43), (48), and (54), we get lim_𝑡→0𝑥_𝑡−𝕌𝑥_𝑡=0. (55) Next we show that the net {𝑥_𝑡} is relatively norm-compact as 𝑡→0⁺. Assume that {𝑡_𝑛}⊂(0,1) is such that 𝑡_𝑛→0⁺ as 𝑛→∞. Put 𝑥_𝑛∶=𝑥_𝑡𝑛 and 𝑣_𝑛∶=𝑣_𝑡𝑛.
From (45), we have 𝑥_𝑡−𝑝^†2≤1−𝑡𝛾²𝑢_𝑡−𝑝^†2+2𝑡𝜁𝒞𝑥_𝑡−𝔹𝑝^†,𝑥_𝑡−𝑝^†≤1−𝑡𝛾²𝑥_𝑡−𝑝^†2+2𝑡𝜁𝒞𝑥_𝑡−𝒞𝑝^†,𝑥_𝑡−𝑝^†+2𝑡𝜁𝒞𝑝^†−𝐵𝑝^†,𝑥_𝑡−𝑝^†≤1−𝑡𝛾²𝑥_𝑡−𝑝^†2+2𝑡𝜁𝜅𝑥_𝑡−𝑝^†2+2𝑡𝜁𝒞𝑝^†−𝔹𝑝^†,𝑥_𝑡−𝑝^†. (56) It follows that 𝑥_𝑡−𝑝^†2≤1𝑝𝛾−𝜁𝜅𝜁𝒞^†−𝔹𝑝^†,𝑥_𝑡−𝑝^†+𝛾²𝑥_𝑡−𝑝^†22𝑡≤1𝛾−𝜁𝜅𝑝𝛾−𝜁𝜅𝜁𝒞^†−𝔹𝑝^†,𝑥_𝑡−𝑝^†+𝑀𝑡, (57) where 𝑀>0 is a constant satisfying sup_𝑡∈(0,1)(𝛾²‖𝑥_𝑡−𝑝^†‖²/2(𝛾−𝜁𝜅))≤𝑀. In particular, we have 𝑥_𝑛−𝑝^†2≤1𝑝𝛾−𝜁𝜅𝜁𝒞^†−𝔹𝑝^†,𝑥_𝑛−𝑝^†+𝑀𝑡_𝑛, (58) since {𝑥_𝑛} is bounded, without loss of generality, we may assume that {𝑥_𝑛} converges weakly to a point 𝑞^†∈𝐶. We deduce from the above results that 𝑣_𝑛⇀𝑞^†,𝑢_𝑛⇀𝑞^†,𝔸𝑥_𝑛⇀𝑞^†,𝑟_𝑛⇀𝔸𝑞^†. (59) By the demiclosed principle of the nonexpansive mappings 𝕍 and 𝕌 (see Lemma 5), we deduce 𝑞^†∈Fix(𝕌) and 𝔸𝑞^†∈Fix(𝕍). Note that 𝑢_𝑛𝑖=proj_ℂ𝑣_𝑛𝑖∈ℂ and 𝑟_𝑛𝑖=proj_ℚ𝔸𝑥_𝑛𝑖∈ℚ. From (58), we deduce 𝑞^†∈ℂ and 𝔸𝑞^†∈ℚ. To this end, we deduce 𝑞^†∈ℂ∩Fix(𝕌) and 𝔸𝑞^†∈ℚ∩Fix(𝕍). So, 𝑞^†∈Ω. We substitute 𝑝^† for 𝑞^† in (58) to obtain 𝑥_𝑛−𝑞^†2≤1𝑞𝛾−𝜁𝜅𝜁𝒞^†−𝔹𝑞^†,𝑥_𝑛−𝑞^†+𝑀𝑡_𝑛, (60) since 𝑥_𝑛 weakly converges to 𝑞^†, we deduce that 𝑥_𝑛→𝑞^† strongly. Therefore, the net {𝑥_𝑡} is relatively norm-compact.
In (58), we take the limit as 𝑛→∞ to deduce 𝑞^†−𝑝^†2≤1×𝑝𝛾−𝜁𝜅𝜁𝒞^†−𝔹𝑝^†,𝑞^†−𝑝^†,𝑝^†∈Ω. (61) Hence, 𝑞^† solves the variational inequality 𝑞^†∈Ω,𝜁𝒞𝑝^†−𝔹𝑝^†,𝑞^†−𝑝^†≥0,𝑝^†∈Ω, (62) which is equivalent to its dual variational inequality 𝑞^†∈Ω,𝜁𝒞𝑞^†−𝐵𝑞^†,𝑞^†−𝑝^†𝑝≥0,_†∈Ω. (63)
Therefore, 𝑞^†=proj_Ω(𝜁𝒞+𝐼−𝔹)𝑞^†. That is, 𝑞^† is the unique solution in VI(ℂ,𝔸) of the contraction proj_Ω(𝜁𝒞+𝐼−𝔹). Clearly this is sufficient to deduce that {𝑥_𝑡} converges strongly to 𝑞^† as 𝑡→0⁺. The proof is completed.

Conflict of Interests

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

Acknowledgment

This study was supported by research funds from Dong-A University.