# Weak and Strong Convergence of an Algorithm for the Split Common Fixed-Point of Asymptotically Quasi-Nonexpansive Operators

Authors: Yazheng Dang; Yan Gao
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Inspired by the Moudafi (2010), we propose an algorithm for solving the split common fixed-point problem for a wide class of asymptotically quasi-nonexpansive operators and the weak and strong convergence of the algorithm are shown under some suitable conditions in Hilbert spaces. The algorithm and its convergence results improve and develop previous results for split feasibility problems.


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## Body

## 1. Introduction

Fixed-point problem is a classical problem in nonlinear analysis and it has application in a wide spectrum of fields such as economics, physics, and applied sciences. In this paper, We are concerned with the split common fixed point problem (SCFP). In fact, the SCFP is an extension of the split feasibility problem (SFP) and the convex feasibility problem (CFP), see [1]. The CFP and SCFP have many applications such as approximation theory [2], image reconstruction, radiation therapy [3, 4], and control [5]. The SFP in finite-dimensional space was first introduced by Censor and Elfving [6] for modeling inverse problems, which arise from phase retrievals, and in medical image reconstruction [7]. 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 [8, 9]. The SFP in an infinite-dimensional Hilbert space can be found in [10–13].Throughout this paper, we assume that bothH1 and H2 are real Hilbert spaces, “→” and “⇀” denote strong and weak convergence, respectively, Fix(T) denotes the set of the fixed points of an operator T, that is, and Fix(T):={x∣x=T(x)}. Let U:H1→H1 and T:H2→H2 be two asymptotically quasi-nonexpansive mappings with nonempty fixed-point sets Fix(U)=C and Fix(T)=Q, respectively. The split common fixed point problem for operators U and T is to find
(1)x*∈CsuchthatAx*∈Q,
where A:H1→H2 is bounded linear. Denote the solution set of the two-operator SCFP by
(2)Γ={y∈C∣Ay∈Q}.The split common fixed point problem for quasi-nonexpansive mapping in the setting of Hilbert space was first introduced and studied by Moudafi [14]. However, the algorithm presented in [14] has only weak convergence. The purpose of this paper is to propose an algorithm of split common fixed point problem for asymptotically quasi-nonexpanding operator which includes the quasi-nonexpansive mapping and the weak and strong convergence of the algorithm are shown under some suitable conditions in Hilbert spaces. The algorithm and the convergence results improve and develop previously discussed split feasibility problems.The paper is organized as follows. In Section2, we recall some preliminaries. In Section 3, we present an algorithm and show its weak convergence and strong convergence. Section 4 gives some concluding remarks.


## 2. Preliminaries

Recall that a mappingT:H→H is said to be nonexpansive if
(3)∥T(x)-T(y)∥≤∥x-y∥,(x,y)∈H×H.A mappingT is called asymptotically nonexpansive if there exists a sequence {lk}⊂[1,∞) with lk→1 as k→∞ such that, for all (x,y)∈H×H,
(4)∥Tk(x)-Tk(y)∥≤lk∥x-y∥,∀k≥1.A mappingT is called quasi-nonexpansive if for all (x,z)∈H×F(T)(5)∥T(x)-z∥≤∥x-z∥.A mappingT is called asymptotically quasi-nonexpansive if there exists a sequence {lk}⊂[1,∞) with lk→1 as k→∞ such that, for all (x,z)∈H×Fix(T),
(6)∥Tk(x)-z∥≤lk∥x-z∥,∀k≥1.A mappingT is said to be uniformly L-Lipschitzian if there exists a constant L>0 such that for all (x,y)∈H×H,
(7)∥Tk(x)-Tk(y)∥≤L∥x-y∥,∀k≥1.A mappingT is said to be semicompact if for any bounded sequence {xk}⊂H with limk→∞∥xk-Txk∥=0 there exists a subsequence {xkp} of {xk} such that {xkp} converges strongly to a point x*∈H.LetE be a Banach space. A mapping T:E→E is said to be demiclosed at origin, if for any sequence {xk}⊂E with xk⇀x* and ∥(I-T)xk∥→0, x*=T(x*).Lemma 1.
LetT:H→H be asymptotically quasi-nonexpansive and set Tαk:=(1-α)I+αTk for α∈(0,1]. Then, for any z∈Fix(T) and x∈H, one has the following.(1)
〈x-Tk(x),x-z〉≥(1/2)∥x-Tk(x)∥2-((lk2-1)/2)∥xk-z∥2;(2)
〈x-Tk(x),z-Tk(x)〉≤(1/2)∥x-Tk(x)∥2+((lk2-1)/2)∥xk-z∥2;(3)
∥Tαk(x)-z∥2≤[1+α(lk2-1)]∥x-z∥2-α(1-α)∥x-Tk(x)∥;(4)
〈x-Tαk(x),x-z〉≥(α/2)∥x-Tk(x)∥2-((α(lk2-1))/2)∥xk-z∥2.Lemma 2 (see [15]).
Suppose{ak} is a sequence of nonnegative real numbers such that
(8)ak+1≤(1+δk)ak+bk,k≥0,
if (i)∑k=0∞δk<∞and(ii)∑k=1∞bk<∞. Then, limk→∞ak exists. In particular, if {ak} has a subsequence which converges strongly to zero, then limk→∞ak=0.


## 3. The Algorithm and Its Asymptotic Convergence

We now give a description of an algorithm.Algorithm 3.
Initialization: letx0∈H1 be arbitrary.
Iterative step: fork∈N, set uk=xk+γAT(Tk-I)A(xk), let
(9)xk+1=(1-αk)uk+αkUk(uk),k∈N,
where αk∈[α,1-α],α∈(0,1), and γ∈(0,1/λ),λ=∥A∥2.In what follows, we establish the weak convergence and strong convergence of Algorithm3.Lemma 4 (Opial [16]).
LetH be a Hilbert space and let {xk} be a sequence in H such that there exists a nonempty set S⊂H satisfying the following:(1)
For everyx*, limk∥xk-x*∥ exists.(2)
Any weak cluster point of the sequence{xk} belongs to S.
Then, there existsz∈S such that {xk} weakly converges to z.Theorem 5.
LetU:H1→H1 and T:H2→H2 be two asymptotically quasi-nonexpansive operators with nonempty Fix(U)=C and Fix(T)=Q. Assume that U-I and T-I are demiclosed at 0 and Γ≠∅, ∑k=1∞(lk2-1)<∞, and T and U are uniformly L-Lipschitzian. Then,(I)
any sequence{xk} generated by Algorithm 3 converges weakly to a point x*∈Γ;(II)
ifU is also semi-compact, then both {xk} and {uk} generated by Algorithm 3 converge strongly to a point x*∈Γ.Proof.
First, we prove that for eachz∈Γ, limk→∞∥xk-z∥ and limk→∞∥uk-z∥ exist. Also limk→+∞∥(Tk-I)(Axk)∥=0 and limk→+∞∥(Uk(uk)-uk)∥=0.
By using (1) in Lemma1, from (9) we obtain
(10)∥xk+1-z∥2=∥(1-αk)uk+αkUk(uk)-z∥2≤[1+αk(lk2-1)]∥uk-z∥2-αk(1-αk)∥uk-Uk(uk)∥2.
By deducing, it follows that
(11)∥uk-z∥2=∥xk+γAT(Tk-I)(Axk)-z∥2=∥xk-z∥2+γ2∥AT(Tk-I)(Axk)∥2+2γ〈xk-z,AT(Tk-I)(Axk)〉≤∥xk-z∥2+λγ2∥(Tk-I)(Axk)∥2+2γ〈Axk-Az,(Tk-I)(Axk)〉;
that is,
(12)∥uk-z∥2≤∥xk-z∥2+λγ2∥(Tk-I)(Axk)∥2+2γ〈Axk-Az,(Tk-I)(Axk)〉.
Setting θ:=2γ〈Axk-Az,(T-I)(Axk)〉 and using (1) of Lemma 1, we obtain
(13)θ=2γ〈Axk-Az,(Tk-I)(Axk)〉=2γ〈Axk-Az+(Tk-I)(Axk)-(Tk-I)(Axk),(Tk-I)(Axk)〉=2γ(〈Axk-Az,(T-I)(Axk)〉-∥(T-I)(Axk)∥2)≤2γ(12∥(Tk-I)(Axk)∥2+lk2-12∥Axk-Az∥2-∥(Tk-I)(Axk)∥212∥(Tk-I)(Axk)∥2+lk2-12)=-γ∥(Tk-I)(Axk)∥2+γλ(lk2-1)∥xk-z∥2.
The key inequality mentioned above, combined with (12), yields
(14)∥uk-z∥2≤(1+γλ(lk2-1))∥xk-z∥2-γ(1-λγ)∥(Tk-I)(Axk)∥2.
Substituting (14) into (10) yields
(15)∥xk+1-z∥2≤[1+αk(lk2-1)]×[(1+γλ(lk2-1))∥xk-z∥2-γ(1-λγ)∥(Tk-I)(Axk)∥2(1+γλ(lk2-1))]-αk(1-αk)∥uk-Uk(uk)∥=[1+(αk+γλ)(lk2-1)+αkγλ(lk2-1)2]∥xk-z∥2-[1+αk(lk2-1)]γ(1-λγ)∥(Tk-I)(Axk)∥2-αk(1-αk)∥uk-Uk(uk)∥2=[1+δk]∥xk-z∥2-[1+αk(lk2-1)]γ(1-λγ)×∥(Tk-I)(Axk)∥2-αk(1-αk)∥uk-Uk(uk)∥2+bk,
where δk=(αk+γλ)(lk2-1)+αkγλ(lk2-1)2,bk=0. Since ∑k=1∞‍(lk2-1)<∞, we have ∑k=1∞‍δk<∞, From Lemma 1, it follows that limk→∞∥xk-z∥ exists. By the virtue of (14), we know that limk→∞∥uk-z∥ exists. Therefore, from (15), we have
(16)[1+αk(lk2-1)]γ(1-λγ)∥(Tk-I)(Axk)∥2+αk(1-αk)∥uk-Uk(uk)∥2≤∥xk-z∥2-∥xk+1-z∥2-(αk+γλ)(lk2-1)+αkγλ(lk2-1)2∥xk-z∥2⟶0.
By the assumptions on lk and αk, we get
(17)limk→+∞∥(Tk-I)(Axk)∥=0,limk→+∞∥(Uk(uk)-uk)∥=0.
Step  1. Now, we prove that limk→∞∥xk+1-xk∥=0 and limk→∞∥uk+1-uk∥=0. As a matter of a fact, it follows from (9) that
(18)∥xk+1-xk∥=∥(1-αk)uk+αkUk(uk)-xk∥=∥(1-αk)(xk+γAT(Tk-I)Axk)+αkUk(uk)-xk∥=∥(1-αk)γAT(Tk-I)Axk+αk(Uk(uk)-xk)∥=∥(1-αk)γAT(Tk-I)Axk+αk(Uk(uk)-uk)+αk(uk-xk)∥=∥(1-αk)γAT(Tk-I)Axk+αk(Uk(uk)-uk)+αkγAT(Tk-I)Axk∥=∥γAT(Tk-I)Axk+αk(Uk(uk)-uk)∥.
In view of (17), we have that
(19)limk→∞∥xk+1-xk∥=0.
Similarly, it follows from (9), (17), and (19) that
(20)∥uk+1-uk∥=∥(xk+1+γAT(Tk+1-I)Axk+1)-(xk+γAT(Tk-I)Axk)∥≤∥xk+1-xk∥+γ∥AT(Tk+1-I)Axk+1∥+γ∥AT(Tk-I)Axk∥⟶0,(k⟶∞).
Step  2. We prove that limk→∞∥Axk-T(Axk)∥=0 and limk→∞∥uk-U(uk)∥=0. Setting ηk:=∥uk-Uk(uk)∥, since U is uniformly L-Lipschitzian continuous, it follows from (14) and (17) that
(21)∥uk-U(uk)∥≤∥uk-Uk(uk)∥+∥Uk(uk)-U(uk)∥≤ηk+L∥Uk-1(uk)-uk∥≤ηk+L(∥Uk-1(uk)-Uk-1(uk-1)∥+∥Uk-1(uk-1)-uk∥)≤ηk+L2∥uk-uk-1∥+L(∥Uk-1(uk-1)-uk-1∥+∥uk-1-uk∥)≤ηk+L(L+1)∥uk-uk-1∥+Lηk-1⟶0,(k⟶∞);
in other words,
(22)limk→∞∥uk-U(uk)∥=0.
Similarly, we have
(23)limk→∞∥Axk-T(Axk)∥=0.
Step  3. Finally, we prove that xk⇀x* and xk⇀x*, where x*∈Γ. Denote by x* a weak-cluster point of {xk}, and denote by {xkσ} a subsequence of {xk}. Obviously,
(24)w-limσykσ=w-limσxkσ=x*.
Then, from (23) and the demiclosedness of T-I at 0, we obtain
(25)T(Ax*)=Ax*;
it follows that,Ax*∈Q.
Noticinguk=xk+γA*(T-I)(Axk), it follows that w-limσukσ=x*. By the demiclosedness of U-I at 0, we have
(26)U(x*)=x*.
Hence, x*∈C, and therefore x*∈Γ.
Since there is no more than one weak cluster point, the weak convergence of the whole sequence{xk} follows by applying Lemma 4 with S=Γ.
The Proof of Conclusion (II). Since U is semi-compact, it follows from (22) that there exists a subsequence {ukh} of {uk} such that {ukh}→u*∈H (some point in H). Since xk=uk-γAT(Tk-I)A(xk){ukh}⇀x*, this implies that x*=u*. Therefore, {ukh}→x*∈Γ as h→∞. Since for any z∈Γ,limk→∞∥xk-z∥ and limk→∞∥uk-z∥ exist, we know that limk→∞∥xk-x*∥=limk→∞∥uk-x*∥=0. This implies that {xk} and {uk} both converge strongly to a point x*∈Γ. The proof is completed.


## 4. Concluding Remarks

In this paper, we have proposed an algorithm for solving the SCFP in the wide class of asymptotically quasi-nonexpansive operators and obtained its weak and strong convergence in general Hilbert spaces in a new way. Next, we will improve the algorithm to solve the multiple split common fixed point problem in infinite Hilbert spaces.



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