Parallel and Cyclic Algorithms for Quasi-Nonexpansives in Hilbert Space

Abstract

Let {𝑇}^𝑁_𝑖=1 be N quasi-nonexpansive mappings defined on a closed convex subset C of a real Hilbert space H. Consider the problem of finding a common fixed point of these mappings and introduce the parallel and cyclic algorithms for solving this problem. We will prove the strong convergence of these algorithms.

1. Introduction

Throughout this paper, we always assume that 𝐶 is a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝐴∶𝐶→𝐻 be a nonlinear mapping. Recall the following definitions.(1)𝐴is said to be monotone if 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶. (1.1)(2)𝐴 is said to be strongly positive if there exists a constant 𝛾>0 such that ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐶. (1.2)(3)𝐴 is said to be strongly monotone if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (1.3)For such a case, 𝐴 is said to be 𝛼-strongly monotone.(4)𝐴 is said to be inverse strongly if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (1.4)For such a case, 𝐴 is said to be 𝛼-inverse-strongly-monotone (𝛼-ism).

Assume 𝐴 is strongly positive operator, that is, there is a constant 𝛾 with the property ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻. (1.5)

Remark 1.1. Let 𝐹=𝐴−𝛾𝑓, where 𝐴 is strongly positive operator, and 𝑓 is contraction mapping with coefficient 𝛽∈(0,1). It is a simple matter to see that the operator 𝐹 is (𝛾−𝛾𝛽)-strongly monotone over 𝐶, that is, 𝐹𝑥−𝐹𝑦,𝑥−𝑦≥𝛾−𝛾𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐶×𝐶. (1.6)

The classical variational inequality which is denoted by VI(𝐴,𝐶) is to find 𝑥∈𝐶 such that 𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (1.7)

The variational inequality has been extensively studied in literature; see, for example, [1, 2] and the reference therein. A mapping 𝑇∶𝐶→𝐶 is said to be a strict pseudocontraction [3] if there exists a constant 0≤𝑘<1 such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦², (1.8) for all 𝑥,𝑦∈𝐶 (If (1.8) holds, we also say that 𝑇 is a 𝑘-strict pseudo-contraction). These mappings are extensions of nonexpansive mappings which satisfy the inequality (1.8) with 𝑘=0. That is, 𝑇∶𝐶→𝐶 is nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.9)

In [4], Xu proved that the sequence {𝑥_𝑛} defined by the iterative method below with the initial guess 𝑥₀∈𝐻 chosen arbitrarily, 𝑥_𝑛+1=𝛼_𝑛𝑏+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,𝑛≥0, (1.10) where the sequence {𝛼_𝑛} satisfies certain conditions, he proved the sequence {𝑥_𝑛} converges strongly to the unique solution of the following minimization problem: min_𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩. (1.11) In [5], Marino and Xu considered the following general iterative method: 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,𝑛≥0, (1.12) they proved that if the sequence {𝛼_𝑛} of parameters satisfies appropriate conditions, then the sequence {𝑥_𝑛} generated by (1.12) converges strongly to the unique solution of the variational inequality 𝛾𝑓−𝐴𝑥^∗,𝑥−𝑥^∗≤0,𝑥∈𝐶, (1.13) which is the optimality condition for the minimization problem min_𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), (1.14) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ′(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻). Some people also study the applications of the iterative method (1.12) [6, 7].

Acedo and Xu [8] consider the following parallel and cyclic algorithms:

Parallel Algorithm
The sequence {𝑥_𝑛} was generated by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑇_𝑖𝑥_𝑛, (1.15) where {𝑇_𝑖}^𝑁_𝑖=1 are 𝑁 strict pseudocontractions defined on a closed convex subset 𝐶 of a Hilbert space 𝐻. Under the following assumptions on the sequences of the weights {𝜆_𝑖^(𝑛)}^𝑁_𝑖=1: (a1)∑^𝑁_𝑖=1𝜆_𝑖^(𝑛)=1 for all 𝑛 and inf_𝑛≥1𝜆^(𝑛)>0, for all 1≤𝑖≤𝑁, (a2)∑^𝑁_𝑖=1∑^𝑁_𝑖=1|𝜆_𝑖^(𝑛+1)−𝜆_𝑖^(𝑛)|<∞.

By (1.15), they will prove the weak convergence to a solution of the problem 𝑥∈⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝑖).

Cyclic Algorithm
They define the sequence {𝑥_𝑛} cyclically by 𝑥₁=𝛼₀𝑥₀+1−𝛼₀𝑇₀𝑥₀;𝑥₂=𝛼₁𝑥₁+1−𝛼₁𝑇₁𝑥₁;⋮𝑥_𝑁=𝛼_𝑁−1𝑥_𝑁−1+1−𝛼_𝑁−1𝑇_𝑁−1𝑥_𝑁−1;𝑥_𝑁+1=𝛼_𝑁𝑥_𝑁+1−𝛼_𝑁𝑇₀𝑥_𝑁;⋮ (1.16) In a more compact form, they are rewritten 𝑥_𝑛+1 as 𝑥_𝑁+1=𝛼_𝑁𝑥_𝑁+1−𝛼_𝑁𝑇_𝑁𝑥_𝑛, (1.17) where {𝑇_𝑖}^𝑁_𝑖=1 are 𝑘_𝑖-strict pseudo-contractions and 𝑇_𝑁=𝑇_𝑖 with 𝑖=𝑛 (mod 𝑁), 0≤𝑖≤𝑁−1. They show that this cyclic algorithm (1.17) is weakly convergent if the sequence {𝛼_𝑛} of parameters is appropriately chosen. On the other hand, Osilike and Shehu [9] also consider the cyclic algorithm (1.17), under appropriate assumptions on the sequences of {𝛼_𝑛}, some strong convergence theorems are proved.

In this paper, we are concerned with the problem of finding a point 𝑥 such that 𝑥∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖,𝑁≥1, (1.18) where 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, {𝜔_𝑖}^𝑁_𝑖=1∈(0,1] and {𝑇_𝑖}^𝑁_𝑖=1 are quasi-nonexpansive mappings defined on a closed convex subset 𝐶 of a Hilbert space 𝐻. Here 𝐹_𝑖𝑥(𝑇_𝜔𝑖)={𝑞∈𝐶∶𝑇_𝜔𝑖𝑞=𝑞} is the set of fixed points of 𝑇_𝑖, 1≤𝑖≤𝑁.

Let 𝑇 be defined by 𝑇=^𝑁_𝑖=1𝜆_𝑖𝑇_𝜔𝑖, (1.19) where 𝜆_𝑖>0 for all 𝑖∈(0,1) such that ∑^𝑁_𝑖=1𝜆_𝑖=1. Motivated and inspired by Acedo and Xu [8], we consider the following two general iterative algorithms for a family of quasi-nonexpansive mappings.

Algorithm 1.2. 𝑇=^𝑁_𝑖=1𝜆_𝑖𝑇_𝜔𝑖,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛. (1.20)

Algorithm 1.3. 𝑇=^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑇_𝜔𝑖,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛. (1.21) In (1.20), the weights {𝜆_𝑖}^𝑁_𝑖=1 are constant in the sense that they are independent of 𝑛, the number of steps of the iteration process. In (1.21), we consider a more general case by allowing the weights {𝜆_𝑖^(𝑛)}^𝑁_𝑖=1. Under appropriate assumptions on the sequences of the wights {𝜆_𝑖^(𝑛)}^𝑁_𝑖=1,{𝜆_𝑖}^𝑁_𝑖=1,{𝛼_𝑛} and {𝛽_𝑛}. From (1.20) and (1.21), we will prove some strong convergence to a solution of the problem (1.18). In addition, we can also know that the condition ∑^𝑁_𝑖=1∑^𝑁_𝑖=1|𝜆_𝑖^(𝑛+1)−𝜆_𝑖^(𝑛)|<∞ in [8] is superfluous.

Another approach to the problem (1.18) is the cyclic algorithm (for convenience, we relabel the mappings {𝑇_𝜔𝑖}^𝑁_𝑖=1 as {𝑇_𝜔𝑖}^𝑁−1_𝑖=0). This means that beginning with an 𝑥₀∈𝐶, we define the sequence {𝑥_𝑛} cyclically by 𝑥₁=𝛼₀𝛾𝑓𝑥₀+𝛽₀𝑥₀+𝐼−𝛽₀𝐼−𝛼₀𝐴𝑇_𝜔0𝑥₀,𝑥₂=𝛼₁𝛾𝑓𝑥₁+𝛽₁𝑥₁+𝐼−𝛽₁𝐼−𝛼₁𝐴𝑇_𝜔1𝑥₁,⋮𝑥_𝑁=𝛼_𝑁−1𝛾𝑓𝑥_𝑁−1+𝛽_𝑁−1𝑥_𝑁−1+𝐼−𝛽_𝑁−1𝐼−𝛼_𝑁−1𝐴𝑇_𝜔𝑁−1𝑥_𝑁−1,𝑥_𝑁+1=𝛼_𝑁𝛾𝑓𝑥_𝑁+𝛽_𝑁𝑥_𝑁+𝐼−𝛽_𝑁𝐼−𝛼_𝑁𝐴𝑇_𝜔𝑁𝑥_𝑁,⋮ (1.22)(1.23) In a more compact from, 𝑥_𝑛+1 can be written as 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇_[𝑛]𝑥_𝑛, (1.24) where 𝑇_[𝑛]=𝑇_𝜔𝑖, 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, {𝜔_𝑖}^𝑁_𝑖=1∈(0,1], with 𝑖=𝑛 (mod 𝑁), 0≤𝑖≤𝑁−1.

We will show that this cyclic algorithm (1.24) is also strongly convergent if the sequence {𝛼_𝑛} of parameters is appropriately chosen.

2. Preliminaries

Throughout this paper, we write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥. 𝑥_𝑛→𝑥 implies that {𝑥_𝑛} converges strongly to 𝑥. The following definitions and lemmas are useful for main results.

Definition 2.1. An operator 𝑇∶𝐻→𝐻 is said to be quasi-nonexpansive if 𝐹_𝑖𝑥(𝑇)≠0andif‖𝑇𝑥−𝑧‖≤‖𝑥−𝑧‖,∀𝑧∈𝐹_𝑖𝑥(𝑇),∀𝑥∈𝐻. (2.1)
Iterative methods for quasi-nonexpansive mappings have been extensively investigated; see [10, 11].

Remark 2.2. From the above definitions, It is easy to see that(i)a nonexpansive mapping is a quasi-nonexpansive mapping; (ii)the set of fixed points of 𝑇 is the set 𝐹_𝑖𝑥(𝑇)={𝑥∈𝐻∶𝑇𝑥=𝑥}. We assume that 𝐹_𝑖𝑥(𝑇)≠∅, it is well know that 𝐹_𝑖𝑥(𝑇) is closed and convex.

Remark 2.3 (see [10]). Let 𝑇_𝛼=(1−𝛼)𝐼+𝛼𝑇, where 𝑇 is a quasi-nonexpansive on 𝐻, 𝐹_𝑖𝑥(𝑇)≠∅ and 𝛼∈(0,1]. Then the following statements are reached: (i)𝐹_𝑖𝑥(𝑇)=𝐹_𝑖𝑥(𝑇_𝛼); (ii)𝑇_𝛼 is quasi-nonexpansive; (iii)‖𝑇_𝛼𝑥−𝑞‖²≤‖𝑥−𝑞‖²−𝛼(1−𝛼)‖𝑇𝑥−𝑥‖², for all (𝑥,𝑞)∈𝐻×𝐹_𝑖𝑥(𝑇); (iv)⟨𝑥−𝑇_𝛼𝑥,𝑥−𝑞⟩≥(𝛼/2)‖𝑇𝑥−𝑥‖², for all (𝑥,𝑞)∈𝐻×𝐹_𝑖𝑥(𝑇).

Example 2.4. Let 𝑋=𝑙² with the norm ‖⋅‖ defined by ‖𝑋‖=^∞_𝑖=1𝑥²_𝑖,∀𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛,…∈𝑋, (2.2) and 𝐶={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛,…)∣𝑥₁≤0,𝑥_𝑖∈𝑅¹,𝑖=2,3,…}. Then 𝐶 is a nonempty subset of 𝑋.
Now, for any 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛,…)∈𝐶, define a mapping 𝑇∶𝐶→𝐶 as follows: 𝑇(𝑥)=0,4𝑥₁,0,…,0,…. (2.3) It is easy to see that 𝑇 is a quasi-nonexpansive mapping. In fact, for any 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛,…)∈𝑋, taking 𝑇(𝑥)=𝑥, that is, 0,4𝑥₁,0,…,0,…=𝑥₁,𝑥₂,…,𝑥_𝑛,…, (2.4) we have 𝐹(𝑇)={0} and ‖𝑇(𝑥)−0‖=0,4𝑥₁,0,…,0,…−(0,0,0,…,0,…)=4𝑥₁≤4^∞_𝑖=1𝑥²_𝑖=𝑥₁,𝑥₂,…,𝑥_𝑛−(=‖‖.,…0,0,0,…,0,…)𝑥−0 (2.5)

Lemma 2.5. Assume 𝐶 is a closed convex subset of a Hilbert space 𝐻. (i)Given an integer 𝑁≥1, assume, for all 1≤𝑖≤𝑁, 𝑇_𝑖∶𝐶→𝐶 is a quasi-nonexpansive. Let {𝜆_𝑖}^𝑁_𝑖=1 be a positive sequence such that ∑^𝑁_𝑖=1𝜆_𝑖=1. Then ∑^𝑁_𝑖=1𝜆_𝑖𝑇_𝑖 is a quasi-nonexpansive.(ii)Let {𝑇_𝑖}^𝑁_𝑖=1 and ∑^𝑁_𝑖=1𝜆_𝑖=1 be given as in (i) above. Suppose that {𝑇_𝑖}^𝑁_𝑖=1 has a common fixed point. Then 𝐹_𝑖𝑥^𝑁_𝑖=1𝜆_𝑖𝑇_𝑖=^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝑖. (2.6)(iii)Assume 𝑇_𝑖∶𝐶→𝐶 be quasi-nonexpansives, let 𝑇_𝛼𝑖=(1−𝛼_𝑖)𝐼+𝛼_𝑖𝑇_𝑖, 1≤𝑖≤𝑁. If ⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝑖)≠∅, then 𝐹_𝑖𝑥𝑇_𝛼1𝑇_𝛼2⋯𝑇_𝛼𝑁=^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝛼𝑖. (2.7)

Proof. To prove (i) we only need to consider the case of 𝑁=2 (the general case can be proved by induction). Set 𝑇=(1−𝜆)𝑇₁+𝜆𝑇₂, where 𝜆∈(0,1) and for 𝑖=1,2, 𝑇_𝑖 is a quasi-nonexpansive. We verify directly the following inequality: for all 𝑧∈𝐹_𝑖𝑥(𝑇₁)∩𝐹_𝑖𝑥(𝑇₂), ‖𝑇𝑥−𝑧‖=(1−𝜆)𝑇₁+𝜆𝑇₂≤(𝑇𝑥−𝑧1−𝜆)₁𝑥−𝑧+𝜆𝑇₂≤(‖‖‖≤‖‖,𝑥−𝑧1−𝜆)‖𝑥−𝑧+𝜆𝑥−𝑧𝑥−𝑧 (2.8) that is, 𝑇 is a quasi-nonexpansive.
To prove (ii) again we can assume 𝑁=2. It suffices to prove that 𝐹_𝑖𝑥(𝑇)⊂𝐹_𝑖𝑥(𝑇₁)∩𝐹_𝑖𝑥(𝑇₂), where 𝑇=(1−𝜆)𝑇₁+𝜆𝑇₂ with 𝜆∈(0,1). Let 𝑥∈𝐹_𝑖𝑥(𝑇).
Taking 𝑧∈𝐹_𝑖𝑥(𝑇₁)∩𝐹_𝑖𝑥(𝑇₂) to deduce that ‖𝑧−𝑥‖=(1−𝜆)𝑧−𝑇₁𝑥+𝜆𝑧−𝑇₂𝑥≤(1−𝜆)𝑧−𝑇₁𝑥+𝜆𝑧−𝑇₂𝑥≤(‖‖‖≤‖‖.1−𝜆)‖𝑧−𝑥+𝜆𝑧−𝑥𝑧−𝑥 (2.9) By the strict convexity of 𝐻, it follows that 𝑇₁(𝑥)−𝑧=𝑇₂(𝑥)−𝑧=𝑥−𝑧; that is, 𝑇₁(𝑥)=𝑇₂(𝑥)=𝑥, hence 𝑥∈𝐹_𝑖𝑥(𝑇₁)∩𝐹_𝑖𝑥(𝑇₂). According to induction, we can easily claim that (2.6) is holds.
To prove (iii) by induction, for 𝑁=2, set 𝑇_𝛼𝑖=(1−𝛼_𝑖)𝐼+𝛼_𝑖𝑇_𝑖 for all 𝑖=1,2. Obviously 𝐹_𝑖𝑥𝑇_𝛼1𝐹_𝑖𝑥𝑇_𝛼2⊂𝐹_𝑖𝑥𝑇_𝛼1𝑇_𝛼2. (2.10) Now we prove 𝐹_𝑖𝑥𝑇_𝛼1𝑇_𝛼2⊂𝐹_𝑖𝑥𝑇_𝛼1𝐹_𝑖𝑥𝑇_𝛼2. (2.11) For all 𝑞∈𝐹_𝑖𝑥(𝑇_𝛼1𝑇_𝛼2), 𝑇_𝛼1𝑇_𝛼2𝑞=𝑞, if 𝑇_𝛼2𝑞=𝑞, then 𝑇_𝛼1𝑞=𝑞, the conclusion holds. In fact, we can claim that 𝑇_𝛼2𝑞=𝑞. From Remark 2.3, we know that 𝑇_𝛼2 is quasi-nonexpansive and 𝐹_𝑖𝑥(𝑇_𝛼1)⋂𝐹_𝑖𝑥(𝑇_𝛼2)=𝐹_𝑖𝑥(𝑇₁)⋂𝐹_𝑖𝑥(𝑇₂)≠∅. Take 𝑝∈𝐹_𝑖𝑥(𝑇_𝛼1)⋂𝐹_𝑖𝑥(𝑇_𝛼2), then 𝑝−𝑞²=𝑝−𝑇_𝛼1𝑇_𝛼2𝑞²=𝑝−1−𝛼₁𝑇_𝛼2𝑞+𝛼₁𝑇₁𝑇_𝛼2𝑞²=1−𝛼₁𝑝−𝑇_𝛼2𝑞+𝛼₁𝑝−𝑇₁𝑇_𝛼2𝑞²=1−𝛼₁𝑝−𝑇_𝛼2𝑞²+𝛼₁𝑝−𝑇₁𝑇_𝛼2𝑞²−𝛼₁1−𝛼₁𝑇_𝛼2𝑞−𝑇₁𝑇_𝛼2𝑞²≤1−𝛼₁𝑝−𝑇_𝛼2𝑞²+𝛼₁𝑝−𝑇_𝛼2𝑞²−𝛼₁1−𝛼₁𝑇_𝛼2𝑞−𝑇₁𝑇_𝛼2𝑞²=𝑝−𝑇_𝛼2𝑞²−𝛼₁1−𝛼₁𝑇_𝛼2𝑞−𝑇₁𝑇_𝛼2𝑞². (2.12) From (2.12), we have 𝑇_𝛼2𝑞−𝑇₁𝑇_𝛼2𝑞²≤0, (2.13) namely, 𝑇_𝛼2𝑞=𝑇₁𝑇_𝛼2𝑞, that is, 𝑇_𝛼2𝑞∈𝐹_𝑖𝑥𝑇₁=𝐹_𝑖𝑥𝑇_𝛼1,𝑇_𝛼2𝑞=𝑇_𝛼1𝑇_𝛼2𝑞=𝑞. (2.14)
Suppose that the conclusion holds for 𝑁=𝑘, we prove that 𝐹_𝑖𝑥𝑇_𝛼1𝑇_𝛼2⋯𝑇_𝛼𝑘+1=^𝑘+1_𝑖=1𝐹_𝑖𝑥𝑇_𝛼𝑖. (2.15) It suffices to verify 𝐹_𝑖𝑥𝑇_𝛼1𝑇_𝛼2⋯𝑇_𝛼𝑘+1⊂^𝑘+1_𝑖=1𝐹_𝑖𝑥𝑇_𝛼𝑖, (2.16) for all 𝑞∈𝐹_𝑖𝑥(𝑇_𝛼1𝑇_𝛼2⋯𝑇_𝛼𝑘+1), that is, 𝑇_𝛼1𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞=𝑞. Using Remark 2.3 again, take 𝑝∈⋂^𝑘+1_𝑖=1𝐹_𝑖𝑥(𝑇_𝛼𝑖), we obtain 𝑝−𝑞²=𝑝−𝑇_𝛼1𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞²=𝑝−1−𝛼₁𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞−𝛼₁𝑇₁𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞²=1−𝛼₁𝑝−𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞−𝛼₁𝑝−𝑇₁𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞²=1−𝛼₁𝑝−𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞²+𝛼₁𝑝−𝑇₁𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞²−𝛼₁1−𝛼₁𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞−𝑇₁𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞²≤𝑝−𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞²−𝛼₁1−𝛼₁𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞−𝑇₁𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞². (2.17) From (2.17), we obtain 𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞−𝑇₁𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞²≤0, (2.18) this implies that 𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞∈𝐹_𝑖𝑥𝑇₁=𝐹_𝑖𝑥𝑇_𝛼1, (2.19) namely, 𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞=𝑇_𝛼1𝑇_𝛼2⋯𝑇_𝛼𝑘+1𝑞=𝑞. (2.20) From (2.20) and inductive assumption, we have 𝑞∈𝐹_𝑖𝑥𝑇_𝛼1𝑇_𝛼2⋯𝑇_𝛼𝑘+1⊂^𝑘+1_𝑖=2𝐹_𝑖𝑥𝑇_𝛼𝑖, (2.21) therefore 𝑇_𝛼𝑖𝑞=𝑞,𝑖=2,3,…,𝑘+1. (2.22) Substituting it into (2.20), we obtain 𝑇_𝛼1𝑞=𝑞. Thus we assert that 𝑞∈^𝑘+1_𝑖=1𝐹_𝑖𝑥𝑇_𝛼𝑖. (2.23)

Definition 2.6. A mapping 𝑇 is said to be demiclosed, if for any sequence {𝑥_𝑛} weakly converges to 𝑦, and if the sequence {𝑇𝑥_𝑛} strongly converges to 𝑧, then 𝑇(𝑦)=𝑧.

Lemma 2.7 (see [5]). Assume 𝐴 is a strong positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹, then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Lemma 2.8 (see [12]). Let 𝐻 be a Hilbert space, 𝐾 a closed convex subset of 𝐻, and 𝑇∶𝐾→𝐾 a nonexpansive mapping with 𝐹_𝑖𝑥(𝑇)≠∅, if {𝑥_𝑛} is a sequence in 𝐾 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦.

Lemma 2.9 (see [13]). Let {𝒯_𝑛} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {𝒯_𝑛𝑗}_𝑗≥0 of {𝒯_𝑛} which satisfies 𝒯_𝑛𝑗<𝒯_𝑛𝑗+1 for all 𝑗≥0. Also consider the sequence of integers {𝛿(𝑛)}_𝑛≥𝑛0 defined by 𝛿(𝑛)=max𝑘≤𝑛∣𝒯_𝑘<𝒯_𝑘+1. (2.24) Then {𝛿(𝑛)}_𝑛≥𝑛0 is a nondecreasing sequence verifying lim_𝑛→∞𝛿(𝑛)=∞, for all 𝑛≥𝑛₀, it holds that 𝒯_𝛿(𝑛)<𝒯_𝛿(𝑛)+1 and one has 𝒯_𝑛<𝒯_𝛿(𝑛)+1. (2.25)

Lemma 2.10. Let 𝐾 be a closed convex subset of a real Hilbert space 𝐻, given 𝑥∈𝐻 and 𝑦∈𝐾. Then 𝑦=𝑃_𝐾𝑥 if and only if there holds the inequality 𝑥−𝑦,𝑦−𝑧≥0,∀𝑧∈𝐾. (2.26)

3. Parallel Algorithm

In this section, we discuss the parallel algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite quasi-nonexpansives.

Before stating our main convergence result, we establish the boundedness of the iterates given by following algorithm: 𝑇=^𝑁_𝑖=1𝜆_𝑖𝑇_𝜔𝑖,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛. (3.1)

In (3.1), the weight {𝜆_𝑖}^𝑁_𝑖=1 are constant in the sense that they are independent of 𝑛, the number of steps of the iteration process. Below we consider a more general case by allowing the weights {𝜆_𝑖}^𝑁_𝑖=1 to be step dependent. That is, initializing with 𝑥₀, we define {𝑥_𝑛} by the algorithm 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑇_𝜔𝑖𝑥_𝑛. (3.2) From (3.1) and (3.2), the sequence {𝑥_𝑛} which converges strongly to the unique solution of variational inequality problem VI(𝛾𝑓−𝐴,⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖)): find 𝑥^∗ in ⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖) such that ∀𝑣∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖,𝛾𝑓−𝐴𝑥^∗,𝑣−𝑥^∗≤0, (3.3) or equivalently 𝑥^∗=𝑃_⋂^𝑁_{𝑖=1𝐹𝑖𝑥(𝑇𝜔𝑖)}⋅𝐶𝑥^∗, (3.4) where 𝑃_⋂^𝑁_{𝑖=1𝐹𝑖𝑥(𝑇𝜔𝑖)} denotes the metric projection from 𝐻 onto ⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖) (see, [14] for more details on the metric projection).

Lemma 3.1. The sequence {𝑥_𝑛} is generated by (3.2), where {𝛼_𝑛} and {𝛽_𝑛} are sequence in [0,1], and {𝑇_𝑖}^𝑁_𝑖=1 is a quasi-nonexpansive mapping on 𝐻, is bounded and satisfies 𝑥_𝑛−𝑣≤max𝑥₁,(𝑣)−𝑣𝛾𝑓−𝐴𝑣𝛾−𝛾𝛽,∀𝑛≥1, (3.5) where 𝑣 is any element in 𝐹_𝑖𝑥(𝑇_𝑖), 1≤𝑖≤𝑁.

Proof. Since lim_𝑛→∞𝛼_𝑛=0, we shall assume that 𝛼_𝑛≤(1−𝛽_𝑛)‖𝐴‖⁻¹ and 1−𝛼_𝑛(𝛾−𝛾𝛽)>0. Observe that if ‖𝑢‖=1, then 𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑢,𝑢=1−𝛽_𝑛−𝛼_𝑛⟨𝐴𝑢,𝑢⟩≥1−𝛽_𝑛−𝛼_𝑛‖𝐴‖≥0. (3.6) By Lemma 2.7, we obtain 𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴≤1−𝛽_𝑛−𝛼_𝑛𝛾. (3.7) Let 𝐵_𝑛=∑^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑇_𝜔𝑖, for all 𝑛≥1. By Lemma 2.5, each 𝐵_𝑛 is a quasi-nonexpansive mapping on 𝐻, and in light of Remark 2.3. Taking 𝑣∈𝐹_𝑖𝑥(𝑇), we have ^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑇_𝜔𝑖𝑥_𝑛−𝑣≤^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑇_𝜔𝑖𝑥_𝑛−𝑣≤^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑥_𝑛−𝑣≤𝑥_𝑛−𝑣. (3.8) From (3.1), we have 𝑥_𝑛+1−𝑣=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑣+𝛽_𝑛𝑥_𝑛−𝑣+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑣≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑣+𝛽_𝑛𝑥_𝑛−𝑣+1−𝛽_𝑛−𝛼_𝑛𝛾𝑥_𝑛−𝑣≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓(𝑣)+𝛼_𝑛𝛾𝑓(𝑣)−𝐴𝑣+1−𝛼_𝑛𝛾𝑥_𝑛=−𝑣1−𝛼_𝑛𝛾−𝛾𝛽𝑥_𝑛−𝑣+𝛼_𝑛𝛾𝑓(𝑣)−𝐴𝑣. (3.9) By simple inductions, we obtain 𝑥_𝑛−𝑣≤max𝑥₁,(𝑣)−𝑣𝛾𝑓−𝐴𝑣𝛾−𝛾𝛽,∀𝑛≥1, (3.10) which gives that the sequence {𝑥_𝑛} is bounded.

Lemma 3.2. Assume that {𝑥_𝑛} is defined by (3.2), if 𝑥^∗ is solution of (3.3) with 𝑇∶𝐶→𝐶 demiclosed and {𝑦_𝑛}⊂𝐻 is a bounded sequence such that ‖𝑇𝑦_𝑛−𝑦_𝑛‖→0, then liminf_𝑛→∞𝐴−𝛾𝑓𝑥^∗,𝑦_𝑛−𝑥^∗≥0. (3.11)

Proof. Clearly, by ‖𝑇𝑦_𝑛−𝑦_𝑛‖→0 and 𝑇∶𝐻→𝐻 demi-closed, we know that any weak cluster point of {𝑦_𝑛} belongs to 𝐹_𝑖𝑥(𝑇). It is also a simple matter to see that there exist 𝑦 and a subsequence {𝑦_𝑛𝑗} of {𝑦_𝑛} such that lim_𝑗→∞𝑦_𝑛𝑗⇀𝑦 (hence 𝑦∈𝐹_𝑖𝑥(𝑇)) and such that liminf_𝑛→∞𝐴−𝛾𝑓𝑥^∗,𝑦_𝑛−𝑥^∗=lim_𝑗→∞𝐴−𝛾𝑓𝑥^∗,𝑦_𝑛𝑗−𝑥^∗, (3.12) it follows from (3.3), we can derive that liminf_𝑛→∞𝐴−𝛾𝑓𝑥^∗,𝑦_𝑛−𝑥^∗=𝐴−𝛾𝑓𝑥^∗,𝑦−𝑥^∗≥0, (3.13) that is the desired result.

Theorem 3.3. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 and let 𝑇_𝑖∶𝐶→𝐶 be a quasi-nonexpansive for 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, 𝜔_𝑖∈(0,1), 𝑖∈(1,…,𝑁) such that ⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖)≠∅, 𝑓 be a contraction with coefficient 𝛽∈(0,1), and 𝜆_𝑖 a positive constant such that ∑^𝑁_𝑖=1𝜆_𝑖^(𝑛)=1 for all 𝑛 and inf_𝑛≥1𝜆_𝑖^(𝑛)>0 for all 𝑖∈[1,𝑁]. Let 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾. Given the initial guess 𝑥₀∈𝐻 chosen arbitrarily and given sequences {𝛼_𝑛} and {𝛽_𝑛} in (0,1), satisfying the following conditions: (c1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞, (c2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Let {𝑥_𝑛} be the sequence generated by (3.2). Then {𝑥_𝑛} converges strongly to the unique a element 𝑥^∗∈⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖),𝑁≥1 verifying 𝑥^∗=𝑃_⋂^𝑁_{𝑖=1𝐹𝑖𝑥(𝑇𝜔𝑖)}⋅𝑓𝑥^∗ (3.14) which equivalently solves the following variational inequality problem: 𝑥^∗∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖,𝛾𝑓−𝐴𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖. (3.15)

Proof. Taking 𝐵_𝑛=∑^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑇_𝜔𝑖, for all 𝑛≥1. By Lemma 2.5(i), each 𝐵_𝑛 is a quasi-nonexpansive mapping on 𝐶, and (3.2) can be rewritten as 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝐵_𝑛𝑥_𝑛. (3.16) Denote by Ω the common fixed point of the mappings {𝑇_𝜔𝑖}^𝑁_𝑖=1 (by Lemma 2.5(ii), we can easily know that Ω=⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖)=⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝑖)) and take 𝑥^∗∈Ω and from (3.16) we deduce that 𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝐴𝑥_𝑛−𝛾𝑓𝑥_𝑛=𝐼−𝛽_𝑛−𝛼_𝑛𝐴𝐵_𝑛𝑥_𝑛−𝑥_𝑛, (3.17) and hence 𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝐴𝑥_𝑛−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗=1−𝛽_𝑛−𝛼_𝑛𝐴𝐵_𝑛𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑥^∗=1−𝛽_𝑛−𝛼_𝑛𝐵_𝑛𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑥^∗+𝛼_𝑛(𝐼−𝐴)𝐵_𝑛−𝐼𝑥_𝑛,𝑥_𝑛−𝑥^∗. (3.18) Moreover, by 𝑥^∗∈⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖) and using Remark 2.3(iv), we obtain 𝑥_𝑛−𝐵_𝑛𝑥_𝑛,𝑥_𝑛−𝑥^∗≥𝑥_𝑛−^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑇_𝜔𝑖𝑥_𝑛,𝑥_𝑛−𝑥^∗≥^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑥_𝑛−𝑇_𝜔𝑖𝑥_𝑛,𝑥_𝑛−𝑥^∗≥^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝜔_𝑖2𝑥_𝑛−𝑇_𝑖𝑥_𝑛², (3.19) which combined with the (3.18) entails 𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝐴−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗≤−1−𝛽_𝑛−𝛼_𝑛2^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝜔_𝑖𝑥_𝑛−𝑇_𝑖𝑥_𝑛²+𝛼_𝑛(𝐼−𝐴)𝐵_𝑛−𝐼𝑥_𝑛,𝑥_𝑛−𝑥^∗, (3.20) or equivalently −𝑥_𝑛−𝑥_𝑛+1,𝑥_𝑛−𝑥^∗≤−𝛼_𝑛𝐴−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗−1−𝛽_𝑛−𝛼_𝑛2^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝜔_𝑖𝑥_𝑛−𝑇_𝑖𝑥_𝑛²+𝛼_𝑛(𝐼−𝐴)𝐵_𝑛−𝐼𝑥_𝑛,𝑥_𝑛−𝑥^∗. (3.21) Furthermore, using the following classical equality: ⟨⟩=1𝑢,𝑣2‖𝑢‖²−12‖𝑢−𝑣‖²+12‖𝑣‖²,∀𝑢,𝑣∈𝐶, (3.22) and setting 𝒯_𝑛=(1/2)‖𝑥_𝑛−𝑥^∗‖², we have 𝑥_𝑛−𝑥_𝑛+1,𝑥_𝑛−𝑥^∗=𝒯_𝑛−𝒯_𝑛+1+12𝑥_𝑛−𝑥_𝑛+1². (3.23) So that (3.21) can be equivalently rewritten as 𝒯_𝑛+1−𝒯_𝑛−12𝑥_𝑛−𝑥_𝑛+1²≤−𝛼_𝑛𝐴−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗−1−𝛽_𝑛−𝛼_𝑛2^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝜔_𝑖𝑥_𝑛−𝑇_𝑖𝑥_𝑛²+𝛼_𝑛(𝐼−𝐴)𝐵_𝑛−𝐼𝑥_𝑛,𝑥_𝑛−𝑥^∗. (3.24) Now using (3.16) again, we have 𝑥_𝑛+1−𝑥_𝑛²=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛+𝐼−𝛽_𝑛−𝛼_𝑛𝐴𝐵_𝑛𝑥_𝑛−𝑥_𝑛². (3.25) Since 𝐴∶𝐻→𝐻 is a strongly positive bounded linear operator with coefficient 𝛾>0, hence it is a classical matter to see that 𝑥_𝑛+1−𝑥_𝑛²≤2𝛼²_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛²+21−𝛽_𝑛−𝛼_𝑛𝛾²𝐵_𝑛𝑥_𝑛−𝑥_𝑛². (3.26) from 𝐵_𝑛𝑥_𝑛−𝑥_𝑛²=^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑇_𝜔𝑖𝑥_𝑛−𝑥_𝑛²=^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑇_𝜔𝑖𝑥_𝑛−𝑥_𝑛²≤2^𝑁_𝑖=1𝜆_𝑖^(𝑛)2𝜔²_𝑖𝑥_𝑛−𝑇_𝑖𝑥_𝑛²≤2^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝜔_𝑖𝑥_𝑛−𝑇_𝑖𝑥_𝑛² (3.27) and (1−𝛽_𝑛−𝛼_𝑛𝛾)²≤(1−𝛽_𝑛−𝛼_𝑛𝛾) yields 12𝑥_𝑛+1−𝑥_𝑛²≤𝛼²_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛²+1−𝛽_𝑛−𝛼_𝑛𝛾^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝜔_𝑖𝑥_𝑛−𝑇_𝑖𝑥_𝑛². (3.28) Then from (3.24) and (3.28), we have 𝒯_𝑛+1−𝒯_𝑛+1−𝛽_𝑛−𝛼_𝑛2−1−𝛽_𝑛−𝛼_𝑛𝛾^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝜔_𝑖𝑥_𝑛−𝑇_𝑖𝑥_𝑛²≤𝛼_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛²−𝐴−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗+(𝐼−𝐴)𝐵_𝑛−𝐼𝑥_𝑛,𝑥_𝑛−𝑥^∗. (3.29) The rest of the proof will be divided into two parts.
Case 1. Suppose that there exists 𝑛₀ such that {𝒯_𝑛}_𝑛≥𝑛0 is nonincreasing. In this situation, {𝒯_𝑛} is then convergent because it is also nonnegative (hence it is bounded from below), so that lim_𝑛→∞(𝒯_𝑛+1−𝒯_𝑛)=0; hence, in light of (3.29) together with lim_𝑛→∞𝛼_𝑛=0, 0<liminf_𝑛→∞𝛽_𝑛≤liminf_𝑛→∞𝛽_𝑛<1, and the boundedness of {𝑥_𝑛}, we obtain lim^𝑁_{𝑛→∞𝑖=1}𝜆_𝑖^(𝑛)𝜔_𝑖𝑥_𝑛−𝑇_𝑖𝑥_𝑛²=0. (3.30) By (3.27) and (3.30), we can easily claim that lim_𝑛→∞𝐵_𝑛𝑥_𝑛−𝑥_𝑛=0. (3.31) It also follows from (3.29) that 𝒯_𝑛−𝒯_𝑛+1≥𝛼_𝑛−𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛²+𝐴−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗+(𝐼−𝐴)𝐵_𝑛−𝐼𝑥_𝑛,𝑥_𝑛−𝑥^∗. (3.32) Then, by ∑^∞_𝑛=0𝛼_𝑛=∞, we obviously deduce that liminf_𝑛→∞−𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛²+𝐴−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗+(𝐼−𝐴)𝐵_𝑛−𝐼𝑥_𝑛,𝑥_𝑛−𝑥^∗≤0. (3.33) Since {𝑓(𝑥_𝑛)} and {𝑥_𝑛} are both bounded, lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞‖𝐵_𝑛𝑥_𝑛−𝑥_𝑛‖=0, we obtain liminf_𝑛→∞𝐴−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗≤0. (3.34) Moreover, by Remark 1.1, we have 2𝛾−𝛾𝛽𝒯_𝑛+𝐴−𝛾𝑓𝑥^∗,𝑥_𝑛−𝑥^∗≤𝐴−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗, (3.35) which by (3.34) entails liminf_𝑛→∞2𝛾−𝛾𝛽𝒯_𝑛+𝐴−𝛾𝑓𝑥^∗,𝑥_𝑛−𝑥^∗≤0, (3.36) hence, recalling that lim_𝑛→∞𝒯_𝑛 exists, we equivalently obtain 2𝛾−𝛾𝛽lim_𝑛→∞𝒯_𝑛+liminf_𝑛→∞𝐴−𝛾𝑓𝑥^∗,𝑥_𝑛−𝑥^∗≤0, (3.37) namely, 2𝛾−𝛾𝛽lim_𝑛→∞𝒯_𝑛≤−liminf_𝑛→∞𝐴−𝛾𝑓𝑥^∗,𝑥_𝑛−𝑥^∗. (3.38) From (3.30) and invoking Lemma 3.2, we obtain liminf_𝑛→∞𝐴−𝛾𝑓𝑥^∗,𝑥−𝑥^∗≥0,𝑥∈^𝑁_𝑛=1𝐹_𝑖𝑥𝑇_𝜔𝑖, (3.39) which by (3.38) yields lim_𝑛→∞𝒯_𝑛=0, so that {𝑥_𝑛} converges strongly to 𝑥^∗.
Case 2. Suppose there exists a subsequence {𝒯_𝑛𝑘}_𝑘≥0 of {𝒯_𝑛}_𝑛≥0 such that 𝒯_𝑛𝑘≤𝒯_𝑛𝑘+1 for all 𝑘≥0. In this situation, we consider the sequence of indices {𝛿(𝑛)} as defined in Lemma 2.9. It follows that 𝒯_𝛿(𝑛+1)−𝒯_𝛿(𝑛)>0, which by (3.29) amounts to 1−𝛽_𝛿(𝑛)−𝛼_𝛿(𝑛)2−1−𝛽_𝛿(𝑛)−𝛼_𝛿(𝑛)𝛾^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝜔_𝑖𝑥_𝛿(𝑛)−𝑇_𝑖𝑥_𝛿(𝑛)²≤𝛼_𝛿(𝑛)𝛼_𝛿(𝑛)𝛾𝑓𝑥_𝛿(𝑛)−𝐴𝑥_𝛿(𝑛)²−𝐴−𝛾𝑓𝑥_𝛿(𝑛),𝑥_𝑛−𝑥^∗, (3.40) hence, by the boundedness of {𝑥_𝑛} and lim_𝑛→∞𝛼_𝑛=0, we immediately obtain lim^𝑁_{𝑛→∞𝑖=1}𝜆_𝑖^(𝑛)𝜔_𝑖𝑥_𝛿(𝑛)−𝑇_𝑖𝑥_𝛿(𝑛)²=0. (3.41) From (3.28) we have 12𝑥_𝛿(𝑛)+1−𝑥_𝛿(𝑛)²≤𝛼²_𝛿(𝑛)𝛾𝑓𝑥_𝛿(𝑛)−𝐴𝑥_𝛿(𝑛)²+1−𝛽_𝛿(𝑛)−𝛼_𝛿(𝑛)𝛾^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝜔_𝑖𝑥_𝛿(𝑛)−𝑇_𝑖𝑥_𝛿(𝑛)²≤𝛼_𝛿(𝑛)𝛾𝑓𝑥_𝛿(𝑛)−𝐴𝑥_𝛿(𝑛)²+1−𝛽_𝛿(𝑛)−𝛼_𝛿(𝑛)𝛾^𝑁_𝑖=1𝜆_𝑖^(𝑛)𝑥_𝛿(𝑛)−𝑇_𝑖𝑥_𝛿(𝑛)², (3.42) which together with (3.41), lim_𝑛→∞𝛼_𝑛=0 and 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 yields lim_𝑛→∞𝑥_𝛿(𝑛)+1−𝑥_𝛿(𝑛)=0. (3.43) Now by (3.40), we clearly have 𝛼_𝛿(𝑛)𝛾𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥_𝛿(𝑛)²≥𝐴−𝛾𝑓𝑥_𝛿(𝑛),𝑥_𝛿(𝑛)−𝑥^∗, (3.44) which in the light of (3.38) yields 2𝛾−𝛾𝛽𝒯_𝛿(𝑛)+𝐴−𝛾𝑓𝑥^∗,𝑥_𝛿(𝑛)−𝑥^∗≤𝛼_𝛿(𝑛)𝛾𝑓𝑥_𝛿(𝑛)−𝐴𝑥_𝛿(𝑛)², (3.45) hence (as lim_𝑛→∞𝛼_𝛿(𝑛)‖𝛾𝑓(𝑥_𝛿(𝑛))−𝐴𝑥_𝛿(𝑛)‖²=0) it follows that 2𝛾−𝛾𝛽limsup_𝑛→∞𝒯_𝛿(𝑛)≤−liminf_𝑛→∞𝐴−𝛾𝑓𝑥^∗,𝑥_𝛿(𝑛)−𝑥^∗. (3.46) From (3.41) and invoking Lemma 3.2, we obtain lim_𝑛→∞𝐴−𝛾𝑓𝑥^∗,𝑥−𝑥^∗≥0,𝑥∈^𝑁_𝑛=1𝐹_𝑖𝑥𝑇_𝜔𝑖, (3.47) which by (3.46) yields limsup_𝑛→∞𝒯_𝛿(𝑛)=0, so that lim_𝑛→∞𝒯_𝛿(𝑛)=0. Combining (3.43), we have lim_𝑛→∞𝒯_𝛿(𝑛)+1=0. Then, recalling that 𝒯_𝑛<𝒯_𝛿(𝑛)+1 (by Lemma 2.9), we get lim_𝑛→∞𝒯_𝑛=0, so that 𝑥_𝑛→𝑥^∗ strongly. In addition, the variational inequality (3.39) and (3.47) can be written as 𝐼−𝐴+𝛾𝑓𝑥^∗−𝑥^∗,𝑥−𝑥^∗≤0,𝑥∈^𝑁_𝑛=1𝐹_𝑖𝑥𝑇_𝜔𝑖. (3.48) So, by the Lemma 2.10, it is equivalent to the fixed point equation 𝑥^∗=𝑃_⋂^𝑁_{𝑛=1𝐹𝑖𝑥(𝑇𝜔𝑖)}𝐼−𝐴+𝛾𝑓𝑥^∗=𝑃_⋂^𝑁_{𝑛=1𝐹𝑖𝑥(𝑇𝜔𝑖)}⋅𝑓𝑥^∗. (3.49)

If the sequences of the weights {𝜆_𝑖^(𝑛)}^𝑁_𝑖=1={𝜆_𝑖}^𝑁_𝑖=1 in (3.2), according to the proof of Theorem 3.3, we can obtain the following corollary.

Corollary 3.4. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 and let 𝑇_𝑖∶𝐶→𝐶 be a quasi-nonexpansive for 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, 𝜔_𝑖∈(0,1), 𝑖∈(1,…,𝑁) such that ⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖)≠∅, 𝑓 is a contraction with coefficient 𝛽∈(0,1) and 𝜆_𝑖 is a positive constant such that ∑^𝑁_𝑖=1𝜆_𝑖=1. Let 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾. Given the initial guess 𝑥₀∈𝐻 chosen arbitrarily and given sequences {𝛼_𝑛} and {𝛽_𝑛} in (0,1), satisfying the following conditions: (c1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞; (c2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Let {𝑥_𝑛} be the sequence generated by (3.1). Then {𝑥_𝑛} converges strongly to the unique a element 𝑥^∗∈⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖),𝑁≥1 verifying 𝑥^∗=𝑃_⋂^𝑁_{𝑖=1𝐹𝑖𝑥(𝑇𝜔𝑖)}⋅𝑓𝑥^∗, (3.50) which equivalently solves the following variational inequality problem: 𝑥^∗∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖,𝛾𝑓−𝐴𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖. (3.51)

4. Cyclic Algorithm

In this section, we discuss the cyclic algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite quasi-nonexpansives and introduce quasi-shrinking mapping and quoted its definition from [11]. Hereafter, for nonempty closed set 𝑆⊂𝐻 and 𝑟≥0, we use the notations 𝑑_𝑆∶𝐻∋𝑢↦𝑑(𝑢,𝑆)∶=inf_𝑥∈𝑆‖𝑢−𝑥‖, ⋄(𝑆,𝑟)∶={𝑢∈𝐻∣𝑑(𝑢,𝑆)=𝑟}, ⊲(𝑆,𝑟)∶={𝑢∈𝐻∣𝑑(𝑢,𝑆)≤ 𝑟} and ⊳(𝑆,𝑟)∶={𝑢∈𝐻∣𝑑(𝑢,𝑆)≥𝑟}. In this case, by the upper semicontinuity of 𝑑_𝑠 (see e.g., [14, Theorem 1.3.3]), ⊳(𝑆,𝑟) is closed. Moreover, for a nonempty bounded closed convex set 𝐶⊂𝐻 and 𝑟≥0, it is not hard to verify that (i) ⋄(𝐶,𝑟) and ⊲(𝐶,𝑟) are also closed; (ii) ⋄(𝐶,𝑟) and ⊲(𝐶,𝑟) are bounded; (iii) ⊲(𝐶,𝑟) is convex.

Definition 4.1 (see [11]). Suppose that 𝑇∶𝐻→𝐻 is quasi-nonexpansive with 𝐹_𝑖𝑥(𝑇)⋂𝐶≠∅ for some closed convex set 𝐶. Then 𝑇∶𝐻→𝐻 is called quasi-shrinking on 𝐶 if 𝐷∶𝑟∈[0,∞)↦inf_{𝐹𝑢∈⊳𝑖𝑥(𝑇),𝑟∩𝐶}𝑑𝑢,𝐹_𝑖𝑥(𝑇)−𝑑𝑇(𝑢),𝐹_𝑖𝑥(𝑇),if𝑢∈⊳𝐹_𝑖𝑥(𝑇),𝑟∩𝐶≠∅,∞otherwise (4.1) satisfies 𝐷(𝑟)=0⇔𝑟=0. In particular, if 𝑇 is quasi-shrinking on 𝐻, then 𝑇 is just called quasi-shrinking.

Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 and let {𝑇_𝑖}^𝑁−1_𝑖=1 be quasi-nonexpansives defined on 𝐶 such that the common fixed point set 𝐹∶=^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖,𝑁≥1, (4.2) where 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, {𝜔_𝑖}^𝑁_𝑖=1∈(0,1). Let 𝑥₀∈𝐶, let {𝛼_𝑛}^∞_𝑛=0, and {𝛽_𝑛}^∞_𝑛=0 sequences in (0,1). The cyclic algorithm generates a sequence {𝑥_𝑛}^∞_𝑛=1 in the following way: 𝑥₁=𝛼₀𝛾𝑓𝑥₀+𝛽₀𝑥₀+𝐼−𝛽₀𝐼−𝛼₀𝐴𝑇_𝜔0𝑥₀,𝑥₂=𝛼₁𝛾𝑓𝑥₁+𝛽₁𝑥₁+𝐼−𝛽₁𝐼−𝛼₁𝐴𝑇_𝜔1𝑥₁,⋮𝑥_𝑁=𝛼_𝑁−1𝛾𝑓𝑥_𝑁−1+𝛽_𝑁−1𝑥_𝑁−1+𝐼−𝛽_𝑁−1𝐼−𝛼_𝑁−1𝐴𝑇_𝜔𝑁−1𝑥_𝑁−1,𝑥_𝑁+1=𝛼_𝑁−1𝛾𝑓𝑥_𝑁−1+𝛽_𝑁𝑥_𝑁+𝐼−𝛽_𝑁𝐼−𝛼_𝑁𝐴𝑇_𝜔𝑁𝑥_𝑁,⋮ (4.3) In general, 𝑥_𝑛+1 is defined by 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇_[𝑛]𝑥_𝑛, (4.4) where 𝑇_[𝑛]=𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, with 𝑖=𝑛 (mod 𝑁), 0≤𝑖≤𝑁−1.

Lemma 4.2 (see [11]). Let 𝜑(𝑥)∶[0,∞)→[0,∞) satisfy (i)𝑥₁>𝑥₂⇒𝜑(𝑥₁)>𝜑(𝑥₂), (ii)𝜑(𝑥)=0⇔𝑥=0.
Let {𝑧_𝑛}_𝑛≥1⊂[0,∞) satisfy lim_𝑛→∞𝑧_𝑛=0. Then any sequence {𝑏_𝑛}_𝑛≥1⊂[0,∞) satisfying 𝑏_𝑛+1≤𝑏_𝑛−𝜑𝑏_𝑛+𝑧_𝑛+1,𝑛=0,1,2,… (4.5) converges to 0.

Lemma 4.3 (see [15]). Assume that {𝛼_𝑛}^∞_𝑛=0 is a sequence of nonnegative real numbers such that 𝛼_𝑛+1≤1−𝛾_𝑛𝛼_𝑛+𝛾_𝑛𝛿_𝑛,𝑛≥0, (4.6) where {𝛾_𝑛}^∞_𝑛=0⊂[0,1] and {𝛿_𝑛}^∞_𝑛=0 satisfy the following conditions: (i)∑^∞_𝑛=0𝛾_𝑛=∞ and lim_𝑛→∞𝛾_𝑛=0, (ii)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞_𝑛=0|𝛾_𝑛𝛿_𝑛|<∞.
Then lim_𝑛→∞𝛼_𝑛=0.

Theorem 4.4. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 and let 𝑇_𝑖∶𝐻→𝐻 be quasi-nonexpansives for 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, 𝜔_𝑖∈(0,1), 𝑖∈(1,2,…,𝑁) such that 𝐹∶=⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖)≠∅ and 𝑓 a contraction with coefficient 𝛽∈(0,1). Let 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾. Given the initial guess 𝑥₀∈𝐻 chosen arbitrarily and given sequences {𝛼_𝑛} and {𝛽_𝑛} in (0,1), satisfying the following conditions: (4.1a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞; (4.1b)∑^∞_𝑛=0‖𝛼_𝑛+1−𝛼_𝑛‖<∞ or lim_𝑛→∞𝛼_𝑛/𝛼_𝑛+1=1; (4.1c)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Let {𝑥_𝑛} be the sequence generated by (4.4). Then {𝑥_𝑛} converges strongly to the unique a element 𝑥^∗∈𝐹∶=⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖),𝑁≥1 verifying 𝑥^∗=𝑃_⋂^𝑁_{𝑖=1𝐹𝑖𝑥(𝑇𝜔𝑖)}⋅𝑓𝑥^∗, (4.7) which equivalently solves the following variational inequality problem: 𝑥^∗∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖,𝛾𝑓−𝐴𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖. (4.8)

Proof. Take a 𝑝∈𝐹∶=⋂^𝑁_𝑖=1𝐹_𝑖𝑥(𝑇_𝜔𝑖). We break the proof process into several steps.
Step 1. {𝑥_𝑛} is bounded. In light of the Remark 2.3, we obtain 𝑇_[𝑛]𝑥_𝑛−𝑝=𝑇_𝜔𝑖𝑥_𝑛−𝑝≤𝑥_𝑛−𝑝. (4.9) From (4.4), we have 𝑥_𝑛+1−𝑝=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇_[𝑛]𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛−𝛼_𝑛𝛾𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑝+𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝+1−𝛼_𝑛𝛾𝑥_𝑛=−𝑝1−𝛼_𝑛𝛾−𝛾𝛽𝑥_𝑛−𝑝+𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝. (4.10) By simple inductions, we obtain 𝑥_𝑛−𝑝≤max𝑥₁,𝑝−𝑝𝛾𝑓−𝐴𝑝𝛾−𝛾𝛽,∀𝑛≥1, (4.11) which gives that the sequence {𝑥_𝑛} is bounded; we also know that {𝑇_[𝑛]𝑥_𝑛} and {𝑓(𝑥_𝑛)} are bounded.
Step 2. Moreover if 𝑇_[𝑛]∶𝐻→𝐻 is quasi-shrinking on the set 𝐶, we obtain the following statements: (a)lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0; (b)lim_𝑛→∞‖𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛‖=0; (c)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
By the boundedness of {𝑥_𝑛}, {𝑇_[𝑛]𝑥_𝑛}, and {𝑓(𝑥_𝑛)}, there exists 𝑀>0 satisfying max_𝑛≥0𝑥_𝑛,𝑇_[𝑛]𝑥_𝑛,𝑓𝑥_𝑛≤𝑀. (4.12) By a simple inspection, we deduce 𝑑𝑥_𝑛+1,𝐹≤𝑥_𝑛+1−𝑃_𝐹𝑇_[𝑛]𝑥_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇_[𝑛]𝑥_𝑛−𝑃_𝐹𝑇_[𝑛]𝑥_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑇_[𝑛]𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛+𝑇_[𝑛]𝑥_𝑛−𝑃_𝐹𝑇_[𝑛]𝑥_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐴𝑇_[𝑛]𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝑇_[𝑛]𝑥_𝑛+𝑇_[𝑛]𝑥_𝑛−𝑃_𝐹𝑇_[𝑛]𝑥_𝑛≤𝑑𝑇_[𝑛]𝑥_𝑛,𝐹+2𝛼_𝑛+𝛽_𝑛𝑀. (4.13) By {𝑥_𝑛}_𝑛≥0⊂𝐶, we can assume the boundedness of the sequence 𝑏_𝑛∶=𝑑(𝑥_𝑛,𝐹)≥0(𝑛∈ℕ). Moreover, by Definition 4.1 and (4.13), it follows that 𝐷𝑏_𝑛≤𝑏_𝑛−𝑑𝑇_[𝑛]𝑥_𝑛,𝐹≤𝑏_𝑛−𝑏_𝑛+1+2𝛼_𝑛+𝛽_𝑛𝑀,∀𝑛≥0. (4.14) Now application of Lemma 4.2 to (4.14) yields lim_𝑛→∞𝑏_𝑛=0, hence (a) is proved.
The statements (b) and (c) are verified by 𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛=𝑇_[𝑛]𝑥_𝑛−𝑃_𝐹𝑥_𝑛+𝑃_𝐹𝑥_𝑛−𝑥_𝑛≤𝑇_[𝑛]𝑥_𝑛−𝑃_𝐹𝑥_𝑛+𝑃_𝐹𝑥_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑃_𝐹𝑥_𝑛+𝑃_𝐹𝑥_𝑛−𝑥_𝑛=2𝑑𝑥_𝑛(),𝑥,𝐹→0,𝑛→∞_𝑛+1−𝑥_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛+1−𝛽_𝑛−𝛼_𝑛𝛾𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛→0,(𝑛→∞). (4.15)(4.16)
Step 3. lim_𝑛→∞‖𝑥_𝑛+𝑁−𝑥_𝑛‖=0. From (4.4) and (4.16), we obtain 𝑥_𝑛+𝑁+1−𝑥_𝑛+1=𝛼_𝑛+𝑁𝛾𝑓𝑥_𝑛+𝑁+𝛽_𝑛+𝑁𝑥_𝑛+𝑁+𝐼−𝛽_𝑛+𝑁𝐼−𝛼_𝑛+𝑁𝐴𝑇_[𝑛+𝑁]𝑥_𝑛+𝑁−𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛽_𝑛𝑥_𝑛−𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇_[𝑛]𝑥_𝑛=𝛼_𝑛+𝑁𝛾𝑓𝑥_𝑛+𝑁−𝛼_𝑛+𝑁𝛾𝑓𝑥_𝑛+𝛽_𝑛+𝑁𝑥_𝑛+𝑁−𝑇_[𝑛+𝑁]𝑥_𝑛+𝑁+𝐼−𝛼_𝑛+𝑁𝐴𝑇_[𝑛+𝑁]𝑥_𝑛+𝑁−𝐼−𝛼_𝑛+𝑁𝐴𝑇_[𝑛]𝑥_𝑛+𝐼−𝛼_𝑛+𝑁𝐴𝑇_[𝑛]𝑥_𝑛−𝐼−𝛼_𝑛𝐴𝑇_[𝑛]𝑥_𝑛+𝛼_𝑛+𝑁𝛾𝑓𝑥_𝑛−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛≤𝛼_𝑛+𝑁𝛾𝛽𝑥_𝑛+𝑁−𝑥_𝑛+𝛽_𝑛+𝑁𝑥_𝑛+𝑁−𝑇_[𝑛+𝑁]𝑥_𝑛+𝑁+1−𝛼_𝑛+𝑁𝛾𝑇_[𝑛]𝑥_𝑛−𝑇_[𝑛+𝑁]𝑥_𝑛+𝑁+𝛼_𝑛+𝛼_𝑛+𝑁𝛾𝑇_[𝑛]𝑥_𝑛+𝛼_𝑛−𝛼_𝑛+𝑁𝛾𝛽𝑥_𝑛−𝛽_𝑛𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛=𝛼_𝑛+𝑁𝛾𝛽𝑥_𝑛+𝑁−𝑥_𝑛+𝛽_𝑛+𝑁𝑥_𝑛+𝑁−𝑇_[𝑛+𝑁]𝑥_𝑛+𝑁+1−𝛼_𝑛+𝑁𝛾^𝑁−1_𝑗=𝑛𝑇_[𝑗]𝑥_𝑗−𝑇_[𝑗+1]𝑥_𝑗+1+𝛼_𝑛−𝛼_𝑛+𝑁𝛾𝑇_[𝑛]𝑥_𝑛+𝛼_𝑛−𝛼_𝑛+𝑁𝛾𝛽𝑥_𝑛−𝛽_𝑛𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛≤𝛼_𝑛+𝑁𝛾𝛽𝑥_𝑛+𝑁−𝑥_𝑛+𝛽_𝑛+𝑁𝑥_𝑛+𝑁−𝑇_[𝑛+𝑁]𝑥_𝑛+𝑁+1−𝛼_𝑛+𝑁𝛾^𝑁−1_𝑗=𝑛𝑥_𝑗+1−𝑇_[𝑗+1]𝑥_𝑗+1+𝑥_𝑗−𝑥_𝑗+1+𝑥_𝑗−𝑇_[𝑗]𝑥_𝑗+𝛼_𝑛−𝛼_𝑛+𝑁𝛾𝑇_[𝑛]𝑥_𝑛+𝛼_𝑛−𝛼_𝑛+𝑁𝛾𝛽𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛. (4.17) By conditions (4.1a), (4.1b), (4.1c), (4.15), and (4.16), {𝑥_𝑛} and {𝑇_[𝑛]𝑥_𝑛} are bounded we obtain that lim_𝑛→∞𝑥_𝑛+𝑁−𝑥_𝑛=0. (4.18)
Step 4. lim_𝑛→∞‖𝑥_𝑛−𝑇_[𝑛+𝑁]⋯𝑇_[𝑛+1]𝑥_𝑛‖=0.
From (4.4), we observe that 𝑥_𝑛+1−𝑇_[𝑛]𝑥_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑇_[𝑛]𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛. (4.19) It follows from the condition (4.1a), (4.1c), (4.15), and the boundedness of {𝑓(𝑥_𝑛)} and {𝑇_[𝑛]𝑥_𝑛} that 𝑥_𝑛+1−𝑇_[𝑛]𝑥_𝑛→0,(𝑛→∞). (4.20) Recursively, 𝑥_𝑛+𝑁−𝑇_[𝑛+𝑁]𝑥_{𝑛+𝑁−1}(),𝑥→0,𝑛→∞_{𝑛+𝑁−1}−𝑇_{[𝑛+𝑁−1]}𝑥_{𝑛+𝑁−2}→0,(𝑛→∞). (4.21) By Remark 2.3, 𝑇_[𝑛+𝑁] is quasi-nonexpansive, we obtain 𝑇_[𝑛+𝑁]𝑥_{𝑛+𝑁−1}−𝑇_[𝑛+𝑁]𝑇_{[𝑛+𝑁−1]}𝑥_{𝑛+𝑁−2}→0,(𝑛→∞). (4.22) Proceeded accordingly, we obtain 𝑇_[𝑛+𝑁]𝑇_{[𝑛+𝑁−1]}𝑥_{𝑛+𝑁−2}−𝑇_[𝑛+𝑁]𝑇_{[𝑛+𝑁−1]}𝑇_{[𝑛+𝑁−2]}𝑥_{𝑛+𝑁−3}(),⋮𝑇→0,𝑛→∞_[𝑛+𝑁]⋯𝑇_[𝑛+2]𝑥_𝑛+1−𝑇_[𝑛+𝑁]⋯𝑇_[𝑛+1]𝑥_𝑛→0,(𝑛→∞). (4.23) Note that 𝑥_[𝑛+𝑁]−𝑇_[𝑛+𝑁]⋯𝑇_[𝑛+1]𝑥_𝑛≤𝑥_𝑛+𝑁−𝑇_[𝑛+𝑁]𝑥_{𝑛+𝑁−1}+𝑇_[𝑛+𝑁]𝑥_{𝑛+𝑁−1}−𝑇_[𝑛+𝑁]𝑇_{[𝑛+𝑁−1]}𝑥_{𝑛+𝑁−2}⋮+𝑇_[𝑛+𝑁]⋯𝑇_[𝑛+2]𝑥_𝑛+1−𝑇_[𝑛+𝑁]⋯𝑇_[𝑛+1]𝑥_𝑛. (4.24) From all the expressions above, we have 𝑥_[𝑛+𝑁]−𝑇_[𝑛+𝑁]⋯𝑇_[𝑛]𝑥_𝑛→0,(𝑛→∞). (4.25) Since 𝑥_𝑛−𝑇_[𝑛+𝑁]⋯𝑇_[𝑛+1]𝑥_𝑛≤𝑥_𝑛+𝑥_𝑛+𝑁+𝑥_𝑛+𝑁−𝑇_[𝑛+𝑁]⋯𝑇_[𝑛]𝑥_𝑛, (4.26) it is concluded that lim_𝑛→∞𝑥_𝑛−𝑇_[𝑛+𝑁]⋯𝑇_[𝑛+1]𝑥_𝑛=0. (4.27)
Step 5. 𝜔_𝑤(𝑥_𝑛)⊂⋂^𝑁_𝑛=1𝐹_𝑖𝑥(𝑇_[𝑛])=⋂^𝑁_𝑛=1𝐹_𝑖𝑥(𝑇_𝜔𝑖),𝑖=1,2,…,𝑁. Assume that {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑥, we prove 𝑥∈⋂^𝑁_𝑛=1𝐹_𝑖𝑥(𝑇_[𝑛]). By the conclusion of step 4, we obtain lim_𝑗→∞𝑥_𝑛𝑗−𝑇_{[𝑛𝑗+𝑁]}⋯𝑇_[𝑛𝑗+1]𝑥_𝑛𝑗=0. (4.28) Observe that, for each 𝑛_𝑗, 𝑇_{[𝑛𝑗+𝑁]},…,𝑇_[𝑛𝑗+1] is some permutation of the mappings 𝑇_[1],…,𝑇_[𝑁], since 𝑇_[1],…,𝑇_[𝑁] are finite, all the full permutation are 𝑁!, there must be some permutation that appears infinite times. Without loss of generality, suppose that this permutation is 𝑇_[1],…,𝑇_[𝑁], we can take a subsequence {𝑥_𝑛𝑗𝑘}⊂{𝑥_𝑛𝑗} such that lim_𝑗→∞𝑥_𝑛𝑗−𝑇_[1]⋯𝑇_[𝑁]𝑥_𝑛𝑗=0. (4.29) It is easy to prove that 𝑇_[1],…,𝑇_[𝑁] is quasi-nonexpansive. By Lemma 2.5, we have 𝑥=𝑇_[1]⋯𝑇_[𝑁]𝑥. (4.30) Using Remark 2.3 and Lemma 2.5, we obtain 𝑥∈𝐹_𝑖𝑥𝑇_[1]⋯𝑇_[𝑁]=^𝑁_𝑛=1𝐹_𝑖𝑥𝑇_[𝑛]=^𝑁_𝑛=1𝐹_𝑖𝑥𝑇_𝑛. (4.31)
Step 6. liminf_𝑛→∞⟨(𝛾𝑓−𝐴)𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0. Indeed, there exists a subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that liminf_𝑛→∞𝛾𝑓−𝐴𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑗→∞𝛾𝑓−𝐴𝑥^∗,𝑥_𝑛𝑗−𝑥^∗. (4.32) Without loss of generality, we may further assume that 𝑥_𝑛𝑗⇀𝑥. It follows from (4.31) that 𝑥∈⋂^𝑁_𝑛=1𝐹(𝑇_𝑛). Since 𝑥^∗ is the unique solution of (4.8), we have liminf_𝑛→∞𝛾𝑓−𝐴𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑗→∞𝛾𝑓−𝐴𝑥^∗,𝑥_𝑛𝑗−𝑥^∗=𝛾𝑓−𝐴𝑥^∗,𝑥−𝑥^∗≤0. (4.33) In addition, the variational inequality (4.33) can be written as 𝐼−𝐴+𝛾𝑓𝑥^∗−𝑥^∗,𝑥−𝑥^∗≤0,𝑥∈^𝑁_𝑛=1𝐹_𝑖𝑥𝑇_𝜔𝑖. (4.34) So, by the Lemma 2.10, it is equivalent to the fixed point equation 𝑥^∗=𝑃_⋂^𝑁_{𝑛=1𝐹𝑖𝑥(𝑇𝜔𝑖)}𝐼−𝐴+𝛾𝑓𝑥^∗=𝑃_⋂^𝑁_{𝑛=1𝐹𝑖𝑥(𝑇𝜔𝑖)}⋅𝑓𝑥^∗. (4.35)
Step 7. lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖=0. From (4.4), we obtain 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇_[𝑛]𝑥_𝑛−𝑥^∗2=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+𝐼−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑇_[𝑛]𝑥_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗+1−𝛽_𝑛−𝛼_𝑛𝑇_[𝑛]𝑥_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛼_𝑛(𝐼−𝐴)𝑇_[𝑛]𝑥_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗+(𝐼−𝐴)𝑇_[𝑛]𝑥_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+1−𝛽_𝑛−𝛼_𝑛𝛾𝑇_[𝑛]𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝛾𝛽𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗𝛼_𝑛1−𝛾𝑇_[𝑛]𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗+1−𝛽_𝑛−𝛼_𝑛𝛾𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗≤1−𝛾−𝛾𝛽2𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+𝛼_𝑛𝑀≤1−𝛾−𝛾𝛽2𝑥_𝑛−𝑥^∗2+12𝑥_𝑛+1−𝑥^∗2+𝛼_𝑛𝑀, (4.36) where 𝑀=(1−𝛾)‖𝑇_[𝑛]𝑥_𝑛−𝑥^∗‖‖𝑥_𝑛+1−𝑥^∗‖+⟨𝛾𝑓(𝑥^∗)−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩. It follows that 𝑥_𝑛+1−𝑥^∗2≤1−𝛾−𝛾𝛽𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑀. (4.37) By using Lemma 4.3, we can obtain the desired conclusion easily.

5. Application

In this section, we constructed a numerical example to compare the parallel algorithm and cyclic algorithm which is simple.

Let 𝑥=(𝑥₁,𝑥₂)∈𝑅² and 𝑓(𝑥)=(1/2)(sin(𝑥₁),cos(𝑥₂)) be a contraction mapping with coefficient 1/2. Let 𝑇₁(𝑥)=(0,4𝑥₁) and 𝑇₂(𝑥)=(4𝑥₂,0) be quasi-nonexpansive mappings. Let 𝛼_𝑛=𝛽_𝑛=1/3, 𝐴=𝐼 and 𝛾=𝜆₁=𝜆₂=1/2. According to (1.20) and (1.24), we can obtain the following parallel algorithm and cyclic algorithm:

Parallel Algorithm
𝑥_𝑛+1=16𝑓𝑥_𝑛+12𝑥_𝑛+1𝑇12¹+𝑇₂𝑥_𝑛. (5.1)

Cyclic Algorithm
𝑥₁=16𝑓𝑥₀+12𝑥₀+16𝑇₁𝑥₀;𝑥₂=16𝑓𝑥₁+12𝑥₁+16𝑇₂𝑥₁;𝑥₃=16𝑓𝑥₂+12𝑥₂+16𝑇₁𝑥₂;⋮𝑥_𝑛−1=16𝑓𝑥_𝑛−2+12𝑥_𝑛−2+16𝑇₂𝑥_𝑛−2;𝑥_𝑛=16𝑓𝑥_𝑛−1+12𝑥_𝑛−1+16𝑇₁𝑥_𝑛−1. (5.2) Form Theorems 3.3 and 4.4, we can easily know that parallel algorithm (5.1) and cyclic algorithm (5.2) are converge to the unique point in 𝑅². Let 𝑥₀=(5,2) and |𝑥_𝑛+1−𝑥_𝑛|²≤10⁻⁹, and let 𝑥^∗_𝑃 and 𝑥^∗_𝑋 be the fixed point of the parallel algorithm and cyclic algorithm. Using the software of MATLAB, we obtain 𝑥^∗_𝑃=𝑥₁₃₃=(0.6821,0.7080) and 𝑥^∗_𝑋=𝑥₁₆₆=(1.9325,0.8729). From the computed results of 𝑥^∗_𝑃 and 𝑥^∗_𝑋, we can easily know that parallel algorithm (5.1) is simpler than cyclic algorithm (5.2). On the other hand, we need to explain that those algorithms do not converge a common fixed point, because parallel algorithm (5.1) and cyclic algorithm (5.2) have the different algorithm structure.

Acknowledgment

This work is supported in part by China Postdoctoral Science Foundation (Grant no. 20100470783).