# A Modified Halpern's Iterative Scheme for Solving Split Feasibility Problems

Authors: Jitsupa Deepho; Poom Kumam
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The purpose of this paper is to introduce and study a modified Halpern’s iterative scheme for solving the split feasibility problem (SFP) in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions a strong convergence theorem is established. The main result presented in this paper improves and extends some recent results done by Xu (Iterative methods for the split feasibility problem in infinite-dimensional Hilbert space, Inverse Problem 26 (2010) 105018) and some others.


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## Body

## 1. Introduction

LetC and Q be nonempty-closed convex subsets of real Hilbert spaces H1 and H2, respectively. Let A be a linear-bounded operator from H1 to H2. The split feasibility problem (SFP) is finding a point x^ satisfying the following property:
(1.1)x^∈C,Ax^∈Q.The SFP was introduced by Censor and  Elfving [1] for modeling inverse problems which arise from phase retrievals and medical image reconstruction [2], and very well-known iterative algorithms have been invented to solve it [2].We useΓ to denote the solution set of SFP:
(1.2)Γ={x^∈C:Ax^∈Q},
and assume that the SFP (1.1) is consistent (i.e., (1.1) has a solution) so that Γ is closed, convex, and nonempty, it is not hard to see that x∈C solves (1.1) if and only if it solves the following fixed point equation;
(1.3)x=PC(I-γA*(I-PQ)A)x,x∈C,
where PC and PQ are the (orthogonal) projections onto C and Q, respectively, γ>0 is any positive constant and A* denotes the adjoint of A. Moreover, for sufficiently small γ>0, the operator PC(I-γA*(I-PQ)A) which defines the fixed point equation in (1.3) is nonexpansive.To solve the SFP (1.1), Byrne [2] proposed his CQ algorithm (see also [3]) which generates a sequence {xn} by
(1.4)xn+1=PC(I-γA*(I-PQ)A)xn,n≥0,
where γ∈(0,2/λ) with λ being the spectral radius of the operator A*A.Very recently, Xu [4] has viewed the CQ algorithm for averaged mappings and applied Mann's algorithm to solving the SFP, and he also proved that an averaged CQ algorithm is weakly convergent to a solution of the SFP.In this paper, we also regard theCQ algorithm as a fixed point algorithm for averaged mappings and try to study the SFP by the following modified Halpern's iterative scheme;(1.5)xn+1=αnu+βnxn+γnPC(I-ξA*(I-PQ)A)xn,n≥0,
where {αn}, {βn}, and {γn} are three sequences in [0,1] satisfying αn+βn+γn=1. Furthermore, our result extends and improves the result of Xu [4] from weak to strong convergence theorems.


## 2. Preliminaries

Throughout the paper, we adopt the following notation.Letxn be a sequence and x be a point in a normed space X. We use xn→x and xn⇀x to denote strong and weak convergence to x of the sequence {xn}, respectively. In addition, we use ωw(xn) to denote the weak ω-limit set of the sequence {xn}; namely,
(2.1)ωw(xn):={x:xni⇀xforsomesubsequence{xni}of{xn}}.LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively, and let K be a nonempty-closed convex subset of H. For every point x∈H, there exists a unique nearest point in K, denoted by PKx, such that
(2.2)∥x-PKx∥≤∥x-y∥,∀y∈K,P
K is called the metric projection of H onto K. It is well known that PK is a nonexpansive mapping of H onto K and satisfies
(2.3)〈x-y,PKx-PKy〉≥∥PKx-PKy∥2,
for every x,y∈H. Moreover, PKx is characterized by the following properties: PKx∈K and
(2.4)〈x-PKx,y-PKx〉≤0,∥x-y∥2≥∥x-PKx∥2+∥y-PKx∥2,
for all x∈H,y∈K.Some important properties of projections are gathered in the following proposition.Proposition 2.1.
Givenx∈H and z∈K. Then z=PKx if and only if
(2.5)〈x-z,y-z〉≤0,∀y∈K.
One also needs other sorts of nonlinear operators which are introduced below.
LetT,A:H→H be the nonlinear operators. (1)
Tis nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x, y∈H.(2)
T is firmly nonexpansive if 2T-I is nonexpansive. Equivalent, T=(I+S)/2, where S:H→H is nonexpansive. Alternatively, T is firmly nonexpansive if and only if
(2.6)〈x-y,Tx-Ty〉≥∥Tx-Ty∥2,x,y∈H.(3)
T is averaged if T=(1-α)I+αS, where α∈(0,1) and S:H→H is nonexpansive. In this case, one also says that T is α-averaged. A firmly nonexpansive mapping is (1/2)-averaged.(4)
A is monotone if 〈Ax-Ay,x-y〉≥0 for x,y∈H.(5)
A is β-strongly monotone, with β>0, if
(2.7)〈x-y,Ax-Ay〉≥β∥x-y∥2,x,y∈H.(6)
A is ν-inverse strongly monotone (ν-ism), with ν>0, if
(2.8)〈x-y,Ax-Ay〉≥ν∥Ax-Ay∥2,x,y∈H.It is well known that both PK and I-PK are firmly nonexpansive and (1/2)-ism.
Denote byFix(T) the set of fixed points of a self-mapping T defined on H, (i.e., Fix(T)=x∈H:Tx=x).Proposition 2.2 (see [2, 5]).
One has the following assertions.(1)
T is nonexpansive if and only if the complement I-T is (1/2)-ism.(2)
IfT is ν-ism and γ>0, then γT is (ν/γ)-ism.(3)
T is averaged if and only if the complement I-T is ν-ism, for some ν>(1/2).Indeed, forα∈(0,1),T is α-averaged if and only if I-T is (1/2α)-ism.(4)
IfT1 is α1-averaged and T2 is α2-averaged, where α1,α2∈(0,1), then the composite T1T2 is α-averaged, where α=α1+α2-α1α2.(5)
IfT1 and T2 are averaged and have a common fixed point, then Fix(T1T2)=Fix(T1)∩Fix(T2).Lemma 2.3 (see [6]).
LetK be a nonempty-closed convex subset of a real Hilbert space H and T be nonexpansive mapping on K with Fix(T)≠∅. If {xn} is a sequence in K which converges weakly to x and if {(I-T)xn} converges strongly to y, then y=(I-T)x. In particular, if y=0, then x∈Fix(T).Lemma 2.4 (see [7]).
Let(E,〈·,·〉) be an inner product space. Then for all x,y,z∈E and αn,βn,γn∈[0,1] with αn+βn+γn=1, one has
(2.9)∥αx+βy+γz∥2=α∥x∥2+β∥y∥2+γ∥z∥2-αβ∥x-y∥2-αγ∥x-z∥2-βγ∥y-z∥2.Lemma 2.5 (see [8]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.10)an+1≤(1-γn)an+δn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that (1)
∑n=0∞‍γn=∞;(2)
limsupn→∞δn/γn≤0 or ∑n=0∞‍|δn|<∞.
Then,limn→∞an=0.


## 3. Main Result

LetC be a nonempty closed and convex subset of a Hilbert space H. For any u,x0∈C, we define the sequence {xn} by
(3.1)xn+1=αnu+βnxn+γnPC(I-ξA*(I-PQ)A)xn,n≥0,
where {αn}, {βn}, and {γn} are three sequences in [0,1] and satisfy αn+βn+γn=1.Theorem 3.1.
Suppose that the SFP is consistent and0<ξ<(2/∥A∥2). Let {xn} be a sequence defined as in (3.1). If the following assumptions are satisfied:(C1)
limn→∞αn=0 but ∑n=1∞‍αn=∞,(C2)
limsupn→∞βn<1,(C3)
the sums∑n=1∞‍|αn+1-αn|, ∑n=1∞‍|βn+1-βn| and ∑n=1∞‍|γn+1-γn| are finite. Then {xn} converges strongly to a solution of the SFP (1.1).Proof.
We firstly show that the sequence{xn} is bounded. For our convenience, we take T:=PC(I-ξA*(I-PQ)A). Then, for any x*∈Γ, we have Tx*=x*. Now, we observe that
(3.2)∥xn+1-x*∥≤∥αnu+βnxn+γnTxn-x*∥≤αn∥u-x*∥+βn∥xn-x*∥+γn∥Txn-x*∥.
Now, we note that the condition 0<ξ<(2/∥A∥2) implies that the operator PC(I-ξA*(I-PQ)A) is averaged. Since I-PQ is firmly nonexpansive mappings and so is (1/2)-average, which is 1-ism. Also observe that A*(I-PQ)A is (1/∥A∥2)-ism so that ξA*(I-PQ)A is (1/ξ∥A∥2)-ism. Further, from the fact that I-ξA*(I-PQ)A is (ξ∥A∥2/2)-averaged and PC is (1/2)-averaged, we may obtain that PC(I-ξA*(I-PQ)A) is χ-averaged, where
(3.3)χ=12+ξ∥A∥22-12·ξ∥A∥22=2+ξ∥A∥24.
This implies thatT=χI+(1-χ)S, where χ=(2+ξ∥A∥2/4)∈(0,1) for some nonexpansive mappings S. Note that T is also nonexpansive mappings. Hence, we have
(3.4)∥Txn-x*∥=∥Txn-Tx*∥≤∥xn-x*∥.
From the inequalities (3.2) and (3.4), we have
(3.5)∥xn+1-x*∥≤αn∥u-x*∥+(1-αn)∥xn-x*∥≤max{∥u-x*∥,∥xn-x*∥}.
Continuing inductively, we may obtain that the inequality
(3.6)∥xn+1-x*∥≤max{∥u-x*∥,∥x0-x*∥},
holds for all n≥0. So, {xn} is bounded so does {Txn}.
Next, we will show thatlimn→∞∥xn+1-xn∥=0. Observe that
(3.7)∥xn+1-xn∥=∥(αnu+βnxn+γnTxn)-(αn-1u+βn-1xn-1+γn-1Txn-1)∥≤∥(αnu+βnxn+γnTxn)-(αnu+βnxn-1+γnTxn-1)∥+∥(αnu+βnxn-1+γnTxn-1)-(αn-1u+βn-1xn-1+γn-1Txn-1)∥≤(1-αn)∥xn-xn-1∥+|αn-αn-1|∥u∥+|βn-βn-1|∥xn-1∥+|γn-γn-1|∥Txn-1∥.
Since {xn} and {Txn} are bounded, there exists M=sup(∥u∥,∥xn+1∥,∥Txn-1∥)>0 such that
(3.8)∥xn+1-xn∥≤(1-αn)∥xn-xn-1∥+M(|αn-αn-1|+|βn-βn-1|+|γn-γn-1|).
According to Lemma 2.5 and the condition (C3), we have limn→∞∥xn+1-xn∥=0.
We note that(3.9)∥xn-Txn∥≤∥xn-xn+1∥+∥xn+1-Txn∥≤∥xn-xn+1∥+∥αnu+βnxn+(1-αn-βn)Txn-Txn∥=∥xn-xn+1∥+∥αn(u-Txn)+βn(xn-Txn)∥≤∥xn-xn+1∥+αn∥(u-Txn)∥+βn∥(xn-Txn)∥=11-βn∥xn+1-xn∥+αn1-βn∥u-Txn∥.
Consequently, by the condition (C1) and (C2), we also have limn→∞∥xn-Txn∥=0. Next, we will show that
(3.10)limsupn→∞〈v-z0,xn+1-z0〉≤0,wherez0=PΓv.
To show this, we can choose a subsequence{xnk} of {xn} such that
(3.11)limsupn→∞〈v-z0,Txn-z0〉=limk→∞〈v-z0,Txnk-z0〉.
As{xn} is bounded, there exists a subsequence {xnk} which converges weakly to z. We may assume without loss of generality that xnk⇀z. Since ∥Txn-xn∥→0, we obtain Txnk⇀z as k→∞. By Lemma 2.3, we obtain that z∈Fix(T)=Γ.
Now from (2.4), observe that
(3.12)limsupn→∞〈v-z0,xn-z0〉=limsupn→∞〈v-z0,Txn-z0〉=limk→∞〈v-z0,Txnk-z0〉=〈v-z0,z-z0〉≤0.
Therefore, we compute
(3.13)∥xn+1-z0∥2=〈αnv+βnxn+γnTxn-z0,xn+1-z0〉=αn〈v-z0,xn+1-z0〉+βn〈xn-z0,xn+1-z0〉+γn〈Txn-z0,xn+1-z0〉≤αn〈v-z0,xn+1-z0〉+12βn(∥xn-z0∥2+∥xn+1-z0∥2)+12γn(∥xn-z0∥2+∥xn+1-z0∥2)≤αn〈v-z0,xn+1-z0〉+12(1-αn)(∥xn-z0∥2+∥xn+1-z0∥2)
which implies that
(3.14)∥xn+1-z0∥2≤(1-αn)(∥xn-z0∥2)+2αn〈v-z0,xn+1-z0〉.
Finally, by (3.12), (3.14), and Lemma 2.5, we conclude that {xn} converges to z0. This completes the proof.Lettingβn≡0 of iterative scheme (3.1) in Theorem 3.1, then we obtain the following corollary.Corollary 3.2.
For anyu,x0∈C, one defines the sequence {xn} by
(3.15)xn+1=αnu+(1-αn)PC(I-ξA*(I-PQ)A)xn,n≥0,
where {αn} is a sequence in [0,1]. Suppose that the SFP is consistent and 0<ξ<(2/∥A∥2).
Let{xn} be defined as in (3.15). If the following assumptions are satisfied: (C1)
limn→∞αn=0 but ∑n=1∞‍αn=∞,(C2)
∑n=1∞‍|αn+1-αn|<∞. Then {xn} converges to a solution of the SFP (1.1).Remark 3.3.
Theorem3.1 and Corollary 3.2 extend and improve the result of Xu [4] from weak to strong convergence theorems by using the modified Halpern's iterative scheme.



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*Source: 876069-2012-10-10.xml*