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

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=1𝐶_𝑖,𝐴𝑥^∗∈^𝑟_𝑗=1𝑄_𝑗,(1.2) where {𝐶_𝑖}^𝑝𝑖=1 and {𝑄_𝑗}^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: 𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛,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=1Fix𝑈_𝑖,𝐴𝑥^∗∈^𝑟_𝑗=1Fix𝑇_𝑗,(1.5) where 𝑈_𝑖∶𝐻₁→𝐻₁(𝑖=1,2,…,𝑝) and 𝑇_𝑗∶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: 𝑥_𝑛+1=𝑈𝑥_𝑛−𝛾𝐴^∗(𝐼−𝑇)𝐴𝑥_𝑛,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: 𝑥_𝑛+1=𝑈_[𝑛]𝑥_𝑛−𝛾𝐴^∗𝐼−𝑇_[𝑛]𝐴𝑥_𝑛,𝑛≥0(1.8) where [𝑛]=𝑛mod𝑝 and 𝑝=𝑟. Under some mild conditions, they proved that {𝑥_𝑛} 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 𝑥^∗∈^∞_𝑖=1Fix𝑈_𝑖,𝐴𝑥^∗∈^∞_𝑗=1Fix𝑇_𝑗.(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: 𝑥_𝑛+1𝑦_𝑛=𝑃_𝐶𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,=𝑃_𝐶𝛼^𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑇_𝑖𝑦_𝑛,(1.10) where 𝑆 and {𝑇_𝑖}^∞𝑖=1 are nonexpansive, 𝛼₀=1, {𝛼_𝑛} is strictly decreasing sequence in (0,1), and {𝛽_𝑛} is a sequence in (0,1). Under some certain conditions on parameters, they proved that the sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈⋂^∞𝑖=1𝐹(𝑇_𝑖). 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: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑈_𝑖𝑥_𝑛+𝛾𝐴^∗𝑇_𝑖−𝐼𝐴𝑥_𝑛.(1.11) Under some certain conditions, we prove that the sequence {𝑥_𝑛} generated by (1.11) converges strongly to the solution of the GSCFPP (1.9).

2. Preliminaries

Throughout this paper, we write 𝑥_𝑛⇀𝑥 and 𝑥_𝑛→𝑥 to indicate that {𝑥_𝑛} 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 𝑃_𝐶𝑥, such that ‖𝑥−𝑃_𝐶𝑥‖≤‖𝑥−𝑦‖ for every 𝑦∈𝐶. 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is known that, for each 𝑥∈𝐻, ⟨𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥⟩≤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 {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥∈𝐶 and {(𝐼−𝑇)𝑥_𝑛} 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 {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛,𝑛≥0,(2.2) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that (i)∑^∞𝑛=1𝛾_𝑛=∞,(ii)limsup_𝑛→∞𝛿_𝑛/𝛾_𝑛≤0𝑜𝑟∑^∞𝑛=1|𝛿_𝑛|<∞.
Then lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

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

Theorem 3.1. Let {𝑈_𝑛} and {𝑇_𝑛} 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 {𝛼_𝑛}⊂(0,1] be a strictly decreasing sequence satisfying the following conditions: (i)lim_𝑛→∞𝛼_𝑛=0; (ii)∑^∞𝑛=1𝛼_𝑛=∞; (iii)∑^∞𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞. Then the sequence {𝑥_𝑛} generated by (1.11) converges strongly to 𝑤∈Ω, where 𝑤=𝑃_Ω𝑓(𝑤).

Proof. We proceed with the following steps.
Step 1. First show that there exists 𝑤∈Ω such that 𝑤=𝑃_Ω𝑓(𝑤).
In fact, since 𝑓 is a contraction with coefficient 𝜌, we have ‖‖𝑃_Ω𝑓(𝑥)−𝑃_Ω𝑓(𝑦)‖‖≤‖𝑓(𝑥)−𝑓(𝑦)‖≤𝜌‖𝑥−𝑦‖(3.1) for every 𝑥,𝑦∈𝐻₁. Hence 𝑃_Ω𝑓 is also a contraction of 𝐻₁ into itself. Therefore, there exists a unique 𝑤∈𝐻₁ such that 𝑤=𝑃_Ω𝑓(𝑤). At the same time, we note that 𝑤∈Ω.
Step 2. Now we show that {𝑥_𝑛} is bounded.
For simplicity, we set 𝑉_𝑖=𝐼+𝛾𝐴^∗(𝑇_𝑖−𝐼)𝐴. Then we can rewrite (1.11) to 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑈_𝑖𝑉_𝑖𝑥_𝑛.(3.2) Observe that ‖‖(𝑇_𝑖−𝐼)𝐴𝑥−(𝑇_𝑖−𝐼)𝐴𝑦‖‖²≤=2‖‖𝐴𝐴𝑥𝑥−−𝐴𝐴𝑦‖𝑦‖²₂+‖‖𝑇_𝑖𝐴𝑥−𝑇_𝑖𝐴𝑦‖‖²−2⟨𝐴𝑥−𝐴𝑦,𝑇_𝑖𝐴𝑥−𝑇_𝑖𝐴𝑦⟩=−2⟨𝐴𝑥−𝐴𝑦,−𝑇_𝑖2⟨𝐴𝑥−𝐴𝑦,𝑇_𝑖𝐴𝑥−𝑇_𝑖𝐴𝑦⟩−𝐼𝐴𝑥−𝑇_𝑖−𝐼𝐴𝑦⟩(3.3) for all 𝑥,𝑦∈𝐻₁. Thus it follows that ‖‖𝑉_𝑖𝑥−𝑉_𝑖𝑦‖‖=‖‖𝐼+𝛾𝐴^∗𝑇_𝑖−𝐼𝐴𝑥−𝐼+𝛾𝐴^∗𝑇_𝑖−𝐼𝐴𝑦‖‖²≤‖𝑥−𝑦‖²+𝛾²‖𝐴‖²‖‖(𝑇_𝑖−𝐼)𝐴𝑥−(𝑇_𝑖−𝐼)𝐴𝑦‖‖²+≤2‖𝛾𝑥−𝐴𝑦𝑥‖−₂𝐴𝑦,𝑇_𝑖−𝐼𝐴𝑥−𝑇_𝑖−𝐼𝐴𝑦+𝛾𝛾‖𝐴‖²−1‖‖𝑇_𝑖−𝐼𝐴𝑥−𝑇_𝑖−𝐼𝐴𝑦‖‖².(3.4) For 0<𝛾<1/‖𝐴‖², we can immediately obtain that 𝑉_𝑖 is a nonexpansive operator for every 𝑖∈ℕ.
Let 𝑝∈Ω, then 𝑈_𝑖𝑝=𝑝 and 𝑇_𝑖𝐴𝑝=𝐴𝑝 for every 𝑖≥1. Thus (𝑇_𝑖−𝐼)𝐴𝑝=0, which implies that 𝑉_𝑖𝑝=𝑝. Since ∑^𝑛𝑖=1(𝛼_𝑖−1−𝛼_𝑖)=1−𝛼_𝑛, we have ‖‖𝑥_𝑛+1−𝑝‖‖≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑝‖‖+^𝑛_𝑖=1𝛼_𝑖−1≤𝛼_𝑛−𝛼_𝑖‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑝‖‖‖‖𝑓𝑥_𝑛−𝑓(𝑝)‖‖+𝛼_𝑛‖𝑓(𝑝)−𝑝‖+^𝑛_𝑖=1𝛼_𝑖−1≤𝛼_𝑛−𝛼_𝑖‖‖𝑉_𝑖𝑥_𝑛−𝑝‖‖=1𝜌−‖‖𝛼𝑥_𝑛𝑛−𝑝‖‖+𝛼_𝑛‖𝑓(𝑝)−𝑝‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖(1−𝜌)‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛(1−𝜌)1≤max‖‖𝑥_𝑛1−𝜌‖𝑓(𝑝)−𝑝‖−𝑝‖‖,11−𝜌‖𝑓(𝑝)−𝑝‖.(3.5) Then it follows that ‖‖𝑈_𝑛𝑉_𝑛𝑥_𝑛−1−𝑝‖‖≤‖‖𝑉_𝑛𝑥_𝑛−1−𝑝‖‖≤‖‖𝑥_𝑛−1−𝑝‖‖≤max‖‖𝑥₁−𝑝‖‖,11−𝜌‖𝑓(𝑝)−𝑝‖(3.6) for every 𝑛∈ℕ. This shows that {𝑥_𝑛} and {𝑈_𝑛𝑉_𝑛𝑥_𝑛−1} is bounded. Hence, {𝑓(𝑥_𝑛)} is also bounded.
Step 3. We show lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
From (3.2), we have ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=‖‖‖‖𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−^𝑛−1_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑈_𝑖𝑉_𝑖𝑥_𝑛−1≤𝛼_𝑛‖‖‖‖+_𝑛‖‖𝑓𝑥_𝑛−𝑓𝑥_𝑛−1‖‖+𝛼_𝑛−1−𝛼_𝑛‖‖𝑓𝑥_𝑛−1‖‖_𝑖=1𝛼_𝑖−1≤𝛼_𝑛−𝛼_𝑖‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑈_𝑖𝑉_𝑖𝑥_𝑛−1‖‖+𝛼_𝑛−1−𝛼_𝑛‖‖𝑈_𝑛𝑉_𝑛𝑥_𝑛−1‖‖≤1𝜌−‖‖𝑥𝛼_𝑛𝑛−𝑥_𝑛−1‖‖+𝛼_𝑛−1−𝛼_𝑛‖‖𝑓𝑥_𝑛−1‖‖+‖‖𝑈_𝑛𝑉_𝑛𝑥_𝑛−1‖‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖(1−𝜌)‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝛼_𝑛−1−𝛼_𝑛𝑀,(3.7) where 𝑀 is a constant such that 𝑀=sup_𝑛≥1‖‖𝑓𝑥_𝑛−1‖‖+‖‖𝑈_𝑛𝑉_𝑛𝑥_𝑛−1‖‖.(3.8) From (i), (ii), (iii), and Lemma 2.2, it follows that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0
Step 4. We show lim_𝑛→∞‖𝑈_𝑖𝑥_𝑛−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖=0 for 𝑖∈ℕ.
We first show lim_𝑛→∞‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖=0 for 𝑖∈ℕ. Since 𝑝∈Ω, we note that ‖‖𝑥_𝑛−𝑝‖‖²≥‖‖𝑉_𝑖𝑥_𝑛−𝑉_𝑖𝑝‖‖²≥‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑈_𝑖𝑉_𝑖𝑝‖‖²=‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑝‖‖²=‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑝‖‖²=‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖‖²+‖‖𝑥_𝑛−𝑝‖‖²+2⟨𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑝⟩,(3.9) which implies that 12‖‖𝑈^𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖‖²≤⟨𝑥_𝑛−𝑈_𝑖𝑉_𝑖𝑥_𝑛,𝑥_𝑛−𝑝⟩.(3.10) Using (3.2) and (3.10), we deduce 12^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖‖²≤^𝑛_𝑖=1𝛼_𝑖−1=1−𝛼_𝑛−𝛼_𝑖⟨𝑥_𝑛−𝑈_𝑖𝑉_𝑖𝑥_𝑛,𝑥_𝑛−𝑝⟩=1−𝛼_𝑛𝑥_𝑛−^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑈_𝑖𝑉_𝑖𝑥_𝑛,𝑥_𝑛−𝑝=𝑥𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑓𝑥_𝑛,𝑥_𝑛−𝑝_𝑛≤‖‖𝑥−𝑥_𝑛+1,𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑝_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖.(3.11) Noting that lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0 and lim_𝑛→∞𝛼_𝑛=0, then we immediately obtain ^∞_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖‖²=lim_{𝑛𝑛→∞}_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖‖²=0.(3.12) Since {𝛼_𝑛} is strictly decreasing, it follows that lim_𝑛→∞‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖‖=0,forevery𝑖∈ℕ.(3.13)
Next we show lim_𝑛→∞‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛‖=0, for every 𝑖∈ℕ. Note for every 𝑖∈ℕ, ‖‖𝐴𝑥_𝑛−𝐴𝑝‖‖²≥‖‖𝑇_𝑖𝐴𝑥_𝑛−𝑇_𝑖𝐴𝑝‖‖²=‖‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑝‖‖²=‖‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛+𝐴𝑥_𝑛−𝐴𝑝‖‖²=‖‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛‖‖²+‖‖𝐴𝑥_𝑛−𝐴𝑝‖‖²+2⟨𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛,𝐴𝑥_𝑛−𝐴𝑝⟩,(3.14) which follows that ⟨𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛,𝐴𝑥_𝑛−𝐴𝑝⟩≤−12‖‖𝑇^𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛‖‖²,(3.15) for every 𝑖∈ℕ. From (3.2), we have ‖‖𝑥_𝑛+1−𝑝‖‖²=‖‖‖‖𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑝‖‖‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑝‖‖²+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑝‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑝‖‖²+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑥_𝑛+𝛾𝐴^∗𝑇_𝑖−𝐼𝐴𝑥_𝑛−𝑝‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑝‖‖²+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑥_𝑛−𝑝‖‖²+𝛾²‖𝐴‖²‖‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛‖‖²+2𝛾⟨𝐴𝑥_𝑛−𝐴𝑝,𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛⟩.(3.16) By (3.15), it follows that ‖‖𝑥_𝑛+1−𝑝‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑝‖‖²+1−𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖²+𝛾𝛾‖𝐴‖²−1^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛‖‖².(3.17) Thus, 𝛾1−𝛾‖𝐴‖²^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛‖‖²≤𝛼_𝑛‖‖𝑓𝑥^𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑝‖‖²+‖‖𝑥^𝑛−𝑝‖‖²−‖‖𝑥_𝑛+1−𝑝‖‖²=≤𝛼𝛼_𝑛𝑛‖‖𝑓𝑥^𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑝‖‖²+‖‖𝑥^𝑛−𝑝‖‖+‖‖𝑥_𝑛+1−𝑝‖‖‖‖𝑥_𝑛−𝑝‖‖−‖‖𝑥_𝑛+1−𝑝‖‖‖‖𝑓𝑥^𝑛−𝑝‖‖²−‖‖𝑥_𝑛−𝑝‖‖²+‖‖𝑥^𝑛−𝑝‖‖+‖‖𝑥_𝑛+1−𝑝‖‖‖‖𝑥_𝑛−𝑥_𝑛+1‖‖.(3.18) Using lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0, we have lim_𝑛→∞𝛾1−𝛾‖𝐴‖²^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛‖‖²=0.(3.19) By 0<𝛾<1/‖𝐴‖², there holds ^∞_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛‖‖²=lim_{𝑛𝑛→∞}_𝑖=1𝛼_𝑖−1−𝛼_𝑖‖‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛‖‖²=0.(3.20) Since {𝛼_𝑛} is strictly decreasing, we obtain lim_𝑛→∞‖‖𝑇_𝑖𝐴𝑥_𝑛−𝐴𝑥_𝑛‖‖=0,∀𝑖∈ℕ.(3.21)
Last we show lim_𝑛→∞‖𝑈_𝑖𝑥_𝑛−𝑥_𝑛‖=0 for every 𝑖∈ℕ. In fact, we note that for every 𝑖∈ℕ, ‖‖𝑈_𝑖𝑥_𝑛−𝑥_𝑛‖‖≤≤‖‖‖‖𝑥𝑈_𝑖𝑛𝑥_𝑛−𝑈_𝑖𝑉_𝑖𝑥_𝑛‖‖+‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖‖=‖‖𝑥_𝑛−𝑉_𝑖𝑥_𝑛‖‖+‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖‖≤𝛾‖𝐴−‖𝑥‖‖_𝑛−𝑇_𝑖𝛾𝐴^∗𝑇_𝑖−𝐼𝐴𝑥_𝑛‖‖+‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖‖−𝐼𝐴𝑥_𝑛‖‖+‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑥_𝑛‖‖.(3.22) Then by (3.13) and (3.21), we obtain lim_𝑛→∞‖‖𝑈_𝑖𝑥_𝑛−𝑥_𝑛‖‖=0,∀𝑖∈ℕ.(3.23)
Step 5. Show limsup_𝑛→∞⟨𝑓(𝑤)−𝑤,𝑥_𝑛−𝑤⟩≤0, where 𝑤=𝑃_Ω𝑓(𝑤).
Since {𝑥_𝑛} is bounded, there exist a point 𝑣∈𝐻₁ and a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞⟨𝑓(𝑤)−𝑤,𝑥_𝑛−𝑤⟩=lim_𝑘→∞𝑓(𝑤)−𝑤,𝑥_𝑛𝑘−𝑤(3.24) and 𝑥_𝑛𝑘⇀𝑣. Since 𝐴 is a bounded linear operator, we have 𝐴𝑥_𝑛𝑘⇀𝐴𝑣. Now applying (3.21), (3.23), and Lemma 2.1, we conclude that 𝑣∈Fix(𝑈_𝑖) and 𝐴𝑣∈Fix(𝑇_𝑖) for every 𝑖. Hence, 𝑣∈Ω. Since Ω is closed and convex, by (2.1), we get limsup_𝑛→∞⟨𝑓(𝑤)−𝑤,𝑥_𝑛−𝑤⟩=lim_𝑘→∞𝑓(𝑤)−𝑤,𝑥_𝑛𝑘−𝑤=⟨𝑓(𝑤)−𝑤,𝑣−𝑤⟩≤0.(3.25)
Step 6. Show 𝑥_𝑛→𝑤=𝑃_Ω𝑓(𝑤).
Since 𝑤∈Ω, we have 𝑈_𝑖𝑤=𝑤 and 𝑇_𝑖𝐴𝑤=𝐴𝑤 for every 𝑖∈ℕ. It follows that 𝑉_𝑖𝑤=𝑤. Using (3.2), we have ‖‖𝑥_𝑛+1−𝑤‖‖²=𝛼_𝑛𝑓𝑥_𝑛−𝑤+^𝑛_𝑖=1𝛼_𝑖−1=𝛼_𝑛−𝛼_𝑖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑤,𝑥_𝑛+1−𝑤+𝑓𝑥_𝑛−𝑓(𝑤),𝑥_𝑛+1−𝑤+𝛼_𝑛𝑓(𝑤)−𝑤,𝑥_𝑛+1−𝑤_𝑛_𝑖=1𝛼_𝑖−1≤𝛼_𝑛−𝛼_𝑖⟨𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑤,𝑥_𝑛+1−𝑤⟩+𝜌‖‖𝑥_𝑛−𝑤‖‖‖‖𝑥_𝑛+1−𝑤‖‖+𝛼_𝑛𝑓(𝑤)−𝑤,𝑥_𝑛+1−𝑤_𝑛_𝑖=1𝛼_𝑖−1≤1−𝛼_𝑖‖‖𝑈_𝑖𝑉_𝑖𝑥_𝑛−𝑤‖‖‖‖𝑥_𝑛+1−𝑤‖‖2𝛼^𝑛+1𝜌‖‖𝑥_𝑛−𝑤‖‖²+‖‖𝑥_𝑛+1−𝑤‖‖²+𝛼^𝑛𝑓(𝑤)−𝑤,𝑥_𝑛+1−𝑤21−𝛼^𝑛‖‖𝑥_𝑛≤1−𝑤‖‖²+‖‖𝑥_𝑛+1−𝑤‖‖²21−𝛼^𝑛(1−𝜌)‖‖𝑥_𝑛−𝑤‖‖²+12‖‖𝑥^𝑛+1−𝑤‖‖²+𝛼_𝑛𝑓(𝑤)−𝑤,𝑥_𝑛+1−𝑤,(3.26) which implies that ‖‖𝑥_𝑛+1−𝑤‖‖²≤1−𝛼_𝑛(1−𝜌)‖‖𝑥_𝑛−𝑤‖‖²+2𝛼_𝑛(1−𝜌)11−𝜌𝑓(𝑤)−𝑤,𝑥^𝑛+1−𝑤,(3.27) for every 𝑛∈ℕ. Consequently, according to (3.25), 𝜌∈[0,1), and Lemma 2.2, we deduce that {𝑥_𝑛} converges strongly to 𝑤=𝑃_Ω(𝑤). This completes the proof.

Remark 3.2. If we set 𝛼_𝑛=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 {𝑥_𝑛} by the following algorithm: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑈𝑥_𝑛+𝛾𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑛,(3.28) where 0<𝛾<1/‖𝐴‖², 𝛼₀=1 and {𝛼_𝑛}⊂(0,1] is a strictly decreasing sequence satisfying the following conditions: (i)lim_𝑛→∞𝛼_𝑛=0; (ii)∑^∞𝑛=1𝛼_𝑛=∞; (iii)∑^∞𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞. Then {𝑥_𝑛} converges strongly to 𝑤∈Ω, where 𝑤=𝑃_Ω𝑓(𝑤).

Proof. Set {𝑈_𝑛} and {𝑇_𝑛} to be sequences of operators defined by 𝑈_𝑛=𝑈 and 𝑇_𝑛=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 {𝑈_𝑛} and {𝑇_𝑛} by setting 𝑈_𝑖=𝐼 for 𝑖≥𝑝+1 and 𝑇_𝑗=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).