Strong Convergence of the Viscosity Approximation Process for the Split Common Fixed-Point Problem of Quasi-Nonexpansive Mappings

Abstract

Very recently, Moudafi (2011) introduced an algorithm with weak convergence for the split common fixed-point problem. In this paper, we will continue to consider the split common fixed-point problem. We discuss the strong convergence of the viscosity approximation method for solving the split common fixed-point problem for the class of quasi-nonexpansive mappings in Hilbert spaces. Our results improve and extend the corresponding results announced by many others.

1. Introduction and Preliminary

Throughout this paper, we always assume that 𝐻 is a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Let 𝐼 denote the identity operator on 𝐻. Let 𝐶 and 𝑄 be nonempty closed convex subset of real Hilbert spaces 𝐻₁ and 𝐻₂, respectively. The split feasibility problem (SFP) is to find a point 𝑥∈𝐶suchthat𝐴𝑥∈𝑄,(1.1) 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]. The SFP attracts many authors' attention due to its application in signal processing. Various algorithms have been invented to solve it (see [3–9] and references therein).

Note that the split feasibility problem (1.1) can be formulated as a fixed-point equation by using the fact 𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗=𝑥^∗;(1.2) that is, 𝑥^∗ solves the SFP (1.1) if and only if 𝑥^∗ solves the fixed point equation (1.2) (see [10] for the details). This implies that we can use fixed-point algorithms (see [11–13]) to solve SFP. A popular algorithm that solves the SFP (1.1) is due to Byrne's CQ algorithm [2] which is found to be a gradient-projection method (GPM) in convex minimization. Subsequently, Byrne [3] applied KM iteration to the CQ algorithm, and Zhao and Yang [14] applied KM iteration to the perturbed CQ algorithm to solve the SFP. It is well known that the CQ algorithm and the KM algorithm for a split feasibility problem do not necessarily converge strongly in the infinite-dimensional Hilbert spaces.

The split common fixed-point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP); see [15]. In this paper, we introduce and study the convergence properties of a viscosity approximation algorithm for solving the SCFP for the class of quasi-nonexpansive operators 𝑆 such that 𝐼−𝑆 is demiclosed at the origin.

Now let us first recall the definition of quasi-nonexpansive operators which appear naturally when using subgradient projection operator techniques in solving some feasibility problems, and also some definitions of classes of operators often used in fixed-point theory and which are commonly encountered in the literature.

Let 𝑇∶𝐻→𝐻 be a mapping. A point 𝑥∈𝐻 is said to be a fixed point of 𝑇 provided that 𝑇𝑥=𝑥. In this paper, we use 𝐹(𝑇) to denote the fixed-point set and use → and ⇀ to denote the strong convergence and weak convergence, respectively. We use 𝜔_𝑤(𝑥_𝑘)={𝑥∶∃𝑥_𝑘𝑗⇀𝑥} stand for the weak 𝜔-limit set of {𝑥_𝑘}.(i)A mapping 𝑇∶𝐻→𝐻 belongs to the general class Φ_𝑄 of (possibly discontinuous) quasi-nonexpansive mappings if ‖𝑇𝑥−𝑞‖≤‖𝑥−𝑞‖,∀(𝑥,𝑞)∈𝐻×𝐹(𝑇).(1.3)(ii)A mapping 𝑇∶𝐻→𝐻 belongs to the set Φ_𝑁 of nonexpansive mappings if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀(𝑥,𝑦)∈𝐻×𝐻.(1.4)(iii)A mapping 𝑇∶𝐻→𝐻 belongs to the set Φ_FN of firmly nonexpansive mappings if ‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²−‖(𝑥−𝑦)−(𝑇𝑥−𝑇𝑦)‖²,∀(𝑥,𝑦)∈𝐻×𝐻.(1.5)(iv)A mapping 𝑇∶𝐻→𝐻 belongs to the set Φ_FQ of firmly quasi-nonexpansive mappings if ‖𝑇𝑥−𝑞‖²≤‖𝑥−𝑞‖²−‖𝑥−𝑇𝑥‖²,∀(𝑥,𝑞)∈𝐻×𝐹(𝑇).(1.6)

It is easily observed that Φ_FN⊂Φ_𝑁⊂Φ_𝑄 and that Φ_FN⊂Φ_FQ⊂Φ_𝑄. Furthermore, Φ_FN is well known to include resolvents and projection operators, while Φ_FQ contains subgradient projection operators (see, e.g., [16] and the reference therein).

A mapping 𝑇∶𝐻→𝐻 is called demiclosed at the origin if any sequence {𝑥_𝑛} weakly converges to 𝑥, and if the sequence {𝑇𝑥_𝑛} strongly converges to 0, then 𝑇𝑥=0. A mapping 𝑓∶𝐻→𝐻 is called a contraction of modulus 𝜌∈[0,1) if ‖𝑓𝑥−𝑓𝑦‖≤𝜌‖𝑥−𝑦‖,∀(𝑥,𝑦)∈𝐻×𝐻.(1.7)

In what follows, we will focus our attention on the following general two-operator split common fixed-point problem: find𝑥^∗∈𝐶suchthat𝐴𝑥^∗∈𝑄,(1.8) where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator, 𝑈∶𝐻₁→𝐻₁ and 𝑆∶𝐻₂→𝐻₂ are two quasi-nonexpansive operators with nonempty fixed-point sets 𝐹(𝑈)=𝐶 and 𝐹(𝑆)=𝑄, and denote the solution set of the two-operator SCFP by Γ={𝑦∈𝐶;𝐴𝑦∈𝑄}.(1.9)

Recall that 𝐹(𝑈) and 𝐹(𝑆) are nonempty closed convex subsets of 𝐻₁ and 𝐻₂, respectively. If Γ≠∅, we have Γ which is close convex subset of 𝐻₁. To solve (1.8), Censor and Segal [15] proposed and proved, in infinite-dimensional spaces, the convergence of the following algorithm: 𝑥_𝑘+1=𝑈𝑥_𝑘+𝛾𝐴^𝑡(𝑆−𝐼)𝐴𝑥_𝑘,𝑘∈𝑁,(1.10)

where 𝛾∈(0,2/𝜆), with 𝜆 being the largest eigenvalue of the matrix 𝐴^𝑡𝐴 (𝐴^𝑡 stands for matrix transposition). Very recently, Moudafi [17] introduced the following relaxed algorithm: 𝑥_𝑘+1=1−𝛼_𝑘𝑢_𝑘+𝛼_𝑘𝑈𝑢_𝑘,𝑘∈𝑁,(1.11)

where 𝑢_𝑘=𝑥_𝑘+𝛾𝛽𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘, 𝛽∈(0,1), 𝛼_𝑘∈(0,1), and 𝛾∈(0,1/𝜆𝛽), with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴. Moudafi proved weak convergence result of the algorithm in Hilbert spaces.

Inspired by their work, we introduce the following viscosity approximation algorithm.

Algorithm 1. Initialization: Let 𝑥₀∈𝐻 be arbitrary.
Iterative step: Set 𝑇=𝑈(𝐼+𝛾𝐴^∗(𝑆−𝐼)𝐴). For 𝑘∈𝑁, let 𝑥_𝑘+1=𝛼_𝑘𝑓𝑥_𝑘+1−𝛼_𝑘1−𝜔_𝑘𝑥_𝑘+𝜔_𝑘𝑇𝑥_𝑘,(1.12) where 𝑓∶𝐻→𝐻 is a contraction of modulus 𝜌, 𝜔_𝑘∈(0,1/2), 𝛾∈(0,1/𝜆) with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴, and 𝛼_𝑘∈(0,1).
This paper establishes the strong convergence of the sequence given by (1.12) to the unique solution of the variational inequality problem VIP(𝐼−𝑓,Γ)∶find𝑥^∗∈Γsuchthat⟨(𝐼−𝑓)𝑥^∗,𝑣−𝑥^∗⟩≥0,∀𝑣∈Γ.(1.13) Now we give a series of preliminary results needed for the convergence analysis of algorithm (1.12).

Lemma 1.1. Let 𝐻 be a real Hilbert space and 𝑇∶𝐻→𝐻 a quasi-nonexpansive mapping. Then, the following properties are reached: (i)⟨𝑥,𝑦⟩=−12‖𝑥−𝑦‖²+12‖𝑥‖²+12‖𝑦‖²,∀(𝑥,𝑦)∈𝐻×𝐻;(ii)⟨𝑥−𝑇𝑥,𝑥−𝑞⟩≥12‖𝑥−𝑇𝑥‖² and ⟨𝑥−𝑇𝑥,𝑞−𝑇𝑥⟩≤12‖𝑥−𝑇𝑥‖²,∀(𝑥,𝑞)∈𝐻×𝐹(𝑇).

Remark 1.2. Let 𝐹∶=𝐼−𝑓, where 𝑓 is the contraction defined in (1.7). It is a simple matter to see that the operator 𝐹 is (1−𝜌) strongly monotone over 𝐻; that is,⟨𝐹𝑥−𝐹𝑦,𝑥−𝑦⟩≥(1−𝜌)‖𝑥−𝑦‖²,∀(𝑥,𝑦)∈𝐻×𝐻.(1.14)

The next result is of fundamental importance for the techniques of analysis used in this paper. It was established in [18], and its proof is given for the sake of completeness.

Lemma 1.3 (see [18, 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)=max𝑘≤𝑛∣𝛿_𝑘<𝛿_𝑘+1.(1.15) 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 one has 𝛿_𝑛≤𝛿_𝜏(𝑛)+1.(1.16)

Proof. Clearly, we can see that {𝜏(𝑛)} is a well-defined sequence, and the fact that it is nondecreasing is obvious as well as lim_𝑛→∞𝜏(𝑛)=∞ and 𝛿_𝜏(𝑛)≤𝛿_𝜏(𝑛)+1. Let us prove (1.16). It is easily observed that 𝜏(𝑛)≤𝑛. Consequently, we prove (1.16) by distinguishing the three cases: (c1) 𝜏(𝑛)=𝑛; (c2) 𝜏(𝑛)=𝑛−1; (c3) 𝜏(𝑛)<𝑛−1. In the first case (i.e., 𝜏(𝑛)=𝑛), (1.16) is immediately given by 𝛿_𝜏(𝑛)≤𝛿_𝜏(𝑛)+1. In the second case (i.e., 𝜏(𝑛)=𝑛−1), (1.16) becomes obvious. In the third case (i.e., 𝜏(𝑛)≤𝑛−2), by (1.15) and for any integer 𝑛≥𝑛₀, we easily observe that 𝛿_𝑗≥𝛿_𝑗+1 for 𝜏(𝑛)+1≤𝑗≤𝑛−1; namely, 𝛿_𝜏(𝑛)+1≥𝛿_𝜏(𝑛)+2≥⋯≥𝛿_𝑛−1≥𝛿_𝑛,(1.17) which entails the desired result.

2. Main Results

Theorem 2.1. Given a bounded linear operator 𝐴∶𝐻₁→𝐻₂, let 𝑈∶𝐻₁→𝐻₁ and 𝑆∶𝐻₂→𝐻₂ be quasi-nonexpansive mappings with nonempty fixed-point set 𝐹(𝑈)=𝐶 and 𝐹(𝑆)=𝑄. Assume that 𝑈−𝐼 and 𝑆−𝐼 are demiclosed at origin. Let {𝑥_𝑘} be the sequence given by (1.12) with 𝛾∈(0,1/𝜆), 𝜔_𝑘∈(0,1/2) such that 0<liminf_𝑘→∞𝜔_𝑘≤limsup_𝑘→∞𝜔_𝑘<1/2 and {𝛼_𝑘}⊂(0,1) such that lim_𝑘→∞𝛼_𝑘=0 and ∑_𝑘𝛼_𝑘=∞. If Γ≠∅, then the sequence {𝑥_𝑘} strongly converges to a split common fixed-point 𝑥^∗∈Γ, verifying 𝑥^∗=𝑃_Γ𝑓(𝑥^∗) which equivalently solves the following variational inequality problem: 𝑥^∗∈Γ,⟨(𝐼−𝑓)𝑥^∗,𝑣−𝑥^∗⟩≥0,∀𝑣∈Γ.(2.1)

Proof. Set 𝑇_𝜔𝑘=(1−𝜔_𝑘)𝐼+𝜔_𝑘𝑇. Then 𝑥_𝑘+1=𝛼_𝑘𝑓(𝑥_𝑘)+(1−𝛼_𝑘)𝑇_𝜔𝑘𝑥_𝑘.
Firstly, we prove that {𝑥_𝑘} is bounded. Taking 𝑦∈Γ, that is, 𝑦∈𝐹(𝑈), 𝐴𝑦∈𝐹(𝑆). We have ‖‖𝑥_𝑘+1−𝑦‖‖≤=𝛼‖‖𝛼_𝑘𝑘𝑓𝑥_𝑘−𝑓(𝑦)+𝛼_𝑘(𝑓(𝑦)−𝑦)+1−𝛼_𝑘𝑇_𝜔𝑘𝑥_𝑘−𝑦‖‖≤𝛼_𝑘‖‖𝑓𝑥_𝑘−𝑓(𝑦)‖‖+𝛼_𝑘‖𝑓(𝑦)−𝑦‖+1−𝛼_𝑘‖‖𝑇_𝜔𝑘𝑥_𝑘−𝑦‖‖𝜌‖‖𝑥_𝑘−𝑦‖‖+𝛼_𝑘‖𝑓(𝑦)−𝑦‖+1−𝛼_𝑘‖‖𝑇_𝜔𝑘𝑥_𝑘−𝑦‖‖.(2.2) From the definition of 𝑇_𝜔𝑘, we get ‖‖𝑇_𝜔𝑘𝑥_𝑘−𝑦‖‖²=‖‖1−𝜔_𝑘𝑥_𝑘+𝜔_𝑘𝑇𝑥_𝑘−𝑦‖‖²=‖‖𝑥_𝑘−𝑦+𝜔_𝑘𝑇𝑥_𝑘−𝑥_𝑘‖‖²=‖‖𝑥_𝑘−𝑦‖‖²−2𝜔_𝑘⟨𝑥_𝑘−𝑦,𝑥_𝑘−𝑇𝑥_𝑘⟩+𝜔^2𝑘‖‖𝑇𝑥_𝑘−𝑥_𝑘‖‖².(2.3) On the other hand, we have ‖‖𝑇𝑥_𝑘−𝑦‖‖²=‖‖𝑈𝐼+𝛾𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘−𝑦‖‖²≤‖‖𝐼+𝛾𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘−𝑦‖‖²==‖‖‖‖𝑥𝑥_𝑘𝑘−𝑦‖‖²+𝛾²‖‖𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘‖‖²+2𝛾⟨𝑥_𝑘−𝑦,𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘⟩−𝑦‖‖²+𝛾²⟨(𝑆−𝐼)𝐴𝑥_𝑘,𝐴𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘⟩+2𝛾⟨𝑥_𝑘−𝑦,𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘⟩.(2.4) From the definition of 𝜆, it follows that 𝛾²⟨(𝑆−𝐼)𝐴𝑥_𝑘,𝐴𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘⟩≤=𝜆𝜆𝛾𝛾²₂⟨(𝑆−𝐼)𝐴𝑥_𝑘,(𝑆−𝐼)𝐴𝑥_𝑘⟩‖‖(𝑆−𝐼)𝐴𝑥_𝑘‖‖².(2.5) Now, by using property (ii) of Lemma 1.1, we obtain 2𝛾⟨𝑥_𝑘−𝑦,𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘⟩==22𝛾𝛾𝐴𝐴𝑥𝑥_𝑘𝑘−𝑦,(𝑆−𝐼)𝐴𝑥_𝑘=2𝛾𝑆𝐴−𝑥_𝑘𝑦+(𝑆−𝐼)𝐴𝑥_𝑘−(𝑆−𝐼)𝐴𝑥_𝑘,(𝑆−𝐼)𝐴𝑥_𝑘≤2𝛾1−𝐴𝑦,(𝑆−𝐼)𝐴𝑥_𝑘−‖‖(𝑆−𝐼)𝐴𝑥_𝑘‖‖²2‖‖(𝑆−𝐼)𝐴𝑥^𝑘=−𝛾‖‖(𝑆−𝐼)𝐴𝑥_𝑘‖‖²−‖‖(𝑆−𝐼)𝐴𝑥_𝑘‖‖²‖‖².(2.6) Combining (2.4)–(2.6), we have ‖‖𝑇𝑥_𝑘−𝑦‖‖²≤‖‖𝑥_𝑘−𝑦‖‖²+𝜆𝛾²‖‖(𝑆−𝐼)𝐴𝑥_𝑘‖‖²−𝛾‖‖(𝑆−𝐼)𝐴𝑥_𝑘‖‖²=‖‖𝑥_𝑘−𝑦‖‖²−𝛾(1−𝜆𝛾)‖‖(𝑆−𝐼)𝐴𝑥_𝑘‖‖²≤‖‖𝑥_𝑘−𝑦‖‖².(2.7) From property (i) of Lemma 1.1, we have ⟨𝑥_𝑘−𝑦,𝑥_𝑘−𝑇𝑥_𝑘⟩=−12‖‖𝑇𝑥^𝑘−𝑦‖‖²+12‖‖𝑥^𝑘−𝑦‖‖²+12‖‖𝑥^𝑘−𝑇𝑥_𝑘‖‖²≥12‖‖𝑥^𝑘−𝑇𝑥_𝑘‖‖².(2.8) From (2.3) and (2.8), we have ‖‖𝑇_𝜔𝑘𝑥_𝑘−𝑦‖‖²≤‖‖𝑥_𝑘−𝑦‖‖²−𝜔_𝑘‖‖𝑥_𝑘−𝑇𝑥_𝑘‖‖²+𝜔^2𝑘‖‖𝑥_𝑘−𝑇𝑥_𝑘‖‖²=‖‖𝑥_𝑘−𝑦‖‖²−𝜔_𝑘1−𝜔_𝑘‖‖𝑥_𝑘−𝑇𝑥_𝑘‖‖²≤‖‖𝑥_𝑘−𝑦‖‖²,(2.9) Combining (2.2), (2.3), and (2.9), it follows that ‖‖𝑥_𝑘+1−𝑦‖‖=≤𝛼1_𝑘𝜌−‖‖𝛼𝑥_𝑘𝑘−𝑦‖‖+𝛼_𝑘‖𝑓(𝑦)−𝑦‖+1−𝛼_𝑘‖‖𝑥_𝑘−𝑦‖‖≤max(‖‖1𝑥−_𝑘𝜌)‖‖𝑥_𝑘−𝑦‖‖+𝛼_𝑘‖𝑓(𝑦)−𝑦‖−𝑦‖‖,11−𝜌‖𝑓(𝑦)−𝑦‖.(2.10)It is obviously that ‖‖𝑥_𝑘−𝑦‖‖≤max‖‖𝑥₀−𝑦‖‖,11−𝜌‖𝑓(𝑦)−𝑦‖,(2.11) and hence {𝑥_𝑘} is bounded. Let 𝑥^∗=𝑃_Γ𝑓(𝑥^∗). We have 𝑥_𝑘+1−𝑥_𝑘+𝛼_𝑘𝑥_𝑘−𝑓𝑥_𝑘=1−𝛼_𝑘𝑇_𝜔𝑘𝑥_𝑘−𝑥_𝑘,(2.12) and hence 𝑥_𝑘+1−𝑥_𝑘+𝛼_𝑘(𝐼−𝑓)𝑥_𝑘,𝑥_𝑘−𝑥^∗=−1−𝛼_𝑘𝑥_𝑘−𝑇_𝜔𝑘𝑥_𝑘,𝑥_𝑘−𝑥^∗.(2.13) By (2.9) we obtain that 𝑥_𝑘−𝑇_𝜔𝑘𝑥_𝑘,𝑥_𝑘−𝑥^∗=12‖‖𝑥^𝑘−𝑇_𝜔𝑘𝑥_𝑘‖‖²+12‖‖𝑥^𝑘−𝑥^∗‖‖²−12‖‖𝑇^𝜔𝑘𝑥_𝑘−𝑥^∗‖‖²≥𝜔^2𝑘2‖‖𝑥^𝑘−𝑇𝑥_𝑘‖‖²+12‖‖𝑥^𝑘−𝑥^∗‖‖²−12‖‖𝑥^𝑘−𝑥^∗‖‖²+𝜔^𝑘21−𝜔^𝑘‖‖𝑥_𝑘−𝑇𝑥_𝑘‖‖²=𝜔^𝑘2‖‖𝑥^𝑘−𝑇𝑥_𝑘‖‖².(2.14) It follows from (2.13) that 𝑥_𝑘+1−𝑥_𝑘+𝛼_𝑘(𝐼−𝑓)𝑥_𝑘,𝑥_𝑘−𝑥^∗≤−𝜔2^𝑘1−𝛼^𝑘‖‖𝑥_𝑘−𝑇𝑥_𝑘‖‖²,(2.15) and hence −𝑥_𝑘−𝑥_𝑘+1,𝑥_𝑘−𝑥^∗≤−𝛼_𝑘⟨(𝐼−𝑓)𝑥_𝑘,𝑥_𝑘−𝑥^∗⟩−𝜔^𝑘21−𝛼^𝑘‖‖𝑥_𝑘−𝑇𝑥_𝑘‖‖².(2.16) Setting 𝛿_𝑘=12‖𝑥^𝑘−𝑥^∗‖², we have 𝑥_𝑘−𝑥_𝑘+1,𝑥_𝑘−𝑥^∗=−21‖‖𝑥^𝑘+1−𝑥^∗‖‖²+12‖‖𝑥^𝑘−𝑥^∗‖‖²+12‖‖𝑥^𝑘−𝑥_𝑘+1‖‖²=−𝛿_𝑘+1+𝛿_𝑘+12‖‖𝑥^𝑘−𝑥_𝑘+1‖‖²,(2.17) so that (2.16) can be rewritten as 𝛿_𝑘+1−𝛿_𝑘−12‖‖𝑥^𝑘−𝑥_𝑘+1‖‖²≤−𝛼_𝑘⟨(𝐼−𝑓)𝑥_𝑘,𝑥_𝑘−𝑥^∗⟩−𝜔^𝑘21−𝛼^𝑘‖‖𝑥_𝑘−𝑇𝑥_𝑘‖‖².(2.18) Now using (2.12) again, we have ‖‖𝑥_𝑘+1−𝑥_𝑘‖‖²=‖‖𝛼_𝑘𝑓𝑥_𝑘−𝑥_𝑘+1−𝛼_𝑘(𝑇_𝜔𝑘𝑥_𝑘−𝑥_𝑘‖‖²≤𝛼_𝑘‖‖𝑓𝑥_𝑘−𝑥_𝑘‖‖+1−𝛼_𝑘‖‖𝑇_𝜔𝑘𝑥_𝑘−𝑥_𝑘‖‖²≤2𝛼^2𝑘‖‖𝑓𝑥_𝑘−𝑥_𝑘‖‖²+21−𝛼_𝑘²‖‖𝑇_𝜔𝑘𝑥_𝑘−𝑥_𝑘‖‖²≤2𝛼^2𝑘‖‖𝑓𝑥_𝑘−𝑥_𝑘‖‖²+21−𝛼_𝑘𝜔^2𝑘‖‖𝑇𝑥_𝑘−𝑥_𝑘‖‖²,(2.19) which yields 12‖‖𝑥^𝑘+1−𝑥_𝑘‖‖²≤𝛼^2𝑘‖‖𝑓𝑥_𝑘−𝑥_𝑘‖‖²+1−𝛼_𝑘𝜔^2𝑘‖‖𝑇𝑥_𝑘−𝑥_𝑘‖‖².(2.20) From (2.18) and (2.20), we obtain 𝛿_𝑘+1−𝛿_𝑘+𝜔_𝑘1−𝛼_𝑘12−𝜔^𝑘‖‖𝑇𝑥^𝑘−𝑥_𝑘‖‖²≤𝛼_𝑘𝛼^𝑘‖‖𝑓𝑥_𝑘−𝑥_𝑘‖‖²−⟨(𝐼−𝑓)𝑥_𝑘,𝑥_𝑘−𝑥^∗⟩.(2.21) It follows from Remark 1.2 that ⟨(𝐼−𝑓)𝑥_𝑘−(𝐼−𝑓)𝑥^∗,𝑥_𝑘−𝑥^∗⟩≥(1−𝜌)‖‖𝑥_𝑘−𝑥^∗‖‖²=2(1−𝜌)𝛿_𝑘,(2.22) and hence 2(1−𝜌)𝛿_𝑘+⟨(𝐼−𝑓)𝑥^∗,𝑥_𝑘−𝑥^∗⟩≤⟨(𝐼−𝑓)𝑥_𝑘,𝑥_𝑘−𝑥^∗⟩.(2.23)
The rest of the proof will be divided into two parts.
Case 1. Suppose that there exists 𝑘₀ such that {𝛿_𝑘}_𝑘≥𝑘0 is nonincreasing. In this situation, {𝛿_𝑘} is convergent because it is nonnegative, so that lim_𝑘→∞(𝛿_𝑘+1−𝛿_𝑘)=0; hence, in light of (2.21) together with 𝛼_𝑘→0, the boundedness of {𝑥_𝑘}, and 0<liminf_𝑘→∞𝜔_𝑘≤limsup_𝑘→∞𝜔_𝑘<1/2, we obtain lim_𝑘→∞‖‖𝑥_𝑘−𝑇𝑥_𝑘‖‖=0.(2.24) From (2.21) again, we have 𝛼_𝑘−𝛼^𝑘‖‖𝑓𝑥_𝑘−𝑥_𝑘‖‖²+⟨(𝐼−𝑓)𝑥_𝑘,𝑥_𝑘−𝑥^∗⟩≤𝛿_𝑘−𝛿_𝑘+1.(2.25) By ∑_𝑘𝛼_𝑘=∞, we deduce that liminf_𝑘→∞−𝛼^𝑘‖‖𝑓𝑥_𝑘−𝑥_𝑘‖‖²+⟨(𝐼−𝑓)𝑥_𝑘,𝑥_𝑘−𝑥^∗⟩≤0(2.26) and hence (as 𝛼_𝑘‖𝑓(𝑥_𝑘)−𝑥_𝑘‖²→0) liminf_𝑘→∞⟨(𝐼−𝑓)𝑥_𝑘,𝑥_𝑘−𝑥^∗⟩≤0.(2.27) By (2.23) and (2.27), we have liminf_𝑘→∞2(1−𝜌)𝛿_𝑘+⟨(𝐼−𝑓)𝑥^∗,𝑥_𝑘−𝑥^∗⟩≤0;(2.28) recalling that lim_𝑘→∞𝛿_𝑘 exists, we obtain 2(1−𝜌)lim_𝑘→∞𝛿_𝑘+liminf_𝑘→∞⟨(𝐼−𝑓)𝑥^∗,𝑥_𝑘−𝑥^∗⟩≤0.(2.29) Now we prove that liminf_𝑘→∞⟨(𝐼−𝑓)𝑥^∗,𝑥_𝑘−𝑥^∗⟩≥0.(2.30) It follows from (2.7) and (2.24) that 𝛾(1−𝜆𝛾)‖‖(𝑆−𝐼)𝐴𝑥_𝑘‖‖²≤‖‖𝑥_𝑘−𝑦‖‖²−‖‖𝑇𝑥_𝑘−𝑦‖‖²=≤‖‖‖‖𝑥𝑥_𝑘−𝑦‖‖−‖‖𝑇𝑥_𝑘−𝑦‖‖‖‖𝑥_𝑘−𝑦‖‖+‖‖𝑇𝑥_𝑘−𝑦_𝑘‖‖⟶0−(𝑘𝑇𝑥⟶_𝑘‖‖∞‖‖𝑥)_𝑘,−𝑦‖‖+‖‖𝑇𝑥_𝑘−𝑦‖‖(2.31) and hence lim_𝑘→∞‖‖(𝑆−𝐼)𝐴𝑥_𝑘‖‖=0.(2.32) Taking 𝑦∈𝜔_𝑤(𝑥_𝑘), from the demiclosedness of 𝑆−𝐼 at 0, we obtain 𝑆(𝐴𝑦)=𝐴𝑦.(2.33) Now, by setting 𝑢_𝑘=𝑥_𝑘+𝛾𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘, it follows that 𝑦∈𝜔_𝑤(𝑢_𝑘). On the other hand, ‖‖𝑈𝑢_𝑘−𝑢_𝑘‖‖=‖‖𝑇𝑥_𝑘−𝑥_𝑘−𝛾𝐴^∗(𝑆−𝐼)𝐴𝑥_𝑘‖‖≤‖‖𝑇𝑥_𝑘−𝑥_𝑘‖‖+𝛾‖𝐴^∗‖⋅‖‖(𝑆−𝐼)𝐴𝑥_𝑘‖‖⟶0,(2.34) which, combined with the demiclosedness of 𝑈−𝐼 at 0, yields 𝑈𝑦=𝑦.(2.35) Hence, 𝑦∈𝐶 and 𝑦∈Γ. We can take subsequence {𝑥_𝑘𝑗} of {𝑥_𝑘} such that 𝑥_𝑘𝑗⇀𝑦 as j→∞ and liminf_𝑘→∞⟨(𝐼−𝑓)𝑥^∗,𝑥_𝑘−𝑥^∗⟩=lim_𝑗→∞(𝐼−𝑓)𝑥^∗,𝑥_𝑘𝑗−𝑥^∗,(2.36) which leads to liminf_𝑘→∞⟨(𝐼−𝑓)𝑥^∗,𝑥_𝑘−𝑥^∗⟩=⟨(𝐼−𝑓)𝑥^∗,𝑦−𝑥^∗⟩≥0.(2.37) By (2.29), we have lim_𝑘→∞𝛿_𝑘=0, and hence {𝑥_𝑘} converges strongly to 𝑥^∗.
Case 2. Suppose there exists a subsequence {𝛿_𝑘𝑗}_𝑗≥0 of {𝛿_𝑘} such that 𝛿_𝑘𝑗<𝛿_𝑘𝑗+1 for all 𝑗≥0. In this situation, we consider the sequence of indices {𝜏(𝑘)} as defined in Lemma 1.3. It follows that 𝛿_𝜏(𝑘)+1−𝛿_𝜏(𝑘)>0, which by (2.21) amounts to 𝜔_𝑘1−𝛼_𝜏(𝑘)12−𝜔^𝑘‖‖𝑇𝑥^𝜏(𝑘)−𝑥_𝜏(𝑘)‖‖²≤𝛼_𝜏(𝑘)𝛼^𝜏(𝑘)‖‖𝑓𝑥_𝜏(𝑘)−𝑥_𝜏(𝑘)‖‖²−(𝐼−𝑓)𝑥_𝜏(𝑘),𝑥_𝜏(𝑘)−𝑥^∗.(2.38) By the boundedness of {𝑥_𝑘} and 𝛼_𝑘→0, we immediately obtain lim_𝑘→∞‖‖𝑇𝑥_𝜏(𝑘)−𝑥_𝜏(𝑘)‖‖=0.(2.39) Similar to Case 1, we have liminf_𝑘→∞(𝐼−𝑓)𝑥^∗,𝑥_𝜏(𝑘)−𝑥^∗≥0.(2.40) It follows from (2.38) that (𝐼−𝑓)𝑥_𝜏(𝑘),𝑥_𝜏(𝑘)−𝑥^∗≤𝛼_𝜏(𝑘)‖‖𝑓𝑥_𝜏(𝑘)−𝑥_𝜏(𝑘)‖‖²,(2.41) which in the light of (2.23) yields 2(1−𝜌)𝛿_𝜏(𝑘)+(𝐼−𝑓)𝑥^∗,𝑥_𝜏(𝑘)−𝑥^∗≤𝛼_𝜏(𝑘)‖‖𝑓𝑥_𝜏(𝑘)−𝑥_𝜏(𝑘)‖‖²;(2.42) hence (as 𝛼_𝜏(𝑘)‖𝑓(𝑥_𝜏(𝑘))−𝑥_𝜏(𝑘)‖²→0) it follows that 2(1−𝜌)limsup_𝑘→∞𝛿_𝜏(𝑘)≤−liminf_𝑘→∞(𝐼−𝑓)𝑥^∗,𝑥_𝜏(𝑘)−𝑥^∗.(2.43) From (2.40) we have limsup_𝑘→∞𝛿_𝜏(𝑘)=0, so that lim_𝑘→∞𝛿_𝜏(𝑘)=0, and hence lim_𝑘→∞‖𝑥_𝜏(𝑘)−𝑥^∗‖=0. On the other hand, it follows that ‖‖𝑥_𝜏(𝑘)+1−𝑥_𝜏(𝑘)‖‖=≤𝛼‖‖𝛼_{𝜏(𝑘)𝜏(𝑘)}𝑓𝑥_𝜏(𝑘)−𝑥_𝜏(𝑘)+1−𝛼_𝜏(𝑘)𝑇_𝜔𝑘𝑥_𝜏(𝑘)−𝑥_𝜏(𝑘)‖‖‖‖𝑓𝑥_𝜏(𝑘)−𝑥_𝜏(𝑘)‖‖+1−𝛼_𝜏(𝑘)𝜔_𝑘‖‖𝑇𝑥_𝜏(𝑘)−𝑥_𝜏(𝑘)‖‖,(2.44) which, by (2.39), implies that lim_𝑘→∞‖‖𝑥_𝜏(𝑘)+1−𝑥_𝜏(𝑘)‖‖=0.(2.45) So we have lim_𝑘→∞𝛿_𝜏(𝑘)+1=12‖‖𝑥^𝜏(𝑘)+1−𝑥^∗‖‖=0.(2.46) Then, recalling that 𝛿_𝑘≤𝛿_𝜏(𝑘)+1 (by Lemma 1.3), we get lim_𝑘→∞𝛿_𝑘=0, so that the sequence {𝑥_𝑘} converges strongly to 𝑥^∗.

Theorem 2.2. Given a bounded linear operator 𝐴∶𝐻₁→𝐻₂, let 𝑈∶𝐻₁→𝐻₁ and 𝑆∶𝐻₂→𝐻₂ be quasi-nonexpansive mappings with nonempty fixed-point set 𝐹(𝑈)=𝐶 and 𝐹(𝑆)=𝑄. Assume that 𝑈−𝐼 and 𝑆−𝐼 are demiclosed at origin. Let 𝑥₀∈𝐻 be arbitrary and {𝑥_𝑘} the sequence given by 𝑥_𝑘+1=𝛼_𝑘𝑓𝑥_𝑘+1−𝛼_𝑘(1−𝜔)𝑥_𝑘+𝜔𝑇𝑥_𝑘,(2.47) where 𝑇=𝑈(𝐼+𝛾𝐴^∗(𝑆−𝐼)𝐴), 𝑓∶𝐻→𝐻 a contraction of modulus 𝜌, 𝛾∈(0,1/𝜆), 𝜔∈(0,1/2), and {𝛼_𝑘}⊂(0,1) such that lim_𝑘→∞𝛼_𝑘=0 and ∑_𝑘𝛼_𝑘=∞. If Γ≠∅, then the sequence {𝑥_𝑘} strongly converges to a split common fixed-point 𝑥^∗∈Γ, verifying 𝑥^∗=𝑃_Γ𝑓(𝑥^∗) which equivalently solves the following variational inequality problem: 𝑥^∗∈Γ,⟨(𝐼−𝑓)𝑥^∗,𝑣−𝑥^∗⟩≥0,∀𝑣∈Γ.(2.48)

Acknowledgments

The research was supported by Fundamental Research Funds for the Central Universities (Program No. ZXH2011D005); it was also supported by science research foundation program in Civil Aviation University of China (2011kys02).