MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 638468 10.1155/2013/638468 638468 Research Article Weak and Strong Convergence of an Algorithm for the Split Common Fixed-Point of Asymptotically Quasi-Nonexpansive Operators http://orcid.org/0000-0002-8530-6994 Dang Yazheng 1, 2 Gao Yan 1 Do Val Joao B. R. 1 School of Management University of Shanghai for Science and Technology Shanghai 200093 China usst.edu.cn 2 Henan Polytechnic University Jiaozuo 454000 China hpu.edu.cn 2013 13 11 2013 2013 28 06 2013 10 10 2013 24 10 2013 2013 Copyright © 2013 Yazheng Dang and Yan Gao. 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.

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.

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 . The CFP and SCFP have many applications such as approximation theory , image reconstruction, radiation therapy [3, 4], and control . The SFP in finite-dimensional space was first introduced by Censor and Elfving  for modeling inverse problems, which arise from phase retrievals, and in medical image reconstruction . 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 .

Throughout this paper, we assume that both H1 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):={xx=T(x)}. Let U:H1H1 and T:H2H2 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:H1H2 is bounded linear. Denote the solution set of the two-operator SCFP by (2)Γ={yCAyQ}.

The split common fixed point problem for quasi-nonexpansive mapping in the setting of Hilbert space was first introduced and studied by Moudafi . However, the algorithm presented in  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 Section 2, 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 mapping T:HH is said to be nonexpansive if (3)T(x)-T(y)x-y,(x,y)H×H.

A mapping T is called asymptotically nonexpansive if there exists a sequence {lk}[1,) with lk1 as k such that, for all (x,y)H×H, (4)Tk(x)-Tk(y)lkx-y,k1.

A mapping T is called quasi-nonexpansive if for all (x,z)H×F(T)(5)T(x)-zx-z.

A mapping T is called asymptotically quasi-nonexpansive if there exists a sequence {lk}[1,) with lk1 as k such that, for all (x,z)H×Fix(T), (6)Tk(x)-zlkx-z,k1.

A mapping T 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)Lx-y,k1.

A mapping T is said to be semicompact if for any bounded sequence {xk}H with limkxk-Txk=0 there exists a subsequence {xkp} of {xk} such that {xkp} converges strongly to a point x*H.

Let E be a Banach space. A mapping T:EE is said to be demiclosed at origin, if for any sequence {xk}E with xkx* and (I-T)xk0, x*=T(x*).

Lemma 1.

Let T:HH be asymptotically quasi-nonexpansive and set Tαk:=(1-α)I+αTk for α(0,1]. Then, for any zFix(T) and xH, one has the following.

x-Tk(x),x-z(1/2)x-Tk(x)2-((lk2-1)/2)xk-z2;

x-Tk(x),z-Tk(x)(1/2)x-Tk(x)2+((lk2-1)/2)xk-z2;

Tαk(x)-z2[1+α(lk2-1)]x-z2-α(1-α)x-Tk(x);

x-Tαk(x),x-z(α/2)x-Tk(x)2-((α(lk2-1))/2)xk-z2.

Lemma 2 (see [<xref ref-type="bibr" rid="B15">15</xref>]).

Suppose {ak} is a sequence of nonnegative real numbers such that (8)ak+1(1+δk)ak+bk,k0, if (i)k=0δk<and(ii)k=1bk<. Then, limkak exists. In particular, if {ak} has a subsequence which converges strongly to zero, then limkak=0.

3. The Algorithm and Its Asymptotic Convergence

We now give a description of an algorithm.

Algorithm 3.

Initialization: let x0H1 be arbitrary.

Iterative step: for kN, set uk=xk+γAT(Tk-I)A(xk), let (9)xk+1=(1-αk)uk+αkUk(uk),kN, where αk[α,1-α],α(0,1), and γ(0,1/λ),λ=A2.

In what follows, we establish the weak convergence and strong convergence of Algorithm 3.

Lemma 4 (Opial [<xref ref-type="bibr" rid="B16">16</xref>]).

Let H be a Hilbert space and let {xk} be a sequence in H such that there exists a nonempty set SH satisfying the following:

For every x*, limkxk-x* exists.

Any weak cluster point of the sequence {xk} belongs to S.

Then, there exists zS such that {xk} weakly converges to z.

Theorem 5.

Let U:H1H1 and T:H2H2 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,

any sequence {xk} generated by Algorithm 3 converges weakly to a point x*Γ;

if U 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 each zΓ, limkxk-z and limkuk-z exist. Also limk+(Tk-I)(Axk)=0 and limk+(Uk(uk)-uk)=0.

By using (1) in Lemma 1, from (9) we obtain (10)xk+1-z2=(1-αk)uk+αkUk(uk)-z2[1+αk(lk2-1)]uk-z2-αk(1-αk)uk-Uk(uk)2. By deducing, it follows that (11)uk-z2=xk+γAT(Tk-I)(Axk)-z2=xk-z2+γ2AT(Tk-I)(Axk)2+2γxk-z,AT(Tk-I)(Axk)xk-z2+λγ2(Tk-I)(Axk)2+2γAxk-Az,(Tk-I)(Axk); that is, (12)uk-z2xk-z2+λγ2(Tk-I)(Axk)2+2γAxk-Az,(Tk-I)(Axk). Setting θ:=2γAxk-Az,(T-I)(Axk) and using (1) of Lemma 1, we obtain (13)θ=2γAxk-Az,(Tk-I)(Axk)=2γAxk-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+lk2-12Axk-Az2-(Tk-I)(Axk)212(Tk-I)(Axk)2+lk2-12)=-γ(Tk-I)(Axk)2+γλ(lk2-1)xk-z2. The key inequality mentioned above, combined with (12), yields (14)uk-z2(1+γλ(lk2-1))xk-z2-γ(1-λγ)(Tk-I)(Axk)2. Substituting (14) into (10) yields (15)xk+1-z2[1+αk(lk2-1)]×[(1+γλ(lk2-1))xk-z2-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-z2-[1+αk(lk2-1)]γ(1-λγ)(Tk-I)(Axk)2-αk(1-αk)uk-Uk(uk)2=[1+δk]xk-z2-[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=1δk<, From Lemma 1, it follows that limkxk-z exists. By the virtue of (14), we know that limkuk-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)2xk-z2-xk+1-z2-(αk+γλ)(lk2-1)+αkγλ(lk2-1)2xk-z20. 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 limkxk+1-xk=0 and limkuk+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)limkxk+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+γAT(Tk+1-I)Axk+1+γAT(Tk-I)Axk0,(k).

Step  2. We prove that limkAxk-T(Axk)=0 and limkuk-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+LUk-1(uk)-ukηk+L(Uk-1(uk)-Uk-1(uk-1)+Uk-1(uk-1)-uk)ηk+L2uk-uk-1+L(Uk-1(uk-1)-uk-1+uk-1-uk)ηk+L(L+1)uk-uk-1+Lηk-10,(k); in other words, (22)limkuk-U(uk)=0. Similarly, we have (23)limkAxk-T(Axk)=0.

Step  3. Finally, we prove that xkx* and xkx*, 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.

Noticing uk=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 zΓ,limkxk-z and limkuk-z exist, we know that limkxk-x*=limkuk-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.

Acknowledgments

This work was supported by the National Science Foundation of China (under Grant 11171221), Shanghai Leading Academic Discipline Project (under Grant XTKX2012), Basic and Frontier Research Program of Science and Technology Department of Henan Province (under Grant 112300410277), Innovation Program of Shanghai Municipal Education Commission (under Grant 14YZ094), Doctoral Program Foundation of Institutions of Higher Education of China (under Grant 20123120110004), Doctoral Starting Projection of the University of Shanghai for Science and Technology (under Grant ID-10-303-002), and Young Teacher Training Projection Program of Shanghai for Science and Technology.

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