Relaxed Extragradient Algorithms for the Split Feasibility Problem

Abstract

The purpose of this paper is to introduce a new relaxed extragradient algorithms for the split feasibility problem. Our relaxed extragradient algorithm is new and it generalized some results for solving the split feasibility problem.

1. Introduction

The split feasibility problem has received much attention due to its applications in image denoising, signal processing, and image reconstruction, with particular progress in intensity modulated therapy; see, for instance [1–4]. In this paper, we continue to study the split feasibility problem and its approximation algorithms.

To begin with, let us first recall the notation of the split feasibility problem and some existing algorithms in the literature. Recall that the split feasibility problem introduced by Censor and Elfving [4] can be formulated as finding a point 𝑥^† with the property 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. The original algorithm introduced in [4] involves the computation of the inverse 𝔸⁻¹: 𝑥^𝑘+1=𝔸⁻¹proj_ℚproj_𝔸(ℂ)𝔸𝑥^𝑘,𝑘≥0, (2) where ℂ, ℚ⊂ℝ^𝑛 are closed convex sets and 𝔸 is a full rank 𝑛×𝑛 matrix. It is an interesting problem to construct iterative algorithms for the split feasibility problem to the bounded linear operator.

We use Ξ to denote the solution set of the split feasibility problem; that is, Ξ∶=𝑥^†∈ℂ∶𝔸𝑥^†∈ℚ. (3) In the sequel, we assume that the split feasibility problem is consistent. Let 𝜛>0 and assume that 𝑥^†∈Ξ. Thus, 𝔸𝑥^†∈ℚ. It follows that proj_ℚ𝔸𝑥^†=𝔸𝑥^† which implies 𝜛𝔸^∗(𝐼−proj_ℚ)𝔸𝑥^†=0. Hence, we deduce that the fixed point equation (𝐼−𝜛𝔸^∗(𝐼−proj_ℚ)𝔸)𝑥^†=𝑥^† holds. Note that 𝑥^†∈ℂ. It is obvious that proj_ℂ𝐼−𝜛𝔸^∗𝐼−proj_ℚ𝔸𝑥^†=𝑥^†. (4) By using (4), Byrne [5] presented a more popular CQ algorithm: 𝑥^𝑘+1=proj_ℂ𝑥^𝑘−𝜛𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘,𝑛∈ℕ. (5) The CQ algorithm only needs to compute the projections proj_ℂ and proj_ℚ. It is therefore implementable when proj_ℂ and proj_ℚ have closed-form expressions. Consequently, CQ algorithm has received so much attention; see [6–17]. In particular, in [14], Xu proposed an averaged CQ algorithm and he obtained the weak convergence result.

On the other hand, we note that 𝑥∈Ξ; then there exists 𝑥^†∈ℚ such that 𝔸𝑥−𝑥^†=0. Thus, we consider the distance function 𝑑(𝔸𝑥,𝑥^†)=‖𝔸𝑥−𝑥^†‖ and the minimization problem min_{𝑥∈ℂ,𝑥†∈ℚ}12𝔸𝑥−𝑥^†2. (6) Since 𝑥^†∈ℚ, we then consider the minimization min_𝑥∈ℂ𝜑(𝑥)1∶=2𝔸𝑥−proj_ℚ𝔸𝑥². (7) However, (7) is, in general, ill-posed. We import the well-known Tikhonov’s regularization method min_𝑥∈ℂ𝜑_𝛼1∶=2(𝐼−proj_ℚ)𝔸𝑥²+12𝛼‖𝑥‖², (8) where 𝛼>0 is the regularization parameter. Note that the gradient ∇𝜑_𝛼 of 𝜑_𝛼 is ∇𝜑_𝛼=∇𝜑(𝑥)+𝛼𝐼=𝔸^∗𝐼−proj_ℚ𝔸+𝛼𝐼. (9) Using (9), some regularized methods have been suggested. For example, Xu [14] suggested the following regularized method: 𝑥^𝑘+1=proj_ℂ𝐼−𝜛_𝑘∇𝜑_𝛼𝑘𝑥^𝑘=proj_ℂ1−𝛼_𝑘𝜛_𝑘𝑥^𝑘−𝜛_𝑘𝐴^∗𝐼−proj_ℚ𝔸𝑥^𝑘. (10) He proved that the sequence {𝑥^𝑘} generated by (10) converges to the solution of the split feasibility problem. Ceng et al. [7] presented the following extragradient method: 𝑥₀=𝑥∈ℍ₁𝑦chosenarbitrarily,_𝑘=proj_ℂ𝑥^𝑘−𝜛_𝑘∇𝜑_𝛼𝑘𝑥^𝑘,𝑥^𝑘+1=𝛽_𝑘𝑥^𝑘+1−𝛽_𝑘proj_ℂ𝑥^𝑘−𝜛_𝑘∇𝜑_𝛼𝑘𝑦^𝑘,𝑛≥0. (11) and proved that the sequence {𝑥^𝑘} generated by (11) converges weakly to the solution of the split feasibility problem. Motivated by the above results, in this paper, we construct a new extragradient algorithm for solving the split feasibility problem. Strong convergence result is demonstrated.

2. Preliminaries

Let ℂ be a nonempty closed convex subset of a real Hilbert space ℍ. A mapping Θ∶ℂ→ℂ is called nonexpansive if ‖Θ𝑢−Θ𝑣‖≤‖𝑢−𝑣‖,∀𝑢,𝑣∈ℂ. (12) A mapping 𝑊∶ℂ→ℂ is said to be 𝜈-inverse strong monotone, if there exists a constant 𝜈>0 such that ⟨𝑢−𝑣,𝑊𝑢−𝑊𝑣⟩≥𝜈‖𝑊𝑢−𝑊𝑣‖²,∀𝑢,𝑣∈ℂ. (13) Recall that the metric projection from ℍ onto ℂ denoted by proj_ℂ means that 𝑥^⋆−proj_ℂ𝑥^⋆=inf𝑥^⋆−𝑥∶𝑥∈ℂ. (14)proj_ℂ can be characterized by ⟨𝑢−𝑣,proj_ℂ(𝑢)−proj_ℂ(𝑣)⟩≥proj_ℂ(𝑢)−proj_ℂ(𝑣)², (15) for all 𝑢,𝑣∈𝐻. Hence, it is nonexpansive. It is also known that 2proj_ℂ−𝐼 is nonexpansive.

Let 𝜑∶ℂ→ℍ be a monotone mapping. The variational inequality problem is to find 𝑥^†∈ℂ such that 𝜑𝑥^†,𝑥−𝑥^†≥0,∀𝑥∈ℂ. (16) The solution set of the variational inequality is denoted by VI(ℂ,𝜑). It is well known that 𝑥^♯∈VI(ℂ,𝜑) if and only if x^♯=proj_ℂ(𝑥^♯−𝜆𝜑(𝑥^♯)), ∀𝜆>0. A set-valued mapping 𝑅∶ℍ→2^ℍ is called monotone if, for each 𝑢∈ℍ, 𝑓∈𝑅𝑢, and 𝑔∈𝑅𝑣, we have 𝑢−𝑣,𝑓−𝑔≥0. (17) A monotone mapping 𝑅∶ℍ→2^ℍ is called maximal if its graph 𝐺(𝑅) is not properly contained in the graph of any other monotone mappings. It is known that a monotone mapping 𝑅 is maximal if and only if for each pair (𝑥,𝑓)∈ℍ×ℍ with ⟨𝑢−𝑣,𝑓−𝑔⟩≥0 for every (𝑣,𝑔)∈𝐺(𝑅); then we have 𝑓∈𝑅𝑢. Let 𝑅∶ℂ→ℍ be a monotone and 𝑘-Lipschitz continuous mapping and let 𝑁_ℂ𝑣 be the normal cone to ℂ at 𝑣∈ℂ; that is, 𝑁_𝐾𝑣={𝑤∈ℍ∶⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈ℂ}. (18) Define 𝑊𝑣=𝑅𝑣+𝑁_ℂ𝑣,if𝑣∈ℂ,∅,if𝑣∉ℂ. (19) Then, 𝑊 is maximal monotone and 0∈𝑊𝑣 if and only if 𝑣∈VI(ℂ,𝑅); see [18] for more details.

Lemma 1 (see [14]). One knows the following properties:(i)𝔸^∗(𝐼−proj_ℚ)𝔸 is Lipschitz continuous with Lipschitz constant ‖𝔸‖²;(ii)𝔸^∗(𝐼−proj_ℚ)𝔸 is 1/‖𝔸‖² inverse strong monotone;(iii)𝐼−𝜛𝔸^∗(𝐼−proj_ℚ)𝐴 is nonexpansive for all 𝜛∈(0,2/‖𝔸‖²).

Lemma 2 (see [19]). Let ℂ be a nonempty closed convex subset of a real Hilbert space ℍ. Let the mapping Φ∶ℂ→ℍ be 𝜁-inverse strong monotone and let 𝜚>0 be a constant. Then, we have (𝐼−𝜚Φ)𝑥^†−(𝐼−𝜚Φ)𝑥^‡2≤𝑥^†−𝑥^‡2+𝜚𝜚−2𝜁Φ𝑥^†−Φ𝑥^‡2,∀𝑥^†,𝑥^‡∈𝐶. (20) In particular, if 0≤𝜚≤2𝜁, then 𝐼−𝜚Φ is nonexpansive.

Lemma 3 (see [20]). Let {𝜄^𝑘} and {𝜎^𝑘} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑘} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑘≤limsup_𝑛→∞𝛽_𝑘<1. Suppose that 𝜄^𝑘+1=1−𝛽_𝑘𝜎^𝑘+𝛽_𝑘𝜄^𝑘 (21) for all 𝑛≥0 and limsup_𝑛→∞𝜎^𝑘+1−𝜎^𝑘−𝜄^𝑘+1−𝜄^𝑘≤0. (22) Then, lim_𝑛→∞‖𝜎^𝑘−𝜄^𝑘‖=0.

Lemma 4 (see [21]). Assume that {𝜉_𝑘} is a sequence of nonnegative real numbers such that 𝜉_𝑘+1≤1−𝛾_𝑘𝜉_𝑘+𝜗_𝑘, (23) where {𝛾_𝑘} is a sequence in (0,1) and {𝜗_𝑘} is a sequence such that(1)∑^∞_𝑛=1𝛾_𝑘=∞; (2)limsup_𝑛→∞(𝜗_𝑘/𝛾_𝑘)≤0 or ∑^∞_𝑛=1|𝜗_𝑘|<∞.Then, lim_𝑛→∞𝜉_𝑘=0.

3. Main Results

Let ℍ₁ and ℍ₂ be two Hilbert spaces and let ℂ, ℚ be two nonempty closed and convex subsets of ℍ₁ and ℍ₂, respectively. Suppose that 𝔸∶ℍ₁→ℍ₂ is a bounded linear operator and 𝔽∶ℂ→ℍ₁ is a 𝜄-contraction. We will state and prove our main results in this paper and at the end of this paper we will give an example to show that our results improve the works in the literature.

Algorithm 5. Let 𝑥₀∈ℂ. Let {𝑥^𝑘} be a sequence generated by 𝑦^𝑘=proj_ℂ𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘+𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘,𝑥^𝑘+1=proj_ℂ𝑥^𝑘−𝜆_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+𝜇_𝑘𝑦^𝑘−𝑥^𝑘,𝑛≥ℕ, (24) where {𝛼_𝑘}⊂(0,1) is a sequence; 𝛿_𝑘,𝜆_𝑘∈[𝑎,𝑏]⊂(0,2/‖𝔸‖²) and 𝜇_𝑘∈(0,1) are sequences satisfying 𝜆_𝑘≤2𝜇_𝑘/‖𝔸‖².

Theorem 6. Suppose that Ξ≠∅. Assume lim_𝑛→∞𝛼_𝑘=0, lim_𝑛→∞(𝜆_𝑘+1−𝜆_𝑘)=0, lim_𝑛→∞(𝛿_𝑘+1−𝛿_𝑘)=0, lim_𝑛→∞(𝜇_𝑘+1−𝜇_𝑘)=0, and ∑^∞_𝑛=1𝛼_𝑘=∞. Then the sequence {𝑥^𝑘} generated by (24) converges strongly to 𝜈=𝑃_Ξ(𝔽)𝜈.

Proof. From the assumptions lim_𝑛→∞𝛼_𝑘=0 and 𝛿_𝑘∈(0,2/‖𝔸‖²), we can choose 𝛼_𝑘 such that 𝛼_𝑘<1−(𝛿_𝑘‖𝔸‖²/2) when 𝑛 is sufficiently large. For the convenience and without loss of generality, we assume that 𝛼_𝑘<1−(𝛿_𝑘‖𝔸‖²/2) for all 𝑛∈ℕ. Then, we deduce 𝛿_𝑘/(1−𝛼_𝑘)∈(0,2/‖𝔸‖²) for all 𝑛∈ℕ.
Let 𝑥^†∈Ξ. In the first section, we know that 𝑥^†=proj_ℂ[𝑥^†−𝜉𝔸^∗(𝐼−proj_ℚ)𝔸𝑥^†] for any 𝜉>0. Since 𝛿_𝑘/(1−𝛼_𝑘)∈(0,2/‖𝔸‖²) and 𝜆_𝑘/𝜇_𝑘∈(0,2/‖𝔸‖²), we have 𝑥^†=proj_ℂ𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†1−𝛼_𝑘=proj_ℂ𝛼_𝑘𝑥^†+1−𝛼_𝑘×𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†1−𝛼_𝑘𝑥,∀𝑛≥ℕ,_†=proj_ℂ𝑥^†−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝐴𝑥^†=proj_ℂ𝛼_𝑘𝑥^†+1−𝛼_𝑘×𝑥^†−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†,∀𝑛≥ℕ. (25)(26) From (24) and (25), we have 𝑦^𝑘−𝑥^†=proj_ℂ𝛼_𝑘𝔽𝑥^𝑘+1−𝛼_𝑘𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝑥^†=proj_ℂ𝛼_𝑘𝔽𝑥^𝑘+1−𝛼_𝑘×𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘1−𝛼_𝑘−proj_ℂ𝛼_𝑘𝑥^†+1−𝛼_𝑘×𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†1−𝛼_𝑘≤𝛼_𝑘𝔽𝑥^𝑘−𝑥^†+1−𝛼_𝑘×𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘1−𝛼_𝑘−𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†1−𝛼_𝑘. (27) From Lemma 1, we know that 𝐼−𝛿_𝑘𝔸^∗(𝐼−proj_ℚ)𝔸/(1−𝛼_𝑘) and 𝐼−𝜆_𝑘/𝜇_𝑘𝔸^∗(𝐼−proj_ℚ)𝔸 are nonexpansive. It follows that 𝑦^𝑘−𝑥^†≤1−𝛼_𝑘𝛿𝐼−^𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝛿𝐼−^𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†+𝛼_𝑘𝔽𝑥^𝑘−𝔽𝑥^†+𝛼_𝑘𝔽𝑥^†−𝑥^†≤𝛼_𝑘𝜄𝑥^𝑘−𝑥^†+𝛼_𝑘𝔽𝑥^†−𝑥^†+1−𝛼_𝑘𝑥^𝑘−𝑥^†=1−(1−𝜄)𝛼_𝑘𝑥^𝑘−𝑥^†+𝛼_𝑘𝔽𝑥^†−𝑥^†. (28) Thus, 𝑥^𝑘+1−𝑥^†=proj_ℂ𝑥^𝑘−𝜆_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+𝜇_𝑘𝑦^𝑘−𝑥^𝑘−𝑥^†=proj_ℂ1−𝜇_𝑘𝑥^𝑘+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘−proj_ℂ1−𝜇_𝑘𝑥^†+𝜇_𝑘𝑥^†−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†≤1−𝜇_𝑘𝑥^𝑘−𝑥^†+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘−𝑥^†−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†≤1−𝜇_𝑘𝑥^𝑘−𝑥^†+𝜇_𝑘𝑦^𝑘−𝑥^†≤1−𝜇_𝑘𝑥^𝑘−𝑥^†+𝜇_𝑘𝛼_𝑘𝔽𝑥^†−𝑥^†+𝜇_𝑘1−(1−𝜄)𝛼_𝑘𝑥^𝑘−𝑥^†=1−(1−𝜄)𝜇_𝑘𝛼_𝑘𝑥^𝑘−𝑥^†+𝜇_𝑘𝛼_𝑘𝔽𝑥^†−𝑥^†≤max𝔽𝑥^†−𝑥^†,𝑥1−𝜄⁰−𝑥^†. (29) Hence, {𝑥^𝑘} is bounded. It follows from (28) that the sequence {𝑦^𝑘} is also bounded.
Set 𝕊=2proj_ℂ−𝐼. Note that 𝑆 is nonexpansive. Thus, we can rewrite 𝑥^𝑘+1 in (24) as 𝑥^𝑘+1=𝐼+𝕊21−𝜇_𝑘𝑥^𝑘+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘=1−𝜇_𝑘2𝑥^𝑘+𝜇_𝑘2𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+12𝕊1−𝜇_𝑘𝑥^𝑘+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘=1−𝜇_𝑘2𝑥^𝑘+1+𝜇_𝑘2𝑧^𝑘, (30) where 𝑧^𝑘=𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+𝕊1−𝜇_𝑘𝑥^𝑘+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘×(1+𝜇_𝑘)⁻¹. (31) It follows that 𝑧^𝑘+1−𝑧^𝑘=𝜇_𝑘+1𝑦^𝑘+1−𝜆_𝑘+1𝜇_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+1+𝕊1−𝜇_𝑘+1𝑥^𝑘+1+𝜇_𝑘+1×𝑦^𝑘+1−𝜆_𝑘+1𝜇_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+1×1+𝜇_𝑘+1⁻¹−𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝐴^∗𝐼−proj_ℚ𝐴𝑦^𝑘+𝕊1−𝜇_𝑘𝑥^𝑘+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘×1+𝜇_𝑘⁻¹. (32) So, 𝑧^𝑘+1−𝑧^𝑘≤𝜇_𝑘+11+𝜇_𝑘+1𝑦^𝑘+1−𝜆_𝑘+1𝜇_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+1−𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+𝜇_𝑘+11+𝜇_𝑘+1−𝑢^𝑘1+𝜇_𝑘×𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+11+𝜇_𝑘+1×𝕊1−𝜇_𝑘+1𝑥^𝑘+1+𝜇_𝑘+1×𝑦^𝑘+1−𝜆_𝑘+1𝜇_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+1−𝑆1−𝜇_𝑘𝑥^𝑘+𝜇_𝑘×𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+11+𝜇_𝑘+1−11+𝜇_𝑘×𝕊1−𝜇_𝑘𝑥^𝑘+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘≤𝜇_𝑘+11+𝜇_𝑘+1𝜆𝐼−^𝑘+1𝜇_𝑘+1𝐴^∗𝐼−proj_ℚ𝔸𝑦^𝑘+1−𝜆𝐼−^𝑘+1𝜇_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+𝜇_𝑘+11+𝜇_𝑘+1𝜆_𝑘+1𝜇_𝑘+1−𝜆_𝑘𝜇_𝑘𝔸𝑦^𝑘+𝜇_𝑘+11+𝜇_𝑘+1−𝑢^𝑘1+𝜇_𝑘×𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+11+𝜇_𝑘+11−𝜇_𝑘+1𝑥^𝑘+1+𝜇_𝑘+1×𝑦^𝑘+1−𝜆_𝑘+1𝜇_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+1−1−𝜇_𝑘𝑥^𝑘−𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+11+𝜇_𝑘+1−11+𝜇_𝑘×𝕊1−𝜇_𝑘𝑥^𝑘+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘. (33) It follows that 𝑧^𝑘+1−𝑧^𝑘≤𝜇_𝑘+11+𝜇_𝑘+1𝑦^𝑘+1−𝑦^𝑘+𝜇_𝑘+11+𝜇_𝑘+1𝜆_𝑘+1𝜇_𝑘+1−𝜆_𝑘𝜇_𝑘𝔸𝑦^𝑘+𝜇_𝑘+11+𝜇_𝑘+1−𝑢^𝑘1+𝜇_𝑘×𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+11+𝜇_𝑘+1×1−𝜇_𝑘+1𝑥^𝑘+1−𝑥^𝑘+𝜇_𝑘−𝜇_𝑘+1𝑥^𝑘−𝑦^𝑘+𝜇_𝑘+1𝜆𝐼−^𝑘+1𝜇_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+1−𝜆𝐼−^𝑘+1𝜇_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+𝜆_𝑘−𝜆_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+11+𝜇_𝑘+1−11+𝜇_𝑘×𝕊1−𝜇_𝑘𝑥^𝑘+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘≤2𝜇_𝑘+11+𝜇_𝑘+1𝑦^𝑘+1−𝑦^𝑘+1−𝜇_𝑘+11+𝜇_𝑘+1𝑥^𝑘+1−𝑥^𝑘+𝜇_𝑘+11+𝜇_𝑘+1𝜆_𝑘+1𝜇_𝑘+1−𝜆_𝑘𝜇_𝑘𝔸𝑦^𝑘+𝜇_𝑘−𝜇_𝑘+11+𝜇_𝑘+1𝑥^𝑘−𝑦^𝑘+𝜇_𝑘+11+𝜇_𝑘+1−𝑢^𝑘1+𝜇_𝑘×𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+11+𝜇_𝑘+1−11+𝜇_𝑘×𝕊1−𝜇_𝑘𝑥^𝑘+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+𝜆_𝑘−𝜆_𝑘+11+𝜇_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘. (34) From (24), we have 𝑦^𝑘+1−𝑦^𝑘=proj_ℂ𝛼_𝑘+1𝔽𝑥^𝑘+1+1−𝛼_𝑘+1𝑥^𝑘+1−𝛿_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘+1−proj_ℂ𝛼_𝑘𝔽𝑥^𝑘+1−𝛼_𝑘𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘≤𝛼_𝑘+1𝔽𝑥^𝑘+1+1−𝛼_𝑘+1𝑥^𝑘+1−𝛿_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘+1−𝛼_𝑘𝔽𝑥^𝑘+1−𝛼_𝑘𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘≤𝐼−𝛿_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘+1−𝐼−𝛿_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘+𝛼_𝑘+1𝔽𝑥^𝑘+1−𝑥^𝑘+1+𝛿_𝑘+1−𝛿_𝑘×𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘+𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘≤𝑥^𝑘+1−𝑥^𝑘+𝛿_𝑘+1−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘+𝛼_𝑘+1𝔽𝑥^𝑘+1−𝑥^𝑘+1+𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘. (35) Hence, we deduce 𝑧^𝑘+1−𝑧^𝑘≤𝑥^𝑘+1−𝑥^𝑘+𝛿_𝑘+1−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘+𝛼_𝑘+1𝔽𝑥^𝑘+1−𝑥^𝑘+1+𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘+𝜇_𝑘+11+𝜇_𝑘+1𝜆_𝑘+1𝜇_𝑘+1−𝜆_𝑘𝜇_𝑘𝔸𝑦^𝑘+𝜇_𝑘+11+𝜇_𝑘+1−𝑢^𝑘1+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+𝜇_𝑘−𝜇_𝑘+11+𝜇_𝑘+1𝑥^𝑘−𝑦^𝑘+11+𝜇_𝑘+1−11+𝜇_𝑘×𝕊1−𝜇_𝑘𝑥^𝑘+𝜇_𝑘𝑦^𝑘−𝜆_𝑘𝜇_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘+𝜆_𝑘−𝜆_𝑘+11+𝜇_𝑘+1𝔸^∗𝐼−proj_ℚ𝔸𝑦^𝑘. (36) Since lim_𝑛→∞𝛼_𝑘=0, lim_𝑛→∞(𝜆_𝑘+1−𝜆_𝑘)=0, lim_𝑛→∞(𝛿_𝑘+1−𝛿_𝑘)=0, and lim_𝑛→∞(𝜇_𝑘+1−𝜇_𝑘)=0, we derive that limsup_𝑛→∞𝑧^𝑘+1−𝑧^𝑘−𝑥^𝑘+1−𝑥^𝑘≤0. (37) By Lemma 3, we obtain lim_𝑛→∞𝑧^𝑘−𝑥^𝑘=0. (38) Hence, lim_𝑛→∞𝑥^𝑘+1−𝑥^𝑘=lim_𝑛→∞1+𝜇_𝑘2𝑧^𝑘−𝑥^𝑘=0. (39) By Lemma 1, Lemma 2, and (29), we get 𝑥^𝑘+1−𝑥^†2≤1−𝜇_𝑘𝑥^𝑘−𝑥^†2+𝜇_𝑘𝑦^𝑘−𝑥^†2≤1−𝜇_𝑘𝑥^𝑘−𝑥^†2+𝜇_𝑘𝛼_𝑘𝔽𝑥^𝑘−𝑥^†+1−𝛼_𝑘×𝑥^𝑘−𝛿_𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝑥^†−𝛿_𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†2≤𝜇_𝑘𝛼_𝑘𝔽𝑥^𝑘−𝑥^†2+1−𝛼_𝑘×𝛿𝐼−^𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝛿𝐼−^𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†2+1−𝜇_𝑘𝑥^𝑘−𝑥^†2≤𝜇_𝑘𝛿_𝑘𝛿_𝑘1−𝛼_𝑘−2‖𝔸‖²×𝔸^∗(𝐼−proj_ℚ)𝔸𝑥^𝑘−𝔸^∗(𝐼−proj_ℚ)𝔸𝑥^†2+1−𝛼_𝑘𝜇_𝑘𝑥^𝑘−𝑥^†2+1−𝜇_𝑘𝑥^𝑘−𝑥^†2+𝛼_𝑘𝜇_𝑘𝔽𝑥^𝑘−𝑥^†2≤𝜇_𝑘𝑎𝑏1−𝛼_𝑘−2‖𝔸‖²×𝔸^∗(𝐼−proj_ℚ)𝔸𝑥^𝑘−𝔸^∗(𝐼−proj_ℚ)𝔸𝑥^†2+𝛼_𝑘𝔽𝑥^𝑘−𝑥^†2+𝑥^𝑘−𝑥^†2. (40) Therefore, we have 𝜇_𝑘𝑎2‖𝔸‖²−𝑏1−𝛼_𝑘×𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝐴^∗𝐼−proj_ℚ𝔸𝑥^†2≤𝛼_𝑘𝔽𝑥^𝑘−𝑥^†2+𝑥^𝑘−𝑥^†2−𝑥^𝑘+1−𝑥^†2≤𝛼_𝑘𝔽𝑥^𝑘−𝑥^†2+𝑥_𝑛−𝑥^†+𝑥^𝑘+1−𝑥^†×𝑥^𝑘−𝑥^𝑘+1. (41) Noting that 0<𝑎<𝜆_𝑘<2𝜇_𝑘/‖𝔸‖², we get liminf_𝑛→∞𝜇_𝑘>0. Since lim_𝑛→∞𝛼_𝑘=0 and 𝑏<2/‖𝔸‖², we obtain liminf_𝑛→∞𝜇_𝑘𝑎(2/‖𝔸‖²−𝑏/(1−𝛼_𝑘))>0. Thus, we have lim_𝑛→∞𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝔸^∗𝐼−proj_ℚ𝔸𝑥^†=0. (42)
Using (15), we have 𝑦^𝑘−𝑥^†2=proj_ℂ𝛼_𝑘𝔽𝑥^𝑘+1−𝛼_𝑘𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−proj_ℂ𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†2≤𝛼_𝑘𝔽𝑥^𝑘+1−𝛼_𝑘𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†,𝑦^𝑘−𝑥^†=12𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†+𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘2+𝑦^𝑘−𝑥^†2−𝛼_𝑘𝔽𝑥^𝑘+1−𝛼_𝑘𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†−𝑦^𝑘−𝑥^†2≤12𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†2+2𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘×𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†+𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘+𝑦^𝑘−𝑥^†2−𝑥^𝑘−𝑦^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝔸^∗𝐼−proj_ℚ𝔸𝑥^†+𝛼_𝑘𝐹𝑥^𝑘−𝑥^𝑘2≤12𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†2+𝛼_𝑘𝑀+𝑦^𝑘−𝑥^†2−𝑥^𝑘−𝑦^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝔸^∗𝐼−proj_ℚ𝔸𝑥^†+𝛼_𝑘(𝔽𝑥^𝑘−𝑥^𝑘)²≤12𝑥^𝑘−𝑥^†2+𝛼_𝑘𝑀+𝑦^𝑘−𝑥^†2−𝑥^𝑘−𝑦^𝑘2+2𝛼_𝑘𝑥^𝑘,𝑥^𝑘−𝑦^𝑘+2𝛿_𝑘𝑥^𝑘−𝑦^𝑘,𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝔸^∗𝐼−proj_ℚ𝔸𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝔸^∗𝐼−proj_ℚ𝔸𝑥^†−𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘2≤12𝑥^𝑘−𝑥^†2+𝛼_𝑘𝑀+𝑦^𝑘−𝑥^†2−𝑥^𝑘−𝑦^𝑘2+2𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘𝑥^𝑘−𝑦^𝑘+2𝛿_𝑘𝑥^𝑘−𝑦^𝑘×𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝔸^∗𝐼−proj_ℚ𝔸𝑥^†, (43) where 𝑀>0 is some constant such that sup_𝑘2𝑥^𝑘𝑥^𝑘−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝑥^†−𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^†+𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘≤𝑀. (44) It follows that 𝑦^𝑘−𝑥^†2≤𝑥^𝑘−𝑥^†2+𝛼_𝑘𝑀−𝑥^𝑘−𝑦^𝑘2+2𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘𝑥^𝑘−𝑦^𝑘+2𝛿_𝑘𝑥^𝑘−𝑦^𝑘×𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝔸^∗𝐼−proj_ℚ𝔸𝑥^†, (45) and hence 𝑥^𝑘+1−𝑥^†2≤1−𝜇_𝑘𝑥^𝑘−𝑥^†2+𝜇_𝑘𝑦^𝑘−𝑥^†2≤𝑥^𝑘−𝑥^†2+𝛼_𝑘𝑀−𝜇_𝑘𝑥^𝑘−𝑦^𝑘2+2𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘𝑥^𝑘−𝑦^𝑘+2𝛿_𝑘𝑥^𝑘−𝑦^𝑘×𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝔸^∗𝐼−proj_ℚ𝔸𝑥^† (46) which implies that 𝜇_𝑘𝑥^𝑘−𝑦^𝑘2≤𝑥^𝑘−𝑥^†+𝑥^𝑘+1−𝑥^†𝑥^𝑘+1−𝑥^𝑘+𝛼_𝑘𝑀+2𝛼_𝑘𝔽𝑥^𝑘−𝑥^𝑘𝑥^𝑘−𝑦^𝑘+2𝛿_𝑘𝑥^𝑘−𝑦^𝑘×𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝔸^∗𝐼−proj_ℚ𝔸𝑥^†. (47) Since 𝛼_𝑘→0,‖𝑥_𝑛−𝑥^𝑘+1‖→0 and ‖𝔸^∗(𝐼−proj_ℚ)𝔸𝑥^𝑘−𝔸^∗(𝐼−proj_ℚ)𝔸𝑥^†‖→0, we derive lim_𝑛→∞𝑥^𝑘−𝑦^𝑘=0. (48)
Next we show that limsup_𝑛→∞𝜈−𝔽𝜈,𝜈−𝑦^𝑘≤0, (49) where 𝜈=𝑃_Ξ(𝔽𝜈). To show it, we choose a subsequence {𝑦_𝑛𝑖} of {𝑦^𝑘} such that limsup_𝑛→∞𝜈−𝔽𝜈,𝜈−𝑦^𝑘=lim_𝑖→∞𝜈−𝔽𝜈,𝜈−𝑦_𝑛𝑖. (50) Since {𝑦^𝑘} is bounded and {𝑦_𝑛𝑖} is then bounded, we have the fact that a subsequence {𝑦_𝑛𝑖𝑗} of {𝑦_𝑛𝑖} converges weakly to 𝑧.
Next we show that 𝑧∈Ξ. We define a mapping 𝑊 by 𝔸𝑊𝑣=^∗𝐼−proj_ℚ𝔸𝑣+𝑁_ℂ𝑣,𝑣∈ℂ,∅,𝑣∉ℂ. (51) Then 𝑊 is maximal monotone. Let (𝑣,𝑤)∈𝐺(𝑊). Since 𝑤−𝔸^∗(𝐼−proj_ℚ)𝔸𝑣∈𝑁_ℂ𝑣 and 𝑦^𝑘∈ℂ, we get 𝑣−𝑦^𝑘,𝑤−𝔸^∗𝐼−proj_ℚ𝔸𝑣≥0. (52) On the other hand, from 𝑦^𝑘=proj_ℂ[𝛼_𝑘𝔽𝑥^𝑘+(1−𝛼_𝑘)𝑥^𝑘−𝛿_𝑘𝔸^∗(𝐼−proj_ℚ)𝔸𝑥^𝑘], we have 𝑣−𝑦^𝑘,𝑦^𝑘−𝛼_𝑘𝔽𝑥^𝑘−1−𝛼_𝑘𝑥^𝑘+𝛿_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘≥0. (53) That is, 𝑣−𝑦^𝑘,𝑦^𝑘−𝑥^𝑘𝛿_𝑘+𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘+𝛼_𝑘𝛿_𝑘𝑥^𝑘−𝔽𝑥^𝑘≥0. (54) Therefore, we have ⟨𝑣−𝑦_𝑛𝑖,𝑤⟩≥⟨𝑣−𝑦_𝑛𝑖,𝔸𝑣⟩≥⟨𝑣−𝑦_𝑛𝑖,𝔸^∗𝐼−proj_ℚ−𝔸𝑣⟩𝑣−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝛿_𝑛𝑖+𝔸^∗𝐼−proj_ℚ𝔸𝑥_𝑛𝑖+𝛼_𝑛𝑖𝛿_𝑛𝑖(𝐼−𝔽)𝑥_𝑛𝑖=𝑣−𝑦_𝑛𝑖,𝔸^∗𝐼−proj_ℚ𝔸𝑣−𝔸^∗𝐼−proj_ℚ𝔸𝑥_𝑛𝑖−𝑦_𝑛𝑖−𝑥_𝑛𝑖𝛿_𝑛𝑖−𝛼_𝑛𝑖𝛿_𝑛𝑖(𝐼−𝔽)𝑥_𝑛𝑖=𝑣−𝑦_𝑛𝑖,𝔸^∗𝐼−proj_ℚ𝔸𝑣−𝔸^∗𝐼−proj_ℚ𝔸𝑦_𝑛𝑖−𝑣−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝛿_𝑛𝑖+𝛼_𝑛𝑖𝛿_𝑛𝑖(𝐼−𝔽)𝑥_𝑛𝑖+𝑣−𝑦_𝑛𝑖,𝔸^∗𝐼−proj_ℚ𝔸𝑦_𝑛𝑖−𝔸^∗𝐼−proj_ℚ𝔸𝑥_𝑛𝑖≥𝑣−𝑦_𝑛𝑖,𝔸^∗𝐼−proj_ℚ𝔸𝑦_𝑛𝑖−𝔸^∗𝐼−proj_ℚ𝔸𝑥_𝑛𝑖−𝑣−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝛿_𝑛𝑖+𝛼_𝑛𝑖𝛿_𝑛𝑖(𝐼−𝔽)𝑥_𝑛𝑖. (55) Noting that 𝛼_𝑛𝑖→0, ‖𝑦_𝑛𝑖−𝑥_𝑛𝑖‖→0 and 𝔸^∗(𝐼−proj_ℚ)𝔸 is Lipschitz continuous, we deduce from above ⟨𝑣−𝑧,𝑤⟩≥0. (56) Since 𝑊 is maximal monotone, we have 𝑧∈𝑊⁻¹(0) and hence 𝑧∈VIℂ,𝔸^∗𝐼−proj_ℚ𝔸=Ξ. (57) Therefore, limsup_𝑛→∞𝜈−𝔽𝜈,𝜈−𝑦^𝑘=lim_𝑖→∞𝜈−𝔽𝜈,𝜈−𝑦_𝑛𝑖=⟨𝜈−𝔽𝜈,𝜈−𝑧⟩≤0. (58) Again from (15), we have 𝑦^𝑘−𝜈²=proj_ℂ𝛼_𝑘𝔽𝑥^𝑘+1−𝛼_𝑘×𝑥^𝑘−𝛿_𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−proj_ℂ𝛼_𝑘𝜈+1−𝛼_𝑘×𝛿𝜈−^𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝜈²≤𝛼_𝑘𝔽𝑥^𝑘−𝜈+1−𝛼_𝑘×𝑥^𝑘−𝛿_𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝛿𝜈−^𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝜈,𝑦^𝑘−𝜈≤𝛼_𝑘⟨𝜈−𝔽𝑥^𝑘,𝜈−𝑦^𝑘⟩+1−𝛼_𝑘×𝑥^𝑘−𝛿_𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝑥^𝑘−𝛿𝜈−^𝑘1−𝛼_𝑘𝔸^∗𝐼−proj_ℚ𝔸𝜈𝑦^𝑘−𝜈≤𝛼_𝑘𝜈−𝔽𝜈,𝜈−𝑦^𝑘+𝛼_𝑘𝔽𝜈−𝔽𝑥^𝑘,𝜈−𝑦^𝑘+1−𝛼_𝑘𝑥^𝑘−𝜈𝑦^𝑘−𝜈≤𝛼_𝑘𝜈−𝔽𝜈,𝜈−𝑦^𝑘+𝛼_𝑘𝔽𝜈−𝔽𝑥^𝑘×𝜈−𝑦^𝑘+1−𝛼_𝑘𝑥^𝑘−𝜈𝑦^𝑘−𝜈≤𝛼_𝑘𝜈−𝔽𝜈,𝜈−𝑦^𝑘+1−(1−𝜄)𝛼_𝑘𝑥^𝑘−𝜈𝑦^𝑘−𝜈≤𝛼_𝑘𝜈−𝔽𝜈,𝜈−𝑦^𝑘+1−(1−𝜄)𝛼_𝑘2𝑥^𝑘−𝜈²+12𝑦^𝑘−𝜈². (59) Hence, 𝑦^𝑘−𝜈²≤1−(1−𝜄)𝛼_𝑘𝑥^𝑘−𝜈²+2𝛼_𝑘𝜈−𝔽𝜈,𝜈−𝑦^𝑘. (60) Therefore, 𝑥^𝑘+1−𝜈²≤1−𝜇_𝑘𝑥^𝑘−𝜈²+𝜇_𝑘𝑦^𝑘−𝜈²≤1−𝜇_𝑘(1−𝜄)𝛼_𝑘𝑥^𝑘−𝜈²+2𝜇_𝑘𝛼_𝑘𝜈−𝔽𝜈,𝜈−𝑦^𝑘. (61) We apply Lemma 4 to the last inequality to deduce that 𝑥^𝑘→𝜈. This completes the proof.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referees for the useful comments and suggestions. This study was supported by research funds from Dong-A University.