# Iterative Algorithms for the Split Problem and Its Convergence Analysis

Authors: Zhangsong Yao; Arif Rafiq; Shin Min Kang; Li-Jun Zhu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Now, it is known that the split common fixed point problem is a generalization of the split feasibility problem and of the convex feasibility problem. In this paper, the split common fixed point problem associated with the pseudocontractions is studied. An iterative algorithm has been presented for solving the split common fixed point problem. Strong convergence result is obtained.


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## Body

## 1. Introduction

Now, we know that the convex feasibility problem can be formulated as finding a pointx† such that
(1)x†∈⋂i=1n‍Ci,n∈N,
where Ci(≠∅) is a closed convex subset of a Hilbert space H. The convex feasibility problem has extensive applications in many applied disciplines such as signal processing, biomedical engineering, and communications. For related works, please see [1–3].Ifn=2 in (1), then a special case of (1) is the following split feasibility problem.Problem 1.
The split feasibility problem: letH1 and H2 be two Hilbert spaces. Let C1⊂H1 and C2⊂H2 be two nonempty closed convex sets. Let A:H1→H2 be a bounded linear operator. The split feasibility problem is
(2)findavectorx†∈C1suchthatAx†∈C2.Such problem arises in the intensity-modulated radiation therapy. In the finite-dimensional space, Censor and Elfving [4] firstly constructed the following iterative algorithm to solve (2):
(3)xm+1=A-1projC2(projA(C1)(Axm)),m∈N,
where C1⊂Rn and C2⊂Rn are closed convex sets and A is an n×n matrix.However, we note that calculating inverseA-1 is very time-consuming, if the dimension n is large. For overcoming this problem, Byrne [5] introduced the following more popular algorithm:
(4)xm+1=projC1(xm-ϖAT(I-projC2)Axm),m∈N,
where AT denotes the transposition of A. Consequently, (4) and its variant have been studied extensively. For related results, please refer to [6–13].In the case whereC1 and C2 in (2) are the fixed point sets of nonlinear operators, problem (2) is called by Censor and Segal [14] the split common fixed point problem.Problem 2.
The split common fixed point problem: this problem is to find a fixed pointx† of an operator S in the space H1 such that its image Ax† under a linear transformation A is a fixed point y† of another operator T in the image space H2. Namely, find a vector x†∈H such that
(5)x†∈Fix⁡(S),y†=Ax†∈Fix⁡(T),
where Fix⁡(S) and Fix⁡(T) denote the fixed point sets of nonlinear operators S:H1→H1 and T:H2→H2, respectively.A natural idea is to apply (4) to the split common fixed point problem (5). That is, taking C1=Fix⁡(S) and C2=Fix⁡(T) in (4), we get
(6)xm+1=projFix⁡(S)(xm-ϖATvvvvvvvvvvv×(I-projFix⁡(T))Axm),m∈N.
However, projFix⁡(S) and projFix⁡(T) are generally not easy to calculate. Thus, (6) may fail. We have to find new algorithm to solve (5). In this respect, Censor and Segal [14] proposed the following iterative method: for any initial guess x1∈H1, define a sequence {xm} by
(7)xm+1=S(xm-λA*(I-T)Axm),m∈N,
where S and T are directed operators. Moudafi [15] relaxed (7) to the following form:
(8)ym=xm-ϖA*(I-T)Axm,xm+1=(1-αm)ym+αmS(ym),m∈N,
where U and T are demicontractive operators.Note that (7) and (8) have weak convergence. Some strong convergence results have been given in the literature; see, for instance, [16, 17]. In the present paper, we consider an interesting respect: could we extend the classes of directed and demicontractive operators to the class of pseudocontractive mappings?Our main purpose of this paper is to solve the above problem. We construct an iterative algorithm in which the involved operators are pseudocontractions and show its strong convergence.


## 2. Definitions and Lemmas

In this section, we collect some definitions and lemmas. LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let (H⊃)E≠∅ be a closed convex set.Definition 3.
An operatorT:E→E is called Lipschitzian, if
(9)∥Tx‡-Ty‡∥≤L∥x‡-y‡∥,
for some L>0 and all x‡, y‡∈E.
In this case, we callTL-Lipschitzian continuous. If L=1 in (9), we call T nonexpansive.Definition 4.
An operatorT:E→E is called a directed operator, if
(10)〈Tx†-x†,Tx†-x〉≤0;
equivalently,
(11)∥Tx†-x∥2≤∥x†-x∥2-∥Tx†-x†∥2
for all x†∈E and x∈Fix⁡(T), the fixed points set of T.Definition 5.
An operatorT:E→E is called a demicontractive operator, if
(12)∥Tx†-x∥2≤∥x†-x∥2+κ∥Tx†-x†∥2,vvvvvvvvwwvvvv∀x†∈E,x∈Fix⁡(T),
where κ∈(0,1)From the above definitions, we note that the class of demicontractive operators contains important operators such as the directed operators and the nonexpansive operators with fixed points. Such a class of operators is fundamental because they include many types of nonlinear operators arising in applied mathematics and optimization; see, for example, [18] and references therein.Definition 6.
An operatorT:E→E is called pseudocontractive, if
(13)〈Tx†-Ty†,x†-y†〉≤∥x†-y†∥2;
equivalently,
(14)∥Tx†-Ty†∥2≤∥x†-y†∥2+∥(I-T)x†-(I-T)y†∥2
for all x†, y†∈E.It is obvious that the class of pseudocontractive mappings with fixed points includes the class of demicontractive mappings.Lemma 7 (see [19]).
LetH be a real Hilbert space; let C⊂H be a closed convex set. Let W:C→C be a continuous pseudocontractive mapping. Then(i)
Fix⁡(W) is a closed convex subset of C;(ii)
(I-W) is demiclosed at zero.


## 3. Main Results

LetH1 and H2 be two real Hilbert spaces and let A:H1→H2 be a bounded linear operator with its adjoint A*. This section is devoted to study problem (5), where S:H1→H1 and T:H2→H2 are two L-Lipschitzian pseudocontractive mappings. We denote the solution set of problem (5) by
(15)Γ={x†∈Fix⁡(S);Ax†∈Fix⁡(T)}.
In the sequel, we assume Γ≠∅.In order to solve problem (5), we present the following iterative algorithm.Algorithm 8.
Letϖ,τ,σ,ϱ, and ς be five constants. For u∈H1, arbitrarily, we define the following iterative manner:
(16)C1=H1,x1=projC1(u),zn=xn+ϖA*[(1-τ)I+τT((1-σ)I+σT)-I]Axn,yn=(1-ϱ)zn+ϱS[(1-ς)zn+ςSzn],Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=projCn+1(u),n∈N.Theorem 9.
Assume thatϖ, τ, σ, ϱ, and ς satisfy the following assumptions: 0<ϖ<1/∥A∥2, 0<τ≤σ<1/(1+L2+1), and 0<ϱ<ς<1/(1+L2+1). Then, {xn}, {yn}, and {zn} defined by (16) converge strongly to projΓ(u).Proof.
The outline of our proof details is as follows:(i)
Γ⊂Cn, for all n∈N;(ii)
Cn is closed and convex, for all n∈N;(iii)
xn→x*;(iv)
x*=projΓ(u).
Proof of (i). We show this by induction. (1)  Γ⊂C1=H1 is obvious.
(
2
) Suppose that Γ⊂Ck for some k∈N. For any p∈Γ⊂Ck, we have, from (14), that
(17)∥S((1-ς)zk+ςSzk)-p∥2=∥S((1-ς)zk+ςSzk)-Sp∥2≤∥(1-ς)zk+ςSzk-p∥2+∥(1-ς)zk+ςSzk-S((1-ς)zk+ςSzk)∥2,∥Szk-p∥2≤∥zk-p∥2+∥zk-Szk∥2.
Observing that in any Hilbert space, we have
(18)∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2,wwwwwwwwwwwwwwwiwwwwwwwwwww∀t∈[0,1].
Set vk=(1-ς)zk+ςSzk, for all k∈N. By (17) and (18), we obtain
(19)∥Svk-p∥2≤∥(1-ς)(zk-p)+ς(Szk-p)∥2+∥(1-ς)zk+ςSzk-Svk∥2=∥(1-ς)(zk-Svk)+ς(Szk-Svk)∥2+∥(1-ς)(zk-p)+ς(Szk-p)∥2=(1-ς)∥zk-Svk∥2+ς∥Szk-Svk∥2-ς(1-ς)∥zk-Szk∥2+(1-ς)∥zk-p∥2+ς∥Szk-p∥2-ς(1-ς)∥zk-Szk∥2≤(1-ς)∥zk-p∥2+ς(∥zk-p∥2+∥zk-Szk∥2)-2ς(1-ς)∥zk-Szk∥2+(1-ς)∥zk-Svk∥2+ς∥Szk-Svk∥2.
Since S is L-Lipschitzian and zk-vk=ς(zk-Szk), we have
(20)∥Svk-p∥2≤(1-ς)∥zk-p∥2+ς(∥zk-p∥2+∥zk-Szk∥2)-2ς(1-ς)∥zk-Szk∥2+(1-ς)∥zk-Svk∥2+ς3L2∥zk-Szk∥2=∥zk-p∥2+(1-ς)∥zk-Svk∥2-ς(1-2ς-ς2L2)∥zk-Szk∥2.
Since ς<1/(1+L2+1), we deduce
(21)1-2ς-ς2L2>0.
This together with (20) implies that
(22)∥Svk-p∥2≤∥zk-p∥2+(1-ς)∥zk-Svk∥2.
Hence,
(23)∥yk-p∥2=∥(1-ϱ)zk+ϱSvk-p∥2=∥(1-ϱ)(zk-p)+ϱ(Svk-p)∥2=(1-ϱ)∥zk-p∥2+ϱ∥Svk-p∥2-ϱ(1-ϱ)∥Svk-zk∥2≤(1-ϱ)∥zk-p∥2+ϱ[∥zk-p∥2+(1-ς)∥zk-Svk∥2]-ϱ(1-ϱ)∥Svk-zk∥2=∥zk-p∥2+(1-ϱ)(ϱ-ς)∥Svk-zk∥2.
Noting that ϱ≤ς, we deduce
(24)∥yk-p∥=∥(1-ϱ)zk+ϱS((1-ς)zk+ςSzk)-p∥≤∥zk-p∥.
Similarly, we also have
(25)∥[(1-τ)I+τT((1-σ)I+σT)]Axk-Ap∥≤∥Axk-Ap∥.
In Hilbert spaces, there holds
(26)2〈z†,z〉=∥z†+z∥2-∥z†∥2-∥z∥2.
With the help of (25) and (26), we get
(27)∥zk-p∥2=∥xk-p+ϖA*[(1-τ)I+τT((1-σ)I+σT)-I]hhh×Axk[(1-τ)I+τT((1-σ)I+σT)-I]∥2=2ϖ〈A*[(1-τ)I+τT((1-σ)I+σT)-I]hhhhhhi×Axk,xk-p[(1-τ)I+τT((1-σ)I+σT)-I]〉+ϖ2∥A*[(1-τ)I+τT((1-σ)I+σT)-I]Axk∥2+∥xk-p∥2=2ϖ〈[(1-τ)I+τT((1-σ)I+σT)-I]Axk,hhhhhhAxk-Ap[(1-τ)I+τT((1-σ)I+σT)-I]〉+ϖ2∥A∥2∥[(1-τ)I+τT((1-σ)I+σT)-I]Axk∥2+∥xk-p∥2=∥xk-p∥2+ϖ[∥[(1-τ)I+τT((1-σ)I+σT)]Axk-Ap∥2-∥[(1-τ)I+τT((1-σ)I+σT)-I]Axk∥2-∥Axk-Ap∥2∥[(1-τ)I+τT((1-σ)I+σT)]Axk-Ap∥2]+ϖ2∥A∥2∥[(1-τ)I+τT((1-σ)I+σT)-I]Axk∥2≤ϖ(ϖ∥A∥2-1)∥Axk[(1-τ)I+τT((1-σ)I+σT)-I]hhhhhhhhhhhhhi×Axk[(1-τ)I+τT((1-σ)I+σT)-I]∥2+∥xk-p∥2≤∥xk-p∥2.
By (24) and (27), we have
(28)∥yk-p∥≤∥zk-p∥≤∥xk-p∥.
This shows that p∈Ck+1. Thus, we get Γ⊂Cn, for all n∈N.
Proof of (ii). It is easy to verify that Cn is closed, for all n∈N. Next, we only need to verify that Cn is convex, for all n∈N. In fact, let z†,z‡∈Cn+1; for each ζ∈(0,1), we have
(29)∥yn-(ζz†+(1-ζ)z‡)∥2=∥ζ(yn-z†)+(1-ζ)(yn-z‡)∥2=ζ∥yn-z†∥2+(1-ζ)∥yn-z‡∥2-ζ(1-ζ)∥z†-z‡∥2≤ζ∥zn-z†∥2+(1-ζ)∥zn-z‡∥2-ζ(1-ζ)∥z†-z‡∥2=∥zn-(ζz†+(1-ζ)z‡)∥2;
namely,
(30)∥yn-(ζz†+(1-ζ)z‡)∥≤∥zn-(ζz†+(1-ζ)z‡)∥;
this shows ζz†+(1-ζ)z‡∈Cn+1 and Cn+1 is a convex set, for all n∈N.
Proof of (iii). Since Γ⊂Cn+1⊂Cn and xn+1=PCn+1(u)⊂Cn, we obtain
(31)∥xn+1-u∥≤∥p-u∥∀n∈N,p∈Γ.
It follows that {xn} is bounded.
It is known that the metric projectionprojC can be characterized by
(32)〈x†-projC(x†),projC(x†)-x〉≥0,∀x∈C;
equivalently,
(33)∥x-projC(x†)∥2+∥x†-projC(x†)∥2≤∥x-x†∥2.
With the help of (33), we have
(34)∥xn+1-xn∥2+∥u-xn∥2=∥xn+1-PCn(u)∥2+∥u-PCn(u)∥2≤∥xn+1-u∥2,
which implies that
(35)0≤∥xn+1-xn∥2≤∥xn+1-u∥2-∥u-xn∥2.
It follows that
(36)∥xn-u∥≤∥xn+1-u∥.
Since {∥xn-u∥} is bounded, we get
(37)lim⁡n→∞∥xn-u∥exists.
This together with (35) implies that
(38)lim⁡n→∞∥xn+1-xn∥=0.
The fact that xn+1=PCn+1(u)∈Cn+1 gives
(39)∥yn-xn+1∥≤∥xn-xn+1∥⟶0.
By (38) and (39), we derive
(40)lim⁡n→∞∥yn-xn∥=0.
Next we show that {xn} is a Cauchy sequence. As a matter of fact, for any m,n∈N with m>n, we have
(41)∥xm-xn∥2+∥xn-u∥2=∥xm-PCn(u)∥2+∥PCn(u)-u∥2≤∥xm-u∥2.
It follows that
(42)∥xm-xn∥2≤∥xm-u∥2-∥xn-u∥2.
Note that lim⁡n→∞(∥xm-u∥2-∥xn-u∥2)=0. Therefore, lim⁡n→∞∥xm-xn∥=0. So, {xn} is a Cauchy sequence and hence xn→x*.
Proof of (iv). From (24), (27), and (40), we have
(43)-ϖ(ϖ∥A∥2-1)∥[(1-τ)I+τT((1-σ)I+σT)-I]Axn∥2≤∥xn-p∥2-∥yn-p∥2≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)⟶0.
Hence,
(44)lim⁡n→∞∥[(1-τ)I+τT((1-σ)I+σT)-I]Axn∥=0.
It follows that
(45)lim⁡n→∞∥T((1-σ)I+σT)Axn-Axn∥=0.
Next, we firstly show that
(46)Fix⁡(T)=Fix⁡(T((1-σ)I+σT)).
As a matter of fact, Fix⁡(T)⊂Fix⁡(T((1-σ)I+σT)) is obvious. We only need to show that Fix⁡(T((1-σ)I+σT))⊂Fix⁡(T).
Take anyy†∈Fix⁡(T((1-σ)I+σT)). We have T((1-σ)I+σT)y†=y†. Set R=(1-σ)I+σT. We have TRy†=y†. Write Ry†=y*. Then, Ty*=y†. Next, we show y†=y*. In fact,
(47)∥y†-y*∥=∥Ty*-Ry†∥=∥Ty*-(1-σ)y†-σTy†∥=σ∥Ty*-Ty†∥≤σL∥y*-y†∥.
Since σ<1/(1+L2+1)<1/L, we deduce y*=y†∈Fix⁡(R)=Fix⁡(T). Thus, y†∈Fix⁡(T). Hence, Fix⁡(T((1-σ)I+σT))⊂Fix⁡(T). Therefore, Fix⁡(T((1-σ)I+σT))=Fix⁡(T).
With (46) in hand, we prove that T((1-σ)I+σT)-I is demiclosed at 0. Let the sequence {un}⊂H2 satisfying un⇀y† and un-T((1-σ)I+σT)un→0. Now, we show that y†∈Fix⁡(T((1-σ)I+σT)).
SinceT is L-Lipschitzian, we have
(48)∥un-Tun∥≤∥un-T((1-σ)I+σT)un∥+∥T((1-σ)I+σT)un-Tun∥≤∥un-T((1-σ)I+σT)un∥+σL∥un-Tun∥.
It follows that
(49)∥un-Tun∥≤11-σL∥un-T((1-σ)I+σT)un∥.
Hence,
(50)lim⁡n→∞∥un-Tun∥=0.
Since T-I is demiclosed at 0 by Lemma 7, we immediately deduce y†∈Fix⁡(T)=Fix⁡(T((1-σ)I+σT)). Therefore, T((1-σ)I+σT)-I is demiclosed at 0. Since A is a bounded linear operator, we get ∥Axn-Ax*∥→0. From (45), we deduce Ax*∈Fix⁡(T((1-σ)I+σT))=Fix⁡(T).
By (16), (40), and (44), we deduce
(51)lim⁡n→∞∥zn-yn∥=0.
So,
(52)lim⁡n→∞∥S[(1-ς)zn+ςSzn]-zn∥=lim⁡n→∞1ϱ∥yn-zn∥=0.
Similarly, we can show that x*∈Fix⁡(S). To this end, we have proven that x*∈Fix⁡(S) and Ax*∈Fix⁡(T). Therefore, x*∈Γ. This completes the proof.



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*Source: 897370-2014-05-28.xml*