Research Article | Open Access
Weak and Strong Convergence of an Algorithm for the Split Common Fixed-Point of Asymptotically Quasi-Nonexpansive Operators
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.
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 . The CFP and SCFP have many applications such as approximation theory , image reconstruction, radiation therapy [3, 4], and control . The SFP in finite-dimensional space was first introduced by Censor and Elfving  for modeling inverse problems, which arise from phase retrievals, and in medical image reconstruction . 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 and are real Hilbert spaces, “” and “” denote strong and weak convergence, respectively, denotes the set of the fixed points of an operator , that is, and . Let and be two asymptotically quasi-nonexpansive mappings with nonempty fixed-point sets and , respectively. The split common fixed point problem for operators and is to find where is bounded linear. Denote the solution set of the two-operator SCFP by
The split common fixed point problem for quasi-nonexpansive mapping in the setting of Hilbert space was first introduced and studied by Moudafi . However, the algorithm presented in  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.
Recall that a mapping is said to be nonexpansive if
A mapping is called asymptotically nonexpansive if there exists a sequence with as such that, for all ,
A mapping is called quasi-nonexpansive if for all
A mapping is called asymptotically quasi-nonexpansive if there exists a sequence with as such that, for all ,
A mapping is said to be uniformly -Lipschitzian if there exists a constant such that for all ,
A mapping is said to be semicompact if for any bounded sequence with there exists a subsequence of such that converges strongly to a point .
Let be a Banach space. A mapping is said to be demiclosed at origin, if for any sequence with and , .
Lemma 1. Let be asymptotically quasi-nonexpansive and set for . Then, for any and , one has the following.(1); (2); (3);(4).
Lemma 2 (see ). Suppose is a sequence of nonnegative real numbers such that if and. Then, exists. In particular, if has a subsequence which converges strongly to zero, then .
3. The Algorithm and Its Asymptotic Convergence
We now give a description of an algorithm.
Algorithm 3. Initialization: let be arbitrary.
Iterative step: for , set , let where , and .
In what follows, we establish the weak convergence and strong convergence of Algorithm 3.
Lemma 4 (Opial ). Let be a Hilbert space and let be a sequence in such that there exists a nonempty set satisfying the following:(1)For every , exists.(2)Any weak cluster point of the sequence belongs to .
Then, there exists such that weakly converges to .
Theorem 5. Let and be two asymptotically quasi-nonexpansive operators with nonempty and . Assume that and are demiclosed at and , , and and are uniformly Lipschitzian. Then,(I)any sequence generated by Algorithm 3 converges weakly to a point ;(II)if is also semi-compact, then both and generated by Algorithm 3 converge strongly to a point .
Proof. First, we prove that for each , and exist. Also and .
By using (1) in Lemma 1, from (9) we obtain By deducing, it follows that that is, Setting and using (1) of Lemma 1, we obtain The key inequality mentioned above, combined with (12), yields Substituting (14) into (10) yields where . Since , we have , From Lemma 1, it follows that exists. By the virtue of (14), we know that exists. Therefore, from (15), we have By the assumptions on and , we get
Step 1. Now, we prove that and . As a matter of a fact, it follows from (9) that In view of (17), we have that Similarly, it follows from (9), (17), and (19) that
Step 2. We prove that and . Setting , since is uniformly -Lipschitzian continuous, it follows from (14) and (17) that in other words, Similarly, we have
Step 3. Finally, we prove that and , where . Denote by a weak-cluster point of , and denote by a subsequence of . Obviously, Then, from (23) and the demiclosedness of at , we obtain
it follows that, .
Noticing , it follows that . By the demiclosedness of at , we have Hence, , and therefore .
Since there is no more than one weak cluster point, the weak convergence of the whole sequence follows by applying Lemma 4 with .
The Proof of Conclusion (II). Since is semi-compact, it follows from (22) that there exists a subsequence of such that (some point in ). Since , this implies that . Therefore, as . Since for any and exist, we know that . This implies that and both converge strongly to a point . 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.
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.
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Copyright © 2013 Yazheng Dang and Yan Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.