Modified Block Iterative Algorithm for Solving Convex Feasibility Problems in Banach Spaces

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Volume 2010, Article ID 869684,14pages doi:10.1155/2010/869684

Research Article

Modified Block Iterative Algorithm for Solving

Convex Feasibility Problems in Banach Spaces

Shih-sen Chang,

1

_{Jong Kyu Kim,}

2

_{and Xiong Rui Wang}

1

1_{Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China}

2_{Department of Mathematics Education, Kyungnam University Masan, Kyungnam 631-701, South Korea}

Correspondence should be addressed to Jong Kyu Kim,[email protected]

Received 20 October 2009; Accepted 28 December 2009

Academic Editor: Yeol J. E. Cho

Copyrightq2010 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.

The purpose of this paper is to use the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi-φ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces withKadec-Klee property. The results presented in the paper improve and extend some recent results.

1. Introduction

The problem of finding a point in the intersection of closed and convex subsets{Ci}mi1of a

Banach space is a frequently appearing problem in diverse areas of mathematics and physical sciences. This problem is commonly referred to as theconvex feasibility problemCFP. There is a considerable investigation on CFPin the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning1. The advantage of a Hilbert spaceH is that the projectionPC onto a closed convex subsetCof H is nonexpansive. So projection methods

have dominated in the iterative approaches toCFPin Hilbert space. In 1993, Kitahara and Takahashi2 deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in uniformly convex Banach spacesee also, O’Hara et al.3and Chang et al. 4. It is known that if C is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach space E, then the generalized projection ΠC from E

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Tan 7, Wattanawitoon and Kumam 8, Li and Su 9, and Takahashi and Zembayashi

10 extend the notion from relatively nonexpansive mappings or quasi-φ-nonexpansive mappings to quasi-φ-asymptotically nonexpansive mappings and also prove some weak and strong convergence theorems to approximate a common fixed point of finite or infinite family of quasi-φ-nonexpansive mappings or quasi-φ-asymptotically nonexpansive mappings.

It should be noted that theblock iterative algorithmis a method which often used by many authors to solve the convex feasibility problem see, e.g., Kikkawa and Takahashi

11, Aleyner and Reich12. Recently, some authors by using the block iterative scheme to establish strong convergence theorems for a finite family of relativity nonexpansive mappings in Hilbert space or finite-dimensional Banach space see, e.g., Aleyner and Reich 12, Plubtieng and Ungchittrakool13,14or uniformly smooth and uniformly convex Banach spacessee, e.g., Sahu et al.15and Ceng et al.16–18.

Motivated and inspired by these facts, the purpose of this paper is to use the modified block iterative method to propose an iterative algorithm for solving the convex feasibility

problems for an infinite family of quasi-φ-asymptotically nonexpansive. Under suitable

conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend the corresponding results in Aleyner and Reich 12, Plubtieng and Ungchittrakool13,14, and Chang et al.19.

2. Preliminaries

Throughout this paper we assume thatEis a real Banach space with the dualE∗andJ:E → 2E∗is thenormalized duality mappingdefined by

Jx f∗_∈_E∗_:_{x, f}∗_x2_f∗2_{, x}_∈_E.

2.1

In the sequel, we useFTto denote the set of fixed points of a mappingT,and useRandR to denote the set of all real numbers and the set of all nonnegative real numbers, respectively. We also denote byxn → xandxn xthe strong convergence and weak convergence of a

sequence{xn},respectively.

A Banach spaceEis said to bestrictly convexifxy/2 <1 for allx, y ∈U {z ∈

E:z1}withx /y.Eis said to beuniformly convexif, for each∈0,2, there existsδ >0 such thatxy/2≤1−δfor allx, y∈Uwithx−y ≥. Eis said to besmoothif the limit

lim

t→0

_x_ty₋_x

t 2.2

exists for allx, y ∈ U.Eis said to beuniformly smooth if the above limit exists uniformly in

x, y∈U.

Remark 2.1. The following basic properties can be found in Cioranescu20.

iIfEis a uniformly smooth Banach space, thenJ is uniformly continuous on each bounded subset ofE.

ii IfEis a reflexive and strictly convex Banach space, thenJ−1 _{is hemicontinuous,}

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iiiIfEis a smooth, strictly convex, and reflexive Banach space, then the normalized duality mappingJ:E → 2E∗is single-valued, one-to-one, and onto.

ivA Banach spaceEis uniformly smooth if and only ifE∗is uniformly convex.

vEach uniformly convex Banach spaceEhas theKadec-Klee property, that is, for any sequence{xn} ⊂E,ifxn x∈Eandxn → x,thenxn → x.

Next we assume thatEis a smooth, strictly convex, and reflexive Banach space and

Cis a nonempty closed convex subset ofE. In the sequel we always useφ:E×E → R to denote the Lyapunov functional defined by

φx, yx2₋₂_{x, Jy}_y2_, _{∀x, y}_∈_E. _2.3

It is obvious from the definition ofφthat

x −y2 _≤_φ_{x, y}_≤_x_y2_, _{∀x, y}_∈_E. _2.4

Following Alber21, thegeneralized projectionΠC :E → Cis defined by

ΠCx inf y∈Cφ

y, x, ∀x∈E. _2.5

Lemma 2.2 see 21. Let E be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset ofE. Then the following conclusions hold:

aφx,ΠCy φΠCy, y≤φx, yfor allx∈Candy∈E; bifx∈Eandz∈C, then

z ΠCx⇐⇒z−y, Jx−Jz≥0, ∀y∈C, 2.6

cforx, y∈E,φx, y 0if and only ifxy.

Remark 2.3. If E is a real Hilbert space H, then φx, y x−y2 and ΠC is the metric

projectionPCofHontoC.

LetEbe a smooth, strictly convex, and reflexive Banach space,Ca nonempty closed convex subset ofE,T :C → Ca mapping, andFTthe set of fixed points ofT. A pointp∈C is said to be anasymptotic fixed pointofT if there exists a sequence{xn} ⊂Csuch thatxn p

andxn−Txn → 0.We denoted the set of all asymptotic fixed points ofTbyFT.

Definition 2.4. 1A mappingT :C → Cis said to berelatively nonexpansive5ifFT/∅,

FT FT, and

φp, Tx≤φp, x, ∀x∈C, p∈FT. 2.7

2A mappingT :C → Cis said to beclosedif for any sequence{xn} ⊂Cwithxn → x

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Definition 2.5. 1A mappingT :C → Cis said to bequasi-φ-nonexpansiveifFT/∅and

φp, Tx≤φp, x, ∀x∈C, p∈FT. 2.8

2A mappingT : C → Cis is said to bequasi-φ-asymptotically nonexpansive7, if

FT/∅and there exists a real sequence{kn} ⊂1,∞withkn → 1 such that

φp, Tn_x_≤_k

nφp, x, ∀n≥1, x∈C, p∈FT. 2.9

Remark 2.6. 1 From the definition, it is easy to know that each relatively nonexpansive

mapping is closed.

2The class of quasi-φ-asymptotically nonexpansive mappings contains properly the class of quasi-φ-nonexpansive mappings as a subclass and the class of quasi-φ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.

Next, we give some examples which are closed and quasi-φ-asymptotically nonexpan-sive mappings.

Example 2.7see7. LetEbe a uniformly smooth and strictly convex Banach space andA⊂

E×E∗_{a maximal monotone mapping such that}_A−1₀_{the set of zero points of}_A_{is nonempty.}

Then the mappingJr JrA−1Jis closed and quasi-φ-asymptotically nonexpansive from EontoDAandFJr A−10.

Example 2.8. Let ΠC be the generalized projection from a smooth, strictly convex and

reflexive Banach spaceEonto a nonempty closed convex subsetC ⊂E. ThenΠC is relative

nonexpansive, which in turn is a closed and quasi-φ-nonexpansive mapping, and so it is a closed and quasi-φ-asymptotically nonexpansive mapping.

Lemma 2.9see13,22. LetEbe a uniformly convex Banach space,r >0be a positive number and

Br0be a closed ball ofE. Then, for any given subset{x1, x2, . . . , xN} ⊂Br0and for any positive

numbersλ1, λ2, . . . , λN with

N

n1λn 1, there exists a continuous, strictly increasing, and convex functiong:0,2r → 0,∞withg0 0such that, for anyi, j∈ {1,2, . . . , N}withi < j,

N

n1

λnxn

2

≤N n1

λnxn2−λiλjgxi−xj. 2.10

Lemma 2.10. Let E be a uniformly convex Banach space, r > 0 a positive number andBr0 a

closed ball ofE. Then, for any given sequence{xi}∞i1 ⊂ Br0and for any given sequence{λi}∞i1of positive numbers with∞_n₁λn1,there exists a continuous, strictly increasing, and convex function g:0,2r → 0,∞withg0 0such that for any positive integersi, jwithi < j,

∞

n1

λnxn

2

≤∞ n1

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Proof. Since{xi}∞i1⊂Br0andλi>0 for alli≥1 with

_∞

n1λn1, we have

∞

i1

λixi

≤

∞

i1

λixi ≤r. 2.12

Hence, for any given >0 and any given positive integersi, jwithi < j,it follows from2.12

that there exists a positive integerN > jsuch that∞_i_N₁λixi ≤.LettingσNNi1λi, by

Lemma 2.9, we have

∞

i1

λixi

2 σN N

i1

λixi σN

∞

iN1

λixi

2 ≤ σN N

i1

λixi σN ∞

iN1

λixi

2

≤σ2

N N

i1

λixi σN

2

2₂_σ

N N

i1

λixi σN

≤σ2

N N

i1

λi σNxi

2₋_λ

iλjgxi−xj

2σN

N

i1

λixi σN

≤N i1

λixi2−λiλjgxi−xj

2

N

i1

λixi

≤∞ i1

λixi2−λiλjgxi−xj

2

N

i1

λixi

. 2.13

Since >0 is arbitrary, the conclusion ofLemma 2.10is proved.

Lemma 2.11. LetEbe a real uniformly smooth and strictly convex Banach space with Kadec-Klee

property, andC a nonempty closed convex subset ofE. LetT : C → C be a closed and quasi-φ

-asymptotically nonexpansive mapping with a sequence {kn} ⊂ 1,∞, kn → 1. ThenFT is a

closed convex subset ofC.

Proof. Letting{pn}be a sequence inFTwithpn → pasn → ∞, we prove thatp∈FT.

In fact, from the definition ofT,we have

φpn, Tp≤k1φ

pn, p−→0 as n−→ ∞. 2.14

Therefore we have

lim

n→ ∞φ

pn, Tp_{n→ ∞}limpn2−2pn, JTpTp

p2₋

2p, JTpTpφp, Tp0, 2.15

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Next we prove thatFTis convex. For anyp, q∈FT, t∈0,1, puttingwtp 1−

tq,we prove thatw∈FT.Indeed, in view of the definition ofφx, ywe have

φw, Tn_w _w2₋₂_{w, JT}n_w_Tn_w2

w2₋₂_t_{p, JT}n_w₋₂₁₋_t_{q, JT}n_w_Tn_w2

w2_tφ_{p, T}n_w ₁₋_t_φ_{q, T}n_w₋_t_p2₋

1−tq2

≤ w2_tk

nφp, w 1−tknφq, w−tp2−1−tq2 kn−1

tp2

1−tq2− w2.

2.16

Sincekn → 1, we haveφw, Tnw → 0asn → ∞. From2.4we haveTnw → w.

ConsequentlyJTnw → Jw.This implies that{JTnw}is a bounded sequence. SinceEis reflexive,E∗is also reflexive. So we can assume that

JTn_{w f}

0∈E∗. 2.17

Again sinceEis reflexive, we haveJE E∗. Therefore there existsx∈Esuch thatJxf0.

By virtue of the weakly lower semicontinuity of norm · ,we have

0lim inf

n→ ∞ φw, T

n_w _{lim inf} n→ ∞

w2₋₂_{w, J}_Tn_w_Tn_w2

lim inf

n→ ∞

w2₋₂_{w, J}_Tn_w_J_Tn_w2

≥ w2₋₂_{w, f}

0f02

w2₋₂_{w, Jx}_Jx2

w2₋₂_{w, Jx}_x2 _φ_{w, x}_,

2.18

that is,w xwhich implies that f0 Jw. Thus from2.17 we haveJTnw Jw ∈ E∗.

SinceJTnw → JwandE∗ has the Kadec-Klee property, we haveJTnw → Jw. Since

Eis uniformly smooth and strictly convex, byRemark 2.1iiit yields thatJ−1 _: _E∗ _→ _E_is

hemi-continuous. ThereforeTnw w. Again sinceTnw → w,by using the Kadec-Klee property ofE, we haveTnw → w. This implies thatTTnw Tn1_w _→ _w_{. Since}_T_{is closed,}

we havewTw. This completes the proof ofLemma 2.11.

3. Main Results

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Definition 3.1. 1Let{Si}∞i1:C → Cbe a sequence of mappings.{Si}∞i1is said to bea family of uniformly quasi-φ-asymptotically nonexpansive mappings, if∞_n₁FSn/∅,and there exists a

sequence{kn} ⊂1,∞withkn → 1 such that for eachi≥1

φp, Sn ix

≤knφp, x, ∀p∈ ∞

n1

FSn, x∈C, ∀n≥1. 3.1

2A mappingS:C → Cis said to beuniformlyL-Lipschitz continuous, if there exists a constantL >0 such that

Sn_x₋_Sn_y_≤_L_x₋_y_, _{∀x, y}_∈_C. _3.2

Theorem 3.2. LetEbe a uniformly smooth and strictly convex Banach space with Kleac-Klee property andCa nonempty closed convex subsets ofE. Let{Si}∞i1:C → Cbe an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence{kn}⊂1,∞andkn→1.

Suppose that for eachi≥1, Si is uniformlyLi-Lipschitz continuous and thatF: ∞n1FSiis a nonempty and bounded subset inC.Let{xn}be the sequence generated by

x0∈Cchosen arbitrary, C0C,

ynJ−1

αn,0Jxn ∞

i1

αn,iJSnixn

,

Cn1

v∈Cn:φv, yn≤φv, xn ξn, xn1 ΠCn1x0, ∀n≥0,

3.3

whereξnsup_u∈Fkn−1φu, xn,ΠCn1is the generalized projection ofEonto the setCn1and for eachi≥0,{αn,i}is a sequence in0,1satisfying the following conditions:

a∞_i₀αn,i1for alln≥0;

blim infn→ ∞αn,0·αn,i>0for alli≥1.

Then{xn}converges strongly toΠFx0.

Proof. We divide the proof ofTheorem 3.2into five steps.

Step 1. We first prove thatFandCnboth are closed and convex subset ofCfor alln≥0.

In fact, It follows fromLemma 2.11thatFSi, i≥1,is closed and convex. Therefore Fis a closed and convex subset inC. Furthermore, it is obvious thatC0 Cis closed and

convex. Suppose thatCnis closed and convex for somen≥1. Since the inequalityφv, yn≤ φv, xn ξnis equivalent to

2v, Jxn−Jyn≤ xn2−yn2ξn, 3.4

therefore, we have

Cn1

v∈Cn : 2v, Jxn−Jyn≤ xn2−yn2ξn

. 3.5

This implies thatCn1 is closed and convex. The desired conclusions are proved. These in

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Step 2. We prove that{xn}is a bounded sequence inC.

By the definition ofCn, we havexn ΠCnx0for alln≥0.It follows fromLemma 2.2a

that

φxn, x0 φΠCnx0, x0≤φu, x0−φu,ΠCnx0

≤φu, x0, ∀n≥0, u∈ F.

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This implies that{φxn, x0}is bounded. By virtue of2.4,{xn}is bounded. Denote

Msup

n≥0

{xn}<∞. _3.7

Step 3. Next, we prove thatF:∞_i₁FSi⊂Cnfor alln≥0.

It is obvious thatF ⊂C0C.Suppose thatF ⊂Cnfor somen≥0. SinceEis uniformly

smooth,E∗is uniformly convex. For any givenu∈ F ⊂Cnand for any positive integerj >0,

fromLemma 2.10we have

φu, ynφ

u, J−1

αn,0Jxn ∞

i1

αn,iJSn_ixn

u2₋₂_α

n,0u, Jxn −2 ∞

i1

αn,iu, JSnixn

αn,0Jxn ∞

i1

αn,iJSnixn

2

≤ u2₋₂_α

n,0u, Jxn −2

∞

i1

αn,iu, JSn_ixnαn,0Jxn2

∞ i1

αn,iJSnixn2−αn,0αn,jgJxn−JSnjxn

by Lemma 2.10

u2₋₂_α

n,0u, Jxn −2 ∞

i1

αn,iu, JSnixnαn,0xn2

∞ i1

αn,iSnixn2−αn,0αn,jgJxn−JSnjxn

αn,0φu, xn ∞

i1

αn,iφu, Snixn−αn,0αn,jgJxn−JSnjxn

≤αn,0φu, xn

∞

i1

αn,iknφu, xn−αn,0αn,jgJxn−JSnjxn

≤knφu, xn−αn,0αn,jgJxn−JSnjxn

≤φu, xn sup

u∈Fkn−1φu, xn−αn,0αn,jg

_Jx

n−JSn_jxn

φu, xn ξn−αn,0αn,jgJxn−JSnjxn

∀u∈ F.

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Henceu∈Cn1and soF ⊂Cnfor alln≥0. By the way, from the definition of{ξn},2.4, and 3.7, it is easy to see that

ξnsup

u∈Fkn−1φu, xn≤supu∈Fkn−1uM

2_−→₀ _as_n_{−→ ∞}_.

3.9

Step 4. Now, we prove that{xn}converges strongly to some pointp∈ F:∞i1FSi.

In fact, since{xn} is bounded inCandEis reflexive, we may assume that xn p.

Again sinceCnis closed and convex for eachn≥1, it is easy to see thatp∈Cnfor eachn≥0.

Sincexn ΠCnx0, from the definition ofΠCn, we have

φxn, x0≤φ

p, x0

, ∀n≥0. 3.10

Since

lim inf

n→ ∞ φxn, x0 lim infn→ ∞

xn2−2xn, Jx0x02

≥p2₋

2p, Jx0

x02φ

p, x0

, 3.11

we have

φp, x0

≤lim inf

n→ ∞ φxn, x0≤lim sup_n_{→ ∞} φxn, x0≤φ

p, x0

. _3.12

This implies that limn→ ∞φxn, x0 φp, x0,that is,xn → p.In view of the Kadec-Klee

property ofE, we obtain that

lim

n→ ∞xnp. 3.13

Now we prove thatp∈∞_i₁FSi.In fact, by the construction ofCn,we have thatCn1 ⊂Cn

andxn1 ΠCn1x0∈Cn.Therefore byLemma 2.2awe have

φxn1, xn φxn1,ΠCnx0

≤φxn1, x0−φΠCnx0, x0

φxn1, x0−φxn, x0−→0 as n−→ ∞.

3.14

In view ofxn1∈Cn1and note the construction ofCn1we obtain that

φxn1, yn≤φxn1, xn ξn−→0 as n−→ ∞. 3.15

From2.4it yieldsxn1 − yn2 → 0.Sincexn1 → p,we have

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Hence we have

_Jy_n_−→_Jp _as _n_{−→ ∞}_. _3.17

This implies that{Jyn}is bounded inE∗.SinceEis reflexive, and soE∗ is reflexive, we can

assume thatJyn f0 ∈E∗.In view of the reflexive ofE,we see thatJE E∗. Hence there

existsx∈Esuch thatJxf0. Since

φxn1, yn

xn12−2xn1, Jynyn2

xn12−2

xn1, JynJyn2.

3.18

Taking lim inf_{n→ ∞} on the both sides of equality above and in view of the weak lower semicontinuity of norm · ,it yields that

0≥p2−2p, f0f02p2−2p, JxJx2

p2₋

2p, Jxx2φp, x,

3.19

that is,px.This implies thatf0Jp,and soJyn Jp. It follows from3.17and the

Kadec-Klee property ofE∗thatJyn → Jpasn → ∞. Note thatJ−1:E∗ → Eis hemi-continuous,

it yields thatyn p.It follows from3.16and the Kadec-Klee property ofEthat

lim

n→ ∞ynp. 3.20

From3.13and3.20we have that

xn−yn−→0 as n−→ ∞. 3.21

SinceJis uniformly continuous on any bounded subset ofE, we have

Jxn−Jyn−→0 as n−→ ∞. 3.22

For anyj ≥1 and anyu∈ F, it follows from3.8,3.13, and3.20that

αn,0αn,jgJxn−JSnjxn

≤φu, xn−φu, ynξn−→0 as n−→ ∞. 3.23

In view of conditionblim infn→ ∞αn,0αn,j >0, we see that

gJxn−JSnjxn

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It follows from the property ofgthat

Jxn−JSnjxn−→0 as n−→ ∞. 3.25

Sincexn → pandJis uniformly continuous, it yieadsJxn → Jp.Hence from3.25we have

JSn

jxn−→Jp as n−→ ∞. 3.26

SinceJ−1_:_E∗ _→ _E_{is hemi-continuous, it follows that}

Sn

jxn p, for each j ≥1. 3.27

On the other hand, for eachj≥ 1 we have

Sn

jxn−pJ

Sn

jxn−Jp≤J

Sn jxn

−Jp−→0 as n−→ ∞. 3.28

This together with3.27shows that

Sn

jxn → p for eachj≥1. 3.29

Furthermore, by the assumption that for eachj ≥1,Sjis uniformlyLi-Lipschitz continuous,

hence we have

Sn1

j xn−Snjxn≤Snj1xn−Snj1xn1Sjn1xn1−xn1xn1−xnxn−Snjxn ≤Lj1xn1−xnSnj1xn1−xn1xn−Snjxn.

3.30

This together with3.13and3.29, yieldsSn_j1xn−Sn_jxn → 0 as n → ∞. Hence from 3.29we haveSn1

j xn → p, that is,SjSnxn → p. In view of3.29and the closeness ofSj, it

yields thatSjpp,for allj≥1. This implies thatp∈∞i1FSj.

Step 5. Finally we prove thatxn → p ΠFx0.

Letw ΠFx0. Sincew∈ F ⊂Cnandxn ΠCnx0, we have

φxn, x0≤φw, x0, ∀n≥0. 3.31

This implies that

φp, x0

lim

n→ ∞φxn, x0≤φw, x0. 3.32

In view of the definition ofΠFx0, from3.32we havep w. Therefore,xn → p ΠFx0.

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The following theorem can be obtained fromTheorem 3.2immediately.

Theorem 3.3. Let E be a uniformly smooth and strictly convex Banach space with Kadec-Klee property ,Ca nonempty closed convex subset of E. Let{Si}∞i1 : C → Cbe an infinite family of closed and quasi-φ-nonexpansive mappings. Suppose thatF :_n∞₁FSiis a nonempty subset in C. Let{xn}be the sequence generated by

x0∈Cchosen arbitrary, C0 C,

ynJ−1

αn,0Jxn ∞

i1

αn,iJSixn

,

Cn1

v∈Cn:φv, yn≤φv, xn, xn1 ΠCn1x0, ∀n≥0,

3.33

where{αn,i},for eachi≥0, is a sequence in0,1satisfying the following conditions:

a∞_i₀αn,i1for alln≥0;

blim infn→ ∞αn,0·αn,i>0for alli≥1.

Then{xn}converges strongly toΠFx0.

Proof. Since{Si}∞i1:C → Cis an infinite family of closed quasi-φ-nonexpansive mappings, it

is an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings with sequence {kn 1}. Hence ξn sup_u∈Fkn−1φu, xn 0. Therefore the conditions

appearing inTheorem 3.2: “Fis a bounded subset inC” and “for eachi≥1,Siis uniformly Li-Lipschitz continuous” are of no use here. In fact, by the same methods as given in the

proofs of3.13,3.20and 3.29, we can prove thatxn → p,yn → pand Sjxn → pas n → ∞for eachj ≥1. By virtue of the closeness of mappingSjfor eachj ≥1, it yields that p ∈ FSjfor eachj ≥ 1, that is,p ∈ ∞i1FSi. Therefore all conditions inTheorem 3.2are

satisfied. The conclusion ofTheorem 3.3is obtained fromTheorem 3.2immediately.

Remark 3.4. Theorems3.2and3.3improve and extend the corresponding results in Aleyner

and Reich12, Plubtieng and Ungchittrakool13,14and Chang et al.19in the following aspects.

aFor the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property note that each uniformly convex Banach space must have the Kadec-Klee property.

b For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings or quasi-φ-nonexpansive mapping to an infinite family of quasi-φ-asymptotically mappings;

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Acknowledgment

This work was supported by the Natural Science Foundation of Yibin Universityno. 2009Z3 and the Kyungnam University Research Fund 2009.

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