Damped Algorithms for the Split Fixed Point and Equilibrium Problems

Abstract

The main purpose of this paper is to study the split fixed point and equilibrium problems which includes fixed point problems, equilibrium problems, and variational inequality problems as special cases. A damped algorithm is presented for solving this split common problem. Strong convergence analysis is shown.

1. Introduction

Very recently, the split problems (e.g., the split feasibility problem, the split common fixed points problem, and the split variational inequality problem) have been studied extensively, see, for instance, [1–19]. Now we recall the related history. Let 𝐻₁ and 𝐻₂ be two Hilbert spaces and 𝐶⊂𝐻₁ and 𝑄⊂𝐻₂ two nonempty closed convex subsets. Let 𝐴∶𝐻₁→𝐻₂ be a bounded linear operator. The split feasibility problem is to solve the inclusion: 𝑥∈𝐶∩𝐴⁻¹(𝑄) (1) which arise in the field of intensity-modulated radiation therapy and was presented in [1]. The iteration 𝜌^𝑛+1=proj_𝐶(𝜌^𝑛−𝜍𝐴^∗(𝐼−𝑃_𝑄)𝐴𝜌^𝑛) is popular with 𝜍∈(0,2/‖𝐴‖²). Further, Xu [3] suggested a single step regularized method. Dang and Gao [4] developed a damped projection algorithm. If 𝐶 and 𝑄 are the fixed point sets of mappings 𝑈 and 𝑇, respectively, then (1) becomes a special case of the split common fixed point problem: Find𝑥∈Fix(𝑈)∩𝐴⁻¹(Fix(𝑇)). (2)

Censor and Segal [5] invented a scheme below to solve (2): 𝜌^𝑛+1=𝑈𝜌^𝑛−𝜍𝐴^∗(𝐼−𝑇)𝐴𝜌^𝑛,𝑛∈ℕ. (3)

Cui et al., [6] extended the damped projection algorithm to the split common fixed point problems. Let 𝜓∶𝐶×𝐶→ℝ be a bifunction. The equilibrium problem is to find 𝑥^†∈𝐶 such that 𝜓𝑥^†,𝑥≥0,∀𝑥∈𝐶. (4)

We will indicate with EP(𝜓) the set of solutions of (4).

In the present paper, our main purpose is to study the following split fixed point and equilibrium problem. Findapoint𝑢^§∈Fix(𝑊)∩EP𝜓suchthat𝐴𝑢^§∈Fix(𝑆)∩EP𝜑, (5) where Fix(𝑆) and Fix(𝑊) are the sets of fixed points of two nonlinear mappings 𝑆 and 𝑊, respectively; EP(𝜓) and EP(𝜑) are the solution sets of two equilibrium problems with bifunctions 𝜓 and 𝜑, respectively, and 𝐴 is a bounded linear mapping. Denote the solution set of (5) by Θ=𝑥∈Fix(𝑊)∩EP𝜓∶𝐴𝑥∈Fix(𝑆)∩EP𝜑. (6)

We develop a damped algorithm to solve this split fixed point and equilibrium problem. Strong convergence of the suggested damped algorithm is demonstrated.

2. Concepts and Lemmas

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐻. A mapping 𝑊∶𝐶→𝐶 is called nonexpansive if 𝑊𝑥^†−𝑊𝑥^‡≤𝑥^†−𝑥^‡, (7) for all 𝑥^†,𝑥^‡∈𝐶. We call proj_𝐶∶𝐻→𝐶 the metric projection if for each 𝑥^♭∈𝐻𝑥^♭−proj_𝐶𝑥^♭=inf𝑥^♭−𝑥^†∶𝑥^†∈𝐶. (8) It is well known that the metric projection proj_𝐶∶𝐻→𝐶 is firmly nonexpansive, that is, proj_𝐶𝑥^†−proj_𝐶𝑥^‡2≤𝑥^†−𝑥^‡,proj_𝐶𝑥^†−proj_𝐶𝑥^‡ (9) for all 𝑥^†,𝑥^‡∈𝐻. Hence proj_𝐶 is also nonexpansive.

Lemma 1 (see [20]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝜓∶𝐶×𝐶→ℝ be a bifunction which satisfies the following conditions: (H1) 𝜓(𝑥^‡,𝑥^‡)=0 for all 𝑥^‡∈𝐶; (H2) 𝜓 is monotone, that is, 𝜓(𝑥^‡,𝑥^†)+𝜓(𝑥^†,𝑥^‡)≤0 for all 𝑥^‡,𝑥^†∈𝐶; (H3) for each 𝑥^†,𝑥^‡,𝑥^♮∈𝐶, lim_𝑡↓0𝜓(𝑡𝑥^♮+(1−𝑡)𝑥^†,𝑥^‡)≤𝜓(𝑥^†,𝑥^‡); (H4) for each 𝑥^†∈𝐶, 𝑥^‡↦𝜓(𝑥^†,𝑥^‡) is convex and lower semicontinuous.
Let 𝜛>0 and 𝑥^†∈𝐶. Then, there exists 𝑥^♮∈𝐶 such that 𝜓𝑥^♮,𝑥^‡+1𝜛𝑥^‡−𝑥^♮,𝑥^♮−𝑥^†≥0,∀𝑥^‡∈𝐶. (10)
Further, if 𝑈^𝜓_𝜛(𝑥^†)={𝑥^♮∈𝐶∶𝜓(𝑥^♮,𝑥^‡)+(1/𝜛)⟨𝑥^‡−𝑥^♮,𝑥^♮−𝑥^†⟩≥0,𝑓𝑜𝑟𝑎𝑙𝑙𝑥^‡∈𝐶}, then the following hold: (i)𝑈^𝜓_𝜛 is single-valued and 𝑈^𝜓_𝜛 is firmly nonexpansive, that is, for any 𝑥^†,𝑥^‡∈𝐻, ‖𝑈^𝜓_𝜛𝑥^†−𝑈^𝜓_𝜛𝑥^‡‖²≤⟨𝑈^𝜓_𝜛𝑥^†−𝑈^𝜓_𝜛𝑥^‡,𝑥^†−𝑥^‡⟩; (ii)EP(𝜓) is closed and convex and EP(𝜓)=Fix(𝑈^𝜓_𝜛).

Lemma 2 (see [21]). Let 𝐻 be a Hilbert space and 𝐶⊂𝐻 a closed convex subset. Let 𝑊∶𝐶→𝐶 be a nonexpansive mapping. Then, the mapping 𝐼−𝑊 is demiclosed. That is, if {𝜌^𝑛} is a sequence in 𝐶 such that 𝜌^𝑛→𝜈 weakly and (𝐼−𝑊)𝜌^𝑛→𝑢 strongly, then (𝐼−𝑊)𝜈=𝑢.

Lemma 3 (see [22]). Assume that {𝜂_𝑛} is a sequence of nonnegative real numbers such that 𝜂_𝑛+1≤1−𝜅_𝑛𝜂_𝑛+𝜍_𝑛,𝑛∈ℕ, (11) where {𝜅_𝑛} is a sequence in (0,1) and {𝜍_𝑛} is a sequence such that(1)∑^∞_𝑛=1𝜅_𝑛=∞; (2)limsup_𝑛→∞(𝜍_𝑛/𝜅_𝑛)≤0 or ∑^∞_𝑛=1|𝜍_𝑛|<∞.
Then lim_𝑛→∞𝜂_𝑛=0.

3. Main Results

Let 𝐻₁ and 𝐻₂ be two Hilbert spaces and 𝐶⊂𝐻₁ and 𝑄⊂𝐻₂ two nonempty closed convex subsets. Let 𝐴∶𝐻₁→𝐻₂ be a bounded linear operator with its adjoint 𝐴^∗. Let 𝜓∶𝐶×𝐶→ℝ and let 𝜑∶𝐷×𝐷→ℝ be two bifunctions satisfying the conditions (H1)–(H4) in Lemma 1. Let 𝑆∶𝐷→𝐷 and 𝑊∶𝐶→𝐶 be two nonexpansive mappings.

Algorithm 4. Let 𝑥₀∈𝐻₁. Define a sequence {𝑥_𝑛} as follows: 𝜌^𝑛+1=𝑊𝑈^𝜓_𝜄1−𝜁_𝑛×𝜌^𝑛+𝜍𝐴^∗𝑆𝑈^𝜑_𝜅−𝐼𝐴𝜌^𝑛,∀𝑛∈ℕ, (12) where 𝜄,𝜅, and 𝜍 are three constants satisfying 𝜄∈(0,∞), 𝜅∈(0,∞), 𝜍∈(0,1/‖𝐴‖²), and {𝜁_𝑛} is a real number sequence in (0,1).

In the sequel, we assume that Θ=𝑥∈Fix(𝑊)∩EP𝜓∶𝐴𝑥∈Fix(𝑆)∩EP𝜑≠∅. (13)

Theorem 5. If {𝜁_𝑛} satisfies lim_𝑛→∞𝜁_𝑛=0, ∑^∞_𝑛=1𝜁_𝑛=∞ and lim_𝑛→∞𝜁_𝑛+1/𝜁_𝑛=1, then {𝜌^𝑛} generated by algorithm (12) converges strongly to proj_Θ(0) which is the minimum-norm element in Θ.

Proof. Let 𝑝=proj_Θ(0). Then, 𝑝∈Fix(𝑊)∩EP(𝜓) and 𝐴𝑝∈Fix(𝑆)∩EP(𝜑). Set 𝑧^𝑛=𝑈^𝜑_𝜅𝐴𝜌^𝑛, 𝑦^𝑛=(1−𝜁_𝑛)(𝜌^𝑛+𝜍𝐴^∗(𝑆𝑈^𝜑_𝜅−𝐼)𝐴𝜌^𝑛) and 𝑢^𝑛=𝑈^𝜓_𝜄[(1−𝜁_𝑛)(𝜌^𝑛+𝜍𝐴^∗(𝑆𝑈^𝜑_𝜅−𝐼)𝐴𝜌^𝑛)] for all 𝑛∈ℕ. Then 𝑢^𝑛=𝑈^𝜓_𝜄𝑦^𝑛. From Lemma 1, we know that 𝑈^𝜓_𝜄 and 𝑈^𝜑_𝜅 are firmly nonexpansive. Thus, we have 𝑧^𝑛−𝐴𝑝=𝑈^𝜑_𝜅𝐴𝜌^𝑛−𝐴𝑝≤𝐴𝜌^𝑛,𝑢−𝐴𝑝_𝑛−𝑝=𝑈^𝜓_𝜄𝑦^𝑛−𝑝≤𝑦^𝑛−𝑝,𝑆𝑈^𝜑_𝜅𝐴𝜌^𝑛−𝐴𝑝²=𝑆𝑈^𝜑_𝜅𝐴𝜌^𝑛−𝑆𝑈^𝜑_𝜅≤𝑈𝐴𝑝_𝜑𝜅𝐴𝜌^𝑛−𝑈^𝜑_𝜅𝐴𝑝²≤𝐴𝜌^𝑛−𝐴𝑝²−𝑈^𝜑_𝜅𝐴𝜌^𝑛−𝐴𝜌^𝑛2. (14)(15)(16) Note that 𝑢^𝑛+1−𝑢^𝑛=𝑈^𝜓_𝜄𝑦^𝑛+1−𝑈^𝜓_𝜄𝑦^𝑛≤𝑦^𝑛+1−𝑦^𝑛,𝑧^𝑛+1−𝑧^𝑛=𝑈^𝜑_𝜅𝐴𝜌^𝑛+1−𝑈^𝜑_𝜅𝐴𝜌^𝑛≤𝐴𝜌^𝑛+1−𝐴𝜌^𝑛. (17)(18) From (12) and (15), we have 𝜌^𝑛+1−𝑝=𝑊𝑢^𝑛−𝑝≤𝑢^𝑛−𝑝≤𝑦^𝑛−𝑝. (19) Observe that 𝑦^𝑛−𝑝²=1−𝜁_𝑛×𝜌^𝑛−𝑝+𝜍𝐴^∗𝑆𝑧^𝑛−𝐴𝜌^𝑛−𝜁_𝑛𝑝²≤1−𝜁_𝑛𝜌^𝑛−𝑝+𝜍𝐴^∗𝑆𝑧^𝑛−𝐴𝜌^𝑛2+𝜁_𝑛𝑝²=1−𝜁_𝑛𝜌^𝑛×𝜌−𝑝+2𝜍_𝑛−𝑝,𝐴^∗𝑆𝑧^𝑛−𝐴𝜌^𝑛+𝜍²𝐴^∗𝑆𝑧^𝑛−𝐴𝜌^𝑛2+𝜁_𝑛𝑝². (20) Since 𝐴^∗ is the adjoint of 𝐴, we have 𝜌^𝑛−𝑝,𝐴^∗𝑆𝑧^𝑛−𝐴𝜌^𝑛=𝐴𝜌^𝑛−𝑝,𝑆𝑧^𝑛−𝐴𝜌^𝑛=𝐴𝜌^𝑛−𝐴𝑝+𝑆𝑧^𝑛−𝐴𝜌^𝑛−𝑆𝑧^𝑛−𝐴𝜌^𝑛,𝑆𝑧^𝑛−𝐴𝜌^𝑛=𝑆𝑧^𝑛−𝐴𝑝,𝑆𝑧^𝑛−𝐴𝜌^𝑛−𝑆𝑧^𝑛−𝐴𝜌^𝑛2. (21) Using parallelogram law, we obtain 𝑆𝑧^𝑛−𝐴𝑝,𝑆𝑧^𝑛−𝐴𝜌^𝑛=12𝑆𝑧^𝑛−𝐴𝑝²+𝑆𝑧^𝑛−𝐴𝜌^𝑛2−𝐴𝜌^𝑛−𝐴𝑝². (22) From (16), (21) and (22), we have 𝜌^𝑛−𝑝,𝐴^∗𝑆𝑧^𝑛−𝐴𝜌^𝑛=12𝑆𝑧^𝑛−𝐴𝑝²+𝑆𝑧^𝑛−𝐴𝜌^𝑛2−𝐴𝜌^𝑛−𝐴𝑝²−𝑆𝑧^𝑛−𝐴𝜌^𝑛2≤12𝐴𝜌^𝑛−𝐴𝑝²−𝑧^𝑛−𝐴𝜌^𝑛2+𝑆𝑧^𝑛−𝐴𝜌^𝑛2−𝐴𝜌^𝑛−𝐴𝑝²−𝑆𝑧^𝑛−𝐴𝜌^𝑛21=−2𝑧^𝑛−𝐴𝜌^𝑛2−12𝑆𝑧^𝑛−𝐴𝜌^𝑛2. (23) By (20) and (23), we deduce 𝑦^𝑛−𝑝²≤1−𝜁_𝑛𝜌^𝑛−𝑝²+𝜍²‖𝐴‖²𝑆𝑧^𝑛−𝐴𝜌^𝑛2−1+2𝜍2𝑧^𝑛−𝐴𝜌^𝑛2−12𝑆𝑧^𝑛−𝐴𝜌^𝑛2+𝜁_𝑛𝑝²=1−𝜁_𝑛×𝜌^𝑛−𝑝²+𝜍²‖𝐴‖²−𝜍𝑆𝑧^𝑛−𝐴𝜌^𝑛2−𝜍𝑧^𝑛−𝐴𝜌^𝑛2+𝜁_𝑛𝑝²≤1−𝜁_𝑛𝜌^𝑛−𝑝²+𝜁_𝑛𝑝². (24) It follows from (19), we get 𝜌^𝑛+1−𝑝²≤𝑦^𝑛−𝑝²≤1−𝜁_𝑛𝜌^𝑛−𝑝²+𝜁_𝑛𝑝²≤max𝜌^𝑛−𝑝²,𝑝². (25) The boundedness of the sequence {𝜌^𝑛} yields.
Set 𝑣^𝑛=𝜌^𝑛+𝜍𝐴^∗(𝑆𝑈^𝜑_𝜅−𝐼)𝐴𝜌^𝑛. Then, we have 𝑣^𝑛+1−𝑣^𝑛2=𝜌^𝑛+1−𝜌^𝑛+𝜍𝐴^∗𝑆𝑧^𝑛+1−𝐴𝜌^𝑛+1−𝐴^∗𝑆𝑧^𝑛−𝐴𝜌^𝑛2=𝜌^𝑛+1−𝜌^𝑛2+2𝜍𝜌^𝑛+1−𝜌^𝑛,𝐴^∗𝑆𝑧^𝑛+1−𝐴𝜌^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+𝜍²𝐴^∗𝑆𝑧^𝑛+1−𝐴𝜌^𝑛+1−𝐴^∗𝑆𝑧^𝑛−𝐴𝜌^𝑛2≤𝜌^𝑛+1−𝜌^𝑛2+2𝜍𝐴𝜌^𝑛+1−𝐴𝜌^𝑛,𝑆𝑧^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛+𝜍²‖𝐴‖²𝑆𝑧^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2=𝜌^𝑛+1−𝜌^𝑛2+𝜍²‖𝐴‖²𝑆𝑧^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2+2𝜍𝑆𝑧^𝑛+1−𝑆𝑧^𝑛,𝑆𝑧^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛−2𝜍𝑆𝑧^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2=𝜌^𝑛+1−𝜌^𝑛2+𝜍²‖𝐴‖²𝑆𝑧^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2+𝜍𝑆𝑧^𝑛+1−𝑆𝑧^𝑛2+𝑆𝑧^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2−2𝜍𝑆𝑧^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2=𝜌^𝑛+1−𝜌^𝑛2+𝜍²‖𝐴‖²×−𝜍𝑆𝑧^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2+𝜍𝑆𝑧^𝑛+1−𝑆𝑧^𝑛2−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2≤𝜌^𝑛+1−𝜌^𝑛2+𝜍²‖𝐴‖²×−𝜍𝑆𝑧^𝑛+1−𝑆𝑧^𝑛−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2+𝜍𝑧^𝑛+1−𝑧^𝑛2−𝐴𝜌^𝑛+1−𝐴𝜌^𝑛2. (26) Since 𝜍∈(0,1/‖𝐴‖²), we derive by virtue of (18) and (26) that 𝑣^𝑛+1−𝑣^𝑛≤𝜌^𝑛+1−𝜌^𝑛. (27) According to (17) and (27), we have 𝜌^𝑛+1−𝜌^𝑛=𝑊𝑢^𝑛+1−𝑊𝑢^𝑛≤𝑢^𝑛+1−𝑢^𝑛≤𝑦^𝑛+1−𝑦^𝑛=1−𝜁_𝑛+1𝑣^𝑛+1−1−𝜁_𝑛𝑣^𝑛=1−𝜁_𝑛+1𝑣^𝑛+1−𝑣^𝑛+𝜁_𝑛−𝜁_𝑛+1𝑣^𝑛≤1−𝜁_𝑛+1𝑣^𝑛+1−𝑣^𝑛+𝜁_𝑛+1−𝜁_𝑛𝑣^𝑛≤1−𝜁_𝑛+1𝜌^𝑛+1−𝜌^𝑛+𝜁_𝑛+1−𝜁_𝑛𝑣^𝑛. (28) It follows that 𝜌^𝑛+1−𝜌^𝑛≤𝜁_𝑛+1−𝜁_𝑛𝜁_𝑛+1𝑣^𝑛. (29) Since {𝜌^𝑛} is bounded, we can deduce {𝑣^𝑛} is also bounded. From (29), we have lim_𝑛→∞𝜌^𝑛+1−𝜌^𝑛=0. (30) Hence, lim_𝑛→∞𝜌^𝑛−𝑊𝑢^𝑛=0. (31) Using the firmly-nonexpansivenessity of 𝑈^𝜓_𝜄, we have 𝑢^𝑛−𝑝²=𝑈^𝜓_𝜄𝑦^𝑛−𝑝²≤𝑦^𝑛−𝑝²−𝑈^𝜓_𝜄𝑦^𝑛−𝑦^𝑛2=𝑦^𝑛−𝑝²−𝑢^𝑛−𝑦^𝑛2. (32) Thus, we get 𝜌^𝑛+1−𝑝²≤𝑢^𝑛−𝑝²≤𝑦^𝑛−𝑝²−𝑢^𝑛−𝑦^𝑛2≤1−𝜁_𝑛𝜌^𝑛−𝑝²+𝜁_𝑛𝑝²−𝑢^𝑛−𝑦^𝑛2. (33) It follows that 𝑢^𝑛−𝑦^𝑛2≤𝜌^𝑛−𝑝²−𝜌^𝑛+1−𝑝²+𝜁_𝑛𝑝²≤𝜌^𝑛−𝑝+𝜌^𝑛+1−𝑝²𝜌^𝑛+1−𝑝²+𝜁_𝑛𝑝². (34) This together with (30) and (C1) implies that lim_𝑛→∞𝑢^𝑛−𝑦^𝑛=0. (35) Note that 𝜌^𝑛+1−𝑝²≤𝑦^𝑛−𝑝²≤1−𝜁_𝑛𝜌^𝑛−𝑝²+1−𝜁_𝑛𝜍²‖𝐴‖²−𝜍𝑆𝑧^𝑛−𝐴𝜌^𝑛2−1−𝜁_𝑛𝜍𝑧^𝑛−𝐴𝜌^𝑛2+𝜁_𝑛𝑝². (36) Hence, 1−𝜁_𝑛𝜍−𝜍²‖𝐴‖²𝑆𝑧^𝑛−𝐴𝜌^𝑛2+1−𝜁_𝑛𝜍𝑧^𝑛−𝐴𝜌^𝑛2≤𝜌^𝑛−𝑝²−𝜌^𝑛+1−𝑝²+𝜁_𝑛𝑝²≤𝜌^𝑛−𝑝+𝜌^𝑛+1−𝑝𝜌^𝑛+1−𝜌^𝑛+𝜁_𝑛𝑝², (37) which implies that lim_𝑛→∞𝑆𝑧^𝑛−𝐴𝜌^𝑛=lim_𝑛→∞𝑧^𝑛−𝐴𝜌^𝑛=0. (38) So, we get lim_𝑛→∞𝑆𝑧^𝑛−𝑧^𝑛=0. (39) Since 𝑦^𝑛−𝑝^𝑛=𝜍𝐴^∗𝑆𝑈^𝜑_𝜅−𝐼𝐴𝜌^𝑛+𝜁_𝑛𝑣^𝑛≤𝜍‖𝐴‖𝑆𝑧^𝑛−𝐴𝜌^𝑛+𝜁_𝑛𝑣^𝑛, (40) we get lim_𝑛→∞𝜌^𝑛−𝑦^𝑛=0. (41) From (31), (35), and (41), we get lim_𝑛→∞𝜌^𝑛−𝑊𝜌^𝑛=0. (42) Now, we show that limsup_𝑛→∞𝑝,𝑦^𝑛−𝑝≥0. (43) Choose a subsequence {𝑦^𝑛𝑖} of {𝑦^𝑛} such that limsup_𝑛→∞𝑝,𝑦^𝑛−𝑝=lim_𝑖→∞𝑝,𝑦^𝑛𝑖−𝑝. (44) Notice that {𝑦^𝑛𝑖} is bounded, we can choose {𝑦^𝑛𝑖𝑗} of {𝑦^𝑛𝑖} such that 𝑦^𝑛𝑖𝑗⇀𝑧. Without loss of generality, we assume that 𝑦^𝑛𝑖⇀𝑧. From the above conclusions, we derive that 𝜌^𝑛𝑖⇀𝑧,𝑢_𝑛𝑖⇀𝑧,𝐴𝜌^𝑛𝑖⇀𝐴𝑧,𝑧^𝑛𝑖⇀𝐴𝑧. (45) By Lemma 2, (39), and (41), we deduce 𝑧∈Fix(𝑊) and 𝐴𝑧∈Fix(𝑆).
Next, we show that 𝑧∈EP(𝜓). Since 𝑢^𝑛=𝑈^𝜓_𝜄𝑦^𝑛, we have 𝜓𝑢^𝑛,𝑥^†+1𝜄𝑥^†−𝑢^𝑛,𝑢^𝑛−𝑦^𝑛≥0,∀𝑥^†∈𝐶. (46) By the monotonicity of 𝜓, we have 1𝜄𝑥^†−𝑢^𝑛,𝑢^𝑛−𝑦^𝑛≥𝜓𝑥^†,𝑢^𝑛, (47) and so 𝑥^†−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑦^𝑛𝑖𝜄≥𝜓𝑥^†,𝑢_𝑛𝑖. (48) Since ‖𝑢^𝑛−𝑦^𝑛‖→0, 𝑢_𝑛𝑖⇀𝑧, we obtain (𝑢_𝑛𝑖−𝑦^𝑛𝑖)/𝜄→0. Thus, 0≥𝜓(𝑥^†,𝑧). For 𝑡 with 0<𝑡≤1 and 𝑥^†∈𝐶, let 𝑦^𝑡=𝑡𝑥^†+(1−𝑡)𝑧∈𝐶. We obtain 𝜓(𝑦^𝑡,𝑧)≤0. Hence, 0=𝜓𝑦^𝑡,𝑦^𝑡≤𝑡𝜓𝑦^𝑡,𝑥^†+(1−𝑡)𝜓𝑦^𝑡,𝑧≤𝑡𝜓𝑦^𝑡,𝑥^†. (49) So, 0≤𝜓(𝑦^𝑡,𝑥^†). And, thus, 0≤𝜓(𝑧,𝑥^†). This implies that 𝑧∈EP(𝜓). Similarity, we can prove that 𝐴𝑧∈EP(𝜑). To this end, we deduce 𝑧∈Fix(𝑊)∩EP(𝜓) and 𝐴𝑧∈Fix(𝑆)∩EP(𝜑). That is to say, 𝑧∈Θ. Therefore, limsup_𝑛→∞𝑝,𝑦^𝑛−𝑝=lim_𝑖→∞𝑝,𝑦^𝑛𝑖−𝑝=lim_𝑖→∞𝑝,𝑧−𝑝≥0. (50) Finally, we prove 𝜌^𝑛→𝑝. From (12), we have 𝜌^𝑛+1−𝑝²≤𝑦^𝑛−𝑝²=1−𝜁_𝑛𝑣^𝑛−𝑝−𝜁_𝑛𝑝²≤1−𝜁_𝑛𝑣^𝑛−𝑝²−2𝜁_𝑛𝑝,𝑦^𝑛≤−𝑝1−𝜁_𝑛𝜌^𝑛−𝑝²−2𝜁_𝑛𝑝,𝑦^𝑛−𝑝. (51) Applying Lemma 3 and (50) to (51), we deduce 𝜌^𝑛→𝑝. The proof is completed.

Algorithm 6. Let 𝜌⁰∈𝐻₁ arbitrarily define a sequence {𝜌^𝑛} by the following: 𝜌^𝑛+1=𝑊1−𝜁_𝑛𝜌^𝑛+𝜍𝐴^∗(𝑆−𝐼)𝐴𝜌^𝑛, (52) for all 𝑛∈ℕ, where 𝜍∈(0,1/‖𝐴‖²) and {𝜁_𝑛} is a real number sequence in (0,1).

Corollary 7. Suppose Θ₁={𝑥∈Fix(𝑊)∶𝐴𝑥∈Fix(𝑆)}≠∅. If {𝜁_𝑛} satisfies lim_𝑛→∞𝜁_𝑛=0, ∑^∞_𝑛=1𝜁_𝑛=∞, and lim_𝑛→∞𝜁_𝑛+1/𝜁_𝑛=1, then the sequence {𝜌^𝑛} generated by algorithm (52) converges strongly to 𝑝=proj_Θ1(0) which is the mum-norm element in Θ₁.

Algorithm 8. Let 𝜌⁰∈𝐻₁ arbitrarily define a sequence {𝜌^𝑛} by the following: 𝜌^𝑛+1=𝑈^𝜓_𝜄1−𝜁_𝑛𝜌^𝑛+𝜍𝐴^∗𝑈^𝜑_𝜅−𝐼𝐴𝜌^𝑛, (53) for all 𝑛∈ℕ, where 𝜄,𝜅, and 𝜍 are three constants satisfying 𝜄∈(0,∞),𝜅∈(0,∞), 𝜍∈(0,1/‖𝐴‖²), and {𝜁_𝑛} is a real number sequence in (0,1).

Corollary 9. Suppose Θ₂={𝑥∈EP(𝜓)∶𝐴𝑥∈EP(𝜑)}≠∅. If {𝜁_𝑛} satisfies lim_𝑛→∞𝜁_𝑛=0, ∑^∞_𝑛=1𝜁_𝑛=∞, and lim_𝑛→∞𝜁_𝑛+1/𝜁_𝑛=1, then the sequence {𝜌^𝑛} generated by algorithm (53) converges strongly to 𝑝=proj_Θ2(0) which is the mum-norm element in Θ₂.

Algorithm 10. Let 𝜌⁰∈𝐻₁ arbitrarily define a sequence {𝜌^𝑛} by the following: 𝜌^𝑛+1=proj_𝐶1−𝜁_𝑛𝜌^𝑛+𝜍𝐴^∗proj_𝑄−𝐼𝐴𝜌^𝑛, (54) for all 𝑛∈ℕ, where 𝜍∈(0,1/‖𝐴‖²) and {𝜁_𝑛} is a real number sequence in (0,1).

Corollary 11. Suppose Θ₃={𝑥∈𝐶∶𝐴𝑥∈𝑄}≠∅. If {𝜁_𝑛} satisfies lim_𝑛→∞𝜁_𝑛=0, ∑^∞_𝑛=1𝜁_𝑛=∞, and lim_𝑛→∞𝜁_𝑛+1/𝜁_𝑛=1, then the sequence {𝜌^𝑛} generated by algorithm (54) converges strongly to 𝑝=proj_Θ3(0) which is the mum-norm element in Θ₃.

Acknowledgment

Li-Jun Zhu was supported in part by NNSF of China (61362033), NZ13087, NGY2012097 and 2013xyz023.