# Shrinking Projection Methods for Split Common Fixed-Point Problems in Hilbert Spaces

Authors: Huan-chun Wu; Cao-zong Cheng
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Inspired by Moudafi (2011) and Takahashi et al. (2008), we present the shrinking projection method for the split common fixed-point problem in Hilbert spaces, and we obtain the strong convergence theorem. As a special case, the split feasibility problem is also considered.


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## Body

## 1. Introduction

LetC and Q be nonempty closed convex sets in real Hilbert spaces H
1 and H
2, respectively. The split feasibility problem (SFP) is to find
(1)
x
∈
C
,
such
that
A
x
∈
Q
,
where A
:
H
1
→
H
2 is a bounded linear operator. We use Φ to denote the solution set of the SFP (1). The SFP in finite-dimensional Hilbert space was first introduced by Censor and Elfving [1]. In 2010, Xu [2] considered the SFP in the setting of infinite-dimensional Hilbert space. The SFP has received much attention due to its wide applications in signal processing, image reconstruction, intensity-modulated radiation therapy, and so on (see [3–6]). Several iterative methods can be used to solve the SFP (1). Censor and Elfving [1] constructed the iterative process which involves the computation of the inverse of a matrix. A more popular algorithm that solves the SFP is the CQ algorithm of Byrne [3, 4]; that is, let x
0 be an arbitrary point in H
1:
(2)
x
n
+
1
=
P
C
(
I
-
γ
A
*
(
I
-
P
Q
)
A
)
x
n
,
where γ
>
0 is a parameter and P
C and P
Q are metric projections onto C and Q, respectively.LetK be a nonempty closed convex subset of a real Hilbert space H and let T
:
K
→
K be a mapping. We denote by Fix
⁡
(
T
) the fixed-point set of T; that is, Fix
⁡
(
T
)
=
{
x
∈
K
:
T
x
=
x
}. A mapping T
:
K
→
K is nonexpansive if ∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥ for all x
,
y
∈
K. A mapping T
:
K
→
K is quasinonexpansive if Fix
⁡
(
T
)
≠
∅ and ∥
T
x
-
y
∥
≤
∥
x
-
y
∥ for all x
∈
K and y
∈
Fix
⁡
(
T
). It is known that the fixed-point set of a quasinonexpansive mapping is closed and convex (see [7, 8]). There are some quasinonexpansive mappings which are not nonexpansive (see [9–11]). For example, the level set of a continuous convex function is characterized as the fixed-point set of a nonlinear mapping called the subgradient projection, which is not nonexpansive but quasinonexpansive.Now we focus our attention on the following two-operator split common fixed-point problem (SCFP):(3)
find
x
*
∈
C
,
such
that
A
x
*
∈
Q
,
where A
:
H
1
→
H
2 is a bounded linear operator and U
:
H
1
→
H
1 and T
:
H
2
→
H
2 are two quasinonexpansive mappings with Fix
⁡
(
U
)
=
C and Fix
⁡
(
T
)
=
Q. The solution set of the SCFP (3) is denoted by
(4)
Γ
=
{
x
*
∈
C
:
A
x
*
∈
Q
}
.
As far as we know, the SCFP is introduced by Censor and Segal [12]. By taking U
=
P
C and T
=
P
Q, the SCFP reduces to the SFP. Hence, the SCFP is a generalization of the SFP. Moudafi [13] considered the following algorithm for the SCFP: let x
0
∈
H
1 be arbitrary, u
k
=
x
k
-
γ
β
A
*
(
I
-
T
)
A
x
k and
(5)
x
k
+
1
=
(
1
-
α
k
)
u
k
+
α
k
U
(
u
k
)
,
where β
∈
(
0,1
), α
k
∈
(
0,1
), and γ
∈
(
0
,
1
/
λ
β
), with λ being the spectral radius of the operator A
*
A. He obtained the weak convergence of the algorithm (5).In 2008, Takahashi et al. [14] developed the shrinking projection method for the nonexpansive mapping. Let T be a nonexpansive mapping of K into itself such that Fix
⁡
(
T
)
≠
∅. Let x
0
∈
H, C
1
=
K and u
1
=
P
C
1
x
0;
(6)
y
n
=
α
n
u
n
+
(
1
-
α
n
)
T
u
n
,
C
n
+
1
=
{
z
∈
C
n
:
∥
y
n
-
z
∥
≤
∥
u
n
-
z
∥
}
,
u
n
+
1
=
P
C
n
+
1
x
0
,
where 0
≤
α
n
≤
a
<
1. They proved that the sequence {
u
n
} converges strongly to P
Fix
⁡
(
T
)
x
0.Motivated by the above results, especially by Moudafi [13] and Takahashi et al. [14], in this paper, we present the shrinking projection methods for the split common fixed-point problems. As a special case, the split feasibility problem is also discussed.


## 2. Preliminaries

Throughout this paper, letℕ and ℝ be the sets of positive integers and real numbers, respectively. For any x
∈
H, there exists a unique point P
K
x
∈
K such that
(7)
∥
x
-
P
K
x
∥
≤
∥
x
-
y
∥
∀
y
∈
K
,
where K is a nonempty closed convex subset of a real Hilbert space H. The mapping P
K is called the metric projection of H onto K. Note that P
K is a nonexpansive mapping. For x
∈
H and z
∈
K, we have
(8)
z
=
P
K
x
⟺
〈
x
-
z
,
y
-
z
〉
≤
0
for
every
y
∈
K
.We say that a mappingT
:
K
→
K is demiclosed at zero if for any sequence {
x
n
}
⊂
K which converges weakly to x, the strong convergence of the sequence {
T
x
n
} to zero implies that T
x
=
0. It is well known that I
-
T is demiclosed whenever T is nonexpansive. In fact, this property is satisfied for more general mappings (see [15, 16]).We will use the following notations:(1)
x
n
→
x stands for the strong convergence of {
x
n
} to x;
(2)
x
n
⇀
x stands for the weak convergence of {
x
n
} to x;
(3)
ω
ω
(
x
n
)
=
{
x
:
∃
x
n
j
⇀
x
} denotes the weak ω-limit set of {
x
n
}.Here are two useful lemmas.Lemma 1.
Letx
,
y
∈
H and let λ
∈
ℝ. One has
(9)
∥
λ
x
+
(
1
-
λ
)
y
∥
2
=
λ
∥
x
∥
2
+
(
1
-
λ
)
∥
y
∥
2
-
λ
(
1
-
λ
)
∥
x
-
y
∥
2
.Lemma 2 (see [17]).
LetK be a closed convex subset of a real Hilbert space H and let {
x
n
} be a sequence in H and u
∈
H. Let q
=
P
K
u. If {
x
n
} satisfies the following conditions:(1)
ω
ω
(
x
n
)
⊂
K,
(2)
∥
x
n
-
u
∥
≤
∥
u
-
q
∥ for all n
∈
ℕ,
then one hasx
n
→
q.


## 3. Shrinking Projection Methods

Now we are in a position to give the shrinking projection method for split common fixed-point problem (3).Theorem 3.
LetH
1 and H
2 be real Hilbert spaces and let A
:
H
1
→
H
2 be a bounded linear operator. Let U
:
H
1
→
H
1 and T
:
H
2
→
H
2 be two quasinonexpansive mappings with Fix
⁡
(
U
)
=
C and Fix
⁡
(
T
)
=
Q. Suppose that I
-
U and I
-
T are demiclosed at zero and solution set Γ of the SCFP (3) is nonempty. For u
∈
H
1 chosen arbitrarily, C
1
=
H
1, h
1
=
P
C
1
u, define a sequence {
h
n
} by the following algorithm:
(10)
w
n
=
h
n
-
γ
A
*
(
I
-
T
)
A
h
n
,
y
n
=
α
n
w
n
+
(
1
-
α
n
)
U
w
n
,
C
n
+
1
=
{
z
∈
C
n
:
∥
y
n
-
z
∥
≤
∥
h
n
-
z
∥
}
,
h
n
+
1
=
P
C
n
+
1
u
.
If the following are satisfied:(1)
{
α
n
}
⊂
(
0,1
) and liminf
⁡
n
→
∞
⁡
α
n
(
1
-
α
n
)
>
0
,
(2)
0
<
γ
<
(
1
/
λ
A
A
*
), where λ
A
A
* denotes the spectral radius of the operator A
A
*,then the sequence {
h
n
} converges strongly to P
Γ
u.Proof.
We first show thatΓ
⊂
C
n for all n
∈
ℕ. It is obvious that Γ is contained in C
1
=
H
1. Suppose that Γ
⊂
C
k for some k
∈
ℕ. We have, for any p
∈
Γ
⊂
C
k,
(11)
∥
y
k
-
p
∥
2
=
∥
α
k
w
k
+
(
1
-
α
k
)
U
w
k
-
p
∥
2
≤
α
k
∥
w
k
-
p
∥
2
+
(
1
-
α
k
)
∥
w
k
-
p
∥
2
=
∥
h
k
-
γ
A
*
(
I
-
T
)
A
h
k
-
p
∥
2
=
∥
h
k
-
p
∥
2
-
2
γ
〈
A
*
(
I
-
T
)
A
h
k
,
h
k
-
p
〉
+
γ
2
∥
A
*
(
I
-
T
)
A
h
k
∥
2
=
∥
h
k
-
p
∥
2
+
2
γ
〈
T
A
h
k
-
A
h
k
,
A
h
k
-
A
p
〉
+
γ
2
〈
(
I
-
T
)
A
h
k
,
A
A
*
(
I
-
T
)
A
h
k
〉
≤
∥
h
k
-
p
∥
2
+
γ
2
λ
A
A
*
∥
(
I
-
T
)
A
h
k
∥
2
+
γ
[
∥
T
A
h
k
-
A
p
∥
2
-
∥
T
A
h
k
-
A
h
k
∥
2
-
∥
A
h
k
-
A
p
∥
2
]
≤
∥
h
k
-
p
∥
2
+
γ
2
λ
A
A
*
∥
(
I
-
T
)
A
h
k
∥
2
-
γ
∥
T
A
h
k
-
A
h
k
∥
2
=
∥
h
k
-
p
∥
2
+
γ
(
γ
λ
A
A
*
-
1
)
∥
(
I
-
T
)
A
h
k
∥
2
≤
∥
h
k
-
p
∥
2
.
It follows that p
∈
C
k
+
1. Thus, we get Γ
⊂
C
n for all n
∈
ℕ.
Next we show thatC
n is closed and convex for all n
∈
ℕ. The set C
1
=
H
1 is obviously closed and convex. Suppose that C
k is closed and convex. We see that C
k
+
1 is closed and convex since ∥
y
n
-
z
∥
≤
∥
h
n
-
z
∥ is equivalent to
(12)
∥
y
n
∥
2
-
∥
h
n
∥
2
-
2
〈
y
n
-
h
n
,
z
〉
≤
0
.
It follows that C
n is closed and convex for all n
∈
ℕ. Therefore, we obtain that the sequence {
h
n
} is well defined.
Fromh
n
=
P
C
n
u, we have
(13)
〈
u
-
h
n
,
h
n
-
y
〉
≥
0
∀
y
∈
C
n
.
Recalling that Γ
⊂
C
n, one has
(14)
〈
u
-
h
n
,
h
n
-
p
〉
≥
0
∀
p
∈
Γ
.
Hence,
(15)
0
≤
〈
u
-
h
n
,
h
n
-
p
〉
=
〈
u
-
h
n
,
h
n
-
u
+
u
-
p
〉
≤
-
∥
u
-
h
n
∥
2
+
∥
u
-
h
n
∥
∥
u
-
p
∥
.
This implies that
(16)
∥
u
-
h
n
∥
≤
∥
u
-
p
∥
,
which yields that {
h
n
} is bounded.
Fromh
n
=
P
C
n
u and h
n
+
1
=
P
C
n
+
1
u
∈
C
n
+
1
⊂
C
n, we get
(17)
0
≤
〈
u
-
h
n
,
h
n
-
h
n
+
1
〉
≤
-
∥
u
-
h
n
∥
2
+
∥
u
-
h
n
∥
∥
u
-
h
n
+
1
∥
,
which gives that
(18)
∥
u
-
h
n
∥
≤
∥
u
-
h
n
+
1
∥
.
Hence,
(19)
the
limit
lim
⁡
n
→
∞
∥
u
-
h
n
∥
exists
.
It follows from (17) that
(20)
∥
h
n
-
h
n
+
1
∥
2
=
∥
h
n
-
u
∥
2
+
2
〈
h
n
-
u
,
u
-
h
n
+
1
〉
+
∥
u
-
h
n
+
1
∥
2
=
∥
h
n
-
u
∥
2
+
2
〈
h
n
-
u
,
u
-
h
n
+
h
n
-
h
n
+
1
〉
+
∥
u
-
h
n
+
1
∥
2
=
-
∥
h
n
-
u
∥
2
+
2
〈
h
n
-
u
,
h
n
-
h
n
+
1
〉
+
∥
u
-
h
n
+
1
∥
2
≤
-
∥
h
n
-
u
∥
2
+
∥
u
-
h
n
+
1
∥
2
.
Thus, we get
(21)
lim
⁡
n
→
∞
∥
h
n
-
h
n
+
1
∥
=
0
.
The fact that h
n
+
1
=
P
C
n
+
1
u
∈
C
n
+
1 gives
(22)
∥
y
n
-
h
n
+
1
∥
≤
∥
h
n
-
h
n
+
1
∥
⟶
0
.
The expressions (21) and (22) yield
(23)
lim
⁡
n
→
∞
∥
y
n
-
h
n
∥
=
0
.
We will prove thatω
ω
(
h
n
)
⊂
Γ. Without loss of generality, we assume that h
n
⇀
h
*. It follows from (11) that
(24)
γ
(
1
-
γ
λ
A
A
*
)
∥
(
I
-
T
)
A
h
n
∥
2
≤
∥
h
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
≤
(
∥
h
n
-
p
∥
+
∥
y
n
-
p
∥
)
∥
h
n
-
y
n
∥
.
This together with (23) implies that
(25)
lim
⁡
n
→
∞
∥
(
I
-
T
)
A
h
n
∥
=
0
.
We have A
h
*
∈
Fix
⁡
(
T
)
=
Q since I
-
T is demiclosed at zero.
Using (10) and (25), we get w
n
⇀
h
*. For any p
∈
Γ, one has
(26)
∥
y
n
-
p
∥
2
=
∥
α
n
w
n
+
(
1
-
α
n
)
U
w
n
-
p
∥
2
=
α
n
∥
w
n
-
p
∥
2
+
(
1
-
α
n
)
∥
U
w
n
-
p
∥
2
-
α
n
(
1
-
α
n
)
∥
U
w
n
-
w
n
∥
2
≤
∥
w
n
-
p
∥
2
-
α
n
(
1
-
α
n
)
∥
U
w
n
-
w
n
∥
2
≤
∥
h
n
-
p
∥
2
-
α
n
(
1
-
α
n
)
∥
U
w
n
-
w
n
∥
2
,
which implies that
(27)
α
n
(
1
-
α
n
)
∥
U
w
n
-
w
n
∥
2
≤
∥
h
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
≤
(
∥
h
n
-
p
∥
+
∥
y
n
-
p
∥
)
∥
h
n
-
y
n
∥
.
Therefore, one has
(28)
lim
⁡
n
→
∞
∥
U
w
n
-
w
n
∥
=
0
.
It follows that h
*
∈
Fix
⁡
(
U
)
=
C since I
-
U is demiclosed at zero. Thus, we have obtained ω
ω
(
h
n
)
∈
Γ. According to Lemma 2, we see that h
n
→
P
Γ
u.By Theorem3, we immediately obtain the shrinking projection method for the split feasibility problem.Theorem 4.
LetH
1 and H
2 be real Hilbert spaces and let C and Q be nonempty closed convex subsets of H
1 and H
2, respectively. Let A
:
H
1
→
H
2 be a bounded linear operator. Suppose that the solution set Φ of the SFP (1) is nonempty. For u
∈
H
1 chosen arbitrarily, C
1
=
H
1, h
1
=
P
C
1
u, define a sequence {
h
n
} by the following algorithm:
(29)
w
n
=
h
n
-
γ
A
*
(
I
-
P
Q
)
A
h
n
,
y
n
=
α
n
w
n
+
(
1
-
α
n
)
P
C
w
n
,
C
n
+
1
=
{
z
∈
C
n
:
∥
y
n
-
z
∥
≤
∥
h
n
-
z
∥
}
,
h
n
+
1
=
P
C
n
+
1
u
.
If the following are satisfied: (1)
{
α
n
}
⊂
(
0,1
) and liminf
⁡
n
→
∞
⁡
α
n
(
1
-
α
n
)
>
0,
(2)
0
<
γ
<
1
/
λ
A
A
*, where λ
A
A
* denotes the spectral radius of the operator A
A
*, then the sequence {
h
n
} converges strongly to P
Φ
u.Remark 5.
Lettingu
=
0 in Theorems 3 and 4, we obtain the shrinking projection methods for minimum-norm solutions of corresponding problems.



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*Source: 817367-2014-04-14.xml*