Weak and Strong Convergence of an Algorithm for the Split Common Fixed-Point of Asymptotically Quasi-Nonexpansive Operators

Abstract

Inspired by the Moudafi (2010), we propose an algorithm for solving the split common fixed-point problem for a wide class of asymptotically quasi-nonexpansive operators and the weak and strong convergence of the algorithm are shown under some suitable conditions in Hilbert spaces. The algorithm and its convergence results improve and develop previous results for split feasibility problems.

1. Introduction

Fixed-point problem is a classical problem in nonlinear analysis and it has application in a wide spectrum of fields such as economics, physics, and applied sciences. In this paper, We are concerned with the split common fixed point problem (SCFP). In fact, the SCFP is an extension of the split feasibility problem (SFP) and the convex feasibility problem (CFP), see [1]. The CFP and SCFP have many applications such as approximation theory [2], image reconstruction, radiation therapy [3, 4], and control [5]. The SFP in finite-dimensional space was first introduced by Censor and Elfving [6] for modeling inverse problems, which arise from phase retrievals, and in medical image reconstruction [7]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [8, 9]. The SFP in an infinite-dimensional Hilbert space can be found in [10–13].

Throughout this paper, we assume that both 𝐻₁ and 𝐻₂ are real Hilbert spaces, “→” and “⇀” denote strong and weak convergence, respectively, Fix(𝑇) denotes the set of the fixed points of an operator 𝑇, that is, and Fix(𝑇)∶={𝑥∣𝑥=𝑇(𝑥)}. Let 𝑈∶𝐻₁→𝐻₁ and 𝑇∶𝐻₂→𝐻₂ be two asymptotically quasi-nonexpansive mappings with nonempty fixed-point sets Fix(𝑈)=𝐶 and Fix(𝑇)=𝑄, respectively. The split common fixed point problem for operators 𝑈 and 𝑇 is to find 𝑥^∗∈𝐶suchthat𝐴𝑥^∗∈𝑄, (1) where 𝐴∶𝐻₁→𝐻₂ is bounded linear. Denote the solution set of the two-operator SCFP by Γ=𝑦∈𝐶∣𝐴𝑦∈𝑄. (2)

The split common fixed point problem for quasi-nonexpansive mapping in the setting of Hilbert space was first introduced and studied by Moudafi [14]. However, the algorithm presented in [14] has only weak convergence. The purpose of this paper is to propose an algorithm of split common fixed point problem for asymptotically quasi-nonexpanding operator which includes the quasi-nonexpansive mapping and the weak and strong convergence of the algorithm are shown under some suitable conditions in Hilbert spaces. The algorithm and the convergence results improve and develop previously discussed split feasibility problems.

The paper is organized as follows. In Section 2, we recall some preliminaries. In Section 3, we present an algorithm and show its weak convergence and strong convergence. Section 4 gives some concluding remarks.

2. Preliminaries

Recall that a mapping 𝑇∶𝐻→𝐻 is said to be nonexpansive if 𝑇(𝑥)−𝑇𝑦≤𝑥−𝑦,𝑥,𝑦∈𝐻×𝐻. (3)

A mapping 𝑇 is called asymptotically nonexpansive if there exists a sequence {𝑙_𝑘}⊂[1,∞) with 𝑙_𝑘→1 as 𝑘→∞ such that, for all (𝑥,𝑦)∈𝐻×𝐻, 𝑇^𝑘(𝑥)−𝑇^𝑘𝑦≤𝑙_𝑘𝑥−𝑦,∀𝑘≥1. (4)

A mapping 𝑇 is called quasi-nonexpansive if for all (𝑥,𝑧)∈𝐻×𝐹(𝑇)‖𝑇(𝑥)−𝑧‖≤‖𝑥−𝑧‖. (5)

A mapping 𝑇 is called asymptotically quasi-nonexpansive if there exists a sequence {𝑙_𝑘}⊂[1,∞) with 𝑙_𝑘→1 as 𝑘→∞ such that, for all (𝑥,𝑧)∈𝐻×Fix(𝑇), 𝑇^𝑘(𝑥)−𝑧≤𝑙_𝑘‖𝑥−𝑧‖,∀𝑘≥1. (6)

A mapping 𝑇 is said to be uniformly 𝐿-Lipschitzian if there exists a constant 𝐿>0 such that for all (𝑥,𝑦)∈𝐻×𝐻, 𝑇^𝑘(𝑥)−𝑇^𝑘𝑦≤𝐿𝑥−𝑦,∀𝑘≥1. (7)

A mapping 𝑇 is said to be semicompact if for any bounded sequence {𝑥^𝑘}⊂𝐻 with lim_𝑘→∞‖𝑥^𝑘−𝑇𝑥^𝑘‖=0 there exists a subsequence {𝑥^𝑘𝑝} of {𝑥^𝑘} such that {𝑥^𝑘𝑝} converges strongly to a point 𝑥^∗∈𝐻.

Let 𝐸 be a Banach space. A mapping 𝑇∶𝐸→𝐸 is said to be demiclosed at origin, if for any sequence {𝑥^𝑘}⊂𝐸 with 𝑥^𝑘⇀𝑥^∗ and ‖(𝐼−𝑇)𝑥^𝑘‖→0, 𝑥^∗=𝑇(𝑥^∗).

Lemma 1. Let 𝑇∶𝐻→𝐻 be asymptotically quasi-nonexpansive and set 𝑇^𝑘_𝛼∶=(1−𝛼)𝐼+𝛼𝑇^𝑘 for 𝛼∈(0,1]. Then, for any 𝑧∈Fix(𝑇) and 𝑥∈𝐻, one has the following.(1)⟨𝑥−𝑇^𝑘(𝑥),𝑥−𝑧⟩≥(1/2)‖𝑥−𝑇^𝑘(𝑥)‖²−((𝑙²_𝑘−1)/2)‖𝑥^𝑘−𝑧‖²; (2)⟨𝑥−𝑇^𝑘(𝑥),𝑧−𝑇^𝑘(𝑥)⟩≤(1/2)‖𝑥−𝑇^𝑘(𝑥)‖²+((𝑙²_𝑘−1)/2)‖𝑥^𝑘−𝑧‖²; (3)‖𝑇^𝑘_𝛼(𝑥)−𝑧‖²≤[1+𝛼(𝑙²_𝑘−1)]‖𝑥−𝑧‖²−𝛼(1−𝛼)‖𝑥−𝑇^𝑘(𝑥)‖;(4)⟨𝑥−𝑇^𝑘_𝛼(𝑥),𝑥−𝑧⟩≥(𝛼/2)‖𝑥−𝑇^𝑘(𝑥)‖²−((𝛼(𝑙²_𝑘−1))/2)‖𝑥^𝑘−𝑧‖².

Lemma 2 (see [15]). Suppose {𝑎_𝑘} is a sequence of nonnegative real numbers such that 𝑎_𝑘+1≤1+𝛿_𝑘𝑎_𝑘+𝑏_𝑘,𝑘≥0, (8) if (𝑖)∑^∞_𝑘=0𝛿_𝑘<∞and(𝑖𝑖)∑^∞_𝑘=1𝑏_𝑘<∞. Then, lim_𝑘→∞𝑎_𝑘 exists. In particular, if {𝑎_𝑘} has a subsequence which converges strongly to zero, then lim_𝑘→∞𝑎_𝑘=0.

3. The Algorithm and Its Asymptotic Convergence

We now give a description of an algorithm.

Algorithm 3. Initialization: let 𝑥⁰∈𝐻₁ be arbitrary.
Iterative step: for 𝑘∈𝑁, set 𝑢^𝑘=𝑥^𝑘+𝛾𝐴^𝑇(𝑇^𝑘−𝐼)𝐴(𝑥^𝑘), let 𝑥^𝑘+1=1−𝛼_𝑘𝑢^𝑘+𝛼_𝑘𝑈^𝑘𝑢^𝑘,𝑘∈𝑁, (9) where 𝛼_𝑘∈[𝛼,1−𝛼],𝛼∈(0,1), and 𝛾∈(0,1/𝜆),𝜆=‖𝐴‖².

In what follows, we establish the weak convergence and strong convergence of Algorithm 3.

Lemma 4 (Opial [16]). Let 𝐻 be a Hilbert space and let {𝑥^𝑘} be a sequence in 𝐻 such that there exists a nonempty set 𝑆⊂𝐻 satisfying the following:(1)For every 𝑥^∗, lim_𝑘‖𝑥^𝑘−𝑥^∗‖ exists.(2)Any weak cluster point of the sequence {𝑥^𝑘} belongs to 𝑆.
Then, there exists 𝑧∈𝑆 such that {𝑥^𝑘} weakly converges to 𝑧.

Theorem 5. Let 𝑈∶𝐻₁→𝐻₁ and 𝑇∶𝐻₂→𝐻₂ be two asymptotically quasi-nonexpansive operators with nonempty Fix(𝑈)=𝐶 and Fix(𝑇)=𝑄. Assume that 𝑈−𝐼 and 𝑇−𝐼 are demiclosed at 0 and Γ≠∅, ∑^∞_𝑘=1(𝑙²_𝑘−1)<∞, and 𝑇 and 𝑈 are uniformly 𝐿-Lipschitzian. Then,(I)any sequence {𝑥^𝑘} generated by Algorithm 3 converges weakly to a point 𝑥^∗∈Γ;(II)if 𝑈 is also semi-compact, then both {𝑥^𝑘} and {𝑢^𝑘} generated by Algorithm 3 converge strongly to a point 𝑥^∗∈Γ.

Proof. First, we prove that for each 𝑧∈Γ, lim_𝑘→∞‖𝑥^𝑘−𝑧‖ and lim_𝑘→∞‖𝑢^𝑘−𝑧‖ exist. Also lim_𝑘→+∞‖(𝑇^𝑘−𝐼)(𝐴𝑥^𝑘)‖=0 and lim_𝑘→+∞‖(𝑈^𝑘(𝑢^𝑘)−𝑢^𝑘)‖=0.
By using (1) in Lemma 1, from (9) we obtain 𝑥^𝑘+1−𝑧²=(1−𝛼_𝑘)𝑢^𝑘+𝛼_𝑘𝑈^𝑘(𝑢^𝑘)−𝑧²≤1+𝛼_𝑘𝑙²_𝑘−1𝑢^𝑘−𝑧²−𝛼_𝑘1−𝛼_𝑘𝑢^𝑘−𝑈^𝑘(𝑢^𝑘)². (10) By deducing, it follows that 𝑢^𝑘−𝑧²=𝑥^𝑘+𝛾𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘−𝑧²=𝑥^𝑘−𝑧²+𝛾²𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘2+2𝛾𝑥^𝑘−𝑧,𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘≤𝑥^𝑘−𝑧²+𝜆𝛾²𝑇^𝑘−𝐼𝐴𝑥^𝑘2+2𝛾𝐴𝑥^𝑘−𝐴𝑧,𝑇^𝑘−𝐼𝐴𝑥^𝑘; (11) that is, 𝑢^𝑘−𝑧²≤𝑥^𝑘−𝑧²+𝜆𝛾²(𝑇^𝑘−𝐼)(𝐴𝑥^𝑘)²+2𝛾𝐴𝑥^𝑘−𝐴𝑧,𝑇^𝑘−𝐼𝐴𝑥^𝑘. (12) Setting 𝜃∶=2𝛾⟨𝐴𝑥^𝑘−𝐴𝑧,(𝑇−𝐼)(𝐴𝑥^𝑘)⟩ and using (1) of Lemma 1, we obtain 𝜃=2𝛾𝐴𝑥^𝑘−𝐴𝑧,𝑇^𝑘−𝐼𝐴𝑥^𝑘=2𝛾𝐴𝑥^𝑘−𝐴𝑧+𝑇^𝑘−𝐼𝐴𝑥^𝑘−𝑇^𝑘−𝐼𝐴𝑥^𝑘,𝑇^𝑘−𝐼𝐴𝑥^𝑘=2𝛾𝐴𝑥^𝑘−𝐴𝑧,(𝑇−𝐼)𝐴𝑥^𝑘−(𝑇−𝐼)𝐴𝑥^𝑘21≤2𝛾2(𝑇^𝑘−𝐼)(𝐴𝑥^𝑘)²+𝑙²_𝑘−12𝐴𝑥^𝑘−𝐴𝑧²−𝑇^𝑘−𝐼𝐴𝑥^𝑘2=−𝛾𝑇^𝑘−𝐼𝐴𝑥^𝑘2+𝛾𝜆𝑙²_𝑘−1𝑥^𝑘−𝑧². (13) The key inequality mentioned above, combined with (12), yields 𝑢^𝑘−𝑧²≤1+𝛾𝜆𝑙²_𝑘−1𝑥^𝑘−𝑧²−𝛾1−𝜆𝛾𝑇^𝑘−𝐼𝐴𝑥^𝑘2. (14) Substituting (14) into (10) yields 𝑥^𝑘+1−𝑧²≤1+𝛼_𝑘𝑙²_𝑘×𝑙−11+𝛾𝜆²_𝑘−1𝑥^𝑘−𝑧²−𝛾1−𝜆𝛾𝑇^𝑘−𝐼𝐴𝑥^𝑘2−𝛼_𝑘1−𝛼_𝑘𝑢^𝑘−𝑈^𝑘𝑢^𝑘=1+𝛼_𝑘+𝛾𝜆𝑙²_𝑘−1+𝛼_𝑘𝛾𝜆𝑙²_𝑘−1²𝑥^𝑘−𝑧²−1+𝛼_𝑘𝑙²_𝑘−1𝛾1−𝜆𝛾𝑇^𝑘−𝐼𝐴𝑥^𝑘2−𝛼_𝑘1−𝛼_𝑘𝑢^𝑘−𝑈^𝑘(𝑢^𝑘)²=1+𝛿_𝑘𝑥^𝑘−𝑧²−1+𝛼_𝑘𝑙²_𝑘𝛾×𝑇−11−𝜆𝛾_𝑘−𝐼𝐴𝑥^𝑘2−𝛼_𝑘1−𝛼_𝑘𝑢^𝑘−𝑈^𝑘(𝑢^𝑘)²+𝑏_𝑘, (15) where 𝛿_𝑘=(𝛼_𝑘+𝛾𝜆)(𝑙²_𝑘−1)+𝛼_𝑘𝛾𝜆(𝑙²_𝑘−1)²,𝑏_𝑘=0. Since ∑^∞_𝑘=1(𝑙²_𝑘−1)<∞, we have ∑^∞_𝑘=1𝛿_𝑘<∞, From Lemma 1, it follows that lim_𝑘→∞‖𝑥^𝑘−𝑧‖ exists. By the virtue of (14), we know that lim_𝑘→∞‖𝑢^𝑘−𝑧‖ exists. Therefore, from (15), we have 1+𝛼_𝑘𝑙²_𝑘−1𝛾1−𝜆𝛾𝑇^𝑘−𝐼𝐴𝑥^𝑘2+𝛼_𝑘1−𝛼_𝑘𝑢^𝑘−𝑈^𝑘𝑢^𝑘2≤𝑥^𝑘−𝑧²−𝑥^𝑘+1−𝑧²−𝛼_𝑘+𝛾𝜆𝑙²_𝑘−1+𝛼_𝑘𝛾𝜆𝑙²_𝑘−1²𝑥^𝑘−𝑧²⟶0. (16) By the assumptions on 𝑙_𝑘 and 𝛼_𝑘, we get lim_𝑘→+∞𝑇^𝑘−𝐼𝐴𝑥^𝑘=0,lim_𝑘→+∞𝑈^𝑘𝑢^𝑘−𝑢^𝑘=0. (17)
Step  1. Now, we prove that lim_𝑘→∞‖𝑥^𝑘+1−𝑥^𝑘‖=0 and lim_𝑘→∞‖𝑢^𝑘+1−𝑢^𝑘‖=0. As a matter of a fact, it follows from (9) that 𝑥^𝑘+1−𝑥^𝑘=1−𝛼_𝑘𝑢^𝑘+𝛼_𝑘𝑈^𝑘𝑢^𝑘−𝑥^𝑘=1−𝛼_𝑘𝑥^𝑘+𝛾𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘+𝛼_𝑘𝑈^𝑘𝑢^𝑘−𝑥^𝑘=1−𝛼_𝑘𝛾𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘+𝛼_𝑘𝑈^𝑘𝑢^𝑘−𝑥^𝑘=1−𝛼_𝑘𝛾𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘+𝛼_𝑘𝑈^𝑘𝑢^𝑘−𝑢^𝑘+𝛼_𝑘𝑢^𝑘−𝑥^𝑘=1−𝛼_𝑘𝛾𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘+𝛼_𝑘𝑈^𝑘𝑢^𝑘−𝑢^𝑘+𝛼_𝑘𝛾𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘=𝛾𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘+𝛼_𝑘𝑈^𝑘𝑢^𝑘−𝑢^𝑘. (18) In view of (17), we have that lim_𝑘→∞𝑥^𝑘+1−𝑥^𝑘=0. (19) Similarly, it follows from (9), (17), and (19) that 𝑢^𝑘+1−𝑢^𝑘=𝑥^𝑘+1+𝛾𝐴^𝑇𝑇^𝑘+1−𝐼𝐴𝑥^𝑘+1−𝑥^𝑘+𝛾𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘≤𝑥^𝑘+1−𝑥^𝑘+𝛾𝐴^𝑇𝑇^𝑘+1−𝐼𝐴𝑥^𝑘+1+𝛾𝐴^𝑇𝑇^𝑘−𝐼𝐴𝑥^𝑘⟶0,(𝑘⟶∞). (20)
Step  2. We prove that lim_𝑘→∞‖𝐴𝑥^𝑘−𝑇(𝐴𝑥^𝑘)‖=0 and lim_𝑘→∞‖𝑢^𝑘−𝑈(𝑢^𝑘)‖=0. Setting 𝜂_𝑘∶=‖𝑢^𝑘−𝑈^𝑘(𝑢^𝑘)‖, since 𝑈 is uniformly 𝐿-Lipschitzian continuous, it follows from (14) and (17) that 𝑢^𝑘−𝑈𝑢^𝑘≤𝑢^𝑘−𝑈^𝑘𝑢^𝑘+𝑈^𝑘𝑢^𝑘−𝑈𝑢^𝑘≤𝜂_𝑘+𝐿𝑈^𝑘−1𝑢^𝑘−𝑢^𝑘≤𝜂_𝑘+𝐿𝑈^𝑘−1𝑢^𝑘−𝑈^𝑘−1𝑢^𝑘−1+𝑈^𝑘−1𝑢^𝑘−1−𝑢^𝑘≤𝜂_𝑘+𝐿²𝑢^𝑘−𝑢^𝑘−1+𝐿𝑈^𝑘−1𝑢^𝑘−1−𝑢^𝑘−1+𝑢^𝑘−1−𝑢^𝑘≤𝜂_𝑘+𝐿(𝐿+1)𝑢^𝑘−𝑢^𝑘−1+𝐿𝜂_𝑘−1⟶0,(𝑘⟶∞); (21) in other words, lim_𝑘→∞𝑢^𝑘−𝑈𝑢^𝑘=0. (22) Similarly, we have lim_𝑘→∞𝐴𝑥^𝑘−𝑇𝐴𝑥^𝑘=0. (23)
Step  3. Finally, we prove that 𝑥^𝑘⇀𝑥^∗ and 𝑥^𝑘⇀𝑥^∗, where 𝑥^∗∈Γ. Denote by 𝑥^∗ a weak-cluster point of {𝑥^𝑘}, and denote by {𝑥^𝑘𝜎} a subsequence of {𝑥^𝑘}. Obviously, 𝑤−lim_𝜎𝑦^𝑘𝜎=𝑤−lim_𝜎𝑥^𝑘𝜎=𝑥^∗. (24) Then, from (23) and the demiclosedness of 𝑇−𝐼 at 0, we obtain 𝑇𝐴𝑥^∗=𝐴𝑥^∗; (25)
it follows that, 𝐴𝑥^∗∈𝑄.
Noticing 𝑢^𝑘=𝑥^𝑘+𝛾𝐴^∗(𝑇−𝐼)(𝐴𝑥^𝑘), it follows that 𝑤−lim_𝜎𝑢^𝑘𝜎=𝑥^∗. By the demiclosedness of 𝑈−𝐼 at 0, we have 𝑈𝑥^∗=𝑥^∗. (26) Hence, 𝑥^∗∈𝐶, and therefore 𝑥^∗∈Γ.
Since there is no more than one weak cluster point, the weak convergence of the whole sequence {𝑥^𝑘} follows by applying Lemma 4 with 𝑆=Γ.
The Proof of Conclusion (II). Since 𝑈 is semi-compact, it follows from (22) that there exists a subsequence {𝑢^𝑘ℎ} of {𝑢^𝑘} such that {𝑢^𝑘ℎ}→𝑢^∗∈𝐻 (some point in 𝐻). Since 𝑥^𝑘=𝑢^𝑘−𝛾𝐴^𝑇(𝑇^𝑘−𝐼)𝐴(𝑥^𝑘){𝑢^𝑘ℎ}⇀𝑥^∗, this implies that 𝑥^∗=𝑢^∗. Therefore, {𝑢^𝑘ℎ}→𝑥^∗∈Γ as ℎ→∞. Since for any 𝑧∈Γ,lim_𝑘→∞‖𝑥^𝑘−𝑧‖ and lim_𝑘→∞‖𝑢^𝑘−𝑧‖ exist, we know that lim_𝑘→∞‖𝑥^𝑘−𝑥^∗‖=lim_𝑘→∞‖𝑢^𝑘−𝑥^∗‖=0. This implies that {𝑥^𝑘} and {𝑢^𝑘} both converge strongly to a point 𝑥^∗∈Γ. The proof is completed.

4. Concluding Remarks

In this paper, we have proposed an algorithm for solving the SCFP in the wide class of asymptotically quasi-nonexpansive operators and obtained its weak and strong convergence in general Hilbert spaces in a new way. Next, we will improve the algorithm to solve the multiple split common fixed point problem in infinite Hilbert spaces.

Acknowledgments

This work was supported by the National Science Foundation of China (under Grant 11171221), Shanghai Leading Academic Discipline Project (under Grant XTKX2012), Basic and Frontier Research Program of Science and Technology Department of Henan Province (under Grant 112300410277), Innovation Program of Shanghai Municipal Education Commission (under Grant 14YZ094), Doctoral Program Foundation of Institutions of Higher Education of China (under Grant 20123120110004), Doctoral Starting Projection of the University of Shanghai for Science and Technology (under Grant ID-10-303-002), and Young Teacher Training Projection Program of Shanghai for Science and Technology.