Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 491760, 12 pages
doi:10.1155/2012/491760
Research Article
Multiple-Set Split Feasibility Problems for
Asymptotically Strict Pseudocontractions
Shih-Sen Chang,1 Yeol Je Cho,2 J. K. Kim,3
W. B. Zhang,1 and L. Yang4
1
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming,
Yunnan 650221, China
2
Department of Mathematics Education and RINS, Gyeongsang National University,
Jinju 660-701, Republic of Korea
3
Department of Mathematics Education, Kyungnam University, Masan, Kyungnam 631-701,
Republic of Korea
4
Department of Mathematics, South West University of Science and Technology,
Mianyang Sichuan 621010, China
Correspondence should be addressed to Shih-Sen Chang, changss@yahoo.cn
and Yeol Je Cho, yjcho@gnu.ac.kr
Received 5 October 2011; Revised 5 December 2011; Accepted 7 December 2011
Academic Editor: Khalida Inayat Noor
Copyright q 2012 Shih-Sen Chang et al. 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.
In this paper, we introduce an iterative method for solving the multiple-set split feasibility
problems for asymptotically strict pseudocontractions in infinite-dimensional Hilbert spaces, and,
by using the proposed iterative method, we improve and extend some recent results given by some
authors.
1. Introduction
The split feasibility problem SFP in finite dimensional spaces was first introduced by Censor
and Elfving 1 for modeling inverse problems which arise from phase retrievals and in
medical image reconstruction 2. 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 3–5.
The split feasibility problem in an infinite dimensional Hilbert space can be found in
2, 4, 6–8.
Throughout this paper, we always assume that H1 , H2 are real Hilbert spaces, “ → ”,
“” are denoted by strong and weak convergence, respectively.
2
Abstract and Applied Analysis
The purpose of this paper is to introduce and study the following multiple-set split
feasibility problem for asymptotically strict pseudocontraction MSSFP in the framework of
infinite-dimensional Hilbert spaces. Find x∗ ∈ C such that
Ax∗ ∈ Q,
1.1
where A : H1 → H2 is a bounded linear operator, {Si } and {Ti }, i 1, 2, . . . , M, are the
families of mappings Si : H1 → H1 and Ti : H2 → H2 , respectively, C : M
i1 FSi and
M
Q : i1 FTi , where FSi {xi ∈ H1 : Si xi xi } and FTi {yi ∈ H2 : Ti yi yi } denote
the sets of fixed points of Si and Ti , respectively. In the sequel, we use Γ to denote the set of
solutions of the problem MSSFP, that is,
Γ {x ∈ C : Ax ∈ Q}.
1.2
2. Preliminaries
We first recall some definitions, notations, and conclusions which will be needed in proving
our main results.
Let E be a Banach space. A mapping T : E → E is said to be demiclosed at origin if, for
any sequence {xn } ⊂ E with xn x∗ and I − T xn → 0, we have x∗ T x∗ . A Banach space
E is said to have Opial’s property if, for any sequence {xn } with xn x∗ , we have
lim infxn − x∗ < lim infxn − y
n→∞
2.1
n→∞
for all y ∈ E with y /
x∗ .
Remark 2.1. It is well known that each Hilbert space possesses Opial’s property.
Definition 2.2. Let H be a real Hilbert space.
1 A mapping G : H → H is called a γ, {kn }-asymptotically strict pseudocontraction if
there exists a constant γ ∈ 0, 1 and a sequence {kn } ⊂ 1, ∞ with kn → 1 such
that
n
G x − Gn y2 ≤ kn x − y2 γ I − Gn x − I − Gn y2 ,
∀x, y ∈ H.
2.2
Especially, if kn 1 for each n ≥ 1 in 2.2 and there exists γ ∈ 0, 1 such that
Gx − Gy2 ≤ x − y2 γ I − Gx − I − Gy2 ,
∀x, y ∈ H,
2.3
then G : H → H is called a γ-strict pseudocontraction.
2 A mapping G : H → H is said to be uniformly L-Lipschitzian if there exists a
constant L > 0 such that
n
G x − Gn y ≤ Lx − y,
∀x, y ∈ H, n ≥ 1.
2.4
Abstract and Applied Analysis
3
3 A mapping G : H → H is said to be semicompact if, for any bounded sequence
{xn } ⊂ H with limn → ∞ xn − Gxn 0, there exists a subsequence {xni } ⊂ {xn } such
that xni converges strongly to a point x∗ ∈ H.
Now, we give one example of the γ, {kn }-asymptotically strict pseudocontraction
mapping.
Example 2.3. Let B be the unit ball in a Hilbert space l2 , and define a mapping T : B → B by
T x1 , x2 , . . . 0, x12 , a2 x2 , a3 x3 , . . . ,
2.5
where {ai } is a sequence in 0, 1 such that Π∞
i2 ai 1/2. It is proved in Goebel and Kirk 9
that
a T x T y ≤ 2x − y for all x, y ∈ B,
b T n x − T n y ≤ 2Πnj2 aj for all n ≥ 2 and x, y ∈ B.
Denote by k11/2 2, kn1/2 2Πnj2 aj n ≥ 2 and γ ∈ 0, 1. Then, we have
⎛
⎞2
n
lim kn lim ⎝2 aj ⎠ 1,
n→∞
n→∞
2.6
j2
n
T x − T n y2 ≤ kn x − y2 γ x − y − T n x − T n y2 ,
∀n ≥ 1, x, y ∈ B,
and so the mapping T is a γ, {kn }-asymptotically strict pseudocontraction.
Remark 2.4. 1 If we put γ 0 in 2.2, then the mapping G : H → H is asymptotically
nonexpansive.
2 If we put γ 0 in 2.3, then the mapping G : H → H is nonexpansive.
3 Each γ, {kn }-asymptotically strict pseudocontraction and each γ-strictly pseudocontraction both are demiclosed at origin 10.
Proposition 2.5. Let G : H → H be a (γ, {kn })- asymptotically strict pseudocontraction. If
FG / ∅, then, for any q ∈ FG and x ∈ H, the following inequalities hold and they are equivalent:
n
G x − q2 ≤ kn x − q2 γx − Gn x2 ,
2.7
1−γ
kn − 1 x − q2 ,
x − Gn x, x − q ≥
x − Gn x2 −
2
2
2.8
x − Gn x, q − Gn x ≤
1γ
kn − 1 x − q2 .
x − Gn x2 2
2
2.9
Lemma 2.6 see 11. Let {an }, {bn }, and {δn } be sequences of nonnegative real numbers satisfying
an1 ≤ 1 δn an bn ,
If
∞
i1
δn < ∞ and
∞
i1
∀n ≥ 1.
bn < ∞, then the limit limn → ∞ an exists.
2.10
4
Abstract and Applied Analysis
3. Multiple-Set Split Feasibility Problem
For solving the multiple-set split feasibility problem 1.1, let us assume that the following
conditions are satisfied:
C1 H1 and H2 are two real Hilbert spaces, A : H1 → H2 is a bounded linear operator;
C2 Si : H1 → H1 , i 1, 2, . . . , M, is a uniformly Li -Lipschitzian and βi , {ki,n }asymptotically strict pseudocontraction, and Ti : H2 → H2 , i 1, 2, . . . , M, is a
i -Lipschitzian and μi , {ki,n }-asymptotically strict pseudocontraction
uniformly L
satisfying the following conditions:
a C :
M
i1
FSi /
∅ and Q :
M
i1
FTi /
∅,
b β max1≤i≤M βi < 1 and μ max1≤i≤M μi < 1,
: max1≤i≤M L
i < ∞,
c L : max1≤i≤M Li < ∞ and L
d kn max1≤i≤M {ki,n , ki,n } and ∞ kn − 1 < ∞.
n1
We are now in a position to give the following result.
and {kn } be the same as above. Let {xn }
Theorem 3.1. Let H1 , H2 , A, {Si }, {Ti }, C, Q, β, μ, L, L,
be the sequence generated by
x1 ∈ H1 chosen arbitrarily,
xn1 1 − αn un αn Snn un ,
un xn γA∗ Tnn − IAxn , ∀n ≥ 1,
3.1
n
where Snn Snn mod M , Tnn Tn
for all n ≥ 1, {αn } is a sequence in 0, 1, and γ > 0 is a
mod M
constant satisfying the following conditions.
e αn ∈ δ, 1 − β for all n ≥ 1 and γ ∈ 0, 1 − μ/A2 , where δ ∈ 0, 1 − β is a positive
constant.
1 If Γ /
∅, then the sequence {xn } converges weakly to a point x∗ ∈ Γ.
2 In addition, if there exists a positive integer j such that Sj is semicompact, then the
sequences {xn } and {un } both converge strongly to a point x∗ ∈ Γ.
Proof. 1 The proof is divided into 5 steps as follows.
Step 1. We first prove that, for any p ∈ Γ, the limit
lim xn − p
n→∞
3.2
Abstract and Applied Analysis
exists. In fact, since p ∈ Γ, p ∈ C :
it follows that
5
M
i1
FSi , and Ap ∈ Q :
M
i1
FTi . From 3.1 and 2.8,
xn1 − p2 un − p − αn un − Snn un 2
2
un − p − 2αn un − p, un − Snn un α2n un − Snn un 2
2
2 ≤ un − p − αn 1 − β un − Snn un 2 − kn − 1un − p
3.3
α2n un − Snn un 2
2
1 αn kn − 1un − p − αn 1 − β − αn un − Snn un 2 .
On the other hand, since
un − p2 xn − p γA∗ Tnn − IAxn 2
2
xn − p γ 2 A∗ Tnn − IAxn 2 2γ xn − p, A∗ Tnn − IAxn ,
3.4
γ 2 A∗ Tnn − IAxn 2 γ 2 A∗ Tnn − IAxn , A∗ Tnn − IAxn γ 2 AA∗ Tnn − IAxn , Tnn − IAxn 3.5
≤ γ 2 A2 Tnn Axn − Axn ,
2
2γxn − p, A∗ Tnn − IAxn 2γAxn − Ap, Tnn − IAxn 2γ Axn − Ap Tnn − IAxn − Tnn − IAxn , Tnn − IAxn 2γ Tnn Axn − Ap, Tnn Axn − Axn − Tnn − IAxn 2 .
3.6
Further, letting x Axn , Gn Tnn , q Ap, γ μ in 2.9 and noting Ap ∈ FTn , it follows
that
1μ
kn − 1 Axn − Ap2
Tnn − IAxn 2 2
2
2
1μ
k
n − 1A xn − p2 .
≤
Tnn − IAxn 2 2
2
Tnn Axn − Ap, Tnn Axn − Axn ≤
3.7
Substituting 3.7 into 3.6 and simplifying it, we have
2
2γ xn − p, A∗ Tnn − IAxn ≤ γ μ − 1 Tnn − IAxn 2 kn − 1γA2 xn − p .
Substituting 3.5 and 3.8 into 3.4 and simplifying it, we have
un − p2 ≤ xn − p2 γ 2 A2 Tnn Axn − Axn 2
2
γ μ − 1 Tnn − IAxn 2 kn − 1γA2 xn − p
3.8
6
Abstract and Applied Analysis
2
xn − p − γ 1 − μ − γA2 Tnn Axn − Axn 2
2
kn − 1γA2 xn − p .
3.9
Again, substituting 3.9 into 3.3 and simplifying it, we have
xn1 − p2 ≤ 1 αn kn − 1
2
2 × xn − p − γ 1 − μ − γA2 Tnn Axn − Axn 2 kn − 1γA2 xn − p
− αn 1 − β − αn un − Snn un 2
2
≤ 1 αn kn − 1xn − p − γ 1 − μ − γA2 Tnn Axn − Axn 2
2
1 αn kn − 1kn − 1γA2 xn − p − αn 1 − β − αn un − Snn un 2 .
3.10
By the condition e, we have
xn1 − p2 ≤ 1 αn kn − 1xn − p2 1 αn kn − 1kn − 1γA2 xn − p2
2
≤ 1 Kkn − 1xn − p ,
3.11
where
K sup αn 1 αn kn − 1γA2 < ∞.
n≥1
By the condition d,
limit exists:
n1 kn
3.12
− 1 < ∞; hence, from Lemma 2.6, we know that the following
lim xn − p.
n→∞
3.13
Step 2. We will now prove that, for each p ∈ Γ, the limit
lim un − p
n→∞
3.14
exists. In fact, from 3.10 and 3.13, it follows that
γ 1 − μ − γA2 Tnn − IAxn 2 αn 1 − β − αn un − Snn un 2
2 2
2
≤ xn − p − xn1 − p Kkn − 1xn − p −→ 0 n −→ ∞.
3.15
Abstract and Applied Analysis
7
This together with the condition e implies that
lim un − Snn un 0,
3.16
lim Tnn − IAxn 0.
3.17
n→∞
n→∞
Therefore, it follows from 3.4, 3.13, and 3.17 that the limit limn → ∞ un − p exists.
Step 3. Now, we prove that
lim xn1 − xn 0,
n→∞
lim un1 − un 0.
n→∞
3.18
In fact, it follows from 3.1 that
xn1 − xn 1 − αn un αn Snn un − xn 1 − αn xn γA∗ Tnn − IAxn αn Snn un − xn 1 − αn γA∗ Tnn − IAxn αn Snn un − xn 1 − αn γA∗ Tnn − IAxn αn Snn un − un αn un − xn 1 − αn γA∗ Tnn − IAxn αn Snn un − un αn γA∗ Tnn − IAxn γA∗ Tnn − IAxn αn Snn un − un .
3.19
In view of 3.16 and 3.17, we have
lim xn1 − xn 0.
n→∞
3.20
Similarly, it follows from 3.1, 3.17, and 3.20 that
n1
− I Axn1 − xn γA∗ Tnn − IAxn un1 − un xn1 γA∗ Tn1
n1
≤ xn1 − xn γ A∗ Tn1
− I Axn1 3.21
γA∗ Tnn − IAxn −→ 0 n −→ ∞.
The conclusion 3.18 is proved.
Step 4. Next, we prove that, for each j 1, 2, . . . , M,
uiMj − Sj uiMj −→ 0,
AxiMj − Tj AxiMj −→ 0 i −→ ∞.
3.22
In fact, from 3.16, it follows that
iMj
ζiMj : uiMj − Sj
uiMj −→ 0 i −→ ∞.
3.23
8
Abstract and Applied Analysis
Since Sj is uniformly Lj -Lipschitzian continuous, it follows from 3.18 and 3.23 that
uiMj − Sj uiMj iMj
iMj
uiMj Sj
uiMj − Sj uiMj ≤ uiMj − Sj
iMj−1
≤ ζiMj Lj Sj
uiMj − uiMj iMj−1
iMj−1
iMj−1
≤ ζiMj Lj Sj
uiMj − Sj
uiMj−1 Sj
uiMj−1 − uiMj ≤ ζiMj L2j uiMj − uiMj−1 iMj−1
Lj Sj
uiMj−1 − uiMj−1 uiMj−1 − uiMj ≤ ζiMj Lj 1 Lj uiMj − uiMj−1 Lj ζiMj−1 −→ 0 i −→ ∞.
3.24
Similarly, for each j 1, 2, . . . , M, it follows from 3.17 that
iMj
AxiMj −→ 0
ξiMj : AxiMj − Tj
i −→ ∞.
3.25
j -Lipschitzian continuous, by the same way as above, from 3.18 and
Since Tj is uniformly L
3.25, we can also prove that
AxiMj − Tj AxiMj −→ 0
i −→ ∞.
3.26
Step 5. Finally, we prove that xn x∗ and un x∗ , which is a solution of the problem
MSSFP. In fact, since {un } is bounded, there exists a subsequence {uni } ⊂ {un } such that
uni x∗ ∈ H1 . Hence, for any positive integer j 1, 2, . . . , M, there exists a subsequence
{ni j} ⊂ {ni } with ni jmodM j such that uni j x∗ . Again, from 3.22, it follows that
un j − Sj un j −→ 0 ni j −→ ∞ .
i
i
3.27
Since Sj is demiclosed at zero see Remark 2.4, it follows that x∗ ∈ FSj . By the arbitrariness
of j 1, 2, . . . , M, we have x∗ ∈ C : M
j1 FSj .
Moreover, it follows from 3.1 and 3.17 that
xni uni − γA∗ Tnnii − I Axni x∗ .
3.28
Since A is a linear bounded operator, it follows that Axni Ax∗ . For any positive integer
k 1, 2, . . . , M, there exists a subsequence {ni k} ⊂ {ni } with ni kmodM k such that
Axni k Ax∗ . In view of 3.22, we have
Axn k − Tk Axn k −→ 0 ni k −→ ∞.
i
i
3.29
Since Tk is demiclosed at zero, we have Ax∗ ∈ FTk . By the arbitrariness of k 1, 2, . . . , M, it
∗
∗
∗
follows that Ax∗ ∈ Q : M
k1 FTk . This together with x ∈ C shows that x ∈ Γ, that is, x is
a solution to the problem MSSFP.
Abstract and Applied Analysis
9
Now, we prove that xn x∗ and un x∗ . In fact, assume that there exists another
x∗ . Consequently, by virtue of
subsequence {unj } ⊂ {un } such that unj y∗ ∈ Γ with y∗ /
3.2 and Opial’s property of Hilbert space, we have
lim infuni − x∗ < lim infuni − y∗ ni → ∞
ni → ∞
lim un − y∗ n→∞
lim unj − y∗ nj → ∞
< lim infunj − x∗ 3.30
nj → ∞
lim un − x∗ n→∞
lim infuni − x∗ .
ni → ∞
This is a contradiction. Therefore, un x∗ . By using 3.1 and 3.17, we have
xn un − γA∗ Tnn − IAxn x∗ .
3.31
Therefore, the conclusion I follows.
2 Without loss of generality, we can assume that S1 is semicompact. It follows from
3.27 that
un 1 − S1 un 1 −→ 0
i
i
ni 1 −→ ∞.
3.32
Therefore, there exists a subsequence of {uni 1 } for the sake of convenience, we still denote
it by {uni 1 } such that uni 1 → u∗ ∈ H. Since uni 1 x∗ , x∗ u∗ and so uni 1 → x∗ ∈ Γ. By
virtue of 3.2, we know that
lim un − x∗ 0,
n→∞
lim xn − x∗ 0,
n→∞
3.33
that is, {un } and {xn } both converge strongly to the point x∗ ∈ Γ. This completes the proof.
If we put γ 0 in Theorem 3.1, we can get the following.
Corollary 3.2. Let H, C, L and {kn } be the same as above and {Si } a family of asymptotically
nonexpansive mappings. Let {xn } be the sequence generated by
x1 ∈ H1 chosen arbitrarily,
xn1 1 − αn xn αn Snn xn , ∀n ≥ 1,
3.34
where Snn Snn mod M for all n ≥ 1 and {αn } is a sequence in 0, 1 satisfying the following conditions.
e αn ∈ δ, 1 − β for all n ≥ 1, where δ ∈ 0, 1 − β is a positive constant.
10
Abstract and Applied Analysis
1 If Γ /
∅, then the sequence {xn } converges weakly to a point x∗ ∈ Γ.
2 In addition, if there exists a positive integer j such that Sj is semicompact, then the
sequence {xn } converges strongly to a point x∗ ∈ Γ.
The following theorem can be obtained from Theorem 3.1 immediately.
Theorem 3.3. Let H1 and H2 be two real Hilbert spaces, A : H1 → H2 a bounded linear operator,
Si : H1 → H1 , i 1, 2, . . . , M, a uniformly Li -Lipschitzian and βi -strict pseudocontraction, and Ti :
i -Lipschitzian and μi -strict pseudocontraction satisfying
H2 → H2 , i 1, 2 . . . , M, a uniformly L
the following conditions:
∅ and Q : M
∅,
a C : M
i1 FSi /
i1 FTi /
b β max1≤i≤M βi < 1 and μ sup1≤i≤M μi < 1.
Let {xn } be the sequence generated by
x1 ∈ H1 chosen arbitrarily,
xn1 1 − αn un αn Sn un ,
un xn γA∗ Tn − IAxn ,
3.35
∀n ≥ 1,
where Sn Sn mod M , Tn Tn mod M , {αn } is a sequence in 0, 1, and 0 < γ < 1 is a constant. If
Γ
/ ∅ and the following condition is satisfied:
c αn ∈ δ, 1 − β for all n ≥ 1 and γ ∈ 0, 1 − μ/A2 , where δ ∈ 0, 1 − β is a constant,
then the sequence {xn } converges weakly to a point x∗ ∈ Γ. In addition, if there exists a positive integer
j such that Sj is semicompact, then the sequences {xn } and {un } both converge strongly to the point
x∗ .
Proof. By the same way as given in the proof of Theorem 3.1 and using the case of strict
pseudocontraction with the sequence {kn 1}, we can prove that, for each p ∈ Γ, the limits
limn → ∞ xn − p and limn → ∞ un − p exist,
un − Sn un → 0,
Axn − Tn Axn → 0,
∗
xn x ,
un − un1 → 0,
xn − xn1 → 0,
∗
un x ∈ Γ.
3.36
In addition, if there exists a positive integer j such that Sj is semicompact, we can also
prove that {xn } and {un } both converge strongly to the point x∗ . This completes the proof.
If you put Si Ti or Ti I : the identity mapping for each i 1, 2 . . . , M in
Theorem 3.3, then we have the following.
Corollary 3.4. Let H be a real Hilbert space and Si : H → H, i 1, 2, . . . , M, a uniformly Li Lipschitzian and βi -strict pseudocontraction satisfying the following conditions:
∅,
a C : M
i1 FSi /
b β max1≤i≤M βi < 1.
Abstract and Applied Analysis
11
Let {xn } be the sequence generated by
x1 ∈ H1 chosen arbitrarily,
xn1 1 − αn xn αn Sn xn , ∀n ≥ 1,
3.37
where Sn Sn mod M and {αn } is a sequence in 0, 1. If Γ / ∅ and the following condition is satisfied:
c αn ∈ δ, 1 − β for all n ≥ 1, where δ ∈ 0, 1 − β is a constant,
then the sequence {xn } converges weakly to a point x∗ ∈ Γ. In addition, if there exists a positive integer
j such that Sj is semicompact, then the sequences {xn } converges strongly to the point x∗ .
Remark 3.5. Theorems 3.1 and 3.3 improve and extend the corresponding results of Censor
et al. 1, 4, 5, Byrne 2, Yang 7, Moudafi 12, Xu 13, Censor and Segal 14, Masad and
Reich 15, and others in the following aspects:
1 for the framework of spaces, we extend the space from finite dimension Hilbert
space to infinite dimension Hilbert space;
2 for the mappings, we extend the mappings from nonexpansive mappings,
quasi-nonexpansive mapping or demicontractive mappings to finite families of
asymptotically strictly pseudocontractions;
3 for the algorithms, we propose some new hybrid iterative algorithms which are
different from ones given in 1, 2, 4, 5, 7, 14, 15. And, under suitable conditions,
some weak and strong convergences for the algorithms are proved.
Acknowledgments
The authors would like to express their thanks to the referees for their helpful suggestions
and comments. This work was supported by the Natural Science Foundation of Yunnan
University of Economics and Finance and the Basic Science Research Program through the
National Research Foundation of Korea NRF funded by the Ministry of Education, Science
and Technology Grant no. 2011–0021821.
References
1 Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product
space,” Numerical Algorithms, vol. 8, no. 2-4, pp. 221–239, 1994.
2 C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse
Problems, vol. 18, no. 2, pp. 441–453, 2002.
3 Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “A unified approach for inversion problems in
intensity-modulated radiation therapy,” Physics in Medicine and Biology, vol. 51, no. 10, pp. 2353–2365,
2006.
4 Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its
applications for inverse problem and its applications,” Inverse Problems, vol. 21, no. 6, pp. 2071–2084,
2005.
5 Y. Censor X, A. Motova, and A. Segal, “Perturbed projections and subgradient projections for the
multiple-sets split feasibility problem,” Journal of Mathematical Analysis and Applications, vol. 327, no.
2, pp. 1244–1256, 2007.
6 H. K. Xu, “A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem,”
Inverse Problems, vol. 22, no. 6, pp. 2021–2034, 2006.
12
Abstract and Applied Analysis
7 Q. Yang, “The relaxed CQ algorithm solving the split feasibility problem,” Inverse Problems, vol. 20,
no. 4, pp. 1261–1266, 2004.
8 J. Zhao and Q. Yang, “Several solution methods for the split feasibility problem,” Inverse Problems,
vol. 21, no. 5, pp. 1791–1799, 2005.
9 K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”
Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.
10 T. H. Kim and H. K. Xu, “Convergence of the modified Mann’s iteration method for asymptotically
strict pseudo-contractions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 9, pp. 2828–
2836, 2008.
11 K. Aoyama, W. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a
countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis. Theory, Methods
& Applications, vol. 67, no. 8, pp. 2350–2360, 2007.
12 A. Moudafi, “The split common fixed-point problem for demicontractive mappings,” Inverse Problems,
vol. 26, no. 5, Article ID 055007, 6 pages, 2010.
13 H. K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,”
Inverse Problems, vol. 26, no. 10, Article ID 105018, 17 pages, 2010.
14 Y. Censor and A. Segal, “The split common fixed point problem for directed operators,” Journal of
Convex Analysis, vol. 16, no. 2, pp. 587–600, 2009.
15 E. Masad and S. Reich, “A note on the multiple-set split convex feasibility problem in Hilbert space,”
Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 367–371, 2007.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014