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Journal of Applied Mathematics
Volume 2012, Article ID 438121, 11 pages
http://dx.doi.org/10.1155/2012/438121
Research Article

Strong Convergence Theorems for the Split Common Fixed Point Problem for Countable Family of Nonexpansive Operators

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 2 March 2012; Revised 15 May 2012; Accepted 21 May 2012

Academic Editor: Nazim I. Mahmudov

Copyright © 2012 Cuijie Zhang and Songnian He. 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 a new iterative algorithm for solving the split common fixed point problem for countable family of nonexpansive operators. Under suitable assumptions, we prove that the iterative algorithm strongly converges to a solution of the problem.

1. Introduction

Let 𝐻1 and 𝐻2 be two real Hilbert spaces and let 𝐴𝐻1𝐻2 be a bounded linear operator. The split feasibility problem (SFP), see [1], is to find a point 𝑥 with the property: 𝑥𝐶,𝐴𝑥𝑄,(1.1) where 𝐶 and 𝑄 are nonempty closed convex subsets of 𝐻1 and 𝐻2, respectively. A more general form of the SFP is the so-called multiple-set split feasibility problem (MSSFP) which was recently introduced by Censor et al. [2]. Given integers 𝑝,𝑟1, the MSSFP is to find a point 𝑥 with the property: 𝑥𝑝𝑖=1𝐶𝑖,𝐴𝑥𝑟𝑗=1𝑄𝑗,(1.2) where {𝐶𝑖}𝑝𝑖=1 and {𝑄𝑗}𝑟𝑗=1 are nonempty closed convex subsets of 𝐻1 and 𝐻2, respectively. The SFP (1.1) and the MSSFP (1.2) serve as a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in this operator's ranges. Recently, the SFP (1.1) and the MSSFP (1.2) are widely applied in the image reconstructions [1, 3], the intensity-modulated radiation therapy [4, 5], and many other areas. The problems have been investigated by many researchers, for instance, [613]. The SFP (1.1) can be viewed as a special case of the convex feasibility problem (CFP) since the SFP (1.1) can be rewritten as 𝑥𝐶,𝑥𝐴1𝑄.(1.3) However, the methods for study the SFP (1.1) are actually different from those for the CFP in order to avoid the usage of the inverse 𝐴1. Byrne [6] introduced a so-called CQ algorithm: 𝑥𝑛+1=𝑃𝐶𝑥𝑛𝛾𝐴𝐼𝑃𝑄𝐴𝑥𝑛,𝑛0,(1.4) where the operator 𝐴1 is not relevant.

Censor and Segal in [14] firstly introduced the concept of the split common fixed point problem (SCFPP) in finite-dimensional Hilbert spaces. The SCFPP is a generalization of the convex feasibility problem (CFP) and the split feasibility problem (SFP). The SCFPP considers to find a common fixed point of a family of operators in 𝐻1 such that its image under a linear transformation 𝐴 is a common fixed point of another family of operators in 𝐻2. That is, the SCFPP is to find a point 𝑥 with the property: 𝑥𝑝𝑖=1𝑈Fix𝑖,𝐴𝑥𝑟𝑗=1𝑇Fix𝑗,(1.5) where 𝑈𝑖𝐻1𝐻1(𝑖=1,2,,𝑝) and 𝑇𝑗𝐻2𝐻2(𝑗=1,2,,𝑟) are nonlinear operators. If 𝑝=𝑟=1, the problem (1.5) deduces to the so-called two-set SCFPP, which is to find a point 𝑥 such that 𝑥Fix(𝑈),𝐴𝑥Fix(𝑇),(1.6) where 𝑈𝐻1𝐻1 and 𝑇𝐻2𝐻2 are nonlinear operators.

Censor and Segal in [14] considered the following iterative algorithm for the SCFPP (1.6) for Class- operators in finite-dimensional Hilbert spaces: 𝑥𝑛+1𝑥=𝑈𝑛𝛾𝐴(𝐼𝑇)𝐴𝑥𝑛,𝑛0,(1.7) where 𝑥0𝐻1, 0<𝛾<2/𝐴2 and 𝐼 is the identity operator.

Recently, in the infinite-dimensional Hilbert space, Wang and Xu [15] studied the SCFPP (1.5) and introduced the following iterative algorithm for Class- operators: 𝑥𝑛+1=𝑈[𝑛]𝑥𝑛𝛾𝐴𝐼𝑇[𝑛]𝐴𝑥𝑛,𝑛0(1.8) where [𝑛]=𝑛mod𝑝 and 𝑝=𝑟. Under some mild conditions, they proved that {𝑥𝑛} converges weakly to a solution of the SCFPP (1.5), extended and improved Censor and Segal's results. Moreover, they proved that the SCFPP (1.5) for the Class- operators is equivalent to a common fixed point problem. This is also a classical method. Many problems eventually converted to a common fixed point problem, see [1618]. Very recently, the split common fixed point problems for various types of operators were studied in [1921].

The above-mentioned results are about a finite number of operators; that is, the constraints are finite imposed on the solutions. In this paper, we consider the constraints are infinite, but countable. That is, we consider the generalized case of SCFPP for two countable families of operators (denoted GSCFPP), which is to find a point 𝑥 such that 𝑥𝑖=1𝑈Fix𝑖,𝐴𝑥𝑗=1𝑇Fix𝑗.(1.9) Of course, the GSCFPP is more general and widely used than the SCFPP. This is a novelty of this paper. At the same time, we consider the nonexpansive operator. The nonexpansive operator is important because it includes many types of nonlinear operator arising in applied mathematics. For instance, the projection and the identity operator are nonexpansive. We prove that the GSCFPP (1.9) for the nonexpansive operators is equivalent to a common fixed point problem. Very recently, Gu et al. [22] introduced a new iterative method for dealing with the countable family of operators. They studied the following iterative algorithm: 𝑦𝑛=𝑃𝐶𝛽𝑛𝑆𝑥𝑛+1𝛽𝑛𝑥𝑛,𝑥𝑛+1=𝑃𝐶𝛼𝑛𝑓𝑥𝑛+𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑇𝑖𝑦𝑛,(1.10) where 𝑆 and {𝑇𝑖}𝑖=1 are nonexpansive, 𝛼0=1, {𝛼𝑛} is strictly decreasing sequence in (0,1), and {𝛽𝑛} is a sequence in (0,1). Under some certain conditions on parameters, they proved that the sequence {𝑥𝑛} converges strongly to 𝑥𝑖=1𝐹(𝑇𝑖). On the other hand, from weakly convergence to strongly convergence, the viscosity approximation method is also one of the classical methods, see [2224].

Motivated and inspired by the above results, we introduce the following algorithm: 𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑥𝑛+𝛾𝐴𝑇𝑖𝐼𝐴𝑥𝑛.(1.11) Under some certain conditions, we prove that the sequence {𝑥𝑛} generated by (1.11) converges strongly to the solution of the GSCFPP (1.9).

2. Preliminaries

Throughout this paper, we write 𝑥𝑛𝑥 and 𝑥𝑛𝑥 to indicate that {𝑥𝑛} converges weakly to 𝑥 and converges strongly to 𝑥, respectively.

Let 𝐻 be a real Hilbert space. An operator 𝑇𝐻𝐻 is said to be nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦 for all 𝑥,𝑦𝐻. The set of fixed points of 𝑇 is denoted by Fix(𝑇). It is known that Fix(𝑇) is closed and convex, see [25]. An operator 𝑓𝐻𝐻 is called contraction if there exists a constant 𝜌[0,1) such that 𝑓(𝑥)𝑓(𝑦)𝜌𝑥𝑦 for all 𝑥,𝑦𝐻. Let 𝐶 be a nonempty closed convex subset of 𝐻. For each 𝑥𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃𝐶𝑥, such that 𝑥𝑃𝐶𝑥𝑥𝑦 for every 𝑦𝐶. 𝑃𝐶 is called the metric projection of 𝐻 onto 𝐶. It is known that, for each 𝑥𝐻, 𝑥𝑃𝐶𝑥,𝑦𝑃𝐶𝑥0(2.1) for all 𝑦𝐶.

In order to prove our main results, we collect the following lemmas in this section.

Lemma 2.1 (see [26]). Let 𝐻 be a Hilbert space, 𝐶 a closed convex subset of 𝐻, and 𝑇𝐶𝐶 a nonexpansive operator with Fix(𝑇). If {𝑥𝑛} is a sequence in 𝐶 weakly converging to 𝑥𝐶 and {(𝐼𝑇)𝑥𝑛} converges strongly to 𝑦𝐶, then (𝐼𝑇)𝑥=𝑦. In particular, if 𝑦=0, then 𝑥Fix(𝑇).

Lemma 2.2 (see [23]). Assume {𝑎𝑛} is a sequence of nonnegative real numbers such that 𝑎𝑛+11𝛾𝑛𝑎𝑛+𝛿𝑛,𝑛0,(2.2) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence such that (i)𝑛=1𝛾𝑛=,(ii)limsup𝑛𝛿𝑛/𝛾𝑛0𝑜𝑟𝑛=1|𝛿𝑛|<.
Then lim𝑛𝑎𝑛=0.

3. Main Results

Now we state and prove our main results of this paper.

Theorem 3.1. Let {𝑈𝑛} and {𝑇𝑛} be sequences of nonexpansive operators on real Hilbert spaces 𝐻1 and 𝐻2, respectively. Let 𝑓𝐻1𝐻1 be a contraction with coefficient 𝜌[0,1). Suppose that the solution set Ω of GSCFPP (1.9) is nonempty. Let 𝑥1𝐻1 and 0<𝛾<2/𝐴2. Set 𝛼0=1, and let {𝛼𝑛}(0,1] be a strictly decreasing sequence satisfying the following conditions: (i)lim𝑛𝛼𝑛=0; (ii)𝑛=1𝛼𝑛=; (iii)𝑛=1|𝛼𝑛+1𝛼𝑛|<. Then the sequence {𝑥𝑛} generated by (1.11) converges strongly to 𝑤Ω, where 𝑤=𝑃Ω𝑓(𝑤).

Proof. We proceed with the following steps.
Step 1. First show that there exists 𝑤Ω such that 𝑤=𝑃Ω𝑓(𝑤).
In fact, since 𝑓 is a contraction with coefficient 𝜌, we have 𝑃Ω𝑓(𝑥)𝑃Ω𝑓(𝑦)𝑓(𝑥)𝑓(𝑦)𝜌𝑥𝑦(3.1) for every 𝑥,𝑦𝐻1. Hence 𝑃Ω𝑓 is also a contraction of 𝐻1 into itself. Therefore, there exists a unique 𝑤𝐻1 such that 𝑤=𝑃Ω𝑓(𝑤). At the same time, we note that 𝑤Ω.
Step 2. Now we show that {𝑥𝑛} is bounded.
For simplicity, we set 𝑉𝑖=𝐼+𝛾𝐴(𝑇𝑖𝐼)𝐴. Then we can rewrite (1.11) to 𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛.(3.2) Observe that (𝑇𝑖𝐼)𝐴𝑥(𝑇𝑖𝐼)𝐴𝑦2=𝐴𝑥𝐴𝑦2+𝑇𝑖𝐴𝑥𝑇𝑖𝐴𝑦22𝐴𝑥𝐴𝑦,𝑇𝑖𝐴𝑥𝑇𝑖𝐴𝑦2𝐴𝑥𝐴𝑦22𝐴𝑥𝐴𝑦,𝑇𝑖𝐴𝑥𝑇𝑖𝑇𝐴𝑦=2𝐴𝑥𝐴𝑦,𝑖𝑇𝐼𝐴𝑥𝑖𝐼𝐴𝑦(3.3) for all 𝑥,𝑦𝐻1. Thus it follows that 𝑉𝑖𝑥𝑉𝑖𝑦=𝐼+𝛾𝐴𝑇𝑖𝐴𝐼𝑥𝐼+𝛾𝐴𝑇𝑖𝐴𝑦𝐼2𝑥𝑦2+𝛾2𝐴2(𝑇𝑖𝐼)𝐴𝑥(𝑇𝑖𝐼)𝐴𝑦2𝑇+2𝛾𝐴𝑥𝐴𝑦,𝑖𝑇𝐼𝐴𝑥𝑖𝐼𝐴𝑦𝑥𝑦2+𝛾𝛾𝐴2𝑇1𝑖𝑇𝐼𝐴𝑥𝑖𝐼𝐴𝑦2.(3.4) For 0<𝛾<1/𝐴2, we can immediately obtain that 𝑉𝑖 is a nonexpansive operator for every 𝑖.
Let 𝑝Ω, then 𝑈𝑖𝑝=𝑝 and 𝑇𝑖𝐴𝑝=𝐴𝑝 for every 𝑖1. Thus (𝑇𝑖𝐼)𝐴𝑝=0, which implies that 𝑉𝑖𝑝=𝑝. Since 𝑛𝑖=1(𝛼𝑖1𝛼𝑖)=1𝛼𝑛, we have 𝑥𝑛+1𝑝𝛼𝑛𝑓𝑥𝑛+𝑝𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝑝𝛼𝑛𝑓𝑥𝑛𝑓(𝑝)+𝛼𝑛𝑓(𝑝)𝑝+𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑉𝑖𝑥𝑛𝑝𝛼𝑛𝜌𝑥𝑛𝑝+𝛼𝑛𝑓(𝑝)𝑝+1𝛼𝑛𝑥𝑛=𝑝1𝛼𝑛(𝑥1𝜌)𝑛𝑝+𝛼𝑛(11𝜌)(𝑥1𝜌𝑓𝑝)𝑝max𝑛,1𝑝.1𝜌𝑓(𝑝)𝑝(3.5) Then it follows that 𝑈𝑛𝑉𝑛𝑥𝑛1𝑉𝑝𝑛𝑥𝑛1𝑥𝑝𝑛1𝑥𝑝max1,1𝑝1𝜌𝑓(𝑝)𝑝(3.6) for every 𝑛. This shows that {𝑥𝑛} and {𝑈𝑛𝑉𝑛𝑥𝑛1} is bounded. Hence, {𝑓(𝑥𝑛)} is also bounded.
Step 3. We show lim𝑛𝑥𝑛+1𝑥𝑛=0.
From (3.2), we have 𝑥𝑛+1𝑥𝑛=𝛼𝑛𝑓𝑥𝑛+𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝛼𝑛1𝑓𝑥𝑛1𝑛1𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛1𝛼𝑛𝑓𝑥𝑛𝑥𝑓𝑛1+𝛼𝑛1𝛼𝑛𝑓𝑥𝑛1+𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝑈𝑖𝑉𝑖𝑥𝑛1+𝛼𝑛1𝛼𝑛𝑈𝑛𝑉𝑛𝑥𝑛1𝛼𝑛𝜌𝑥𝑛𝑥𝑛1+𝛼𝑛1𝛼𝑛𝑓𝑥𝑛1+𝑈𝑛𝑉𝑛𝑥𝑛1+1𝛼𝑛𝑥𝑛𝑥𝑛11𝛼𝑛𝑥(1𝜌)𝑛𝑥𝑛1+𝛼𝑛1𝛼𝑛𝑀,(3.7) where 𝑀 is a constant such that 𝑀=sup𝑛1𝑓𝑥𝑛1+𝑈𝑛𝑉𝑛𝑥𝑛1.(3.8) From (i), (ii), (iii), and Lemma 2.2, it follows that lim𝑛𝑥𝑛+1𝑥𝑛=0
Step 4. We show lim𝑛𝑈𝑖𝑥𝑛𝑥𝑛=0 and lim𝑛𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛=0 for 𝑖.
We first show lim𝑛𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛=0 for 𝑖. Since 𝑝Ω, we note that 𝑥𝑛𝑝2𝑉𝑖𝑥𝑛𝑉𝑖𝑝2𝑈𝑖𝑉𝑖𝑥𝑛𝑈𝑖𝑉𝑖𝑝2=𝑈𝑖𝑉𝑖𝑥𝑛𝑝2=𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛+𝑥𝑛𝑝2=𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛2+𝑥𝑛𝑝2+2𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛,𝑥𝑛𝑝,(3.9) which implies that 12𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛2𝑥𝑛𝑈𝑖𝑉𝑖𝑥𝑛,𝑥𝑛𝑝.(3.10) Using (3.2) and (3.10), we deduce 12𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛2𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑥𝑛𝑈𝑖𝑉𝑖𝑥𝑛,𝑥𝑛=𝑝1𝛼𝑛𝑥𝑛𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛,𝑥𝑛=𝑝1𝛼𝑛𝑥𝑛𝑥𝑛+1+𝛼𝑛𝑓𝑥𝑛,𝑥𝑛=𝑥𝑝𝑛𝑥𝑛+1,𝑥𝑛𝑝+𝛼𝑛𝑓𝑥𝑛𝑥𝑛,𝑥𝑛𝑥𝑝𝑛𝑥𝑛+1𝑥𝑛𝑝+𝛼𝑛𝑓𝑥𝑛𝑥𝑛𝑥𝑛.𝑝(3.11) Noting that lim𝑛𝑥𝑛𝑥𝑛+1=0 and lim𝑛𝛼𝑛=0, then we immediately obtain 𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛2=lim𝑛𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛2=0.(3.12) Since {𝛼𝑛} is strictly decreasing, it follows that lim𝑛𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛=0,forevery𝑖.(3.13)
Next we show lim𝑛𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛=0, for every 𝑖. Note for every 𝑖, 𝐴𝑥𝑛𝐴𝑝2𝑇𝑖𝐴𝑥𝑛𝑇𝑖𝐴𝑝2=𝑇𝑖𝐴𝑥𝑛𝐴𝑝2=𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛+𝐴𝑥𝑛𝐴𝑝2=𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛2+𝐴𝑥𝑛𝐴𝑝2+2𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛,𝐴𝑥𝑛𝐴𝑝,(3.14) which follows that 𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛,𝐴𝑥𝑛1𝐴𝑝2𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛2,(3.15) for every 𝑖. From (3.2), we have 𝑥𝑛+1𝑝2=𝛼𝑛𝑓𝑥𝑛+𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝑝2𝛼𝑛𝑓𝑥𝑛𝑝2+𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝑝2𝛼𝑛𝑓𝑥𝑛𝑝2+𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑥𝑛+𝛾𝐴𝑇𝑖𝐼𝐴𝑥𝑛𝑝2𝛼𝑛𝑓𝑥𝑛𝑝2+𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑥𝑛𝑝2+𝛾2𝐴2𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛2+2𝛾𝐴𝑥𝑛𝐴𝑝,𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛.(3.16) By (3.15), it follows that 𝑥𝑛+1𝑝2𝛼𝑛𝑓𝑥𝑛𝑝2+1𝛼𝑛𝑥𝑛𝑝2+𝛾𝛾𝐴21𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛2.(3.17) Thus, 𝛾1𝛾𝐴2𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛2𝛼𝑛𝑓𝑥𝑛𝑝2𝑥𝑛𝑝2+𝑥𝑛𝑝2𝑥𝑛+1𝑝2=𝛼𝑛𝑓𝑥𝑛𝑝2𝑥𝑛𝑝2+𝑥𝑛+𝑥𝑝𝑛+1𝑥𝑝𝑛𝑥𝑝𝑛+1𝑝𝛼𝑛𝑓𝑥𝑛𝑝2𝑥𝑛𝑝2+𝑥𝑛+𝑥𝑝𝑛+1𝑥𝑝𝑛𝑥𝑛+1.(3.18) Using lim𝑛𝛼𝑛=0 and lim𝑛𝑥𝑛+1𝑥𝑛=0, we have lim𝑛𝛾1𝛾𝐴2𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛2=0.(3.19) By 0<𝛾<1/𝐴2, there holds 𝑖=1𝛼𝑖1𝛼𝑖𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛2=lim𝑛𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛2=0.(3.20) Since {𝛼𝑛} is strictly decreasing, we obtain lim𝑛𝑇𝑖𝐴𝑥𝑛𝐴𝑥𝑛=0,𝑖.(3.21)
Last we show lim𝑛𝑈𝑖𝑥𝑛𝑥𝑛=0 for every 𝑖. In fact, we note that for every 𝑖, 𝑈𝑖𝑥𝑛𝑥𝑛𝑈𝑖𝑥𝑛𝑈𝑖𝑉𝑖𝑥𝑛+𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛𝑥𝑛𝑉𝑖𝑥𝑛+𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛=𝑥𝑛𝑥𝑛𝛾𝐴𝑇𝑖𝐼𝐴𝑥𝑛+𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛𝑇𝛾𝐴𝑖𝐼𝐴𝑥𝑛+𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑛.(3.22) Then by (3.13) and (3.21), we obtain lim𝑛𝑈𝑖𝑥𝑛𝑥𝑛=0,𝑖.(3.23)
Step 5. Show limsup𝑛𝑓(𝑤)𝑤,𝑥𝑛𝑤0, where 𝑤=𝑃Ω𝑓(𝑤).
Since {𝑥𝑛} is bounded, there exist a point 𝑣𝐻1 and a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} such that limsup𝑛𝑓(𝑤)𝑤,𝑥𝑛𝑤=lim𝑘𝑓(𝑤)𝑤,𝑥𝑛𝑘𝑤(3.24) and 𝑥𝑛𝑘𝑣. Since 𝐴 is a bounded linear operator, we have 𝐴𝑥𝑛𝑘𝐴𝑣. Now applying (3.21), (3.23), and Lemma 2.1, we conclude that 𝑣Fix(𝑈𝑖) and 𝐴𝑣Fix(𝑇𝑖) for every 𝑖. Hence, 𝑣Ω. Since Ω is closed and convex, by (2.1), we get limsup𝑛𝑓(𝑤)𝑤,𝑥𝑛𝑤=lim𝑘𝑓(𝑤)𝑤,𝑥𝑛𝑘𝑤=𝑓(𝑤)𝑤,𝑣𝑤0.(3.25)
Step 6. Show 𝑥𝑛𝑤=𝑃Ω𝑓(𝑤).
Since 𝑤Ω, we have 𝑈𝑖𝑤=𝑤 and 𝑇𝑖𝐴𝑤=𝐴𝑤 for every 𝑖. It follows that 𝑉𝑖𝑤=𝑤. Using (3.2), we have 𝑥𝑛+1𝑤2=𝛼𝑛𝑓𝑥𝑛+𝑤𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝑤,𝑥𝑛+1𝑤=𝛼𝑛𝑓𝑥𝑛𝑓(𝑤),𝑥𝑛+1𝑤+𝛼𝑛𝑓(𝑤)𝑤,𝑥𝑛+1+𝑤𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝑤,𝑥𝑛+1𝑤𝛼𝑛𝜌𝑥𝑛𝑥𝑤𝑛+1𝑤+𝛼𝑛𝑓(𝑤)𝑤,𝑥𝑛+1+𝑤𝑛𝑖=1𝛼𝑖1𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛𝑥𝑤𝑛+11𝑤2𝛼𝑛𝜌𝑥𝑛𝑤2+𝑥𝑛+1𝑤2+𝛼𝑛𝑓(𝑤)𝑤,𝑥𝑛+1+1𝑤21𝛼𝑛𝑥𝑛𝑤2+𝑥𝑛+1𝑤2121𝛼𝑛𝑥(1𝜌)𝑛𝑤2+12𝑥𝑛+1𝑤2+𝛼𝑛𝑓(𝑤)𝑤,𝑥𝑛+1,𝑤(3.26) which implies that 𝑥𝑛+1𝑤21𝛼𝑛𝑥(1𝜌)𝑛𝑤2+2𝛼𝑛1(1𝜌)1𝜌𝑓(𝑤)𝑤,𝑥𝑛+1,𝑤(3.27) for every 𝑛. Consequently, according to (3.25), 𝜌[0,1), and Lemma 2.2, we deduce that {𝑥𝑛} converges strongly to 𝑤=𝑃Ω(𝑤). This completes the proof.

Remark 3.2. If we set 𝛼𝑛=1/𝑛 and 𝑓(𝑥)=𝑢 for all 𝑥𝐻1, where 𝑢 is an arbitrary point in 𝐻1, it is easily seen that our conditions are satisfied.

Corollary 3.3. Let 𝑈𝐻1𝐻1 and 𝑇𝐻2𝐻2 be nonexpansive operators. Let 𝑓𝐻1𝐻1 be a contraction with coefficient 𝜌[0,1). Suppose that the solution set Ω of SCFPP (1.6) is nonempty. Let 𝑥1𝐻1 and define a sequence {𝑥𝑛} by the following algorithm: 𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑈𝑥𝑛+𝛾𝐴(𝑇𝐼)𝐴𝑥𝑛,(3.28) where 0<𝛾<1/𝐴2, 𝛼0=1 and {𝛼𝑛}(0,1] is a strictly decreasing sequence satisfying the following conditions: (i)lim𝑛𝛼𝑛=0; (ii)𝑛=1𝛼𝑛=; (iii)𝑛=1|𝛼𝑛+1𝛼𝑛|<. Then {𝑥𝑛} converges strongly to 𝑤Ω, where 𝑤=𝑃Ω𝑓(𝑤).

Proof. Set {𝑈𝑛} and {𝑇𝑛} to be sequences of operators defined by 𝑈𝑛=𝑈 and 𝑇𝑛=𝑇 for all 𝑛 in Theorem 3.1. Then by Theorem 3.1 we obtain the desired result.

Remark 3.4. By adding more operators to the families {𝑈𝑛} and {𝑇𝑛} by setting 𝑈𝑖=𝐼 for 𝑖𝑝+1 and 𝑇𝑗=𝐼 for 𝑗𝑟+1, the SCFPP (1.5) can be viewed as a special case of the GSCFPP (1.9).

Acknowledgment

This research is supported by the NSFC Tianyuan Youth Foundation of Mathematics of China (No. 11126136), the Fundamental Research Funds of Science for the Central Universities (Program No. ZXH2012K001), and the science research foundation program in Civil Aviation University of China (07kys09).

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