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

Authors: Cuijie Zhang; Songnian He
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 1. Introduction

LetH1 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:
(1.1)x*∈C,Ax*∈Q,
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:
(1.2)x*∈⋂i=1pCi,Ax*∈‍⋂j=1rQj,
where {Ci}i=1p and {Qj}j=1r 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
(1.3)x*∈C,x*∈A-1Q.
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:
(1.4)xn+1=PC(xn-γA*(I-PQ)Axn),n≥0,
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:
(1.5)x*∈⋂i=1pFix⁡(Ui),Ax*∈⋂j=1rFix⁡(Tj),
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
(1.6)x*∈Fix⁡(U),Ax*∈Fix⁡(T),
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:
(1.7)xn+1=U(xn-γA*(I-T)Axn),n≥0,
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:
(1.8)xn+1=U[n](xn-γA*(I-T[n])Axn),n≥0
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 pointx* such that
(1.9)x*∈⋂i=1∞Fix⁡(Ui),Ax*∈⋂j=1∞Fix⁡(Tj).
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:
(1.10)yn=PC[βnSxn+(1-βn)xn],xn+1=PC[αnf(xn)+∑i=1n(αi-1-αi)Tiyn],
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 x*∈⋂i=1∞‍F(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:(1.11)xn+1=αnf(xn)+∑i=1n(αi-1-αi)Ui(xn+γA*(Ti-I)Axn).
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 writexn⇀x and xn→x to indicate that {xn} converges weakly to x and converges strongly to x, respectively.LetH 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,
(2.1)〈x-PCx,y-PCx〉≤0
for all y∈C.In order to prove our main results, we collect the following lemmas in this section.Lemma 2.1 (see [26]).
LetH 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{an} is a sequence of nonnegative real numbers such that
(2.2)an+1≤(1-γn)an+δn,n≥0,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that  (i)
∑n=1∞γn=∞,(ii)
limsupn→∞δn/γn≤0or∑n=1∞|δn|<∞.
Thenlimn→∞an=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)
limn→∞αn=0;(ii)
∑n=1∞‍αn=∞;(iii)
∑n=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, sincef is a contraction with coefficient ρ, we have
(3.1)‖PΩf(x)-PΩf(y)‖≤‖f(x)-f(y)‖≤ρ‖x-y‖
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 setVi=I+γA*(Ti-I)A. Then we can rewrite (1.11) to
(3.2)xn+1=αnf(xn)+∑i=1n(αi-1-αi)UiVixn.
Observe that
(3.3)‖(Ti-I)Ax-(Ti-I)Ay‖2=‖Ax-Ay‖2+‖TiAx-TiAy‖2-2〈Ax-Ay,TiAx-TiAy〉≤2‖Ax-Ay‖2-2〈Ax-Ay,TiAx-TiAy〉=-2〈Ax-Ay,(Ti-I)Ax-(Ti-I)Ay〉
for all x,y∈H1. Thus it follows that
(3.4)‖Vix-Viy‖=‖(I+γA*(Ti-I)A)x-(I+γA*(Ti-I)A)y‖2≤‖x-y‖2+γ2‖A‖2‖(Ti-I)Ax-(Ti-I)Ay‖2+2γ〈Ax-Ay,(Ti-I)Ax-(Ti-I)Ay〉≤‖x-y‖2+γ(γ‖A‖2-1)‖(Ti-I)Ax-(Ti-I)Ay‖2.
For 0<γ<1/∥A∥2, we can immediately obtain that Vi is a nonexpansive operator for every i∈ℕ.
Letp∈Ω, then Uip=p and TiAp=Ap for every i≥1. Thus (Ti-I)Ap=0, which implies that Vip=p. Since ∑i=1n‍(αi-1-αi)=1-αn, we have
(3.5)‖xn+1-p‖≤αn‖f(xn)-p‖+∑i=1n(αi-1-αi)‖UiVixn-p‖≤αn‖f(xn)-f(p)‖+αn‖f(p)-p‖+∑i=1n(αi-1-αi)‖Vixn-p‖≤αnρ‖xn-p‖+αn‖f(p)-p‖+(1-αn)‖xn-p‖=(1-αn(1-ρ))‖xn-p‖+αn(1-ρ)11-ρ‖f(p)-p‖≤max⁡⁡{‖xn-p‖,11-ρ‖f(p)-p‖}.
Then it follows that
(3.6)‖UnVnxn-1-p‖≤‖Vnxn-1-p‖≤‖xn-1-p‖≤max⁡⁡{‖x1-p‖,11-ρ‖f(p)-p‖}
for every n∈ℕ. This shows that {xn} and {UnVnxn-1} is bounded. Hence, {f(xn)} is also bounded.
Step3. We show limn→∞∥xn+1-xn∥=0.
From (3.2), we have
(3.7)‖xn+1-xn‖=‖αnf(xn)+∑i=1n(αi-1-αi)UiVixn-αn-1f(xn-1)-∑i=1n-1(αi-1-αi)UiVixn-1‖≤αn‖f(xn)-f(xn-1)‖+(αn-1-αn)‖f(xn-1)‖≤+∑i=1n(αi-1-αi)‖UiVixn-UiVixn-1‖+(αn-1-αn)‖UnVnxn-1‖≤αnρ‖xn-xn-1‖+(αn-1-αn)(‖f(xn-1)‖+‖UnVnxn-1‖)+(1-αn)‖xn-xn-1‖≤(1-αn(1-ρ))‖xn-xn-1‖+(αn-1-αn)M,
where M is a constant such that
(3.8)M=sup⁡n≥1{‖f(xn-1)‖+‖UnVnxn-1‖}.
From (i), (ii), (iii), and Lemma 2.2, it follows that limn→∞∥xn+1-xn∥=0
Step
4. We show limn→∞∥Uixn-xn∥=0 and limn→∞∥UiVixn-xn∥=0 for i∈ℕ.
We first showlimn→∞∥UiVixn-xn∥=0 for i∈ℕ. Since p∈Ω, we note that
(3.9)‖xn-p‖2≥‖Vixn-Vip‖2≥‖UiVixn-UiVip‖2=‖UiVixn-p‖2=‖UiVixn-xn+xn-p‖2=‖UiVixn-xn‖2+‖xn-p‖2+2〈UiVixn-xn,xn-p〉,
which implies that
(3.10)12‖UiVixn-xn‖2≤〈xn-UiVixn,xn-p〉.
Using (3.2) and (3.10), we deduce
(3.11)12∑i=1n‍(αi-1-αi)‖UiVixn-xn‖2≤∑i=1n(αi-1-αi)〈xn-UiVixn,xn-p〉=〈(1-αn)xn-∑i=1n(αi-1-αi)UiVixn,xn-p〉=〈(1-αn)xn-xn+1+αnf(xn),xn-p〉=〈xn-xn+1,xn-p〉+αn〈f(xn)-xn,xn-p〉≤‖xn-xn+1‖‖xn-p‖+αn‖f(xn)-xn‖‖xn-p‖.
Noting that limn→∞∥xn-xn+1∥=0 and limn→∞αn=0, then we immediately obtain
(3.12)∑i=1∞(αi-1-αi)‖UiVixn-xn‖2=lim⁡n→∞⁡∑i=1n(αi-1-αi)‖UiVixn-xn‖2=0.
Since {αn} is strictly decreasing, it follows that
(3.13)lim⁡n→∞⁡‖UiVixn-xn‖=0,for everyi∈N.
Next we showlimn→∞∥TiAxn-Axn∥=0, for every i∈ℕ. Note for every i∈ℕ,
(3.14)‖Axn-Ap‖2≥‖TiAxn-TiAp‖2=‖TiAxn-Ap‖2=‖TiAxn-Axn+Axn-Ap‖2=‖TiAxn-Axn‖2+‖Axn-Ap‖2+2〈TiAxn-Axn,Axn-Ap〉,
which follows that
(3.15)〈TiAxn-Axn,Axn-Ap〉≤-12‖TiAxn-Axn‖2,
for every i∈ℕ. From (3.2), we have
(3.16)‖xn+1-p‖2=‖αnf(xn)+∑i=1n(αi-1-αi)UiVixn-p‖2≤αn‖f(xn)-p‖2+∑i=1n(αi-1-αi)‖UiVixn-p‖2≤αn‖f(xn)-p‖2+∑i=1n(αi-1-αi)‖xn+γA*(Ti-I)Axn-p‖2≤αn‖f(xn)-p‖2+∑i=1n(αi-1-αi)[‖xn-p‖2+γ2‖A‖2‖TiAxn-Axn‖2+2γ〈Axn-Ap,TiAxn-Axn〉{‖A‖2}].
By (3.15), it follows that
(3.17)‖xn+1-p‖2≤αn‖f(xn)-p‖2+(1-αn)‖xn-p‖2+γ(γ‖A‖2-1)∑i=1n‍(αi-1-αi)‖TiAxn-Axn‖2.
Thus,
(3.18)γ(1-γ‖A‖2)∑i=1n(αi-1-αi)‖TiAxn-Axn‖2≤αn(‖f(xn)-p‖2-‖xn-p‖2)+‖xn-p‖2-‖xn+1-p‖2=αn(‖f(xn)-p‖2-‖xn-p‖2)+(‖xn-p‖+‖xn+1-p‖)(‖xn-p‖-‖xn+1-p‖)≤αn(‖f(xn)-p‖2-‖xn-p‖2)+(‖xn-p‖+‖xn+1-p‖)‖xn-xn+1‖.
Using limn→∞αn=0 and limn→∞∥xn+1-xn∥=0, we have
(3.19)lim⁡n→∞⁡γ(1-γ‖A‖2)∑i=1n(αi-1-αi)‖TiAxn-Axn‖2=0.
By 0<γ<1/∥A∥2, there holds
(3.20)∑i=1∞(αi-1-αi)‖TiAxn-Axn‖2=lim⁡n→∞⁡∑i=1n(αi-1-αi)‖TiAxn-Axn‖2=0.
Since {αn} is strictly decreasing, we obtain
(3.21)lim⁡n→∞⁡‖TiAxn-Axn‖=0,∀i∈N.
Last we showlimn→∞∥Uixn-xn∥=0 for every i∈ℕ. In fact, we note that for every i∈ℕ,
(3.22)‖Uixn-xn‖≤‖Uixn-UiVixn‖+‖UiVixn-xn‖≤‖xn-Vixn‖+‖UiVixn-xn‖=‖xn-xn-γA*(Ti-I)Axn‖+‖UiVixn-xn‖≤γ‖A‖‖(Ti-I)Axn‖+‖UiVixn-xn‖.
Then by (3.13) and (3.21), we obtain
(3.23)lim⁡n→∞⁡‖Uixn-xn‖=0,∀i∈N.
Step5. Show limsupn→∞〈f(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
(3.24)lim⁡sup⁡n→∞〈f(w)-w,xn-w〉=lim⁡k→∞〈f(w)-w,xnk-w〉
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
(3.25)lim⁡sup⁡n→∞〈f(w)-w,xn-w〉=lim⁡k→∞⁡〈f(w)-w,xnk-w〉=〈f(w)-w,v-w〉≤0.
Step6. Show xn→w=PΩf(w).
Sincew∈Ω, we have Uiw=w and TiAw=Aw for every i∈ℕ. It follows that Viw=w. Using (3.2), we have
(3.26)‖xn+1-w‖2=〈αn(f(xn)-w)+∑i=1n(αi-1-αi)(UiVixn-w),xn+1-w〉=αn〈f(xn)-f(w),xn+1-w〉+αn〈f(w)-w,xn+1-w〉+∑i=1n(αi-1-αi)〈UiVixn-w,xn+1-w〉≤αnρ‖xn-w‖‖xn+1-w‖+αn〈f(w)-w,xn+1-w〉+∑i=1n(αi-1-αi)‖UiVixn-w‖‖xn+1-w‖≤12αnρ(‖xn-w‖2+‖xn+1-w‖2)+αn〈f(w)-w,xn+1-w〉+12(1-αn)(‖xn-w‖2+‖xn+1-w‖2)≤12[1-αn(1-ρ)]‖xn-w‖2+12‖xn+1-w‖2+αn〈f(w)-w,xn+1-w〉,
which implies that
(3.27)‖xn+1-w‖2≤[1-αn(1-ρ)]‖xn-w‖2+2αn(1-ρ)11-ρ〈f(w)-w,xn+1-w〉,
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 setαn=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.
LetU: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:
(3.28)xn+1=αnf(xn)+(1-αn)U(xn+γA*(T-I)Axn),
where 0<γ<1/∥A∥2, α0=1 and {αn}⊂(0,1] is a strictly decreasing sequence satisfying the following conditions:  (i)
limn→∞αn=0;(ii)
∑n=1∞‍αn=∞;(iii)
∑n=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).



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