RETRACTED
This article has been retracted as it is found to contain a substantial amount of material, without referencing, from the paper “The Split Common Fixed Point Problem for Total Asymptotically Strictly Pseudocontractive Mappings,” Shih-sen Chang, Lin Wang, Yong Kun Tang and Li Yang, Journal of Applied Mathematics, vol. 2012, Article ID 385638, 13 pages, 2012. doi:10.1155/2012/385638.

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 768591, 9 pages

doi:10.1155/2012/768591

Research Article

The Split Common Fixed Point Problem for Quasi-Total Asymptotically Nonexpansive Uniformly Lipschitzian Mappings

Jing Na,¹ Lin Wang,¹ and Zhaoli Ma²

¹College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan, Kunming 650221, China

²School of Information Engineering, The College of Arts and Sciences, Yunnan Normal University, Yunnan, Kunming 650222, China

Received 9 February 2012; Accepted 12 March 2012

Academic Editor: Paul Eloe

Copyright © 2012 Jing Na et al. 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.

Abstract.

We introduce an algorithm for solving the split common fixed point problem for quasi-total asymptotically nonexpansive uniformly Lipschitzian mapping in Hilbert spaces. The results presented in this paper improve and extend some recent corresponding results.

1. Introduction and Preliminaries

Let and be real Hilbert spaces with inner product and norm . Let and be nonempty closed convex subsets of and , respectively. The split feasibility problem is formulated as finding a point with the property

where is a bounded linear operator.

The 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]. Recently, it has been found that the can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [1, 3–7]. The in infinite-dimensional Hilbert spaces can be found in [8–16].

The split common fixed point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP), see [4]. Let and be two mappings satisfying and , respectively. The split common fixed point problem for mappings and is to find a point with the property

where is a bounded linear operator from to . We use to denote the set of solutions of (1.2).

We first recall some definitions, notations, and conclusions which will be used in proving our main results.

Let be a Banach space. A mapping is said to be demiclosed at origin, if for any sequence with and , then .

A Banach space is said to satisfy Opial's condition, if for any sequence in , implies that

It is well known that every Hilbert space satisfies Opial's condition.

Definition 1.1. Let be a real Hilbert space, a mapping from into itself, and the fixed point set of nonempty.

(1) is said to be quasi-nonexpansive if for all

(2) is said to be quasi-asymptotically nonexpansive if there exists a sequence with as such that for all

(3) is said to be -quasi-total asymptotically nonexpansive if for all

where is a continuous and strictly increasing function with and and are two nonnegative real sequences satisfying and as . The class of mappings was introduced by Alber et al. [17] in 2006.(4) is said to be uniformly L-Lipschitzian if there exists a constant such that for all

(5) is said to be semicompact if for any bounded sequence with , there exists a subsequence of such that converges strongly to a point .

Remark 1.2. From Definition 1.1 we can see that a quasi-nonexpansive mapping or an asymptotically quasi-nonexpansive mapping is a -quasi-total asymptotically nonexpansive mapping. But the converse does not hold.

In [9], Moudafi proposed the following iterative algorithm for solving split common fixed problem of quasi-nonexpansive mappings: for arbitrarily chosen ,

and proved that converges weakly to a split common fixed point , where and are two quasi-nonexpansive mappings and is a bounded linear operator.

Inspired by the work of Moudafi [9, 10], very recently, Qin et al. [12] introduced the following iterative algorithm to study multiples-sets feasibility problem of a finite family of asymptotically quasi-nonexpansive mappings. For arbitrarily chosen , is defined as follows:

where , , , is a bounded linear operator. They also proved that converges strongly or weakly to a multiple-sets split common fixed point of a finite family of asymptotically quasi-nonexpansive mappings in Hilbert spaces under some suitable conditions.

Motivated and inspired by the work of Moudafi [9, 10] and Qin et al. [12], in this paper, we study the (1.2) of -quasi-total asymptotically nonexpansive mappings in Hilbert spaces and obtain some of the strong and weak convergence of the presented algorithm to some . The results obtained in this paper improve and extend the result of Qin et al. [12], Xu [14], and Yang [16] and others.

By using the well-known equality in Hilbert spaces, we can easily show the following proposition. The proof is omitted.

Proposition 1.3. Let be a -quasi-total asymptotically nonexpansive mapping. Then for each and , the following inequalities hold:

Lemma 1.4 (18). Let , , and be sequences of nonnegative real numbers satisfying

If and , then the exists.

2. Main Results

For solving split common fixed point problem (1.2), we assume that the following conditions are satisfied:

(i) and are two real Hilbert spaces, and is a bounded linear operator;(ii) is uniformly -Lipschitzian and -quasi-total asymptotically nonexpansive mapping and is uniformly -Lipschitzian and -quasi-total asymptotically nonexpansive mapping satisfying the following conditions:

        (a) , ;         (b) , , , and ;        (c) and there exists two positive constant and such that          for all .

Theorem 2.1. Let , and be the same as above. Let be the sequence generated by: arbitrarily chosen

where is a sequence in for some and is a constant satisfying the following condition:

        (d).

If and both are demi-closed at origin and , then

(I)the sequence converges weakly to a split common fixed point ;(II)in addition, if is also semicompact, then and both converge strongly to a .

Proof. (I) The proof will be divided into 4 steps.

Step 1. We prove that and exist, and for each . Since is a continuous and increasing function, as , we can obtain that

Taking , that is and , and using (1.10), (2.1), and (2.2), we have

where

Using (1.11), we have

Substituting (2.5) and (2.7) into (2.4) and simplifying, we obtain

Substituting (2.8) into (2.3) and simplifying, we have

where

Since and , so and . By Condition (4), we have

Therefore, it follows from Lemma 1.4 that exists.We now prove that exists for each . Since exists, from (2.9), we have

This together with Condition (4) implies that

Thus, since exists, it follows from (2.4) and (2.14) that exists and .

Step 2. Now we prove that and .

As a matter of fact, it follows from (2.1) that

In view of (2.13) and (2.14) we have that

Similarly, it follows from (2.1), (2.14), and (2.16) that

Step 3. Next, we prove that and as .

Setting , since is uniformly L-Lipschitzian continuous, it follows from (2.13), (2.16), and (2.17) that

Similarly, we have

Step 4. Finally, we prove that and , where . Since is bounded, there exists a subsequence of such that (some point in ). From (2.18) we have . Since is demi-closed at zero, we know that . Moreover, it follows from (2.1) and (2.14) that

Since is a linear bounded operator, it gets . In view of (2.19) we have .

Again since is demi-closed at zero, we know that . This implies that .

Assume that there exists another subsequence of such that converges weakly to a point with . Using the same argument above, we know that . Since each Hilbert space possesses Opial’s property, we have

which is a contradiction. This implies that converges weakly to the point . Since , we know that converges weakly to . The proof of conclusion (I) is completed.

Proof of Conclusion (II). Since is semi-compact, it follows from Step 4 that there exists a subsequence of (without loss of generality, we still denote it by ) such that (some point in ). Since , this implies that . And so as . Since and exist, and for each , we know that . This implies that and both converge strongly to a . The proof is completed.

Acknowldgments

The authors would like to express their thanks to the referees for helpful suggestions and comments. This work was supported by the Natural Science Foundation of Yunnan Province (Grant no. 2011FB074), P.R. China.

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