Research Article | Open Access

# The Split Common Fixed Point Problem for Total Asymptotically Strictly Pseudocontractive Mappings

**Academic Editor:**Yonghong Yao

#### Abstract

The purpose of this paper is to propose an algorithm for solving *the split common fixed point problems for total asymptotically strictly pseudocontractive mappings* in infinite-dimensional Hilbert spaces. The results presented in the paper improve and extend some recent results of Moudafi (2011 and 2010), Xu (2010 and 2006), Censor and Segal (2009), Censor et al. (2005), Masad and Reich (2007), Censor et al. (2007), Yang (2004), and others.

#### 1. Introduction

Throughout this paper, we always assume that are real Hilbert spaces, “” denote by strong and weak convergence, respectively, and is the fixed point set of a mapping .

The *split common fixed point* problem (SCFP) is a generalization of the *split feasibility* problem (SEP) and the *convex feasibility problem* (CFP). It is worth mentioning that SFP in finite-dimensional 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 SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [3–5].

SEP in an infinite-dimensional Hilbert space can be found in [2, 4, 6–8]. Moreover the convex feasibility formalism is at the core of the modeling of many inverse problems and has been used to model significant real-world problems.

The split common fixed point problems for a class of *quasi-nonexpansive mappings* and *demicontractive mappings* in the setting of Hilbert space were first introduced and studied by Moudafi [9, 10].

The purpose of this paper is to introduce and study the following *split common fixed point problem for a more general class of total asymptotically strict pseudocontraction* (SCFP) in the framework of an infinite-dimensional Hilbert spaces which contains the quasi-nonexpansive mappings and the demicontractive mappings as its special cases:
where is a bounded linear operator, and are mappings , and . In the sequel we use to denote the set of solutions of (SCFP), that is,

#### 2. Preliminaries

We first recall some definitions, notations, and conclusions which will be needed 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 , .

A Banach space is said to have *the Opial property*, if for any sequence with ,

*Remark 2.1. *It is well known that each Hilbert space possesses the Opial property.

*Definition 2.2. *Let be a real Hilbert space, and let be nonempty and closed convex subset of .

(1)A mapping is said to be *-totally asymptotically strictly pseudocontractive*, if there exist a constant and sequences and with and such that for all
where is a continuous and strictly increasing function with .(2)A mapping is said to be *-asymptotically strictly pseudocontractive*, if there exist a constant and a sequence with such that(3)Especially, if there exists such that
then is called a *-strictly pseudocontractive mapping*.(4)A mapping is said to be *uniformly L-Lipschitzian*, if there exists a constant , such that
(5)A mapping is said to be *semicompact*, if for any bounded sequence with , there exists a subsequence such that converges strongly to some point .

*Remark 2.3. *If , and , then a -total asymptotically strictly pseudocontractive mapping is an -asymptotically strict pseudocontractive mapping, where .

Proposition 2.4. *Let be a ()-total asymptotically strictly pseudocontractive mapping. If , then for each and for each , the following inequalities hold and they are equivalent:
*

* Proof. *(I) Inequality (2.6) can be obtained from (2.2) immediately.

(II) (2.6) *⇔* (2.7) In fact, since
from (2.6) we have that
Simplifying it, the inequality (2.7) is obtained.

Conversely, from (2.7) the inequality (2.6) can be obtained immediately.

(III) (2.7) *⇔* (2.8) In fact, since
it follows from (2.7) that
Simplifying it, the inequality (2.8) is obtained.

Conversely, the inequality (2.7) can be obtained from (2.8) immediately.

This completes the proof of Proposition 2.4.

Lemma 2.5 (see [11]). *Let , , and be sequences of nonnegative real numbers satisfying
**
If and , then the limit exists.*

Lemma 2.6 (see [12]). *Let be a real Hilbert space. If is a sequence in weakly convergent to , then
*

Proposition 2.7. *Let be a real Hilbert space and let be a uniformly -Lipschitzian and -total asymptotically strictly pseudocontractive mapping. Then the demiclosedness principle holds for in the sense that if is a sequence in such that , and , then . In particular, if , and , then , that is, is demiclosed at origin.*

*Proof. * Since is bounded, we can define a function on by
Since , it follows from Lemma 2.6 that
In particular, for each ,
On the other hand, since is a ()-total asymptotically strictly pseudocontraction mapping, we get
Taking on both sides and observing the facts that , , and , we derive that
On the other hand, it follows from (2.17) that
Since , this together with (2.19) shows that . That is, ; hence .

#### 3. Split Common Fixed Point Problem

For solving the split common fixed point problem (1.1), let us assume that the following conditions are satisfied.

(1) and are two real Hilbert spaces, and is a bounded linear operator.

(2) is a uniformly -Lipschitzian and -total asymptotically strictly pseudocontractive mapping and is a uniformly -Lipschitzian and -total asymptotically strictly pseudocontractive mapping satisfying the following conditions:(i), ;(ii), and , ;(iii) and there exist two positive constants and such that for all .

We are now in a position to give the following result.

Theorem 3.1. *Let , and be the same as mentiond before. Let be the sequence generated by
**
where is a sequence in and is a constant satisfying the following conditions:*(iv)*for all and , where is a positive constant.(I) If (where is the set of solutions to (SCFP)-(1.1)), then converges weakly to a point .(II)In addition, if is also semicompact, then and both converge strongly to .*

*Proof. *The following is the proof of Theorem 3.1.*The Proof of Conclusion (I)*

(1) First we prove that for each , the following limits exist and

In fact, since is a continuous and increasing function, it results that , if , and , if . In either case, we can obtain that

For any given , hence , and , from (3.1) and (2.7) we have
On the other hand, since
In (2.8) taking and and noting , from (2.8) we have
Substituting (3.8) into (3.7), after simplifying it and then substituting the resultant result into (3.5), we have
Substituting (3.9) into (3.4) and simplifying it we have
where
By condition (iii) we have
By condition (ii), and . Hence it follows from Lemma 2.5 that the following limit exists:
Consequently, from (3.10) and (3.13) we have that
This together with the condition (iii) implies that
It follows from (3.5), (3.13), and (3.16) that the limit exists and
The conclusion (1) is proved.

(2) Next we prove that

In fact, it follows from (3.1) that
In view of (3.15) and (3.16) we have that
Similarly, it follows from (3.1), (3.16), and (3.20) that
The conclusion (3.18) is proved.

(3) Next we prove that

In fact, from (3.15) we have
Since is uniformly -Lipschitzian continuous, it follows from (3.18) and (3.23) that

Similarly, from (3.16) we have
Since is uniformly -Lipschitzian continuous, by the same way as above, from (3.18) and (3.25), we can also prove that

(4) Finally we prove that and which is a solution of (SCFP)-(1.1).

Since is bounded, there exists a subsequence such that (some point in ). From (3.22) we have
By Proposition 2.7, is demiclosed at zero; hence we know that .

Moreover, from (3.1) and (3.16) we have
Since is a linear bounded operator, it gets . In view of (3.22) we have
Again by Proposition 2.7, is demiclosed at zero, and we have . Summing up the above argument, it shows that ; that is, is a solution to the (SCFP)-(1.1).

Now we prove that and .

Suppose to the contrary, if there exists another subsequence such that with , then by virtue of (3.2) and the Opial property of Hilbert space, we have
This is a contradiction. Therefore, . By using (3.1) and (3.16), we have
*The Proof of Conclusion (II)*

By the assumption that is semicompact, it follows from (3.27) 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 . By virtue of (3.2) we know that and ; that is, and both converge strongly to .

This completes the proof of Theorem 3.1.

The following theorem can be obtained from Theorem 3.1 immediately.

Theorem 3.2. *Let and be two real Hilbert spaces, let be a bounded linear operator, let be a uniformly -Lipschitzian and -asymptotically strictly pseudocontractive mapping, and let be a uniformly -Lipschitzian and - asymptotically strictly pseudocontractive mapping satisfying the following conditions:*(i)*, ;*(ii)*, and .**Let be the sequence defined by (3.1), where is a sequence in and is a constant satisfying the following condition:*(iii)*, for all and , where is a constant. If , then the conclusions of Theorem 3.1 still hold.*

From Theorems 3.1 and 3.2 we can obtain the following.

Theorem 3.3. *Let and be two real Hilbert spaces, let be a bounded linear operator, be a uniformly -Lipschitzian and -strictly pseudocontractive mapping, and let be a uniformly -Lipschitzian and -strictly pseudocontractive mapping satisfying the following conditions:*(i)*, ;*(ii)* and both are demiclosed at origin.** Let be the sequence generated by
**
where is a sequence in and is a constant satisfying the following condition:*(iii)*, for all and , where is a constant. If , then the conclusions of Theorem 3.1 still hold.*

*Proof. *By the same way as given in the proof of Theorems 3.1 and 3.2 and noting that in the case of strictly pseudocontractive mapping the sequence in Theorem 3.2. Therefore we can prove that for each , the limits and exist and
In addition, if is also semicompact, we can also prove that and both converge strongly to .

*Remark 3.4. *Theorems 3.1 and 3.2 improve and extend the corresponding results of Censor et al. [4, 5], Yang [7], Moudafi [9, 10], Xu [13], Censor and Segal [14], Masad and Reich [15], and others.

#### Acknowledgments

The authors would like to express their thanks to the editors and the referees for their helpful comments and suggestions. This work was supported by the Natural Science Foundation of Yunnan Province and Yunnan University of Finance and Economics.

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#### Copyright

Copyright © 2012 S. S. Chang 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.