multiple-set split feasibility problems for asymptotically strict

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 491760, 12 pagesdoi:10.1155/2012/491760

Research ArticleMultiple-Set Split Feasibility Problems forAsymptotically Strict Pseudocontractions

Shih-Sen Chang,1 Yeol Je Cho,2 J. K. Kim,3W. 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, [email protected] Yeol Je Cho, [email protected]

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 theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

In this paper, we introduce an iterative method for solving the multiple-set split feasibilityproblems 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 someauthors.

1. Introduction

The split feasibility problem (SFP) in finite dimensional spaces was first introduced by Censorand Elfving [1] for modeling inverse problems which arise from phase retrievals and inmedical image reconstruction [2]. Recently, it has been found that the SFP can also be usedin various disciplines such as image restoration, computer tomograph, and radiation therapytreatment 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 splitfeasibility problem for asymptotically strict pseudocontraction (MSSFP) in the framework ofinfinite-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 thefamilies of mappings Si : H1 → H1 and Ti : H2 → H2, respectively, C :=

⋂Mi=1 F(Si) and

Q :=⋂M

i=1 F(Ti), where F(Si) = {xi ∈ H1 : Sixi = xi} and F(Ti) = {yi ∈ H2 : Tiyi = yi} denotethe sets of fixed points of Si and Ti, respectively. In the sequel, we use Γ to denote the set ofsolutions 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 provingour main results.

Let E be a Banach space. A mapping T : E → E is said to be demiclosed at origin if, forany sequence {xn} ⊂ Ewith xn ⇀ x∗ and ‖(I −T)xn‖ → 0, we have x∗ = Tx∗. A Banach spaceE is said to have Opial’s property if, for any sequence {xn}with xn ⇀ x∗, we have

lim infn→∞

‖xn − x∗‖ < lim infn→∞

∥∥xn − y

∥∥ (2.1)

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 ifthere exists a constant γ ∈ [0, 1) and a sequence {kn} ⊂ [1,∞) with kn → 1 suchthat

∥∥Gnx −Gny

∥∥2 ≤ kn

∥∥x − y

∥∥2 + γ

∥∥(I −Gn)x − (I −Gn)y

∥∥2

, ∀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 −Gy

∥∥2 ≤ ∥

∥x − y∥∥2 + γ

∥∥(I −G)x − (I −G)y

∥∥2

, ∀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 aconstant L > 0 such that

∥∥Gnx −Gny

∥∥ ≤ L

∥∥x − 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} suchthat xni converges strongly to a point x∗ ∈ H.

Now, we give one example of the (γ, {kn})-asymptotically strict pseudocontractionmapping.

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, x2

1, a2x2, a3x3, . . .), (2.5)

where {ai} is a sequence in (0, 1) such that Π∞i=2ai = 1/2. It is proved in Goebel and Kirk [9]

that

(a) ‖Tx = Ty‖ ≤ 2‖x − y‖ for all x, y ∈ B,

(b) ‖Tnx − Tny‖ ≤ 2Πnj=2aj for all n ≥ 2 and x, y ∈ B.

Denote by k1/21 = 2, k1/2

n = 2Πnj=2aj(n ≥ 2) and γ ∈ [0, 1). Then, we have

limn→∞

kn = limn→∞

⎛

⎝2n∏

j=2

aj

⎞

⎠

2

= 1,

∥∥Tnx − Tny

∥∥2 ≤ kn

∥∥x − y

∥∥2 + γ

∥∥x − y − (Tnx − Tny)

∥∥2

, ∀n ≥ 1, x, y ∈ B,

(2.6)

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 asymptoticallynonexpansive.

(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 pseudo-

contraction both are demiclosed at origin [10].

Proposition 2.5. Let G : H → H be a (γ, {kn})- asymptotically strict pseudocontraction. IfF(G)/= ∅, then, for any q ∈ F(G) and x ∈ H, the following inequalities hold and they are equivalent:

∥∥Gnx − q

∥∥2 ≤ kn

∥∥x − q

∥∥2 + γ‖x −Gnx‖2, (2.7)

⟨x −Gnx, x − q

⟩ ≥ 1 − γ

2‖x −Gnx‖2 − kn − 1

2∥∥x − q

∥∥2, (2.8)

〈x −Gnx, q −Gnx〉 ≤ 1 + γ

2‖x −Gnx‖2 + kn − 1

2∥∥x − q

∥∥2

. (2.9)

Lemma 2.6 (see [11]). Let {an}, {bn}, and {δn} be sequences of nonnegative real numbers satisfying

an+1 ≤ (1 + δn)an + bn, ∀n ≥ 1. (2.10)

If∑∞

i=1 δn < ∞ and∑∞

i=1 bn < ∞, then the limit limn→∞an exists.


3. Multiple-Set Split Feasibility Problem

For solving the multiple-set split feasibility problem (1.1), let us assume that the followingconditions are satisfied:

(C1) H1 andH2 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 auniformly L̃i-Lipschitzian and (μi, {k̃i,n})-asymptotically strict pseudocontractionsatisfying the following conditions:

(a) C :=⋂M

i=1 F(Si)/= ∅ and Q :=⋂M

i=1 F(Ti)/= ∅,(b) β = max1≤i≤Mβi < 1 and μ = max1≤i≤Mμi < 1,

(c) L := max1≤i≤MLi < ∞ and L̃ := max1≤i≤ML̃i < ∞,

(d) kn = max1≤i≤M{ki,n, k̃i,n} and∑∞

n=1(kn − 1) < ∞.

We are now in a position to give the following result.

Theorem 3.1. Let H1, H2, A, {Si}, {Ti}, C, Q, β, μ, L, L̃, and {kn} be the same as above. Let {xn}be the sequence generated by

x1 ∈ H1 chosen arbitrarily,xn+1 = (1 − αn)un + αnS

nn(un),

un = xn + γA∗(Tnn − I)Axn, ∀n ≥ 1,

(3.1)

where Snn = Sn

n( mod M), Tnn = Tn

n( mod M) for all n ≥ 1, {αn} is a sequence in [0, 1], and γ > 0 is aconstant satisfying the following conditions.

(e) αn ∈ (δ, 1 − β) for all n ≥ 1 and γ ∈ (0, (1 − μ)/‖A‖2), where δ ∈ (0, 1 − β) is a positiveconstant.

(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 thesequences {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

limn→∞

∥∥xn − p

∥∥ (3.2)


exists. In fact, since p ∈ Γ, p ∈ C :=⋂M

i=1 F(Si), and Ap ∈ Q :=⋂M

i=1 F(Ti). From (3.1) and (2.8),it follows that

∥∥xn+1 − p

∥∥2 =

∥∥un − p − αn(un − Sn

nun)∥∥2

=∥∥un − p

∥∥2 − 2αn〈un − p, un − Sn

nun〉 + α2n‖un − Sn

nun‖2

≤ ∥∥un − p

∥∥2 − αn

{(1 − β

)‖un − Snnun‖2 − (kn − 1)

∥∥un − p

∥∥2}

+ α2n‖un − Sn

nun‖2

= (1 + αn(kn − 1))∥∥un − p

∥∥2 − αn

(1 − β − αn

)‖un − Snnun‖2.

(3.3)

On the other hand, since

∥∥un − p

∥∥2 =

∥∥xn − p + γA∗(Tn

n − I)Axn

∥∥2

=∥∥xn − p

∥∥2 + γ2‖A∗(Tn

n − I)Axn‖2 + 2γ⟨xn − p,A∗(Tn

n − I)Axn

⟩,

(3.4)

γ2‖A∗(Tnn − I)Axn‖2 = γ2〈A∗(Tn

n − I)Axn, A∗(Tn

n − I)Axn〉= γ2〈AA∗(Tn

n − I)Axn, (Tnn − I)Axn〉

≤ γ2‖A‖2‖TnnAxn −Axn‖2,

(3.5)

2γ〈xn − p,A∗(Tnn − I)Axn〉 = 2γ〈Axn −Ap, (Tn

n − I)Axn〉= 2γ〈(Axn −Ap

)+ (Tn

n − I)Axn − (Tnn − I)Axn, (Tn

n − I)Axn〉= 2γ

{⟨TnnAxn −Ap, Tn

nAxn −Axn

⟩ − ‖(Tnn − I)Axn‖2

}.

(3.6)

Further, letting x = Axn, Gn = Tnn , q = Ap, γ = μ in (2.9) and noting Ap ∈ F(Tn), it follows

that

〈TnnAxn −Ap, Tn

nAxn −Axn〉 ≤ 1 + μ

2‖(Tn

n − I)Axn‖2 + kn − 12

∥∥Axn −Ap

∥∥2

≤ 1 + μ

2‖(Tn

n − I)Axn‖2 + (kn − 1)‖A‖22

∥∥xn − p

∥∥2.

(3.7)

Substituting (3.7) into (3.6) and simplifying it, we have

2γ⟨xn − p,A∗(Tn

n − I)Axn

⟩ ≤ γ(μ − 1

)‖(Tnn − I)Axn‖2 + (kn − 1)γ‖A‖2∥∥xn − p

∥∥2. (3.8)

Substituting (3.5) and (3.8) into (3.4) and simplifying it, we have

∥∥un − p

∥∥2 ≤ ∥

∥xn − p∥∥2 + γ2‖A‖2‖Tn

nAxn −Axn‖2

+ γ(μ − 1

)‖(Tnn − I)Axn‖2 + (kn − 1)γ‖A‖2∥∥xn − p

∥∥2


=∥∥xn − p

∥∥2 − γ

(1 − μ − γ‖A‖2

)‖Tn

nAxn −Axn‖2

+ (kn − 1)γ‖A‖2∥∥xn − p∥∥2.

(3.9)

Again, substituting (3.9) into (3.3) and simplifying it, we have

∥∥xn+1 − p

∥∥2 ≤ (1 + αn(kn − 1))

×{∥∥xn − p

∥∥2 − γ

(1 − μ − γ‖A‖2

)‖Tn

nAxn −Axn‖2 + (kn − 1)γ‖A‖2∥∥xn − p∥∥2}

− αn

(1 − β − αn

)‖un − Snnun‖2

≤ (1 + αn(kn − 1))∥∥xn − p

∥∥2 − γ

(1 − μ − γ‖A‖2

)‖Tn

nAxn −Axn‖2

+ (1 + αn(kn − 1))(kn − 1)γ‖A‖2∥∥xn − p∥∥2 − αn

(1 − β − αn

)‖un − Snnun‖2.

(3.10)

By the condition (e), we have

∥∥xn+1 − p

∥∥2 ≤ (1 + αn(kn − 1))

∥∥xn − p

∥∥2 + (1 + αn(kn − 1))(kn − 1)γ‖A‖2∥∥xn − p

∥∥2

≤ (1 +K(kn − 1))∥∥xn − p

∥∥2,

(3.11)

where

K = supn≥1

(αn + (1 + αn(kn − 1))γ‖A‖2

)< ∞. (3.12)

By the condition (d),∑

n=1(kn − 1) < ∞; hence, from Lemma 2.6, we know that the followinglimit exists:

limn→∞

∥∥xn − p

∥∥. (3.13)

Step 2. We will now prove that, for each p ∈ Γ, the limit

limn→∞

∥∥un − p

∥∥ (3.14)

exists. In fact, from (3.10) and (3.13), it follows that

γ(1 − μ − γ‖A‖2

)‖(Tn

n − I)Axn‖2 + αn

(1 − β − αn

)‖un − Snnun‖2

≤ ∥∥xn − p

∥∥2 − ∥

∥xn+1 − p∥∥2 +K(kn − 1)

∥∥xn − p

∥∥2 −→ 0 (n −→ ∞).

(3.15)


This together with the condition (e) implies that

limn→∞

‖un − Snnun‖ = 0, (3.16)

limn→∞

‖(Tnn − I)Axn‖ = 0. (3.17)

Therefore, it follows from (3.4), (3.13), and (3.17) that the limit limn→∞‖un − p‖ exists.

Step 3. Now, we prove that

limn→∞

‖xn+1 − xn‖ = 0, limn→∞

‖un+1 − un‖ = 0. (3.18)

In fact, it follows from (3.1) that

‖xn+1 − xn‖ = ‖(1 − αn)un + αnSnn(un) − xn‖

=∥∥(1 − αn)

(xn + γA∗(Tn

n − I)Axn

)+ αnS

nn(un) − xn

∥∥

=∥∥(1 − αn)γA∗(Tn

n − I)Axn + αn(Snn(un) − xn)

∥∥

=∥∥(1 − αn)γA∗(Tn

n − I)Axn + αn(Snn(un) − un) + αn(un − xn)

∥∥

=∥∥(1 − αn)γA∗(Tn

n − I)Axn + αn(Snn(un) − un) + αnγA∗(Tn

n − I)Axn

∥∥

=∥∥γA∗(Tn

n − I)Axn + αn(Snn(un) − un)

∥∥.

(3.19)

In view of (3.16) and (3.17), we have

limn→∞

‖xn+1 − xn‖ = 0. (3.20)

Similarly, it follows from (3.1), (3.17), and (3.20) that

‖un+1 − un‖ =∥∥∥xn+1 + γA∗

(Tn+1n+1 − I

)Axn+1 −

(xn + γA∗(Tn

n − I)Axn

)∥∥∥

≤ ‖xn+1 − xn‖ + γ∥∥∥A∗

(Tn+1n+1 − I

)Axn+1

∥∥∥

+ γ‖A∗(Tnn − I)Axn‖ −→ 0 (n −→ ∞).

(3.21)

The conclusion (3.18) is proved.

Step 4. Next, we prove that, for each j = 1, 2, . . . ,M,

∥∥uiM+j − SjuiM+j

∥∥ −→ 0,

∥∥AxiM+j − TjAxiM+j

∥∥ −→ 0 (i −→ ∞). (3.22)

In fact, from (3.16), it follows that

ζiM+j :=∥∥∥uiM+j − S

iM+jj uiM+j

∥∥∥ −→ 0 (i −→ ∞). (3.23)


Since Sj is uniformly Lj-Lipschitzian continuous, it follows from (3.18) and (3.23) that

∥∥uiM+j − SjuiM+j

∥∥

≤∥∥∥uiM+j − S

iM+jj uiM+j

∥∥∥ +

∥∥∥S

iM+jj uiM+j − SjuiM+j

∥∥∥

≤ ζiM+j + Lj

∥∥∥S

iM+j−1j uiM+j − uiM+j

∥∥∥

≤ ζiM+j + Lj

{∥∥∥S

iM+j−1j uiM+j − S

iM+j−1j uiM+j−1

∥∥∥ +

∥∥∥S

iM+j−1j uiM+j−1 − uiM+j

∥∥∥}

≤ ζiM+j + L2j

∥∥uiM+j − uiM+j−1

∥∥

+ Lj

∥∥∥S

iM+j−1j uiM+j−1 − uiM+j−1 + uiM+j−1 − uiM+j

∥∥∥

≤ ζiM+j + Lj

(1 + Lj

)∥∥uiM+j − uiM+j−1

∥∥ + LjζiM+j−1 −→ 0 (i −→ ∞).

(3.24)

Similarly, for each j = 1, 2, . . . ,M, it follows from (3.17) that

ξiM+j :=∥∥∥AxiM+j − T

iM+jj AxiM+j

∥∥∥ −→ 0 (i −→ ∞). (3.25)

Since Tj is uniformly L̃j-Lipschitzian continuous, by the same way as above, from (3.18) and(3.25), we can also prove that

∥∥AxiM+j − TjAxiM+j

∥∥ −→ 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 thatuni ⇀ x∗ ∈ H1. Hence, for any positive integer j = 1, 2, . . . ,M, there exists a subsequence{ni(j)} ⊂ {ni}with ni(j)(modM) = j such that uni(j) ⇀ x∗. Again, from (3.22), it follows that

∥∥uni(j) − Sjuni(j)

∥∥ −→ 0

(ni

(j) −→ ∞)

. (3.27)

Since Sj is demiclosed at zero (see Remark 2.4), it follows that x∗ ∈ F(Sj). By the arbitrarinessof j = 1, 2, . . . ,M, we have x∗ ∈ C :=

⋂Mj=1 F(Sj).

Moreover, it follows from (3.1) and (3.17) that

xni = uni − γA∗(Tnini− I

)Axni ⇀ x∗. (3.28)

Since A is a linear bounded operator, it follows that Axni ⇀ Ax∗. For any positive integerk = 1, 2, . . . ,M, there exists a subsequence {ni(k)} ⊂ {ni} with ni(k)(modM) = k such thatAxni(k) ⇀ Ax∗. In view of (3.22), we have

∥∥Axni(k) − TkAxni(k)

∥∥ −→ 0 (ni(k) −→ ∞). (3.29)

Since Tk is demiclosed at zero, we have Ax∗ ∈ F(Tk). By the arbitrariness of k = 1, 2, . . . ,M, itfollows that Ax∗ ∈ Q :=

⋂Mk=1 F(Tk). This together with x∗ ∈ C shows that x∗ ∈ Γ, that is, x∗ is

a solution to the problem (MSSFP).


Now, we prove that xn ⇀ x∗ and un ⇀ x∗. In fact, assume that there exists anothersubsequence {unj} ⊂ {un} such that unj ⇀ y∗ ∈ Γ with y∗ /=x∗. Consequently, by virtue of(3.2) and Opial’s property of Hilbert space, we have

lim infni →∞

‖uni − x∗‖ < lim infni →∞

∥∥uni − y∗∥∥

= limn→∞

∥∥un − y∗∥∥

= limnj →∞

∥∥∥unj − y∗

∥∥∥

< lim infnj →∞

∥∥∥unj − x∗

∥∥∥

= limn→∞

‖un − x∗‖= lim inf

ni →∞‖uni − x∗‖.

(3.30)

This is a contradiction. Therefore, un ⇀ x∗. By using (3.1) and (3.17), we have

xn = un − γA∗(Tnn − I)Axn ⇀ 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

∥∥uni(1) − S1uni(1)

∥∥ −→ 0 (ni(1) −→ ∞). (3.32)

Therefore, there exists a subsequence of {uni(1)} (for the sake of convenience, we still denoteit by {uni(1)}) such that uni(1) → u∗ ∈ H. Since uni(1) ⇀ x∗, x∗ = u∗ and so uni(1) → x∗ ∈ Γ. Byvirtue of (3.2), we know that

limn→∞

‖un − x∗‖ = 0, limn→∞

‖xn − x∗‖ = 0, (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 asymptoticallynonexpansive mappings. Let {xn} be the sequence generated by

x1 ∈ H1 chosen arbitrarily,xn+1 = (1 − αn)xn + αnS

nn(xn), ∀n ≥ 1,

(3.34)

where Snn = Sn

n( 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.


(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 :H2 → H2, i = 1, 2 . . . ,M, a uniformly L̃i-Lipschitzian and μi-strict pseudocontraction satisfyingthe following conditions:

(a) C :=⋂M

i=1 F(Si)/= ∅ and Q :=⋂M

i=1 F(Ti)/= ∅,(b) β = max1≤i≤Mβi < 1 and μ = sup1≤i≤Mμi < 1.

Let {xn} be the sequence generated by

x1 ∈ H1 chosen arbitrarily,

xn+1 = (1 − αn)un + αnSn(un),

un = xn + γA∗(Tn − I)Axn, ∀n ≥ 1,

(3.35)

where Sn = Sn( mod M), Tn = Tn( mod M), {αn} is a sequence in [0, 1], and 0 < γ < 1 is a constant. IfΓ/= ∅ and the following condition is satisfied:

(c) αn ∈ (δ, 1 − β) for all n ≥ 1 and γ ∈ (0, (1 − μ)/‖A‖2), 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 integerj such that Sj is semicompact, then the sequences {xn} and {un} both converge strongly to the pointx∗.

Proof. By the same way as given in the proof of Theorem 3.1 and using the case of strictpseudocontraction with the sequence {kn = 1}, we can prove that, for each p ∈ Γ, the limitslimn→∞‖xn − p‖ and limn→∞‖un − p‖ exist,

‖un − Snun‖ → 0, ‖Axn − TnAxn‖ → 0, ‖un − un+1‖ → 0, ‖xn − xn+1‖ → 0,

xn ⇀ x∗, un ⇀ x∗ ∈ Γ.(3.36)

In addition, if there exists a positive integer j such that Sj is semicompact, we can alsoprove 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 inTheorem 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

i=1 F(Si)/= ∅,(b) β = max1≤i≤Mβi < 1.


Let {xn} be the sequence generated by

x1 ∈ H1 chosen arbitrarily,xn+1 = (1 − αn)xn + αnSn(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 integerj 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 Censoret al. [1, 4, 5], Byrne [2], Yang [7], Moudafi [12], Xu [13], Censor and Segal [14], Masad andReich [15], and others in the following aspects:

(1) for the framework of spaces, we extend the space from finite dimension Hilbertspace 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 ofasymptotically strictly pseudocontractions;

(3) for the algorithms, we propose some new hybrid iterative algorithms which aredifferent 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 suggestionsand comments. This work was supported by the Natural Science Foundation of YunnanUniversity of Economics and Finance and the Basic Science Research Program through theNational Research Foundation of Korea (NRF) funded by the Ministry of Education, Scienceand Technology (Grant no. 2011–0021821).

References

[1] Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a productspace,” 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,” InverseProblems, vol. 18, no. 2, pp. 441–453, 2002.

[3] Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “A unified approach for inversion problems inintensity-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 itsapplications 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 themultiple-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.


[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 asymptoticallystrict 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 acountable 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 demicontractivemappings,” 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 ofConvex 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.

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