Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 438121 | https://doi.org/10.1155/2012/438121

Cuijie Zhang, Songnian He, "Strong Convergence Theorems for the Split Common Fixed Point Problem for Countable Family of Nonexpansive Operators", Journal of Applied Mathematics, vol. 2012, Article ID 438121, 11 pages, 2012. https://doi.org/10.1155/2012/438121

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

Academic Editor: Nazim I. Mahmudov
Received02 Mar 2012
Revised15 May 2012
Accepted21 May 2012
Published02 Jul 2012

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 𝐴∶𝐻1→𝐻2 be a bounded linear operator. The split feasibility problem (SFP), see [1], is to find a point 𝑥∗ with the property: 𝑥∗∈𝐶,𝐴𝑥∗∈𝑄,(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,𝑟≥1, the MSSFP is to find a point 𝑥∗ with the property: 𝑥∗∈𝑝𝑖=1𝐶𝑖,𝐴𝑥∗∈𝑟𝑗=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, [6–13]. 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 𝑥∗∈𝐶,𝑥∗∈𝐴−1𝑄.(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 𝐴−1. Byrne [6] introduced a so-called CQ algorithm: 𝑥𝑛+1=𝑃𝐶𝑥𝑛−𝛾𝐴∗𝐼−𝑃𝑄𝐴𝑥𝑛,𝑛≥0,(1.4) where the operator 𝐴−1 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: 𝑥∗∈𝑝𝑖=1𝑈Fix𝑖,𝐴𝑥∗∈𝑟𝑗=1𝑇Fix𝑗,(1.5) where 𝑈𝑖∶𝐻1→𝐻1(𝑖=1,2,…,𝑝) and 𝑇𝑗∶𝐻2→𝐻2(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 𝑥∗∈Fix(𝑈),𝐴𝑥∗∈Fix(𝑇),(1.6) where 𝑈∶𝐻1→𝐻1 and 𝑇∶𝐻2→𝐻2 are nonlinear operators.

Censor and Segal in [14] considered the following iterative algorithm for the SCFPP (1.6) for Class-ℑ operators in finite-dimensional Hilbert spaces: 𝑥𝑛+1𝑥=𝑈𝑛−𝛾𝐴∗(𝐼−𝑇)𝐴𝑥𝑛,𝑛≥0,(1.7) where 𝑥0∈𝐻1, 0<𝛾<2/‖𝐴‖2 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-ℑ operators: 𝑥𝑛+1=𝑈[𝑛]𝑥𝑛−𝛾𝐴∗𝐼−𝑇[𝑛]𝐴𝑥𝑛,𝑛≥0(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-ℑ 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 [16–18]. Very recently, the split common fixed point problems for various types of operators were studied in [19–21].

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 ğ‘¥âˆ—âˆˆâˆžî™ğ‘–=1𝑈Fix𝑖,ğ´ğ‘¥âˆ—âˆˆâˆžî™ğ‘—=1𝑇Fix𝑗.(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=𝑃𝐶𝛽𝑛𝑆𝑥𝑛+1−𝛽𝑛𝑥𝑛,𝑥𝑛+1=𝑃𝐶𝛼𝑛𝑓𝑥𝑛+𝑛𝑖=1𝛼𝑖−1−𝛼𝑖𝑇𝑖𝑦𝑛,(1.10) where 𝑆 and {𝑇𝑖}âˆžğ‘–=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 ğ‘¥âˆ—âˆˆâ‹‚âˆžğ‘–=1𝐹(𝑇𝑖). On the other hand, from weakly convergence to strongly convergence, the viscosity approximation method is also one of the classical methods, see [22–24].

Motivated and inspired by the above results, we introduce the following algorithm: 𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝑛𝑖=1𝛼𝑖−1−𝛼𝑖𝑈𝑖𝑥𝑛+𝛾𝐴∗𝑇𝑖−𝐼𝐴𝑥𝑛.(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 𝑥𝑛⇀𝑥 and 𝑥𝑛→𝑥 to indicate that {𝑥𝑛} converges weakly to 𝑥 and converges strongly to 𝑥, respectively.

Let 𝐻 be a real Hilbert space. An operator 𝑇∶𝐻→𝐻 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻. 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 𝑓∶𝐻→𝐻 is called contraction if there exists a constant 𝜌∈[0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝜌‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻. Let 𝐶 be a nonempty closed convex subset of 𝐻. For each 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃𝐶𝑥, such that ‖𝑥−𝑃𝐶𝑥‖≤‖𝑥−𝑦‖ for every 𝑦∈𝐶. 𝑃𝐶 is called the metric projection of 𝐻 onto 𝐶. It is known that, for each 𝑥∈𝐻, ⟨𝑥−𝑃𝐶𝑥,𝑦−𝑃𝐶𝑥⟩≤0(2.1) for all 𝑦∈𝐶.

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 𝑇∶𝐶→𝐶 a nonexpansive operator with Fix(𝑇)≠∅. If {𝑥𝑛} is a sequence in 𝐶 weakly converging to 𝑥∈𝐶 and {(𝐼−𝑇)𝑥𝑛} converges strongly to 𝑦∈𝐶, then (𝐼−𝑇)𝑥=𝑦. In particular, if 𝑦=0, then 𝑥∈Fix(𝑇).

Lemma 2.2 (see [23]). Assume {ğ‘Žğ‘›} is a sequence of nonnegative real numbers such that ğ‘Žğ‘›+1≤1âˆ’ğ›¾ğ‘›î€¸ğ‘Žğ‘›+𝛿𝑛,𝑛≥0,(2.2) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence such that (i)âˆ‘âˆžğ‘›=1𝛾𝑛=∞,(ii)limsupğ‘›â†’âˆžğ›¿ğ‘›/𝛾𝑛∑≤0ğ‘œğ‘Ÿâˆžğ‘›=1|𝛿𝑛|<∞.
Then limğ‘›â†’âˆžğ‘Žğ‘›=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 𝑓∶𝐻1→𝐻1 be a contraction with coefficient 𝜌∈[0,1). Suppose that the solution set Ω of GSCFPP (1.9) is nonempty. Let 𝑥1∈𝐻1 and 0<𝛾<2/‖𝐴‖2. Set 𝛼0=1, and let {𝛼𝑛}⊂(0,1] be a strictly decreasing sequence satisfying the following conditions: (i)limğ‘›â†’âˆžğ›¼ğ‘›=0; (ii)âˆ‘âˆžğ‘›=1𝛼𝑛=∞; (iii)âˆ‘âˆžğ‘›=1|𝛼𝑛+1−𝛼𝑛|<∞. Then the sequence {𝑥𝑛} generated by (1.11) converges strongly to 𝑤∈Ω, where 𝑤=𝑃Ω𝑓(𝑤).

Proof. We proceed with the following steps.
Step 1. First show that there exists 𝑤∈Ω such that 𝑤=𝑃Ω𝑓(𝑤).
In fact, since 𝑓 is a contraction with coefficient 𝜌, we have ‖‖𝑃Ω𝑓(𝑥)−𝑃Ω‖‖≤𝑓(𝑦)‖𝑓(𝑥)−𝑓(𝑦)‖≤𝜌‖𝑥−𝑦‖(3.1) for every 𝑥,𝑦∈𝐻1. Hence 𝑃Ω𝑓 is also a contraction of 𝐻1 into itself. Therefore, there exists a unique 𝑤∈𝐻1 such that 𝑤=𝑃Ω𝑓(𝑤). At the same time, we note that 𝑤∈Ω.
Step 2. Now we show that {𝑥𝑛} is bounded.
For simplicity, we set 𝑉𝑖=𝐼+𝛾𝐴∗(𝑇𝑖−𝐼)𝐴. Then we can rewrite (1.11) to 𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝑛𝑖=1𝛼𝑖−1−𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛.(3.2) Observe that ‖‖(𝑇𝑖−𝐼)𝐴𝑥−(𝑇𝑖‖‖−𝐼)𝐴𝑦2=‖𝐴𝑥−𝐴𝑦‖2+‖‖𝑇𝑖𝐴𝑥−𝑇𝑖‖‖𝐴𝑦2−2⟨𝐴𝑥−𝐴𝑦,𝑇𝑖𝐴𝑥−𝑇𝑖𝐴𝑦⟩≤2‖𝐴𝑥−𝐴𝑦‖2−2⟨𝐴𝑥−𝐴𝑦,𝑇𝑖𝐴𝑥−𝑇𝑖𝑇𝐴𝑦⟩=−2⟨𝐴𝑥−𝐴𝑦,𝑖𝑇−𝐼𝐴𝑥−𝑖−𝐼𝐴𝑦⟩(3.3) for all 𝑥,𝑦∈𝐻1. Thus it follows that ‖‖𝑉𝑖𝑥−𝑉𝑖𝑦‖‖=‖‖𝐼+𝛾𝐴∗𝑇𝑖𝐴−𝐼𝑥−𝐼+𝛾𝐴∗𝑇𝑖𝐴𝑦‖‖−𝐼2≤‖𝑥−𝑦‖2+𝛾2‖𝐴‖2‖‖(𝑇𝑖−𝐼)𝐴𝑥−(𝑇𝑖‖‖−𝐼)𝐴𝑦2𝑇+2𝛾𝐴𝑥−𝐴𝑦,𝑖𝑇−𝐼𝐴𝑥−𝑖−𝐼𝐴𝑦≤‖𝑥−𝑦‖2+𝛾𝛾‖𝐴‖2‖‖𝑇−1𝑖𝑇−𝐼𝐴𝑥−𝑖‖‖−𝐼𝐴𝑦2.(3.4) For 0<𝛾<1/‖𝐴‖2, we can immediately obtain that 𝑉𝑖 is a nonexpansive operator for every 𝑖∈ℕ.
Let 𝑝∈Ω, then 𝑈𝑖𝑝=𝑝 and 𝑇𝑖𝐴𝑝=𝐴𝑝 for every 𝑖≥1. Thus (𝑇𝑖−𝐼)𝐴𝑝=0, which implies that 𝑉𝑖𝑝=𝑝. Since ∑𝑛𝑖=1(𝛼𝑖−1−𝛼𝑖)=1−𝛼𝑛, we have ‖‖𝑥𝑛+1‖‖−𝑝≤𝛼𝑛‖‖𝑓𝑥𝑛‖‖+−𝑝𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑈𝑖𝑉𝑖𝑥𝑛‖‖−𝑝≤𝛼𝑛‖‖𝑓𝑥𝑛‖‖−𝑓(𝑝)+𝛼𝑛‖𝑓(𝑝)−𝑝‖+𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑉𝑖𝑥𝑛‖‖−𝑝≤𝛼𝑛𝜌‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛‖𝑓(𝑝)−𝑝‖+1−𝛼𝑛‖‖𝑥𝑛‖‖=−𝑝1−𝛼𝑛(‖‖𝑥1−𝜌)𝑛‖‖−𝑝+𝛼𝑛(11−𝜌)(‖‖𝑥1−𝜌‖𝑓𝑝)−𝑝‖≤max𝑛‖‖,1−𝑝.1−𝜌‖𝑓(𝑝)−𝑝‖(3.5) Then it follows that ‖‖𝑈𝑛𝑉𝑛𝑥𝑛−1‖‖≤‖‖𝑉−𝑝𝑛𝑥𝑛−1‖‖≤‖‖𝑥−𝑝𝑛−1‖‖‖‖𝑥−𝑝≤max1‖‖,1−𝑝‖1−𝜌‖𝑓(𝑝)−𝑝(3.6) for every 𝑛∈ℕ. This shows that {𝑥𝑛} and {𝑈𝑛𝑉𝑛𝑥𝑛−1} is bounded. Hence, {𝑓(𝑥𝑛)} is also bounded.
Step 3. We show limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›+1−𝑥𝑛‖=0.
From (3.2), we have ‖‖𝑥𝑛+1−𝑥𝑛‖‖=‖‖‖‖𝛼𝑛𝑓𝑥𝑛+𝑛𝑖=1𝛼𝑖−1−𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛−𝛼𝑛−1𝑓𝑥𝑛−1−𝑛−1𝑖=1𝛼𝑖−1−𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛−1‖‖‖‖≤𝛼𝑛‖‖𝑓𝑥𝑛𝑥−𝑓𝑛−1‖‖+𝛼𝑛−1−𝛼𝑛‖‖𝑓𝑥𝑛−1‖‖+𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑈𝑖𝑉𝑖𝑥𝑛−1‖‖+𝛼𝑛−1−𝛼𝑛‖‖𝑈𝑛𝑉𝑛𝑥𝑛−1‖‖≤𝛼𝑛𝜌‖‖𝑥𝑛−𝑥𝑛−1‖‖+𝛼𝑛−1−𝛼𝑛‖‖𝑓𝑥𝑛−1‖‖+‖‖𝑈𝑛𝑉𝑛𝑥𝑛−1‖‖+1−𝛼𝑛‖‖𝑥𝑛−𝑥𝑛−1‖‖≤1−𝛼𝑛‖‖𝑥(1−𝜌)𝑛−𝑥𝑛−1‖‖+𝛼𝑛−1−𝛼𝑛𝑀,(3.7) where 𝑀 is a constant such that 𝑀=sup𝑛≥1‖‖𝑓𝑥𝑛−1‖‖+‖‖𝑈𝑛𝑉𝑛𝑥𝑛−1‖‖.(3.8) From (i), (ii), (iii), and Lemma 2.2, it follows that limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›+1−𝑥𝑛‖=0
Step 4. We show limğ‘›â†’âˆžâ€–ğ‘ˆğ‘–ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›â€–=0 and limğ‘›â†’âˆžâ€–ğ‘ˆğ‘–ğ‘‰ğ‘–ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›â€–=0 for 𝑖∈ℕ.
We first show limğ‘›â†’âˆžâ€–ğ‘ˆğ‘–ğ‘‰ğ‘–ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›â€–=0 for 𝑖∈ℕ. Since 𝑝∈Ω, we note that ‖‖𝑥𝑛‖‖−𝑝2≥‖‖𝑉𝑖𝑥𝑛−𝑉𝑖𝑝‖‖2≥‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑈𝑖𝑉𝑖𝑝‖‖2=‖‖𝑈𝑖𝑉𝑖𝑥𝑛‖‖−𝑝2=‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛+𝑥𝑛‖‖−𝑝2=‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛‖‖2+‖‖𝑥𝑛‖‖−𝑝2+2⟨𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛,𝑥𝑛−𝑝⟩,(3.9) which implies that 12‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛‖‖2≤⟨𝑥𝑛−𝑈𝑖𝑉𝑖𝑥𝑛,𝑥𝑛−𝑝⟩.(3.10) Using (3.2) and (3.10), we deduce 12𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛‖‖2≤𝑛𝑖=1𝛼𝑖−1−𝛼𝑖⟨𝑥𝑛−𝑈𝑖𝑉𝑖𝑥𝑛,𝑥𝑛=−𝑝⟩1−𝛼𝑛𝑥𝑛−𝑛𝑖=1𝛼𝑖−1−𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛,𝑥𝑛=−𝑝1−𝛼𝑛𝑥𝑛−𝑥𝑛+1+𝛼𝑛𝑓𝑥𝑛,𝑥𝑛=𝑥−𝑝𝑛−𝑥𝑛+1,𝑥𝑛−𝑝+𝛼𝑛𝑓𝑥𝑛−𝑥𝑛,𝑥𝑛≤‖‖𝑥−𝑝𝑛−𝑥𝑛+1‖‖‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛‖‖𝑓𝑥𝑛−𝑥𝑛‖‖‖‖𝑥𝑛‖‖.−𝑝(3.11) Noting that limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›+1‖=0 and limğ‘›â†’âˆžğ›¼ğ‘›=0, then we immediately obtain âˆžî“ğ‘–=1𝛼𝑖−1−𝛼𝑖‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛‖‖2=limğ‘›ğ‘›â†’âˆžî“ğ‘–=1𝛼𝑖−1−𝛼𝑖‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛‖‖2=0.(3.12) Since {𝛼𝑛} is strictly decreasing, it follows that limğ‘›â†’âˆžâ€–â€–ğ‘ˆğ‘–ğ‘‰ğ‘–ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›â€–â€–=0,forevery𝑖∈ℕ.(3.13)
Next we show limğ‘›â†’âˆžâ€–ğ‘‡ğ‘–ğ´ğ‘¥ğ‘›âˆ’ğ´ğ‘¥ğ‘›â€–=0, for every 𝑖∈ℕ. Note for every 𝑖∈ℕ, ‖‖𝐴𝑥𝑛‖‖−𝐴𝑝2≥‖‖𝑇𝑖𝐴𝑥𝑛−𝑇𝑖‖‖𝐴𝑝2=‖‖𝑇𝑖𝐴𝑥𝑛‖‖−𝐴𝑝2=‖‖𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛+𝐴𝑥𝑛‖‖−𝐴𝑝2=‖‖𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛‖‖2+‖‖𝐴𝑥𝑛‖‖−𝐴𝑝2+2⟨𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛,𝐴𝑥𝑛−𝐴𝑝⟩,(3.14) which follows that ⟨𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛,𝐴𝑥𝑛1−𝐴𝑝⟩≤−2‖‖𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛‖‖2,(3.15) for every 𝑖∈ℕ. From (3.2), we have ‖‖𝑥𝑛+1‖‖−𝑝2=‖‖‖‖𝛼𝑛𝑓𝑥𝑛+𝑛𝑖=1𝛼𝑖−1−𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛‖‖‖‖−𝑝2≤𝛼𝑛‖‖𝑓𝑥𝑛‖‖−𝑝2+𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑈𝑖𝑉𝑖𝑥𝑛‖‖−𝑝2≤𝛼𝑛‖‖𝑓𝑥𝑛‖‖−𝑝2+𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑥𝑛+𝛾𝐴∗𝑇𝑖−𝐼𝐴𝑥𝑛‖‖−𝑝2≤𝛼𝑛‖‖𝑓𝑥𝑛‖‖−𝑝2+𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑥𝑛‖‖−𝑝2+𝛾2‖𝐴‖2‖‖𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛‖‖2+2𝛾⟨𝐴𝑥𝑛−𝐴𝑝,𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛⟩.(3.16) By (3.15), it follows that ‖‖𝑥𝑛+1‖‖−𝑝2≤𝛼𝑛‖‖𝑓𝑥𝑛‖‖−𝑝2+1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝2+𝛾𝛾‖𝐴‖2−1𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛‖‖2.(3.17) Thus, 𝛾1−𝛾‖𝐴‖2𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛‖‖2≤𝛼𝑛‖‖𝑓𝑥𝑛‖‖−𝑝2−‖‖𝑥𝑛‖‖−𝑝2+‖‖𝑥𝑛‖‖−𝑝2−‖‖𝑥𝑛+1‖‖−𝑝2=𝛼𝑛‖‖𝑓𝑥𝑛‖‖−𝑝2−‖‖𝑥𝑛‖‖−𝑝2+‖‖𝑥𝑛‖‖+‖‖𝑥−𝑝𝑛+1‖‖‖‖𝑥−𝑝𝑛‖‖−‖‖𝑥−𝑝𝑛+1‖‖−𝑝≤𝛼𝑛‖‖𝑓𝑥𝑛‖‖−𝑝2−‖‖𝑥𝑛‖‖−𝑝2+‖‖𝑥𝑛‖‖+‖‖𝑥−𝑝𝑛+1‖‖‖‖𝑥−𝑝𝑛−𝑥𝑛+1‖‖.(3.18) Using limğ‘›â†’âˆžğ›¼ğ‘›=0 and limğ‘›â†’âˆžâ€–ğ‘¥ğ‘›+1−𝑥𝑛‖=0, we have limğ‘›â†’âˆžğ›¾î€·1−𝛾‖𝐴‖2𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛‖‖2=0.(3.19) By 0<𝛾<1/‖𝐴‖2, there holds âˆžî“ğ‘–=1𝛼𝑖−1−𝛼𝑖‖‖𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛‖‖2=limğ‘›ğ‘›â†’âˆžî“ğ‘–=1𝛼𝑖−1−𝛼𝑖‖‖𝑇𝑖𝐴𝑥𝑛−𝐴𝑥𝑛‖‖2=0.(3.20) Since {𝛼𝑛} is strictly decreasing, we obtain limğ‘›â†’âˆžâ€–â€–ğ‘‡ğ‘–ğ´ğ‘¥ğ‘›âˆ’ğ´ğ‘¥ğ‘›â€–â€–=0,∀𝑖∈ℕ.(3.21)
Last we show limğ‘›â†’âˆžâ€–ğ‘ˆğ‘–ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›â€–=0 for every 𝑖∈ℕ. In fact, we note that for every 𝑖∈ℕ, ‖‖𝑈𝑖𝑥𝑛−𝑥𝑛‖‖≤‖‖𝑈𝑖𝑥𝑛−𝑈𝑖𝑉𝑖𝑥𝑛‖‖+‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛‖‖≤‖‖𝑥𝑛−𝑉𝑖𝑥𝑛‖‖+‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛‖‖=‖‖𝑥𝑛−𝑥𝑛−𝛾𝐴∗𝑇𝑖−𝐼𝐴𝑥𝑛‖‖+‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛‖‖‖‖𝑇≤𝛾‖𝐴‖𝑖−𝐼𝐴𝑥𝑛‖‖+‖‖𝑈𝑖𝑉𝑖𝑥𝑛−𝑥𝑛‖‖.(3.22) Then by (3.13) and (3.21), we obtain limğ‘›â†’âˆžâ€–â€–ğ‘ˆğ‘–ğ‘¥ğ‘›âˆ’ğ‘¥ğ‘›â€–â€–=0,∀𝑖∈ℕ.(3.23)
Step 5. Show limsupğ‘›â†’âˆžâŸ¨ğ‘“(𝑤)−𝑤,𝑥𝑛−𝑤⟩≤0, where 𝑤=𝑃Ω𝑓(𝑤).
Since {𝑥𝑛} is bounded, there exist a point 𝑣∈𝐻1 and a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} such that limsupğ‘›â†’âˆžâŸ¨ğ‘“(𝑤)−𝑤,𝑥𝑛−𝑤⟩=limğ‘˜â†’âˆžî«ğ‘“(𝑤)−𝑤,𝑥𝑛𝑘−𝑤(3.24) and 𝑥𝑛𝑘⇀𝑣. Since 𝐴 is a bounded linear operator, we have 𝐴𝑥𝑛𝑘⇀𝐴𝑣. Now applying (3.21), (3.23), and Lemma 2.1, we conclude that 𝑣∈Fix(𝑈𝑖) and 𝐴𝑣∈Fix(𝑇𝑖) for every 𝑖. Hence, 𝑣∈Ω. Since Ω is closed and convex, by (2.1), we get limsupğ‘›â†’âˆžâŸ¨ğ‘“(𝑤)−𝑤,𝑥𝑛−𝑤⟩=limğ‘˜â†’âˆžî«ğ‘“(𝑤)−𝑤,𝑥𝑛𝑘−𝑤=⟨𝑓(𝑤)−𝑤,𝑣−𝑤⟩≤0.(3.25)
Step 6. Show 𝑥𝑛→𝑤=𝑃Ω𝑓(𝑤).
Since 𝑤∈Ω, we have 𝑈𝑖𝑤=𝑤 and 𝑇𝑖𝐴𝑤=𝐴𝑤 for every 𝑖∈ℕ. It follows that 𝑉𝑖𝑤=𝑤. Using (3.2), we have ‖‖𝑥𝑛+1‖‖−𝑤2=𝛼𝑛𝑓𝑥𝑛+−𝑤𝑛𝑖=1𝛼𝑖−1−𝛼𝑖𝑈𝑖𝑉𝑖𝑥𝑛−𝑤,𝑥𝑛+1−𝑤=𝛼𝑛𝑓𝑥𝑛−𝑓(𝑤),𝑥𝑛+1−𝑤+𝛼𝑛𝑓(𝑤)−𝑤,𝑥𝑛+1+−𝑤𝑛𝑖=1𝛼𝑖−1−𝛼𝑖⟨𝑈𝑖𝑉𝑖𝑥𝑛−𝑤,𝑥𝑛+1−𝑤⟩≤𝛼𝑛𝜌‖‖𝑥𝑛‖‖‖‖𝑥−𝑤𝑛+1‖‖−𝑤+𝛼𝑛𝑓(𝑤)−𝑤,𝑥𝑛+1+−𝑤𝑛𝑖=1𝛼𝑖−1−𝛼𝑖‖‖𝑈𝑖𝑉𝑖𝑥𝑛‖‖‖‖𝑥−𝑤𝑛+1‖‖≤1−𝑤2𝛼𝑛𝜌‖‖𝑥𝑛‖‖−𝑤2+‖‖𝑥𝑛+1‖‖−𝑤2+𝛼𝑛𝑓(𝑤)−𝑤,𝑥𝑛+1+1−𝑤21−𝛼𝑛‖‖𝑥𝑛‖‖−𝑤2+‖‖𝑥𝑛+1‖‖−𝑤2≤121−𝛼𝑛‖‖𝑥(1−𝜌)𝑛‖‖−𝑤2+12‖‖𝑥𝑛+1‖‖−𝑤2+𝛼𝑛𝑓(𝑤)−𝑤,𝑥𝑛+1,−𝑤(3.26) which implies that ‖‖𝑥𝑛+1‖‖−𝑤2≤1−𝛼𝑛‖‖𝑥(1−𝜌)𝑛‖‖−𝑤2+2𝛼𝑛1(1−𝜌)1−𝜌𝑓(𝑤)−𝑤,𝑥𝑛+1,−𝑤(3.27) for every 𝑛∈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 𝑥∈𝐻1, where 𝑢 is an arbitrary point in 𝐻1, it is easily seen that our conditions are satisfied.

Corollary 3.3. Let 𝑈∶𝐻1→𝐻1 and 𝑇∶𝐻2→𝐻2 be nonexpansive operators. Let 𝑓∶𝐻1→𝐻1 be a contraction with coefficient 𝜌∈[0,1). Suppose that the solution set Ω of SCFPP (1.6) is nonempty. Let 𝑥1∈𝐻1 and define a sequence {𝑥𝑛} by the following algorithm: 𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+1−𝛼𝑛𝑈𝑥𝑛+𝛾𝐴∗(𝑇−𝐼)𝐴𝑥𝑛,(3.28) where 0<𝛾<1/‖𝐴‖2, 𝛼0=1 and {𝛼𝑛}⊂(0,1] is a strictly decreasing sequence satisfying the following conditions: (i)limğ‘›â†’âˆžğ›¼ğ‘›=0; (ii)âˆ‘âˆžğ‘›=1𝛼𝑛=∞; (iii)âˆ‘âˆžğ‘›=1|𝛼𝑛+1−𝛼𝑛|<∞. Then {𝑥𝑛} converges strongly to 𝑤∈Ω, where 𝑤=𝑃Ω𝑓(𝑤).

Proof. Set {𝑈𝑛} and {𝑇𝑛} to be sequences of operators defined by 𝑈𝑛=𝑈 and 𝑇𝑛=𝑇 for all 𝑛∈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 𝑖≥𝑝+1 and 𝑇𝑗=𝐼 for 𝑗≥𝑟+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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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.


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