Strong Convergence Theorems of Modified Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space

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Volume 2009, Article ID 483497,25pages doi:10.1155/2009/483497

Research Article

Strong Convergence Theorems of Modified

Ishikawa Iterations for Countable Hemi-Relatively

Nonexpansive Mappings in a Banach Space

Narin Petrot,

1, 2

_{Kriengsak Wattanawitoon,}

3, 4

and Poom Kumam

2, 3

1_{Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand} 2_{Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand}

3_{Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi}

(KMUTT), Bangmod, Bangkok 10140, Thailand

4_{Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology,}

Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

Correspondence should be addressed to Poom Kumam,[email protected]

Received 17 March 2009; Accepted 12 September 2009

Recommended by Lech G ´orniewicz

We prove some strong convergence theorems for fixed points of modified Ishikawa and Halpern iterative processes for a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space by using the hybrid projection methods. Moreover, we also apply our results to a class of relatively nonexpansive mappings, and hence, we immediately obtain the results announced by Qin and Su’s result2007, Nilsrakoo and Saejung’s result2008, Su et al.’s result2008, and some known corresponding results in the literatures.

Copyrightq2009 Narin Petrot 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.

1. Introduction

LetCbe a nonempty closed convex subset of a real Banach spaceE. A mappingT:C → Cis said to benonexpansiveifTx−Ty ≤ x−yfor allx, y∈C.We denote byFTthe set of fixed points ofT, that isFT {x ∈ C : x Tx}. A mappingT is said to bequasi-nonexpansive

if FT/∅ and Tx−y ≤ x−yfor allx ∈ Cand y ∈ FT. It is easy to see that ifT is nonexpansive withFT/∅, then it is quasi-nonexpansive. Some iterative processes are often used to approximate a fixed point of a nonexpansive mapping. The Mann’s iterative algorithm was introduced by Mann 1 in 1953. This iterative process is now known as Mann’s iterative process, which is defined as

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where the initial guessx0is taken inCarbitrarily and the sequence{αn}∞n0is in the interval

0,1.

In 1976, Halpern2first introduced the following iterative scheme:

x0u∈C, chosen arbitrarily, xn1αnu 1−αnTxn,

1.2

see also Browder3. He pointed out that the conditions limn→ ∞αn0 and∞n1αn ∞are

necessary in the sence that, if the iteration1.2converges to a fixed point ofT, then these conditions must be satisfied.

In 1974, Ishikawa4introduced a new iterative scheme, which is defined recursively by

ynβnxn1−βnTxn,

xn1αnxn 1−αnTyn,

1.3

where the initial guessx0is taken inCarbitrarily and the sequences{αn}and{βn}are in the interval0,1.

Concerning a family of nonexpansive mappings it has been considered by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings; see, for example, 5. The problem of finding an optimal point that minimizes a given cost function over common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importancesee6.

Zhang and Su7introduced the following implicit hybrid method for a finite family of nonexpansive mappings{Ti}Ni1in a real Hilbert space:

x0∈C is arbitrary, ynαnxn 1−αnTnzn,

znβnyn1−βnTnyn,

Cnz∈C:yn−z≤ xn−z,

Qn{z∈C: xn−z, x0−xn ≥0}, xn1PCn∩Qnx0, n0,1,2, . . . ,

1.4

whereTn≡TnmodN,{αn}and{βn}are sequences in0,1and{αn} ⊂0, afor somea∈0,1 and{βn} ⊂b,1for someb∈0,1.

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To find a common fixed point of a family of nonexpansive mappings, Aoyama et al. 10introduced the following iterative sequence. Letx1x∈Cand

xn1αnx 1−αnTnxn, 1.5

for all n ∈ N, whereC is a nonempty closed convex subset of a Banach space, {αn} is a sequence of0,1,and{Tn}is a sequence of nonexpansive mappings. Then they proved that, under some suitable conditions, the sequence{xn}defined by1.5converges strongly to a common fixed point of{Tn}.

In 2008, by using anewhybrid method, Takahashi et al.11proved the following theorem.

Theorem 1.1Takahashi et al. 11. Let H be a Hilbert space and letCbe a nonempty closed

convex subset ofH. Let{Tn}andT be families of nonexpansive mappings ofCinto itself such that

∩∞

n1FTn: FT/∅and letx0 ∈ H. Suppose that{Tn}satisfies the NST-conditionIwithT.

ForC1Candx1PC1x0, define a sequence{xn}ofCas follows:

yn αnxn 1−αnTnxn,

Cn1

z∈Cn:yn−z≤ xn−z,

xn1 PCn1x0, n∈N,

1.6

where0 ≤αn < 1for alln∈Nand{Tn}is said to satisfy the NST-conditionIwithTif for each

bounded sequence{zn} ⊂C,limn→ ∞zn−Tnzn 0implies thatlimn→ ∞zn−Tzn0for all

T ∈ T. Then,{xn}converges strongly toPFTx0.

Note that, recently, many authors try to extend the above result from Hilbert spaces to a Banach space setting.

LetEbe a real Banach space with dualE∗. Denote by ·,·the duality product. The

normalized duality mappingJfromEto 2E∗is defined byJx{f ∈E∗: x, fx2 f2},

for allx∈E. The functionφ:E×E → Ris defined by

φx, yx2₋₂_{x, Jy} _y2_, _∀_{x, y}_∈_E. _1.7

A mappingTis said to behemi-relatively nonexpansivesee12ifFT/∅and

φp, Tx≤φp, x, ∀x∈C, p∈FT. 1.8

A pointpinCis said to be anasymptoticfixed point ofT13ifCcontains a sequence

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On the other hand, Matsushita and Takahashi17introduced the following iteration. A sequence{xn},defined by

xn1 ΠCJ−1αnJxn 1−αnJTxn, n0,1,2, . . . , 1.9

where the initial guess elementx0 ∈ Cis arbitrary,{αn} is a real sequence in0,1,T is a

relatively nonexpansive mapping, andΠC denotes the generalized projection fromEonto a closed convex subsetCofE. Under some suitable conditions, they proved that the sequence

{xn}converges weakly to a fixed point ofT.

Recently, Kohsaka and Takahashi 18 extended iteration 1.9 to obtain a weak convergence theorem for common fixed points of a finite family of relatively nonexpansive mappings{Ti}mi1by the following iteration:

xn1 ΠCJ−1

_m

i1

wn,iαn,iJxn 1−αn,iJTixn

, n0,1,2, . . . , 1.10

whereαn,i⊂0,1andwn,i⊂0,1withmi1wn,i1, for alln∈N. Moreover, Matsushita and

Takahashi14proposed the following modification of iteration1.9in a Banach spaceE:

x0x∈C, chosen arbitrarily,

ynJ−1αnJxn 1−αnJTxn,

Cnz∈C:φz, yn≤φz, xn,

Qn{z∈C: xn−z, Jx−Jxn ≥0},

xn1 ΠCn∩Qnx, n0,1,2, . . . ,

1.11

and proved that the sequence{xn}converges strongly toΠFTx.

Qin and Su 15 showed that the sequence {xn}, which is generated by relatively nonexpansive mappingsTin a Banach spaceE, as follows:

x0∈C, chosen arbitrarily,

ynJ−1αnJxn 1−αnJTzn,

znJ−1βnJxn1−βnJTxn,

Cnv∈C:φv, yn≤αnφv, xn 1−αnφv, zn,

Qn{v∈C: Jx0−Jxn, xn−v ≥0},

xn1 ΠCn∩Qnx0

1.12

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Moreover, they also showed that the sequence{xn}, which is generated by

x0∈C, chosen arbitrarily,

ynJ−1αnJx0 1−αnJTxn,

Cnv∈C:φv, yn≤αnφv, x0 1−αnφv, xn

,

xn1 ΠCn∩Qnx0,

1.13

converges strongly toΠFTx0.

In 2008, Nilsrakoo and Saejung19used the following Mann’s iterative process:

x0∈C is arbitrary,

C−1Q−1C,

ynJ−1αnJxn 1−αnJTnxn,

Cnv∈Cn:φv, yn≤φv, xn,

xn1 ΠCn∩Qnx0, n0,1,2, . . .

1.14

and showed that the sequence {xn} converges strongly to a common fixed point of a countable family of relatively nonexpansive mappings.

Recently, Su et al. 12 extended the results of Qin and Su 15, Matsushita and Takahashi 14 to a class of closed hemi-relatively nonexpansive mapping. Note that, since the hybrid iterative methods presented by Qin and Su 15 and Matsushita and Takahashi 14 cannot be used for hemi-relatively nonexpansive mappings. Thus, as we know, Su et al.12showed their results by using the method as a monotoneCQhybrid method.

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2. Preliminaries

In this section, we will recall some basic concepts and useful well-known results. A Banach spaceEis said to bestrictly convexif

x₂y<1, 2.1

for allx, y ∈ Ewithx y 1 andx /y. It is said to beuniformly convexif for any two sequences{xn} and{yn}inEsuch thatxnyn1 and

lim

n→ ∞xnyn2, 2.2

limn→ ∞xn−yn0 holds.

LetU{x∈E:x 1}be the unit sphere ofE. Then the Banach spaceEis said to

besmoothif

lim t→0

_x_ty₋_x

t 2.3

exists for eachx, y∈U.It is said to beuniformly smoothif the limit is attained uniformly for

x, y ∈E. In this case, the norm ofEis said to beGˆateaux diﬀerentiable. The spaceEis said to

haveuniformly Gˆateaux diﬀerentiableif for eachy∈U, the limit2.3is attained uniformly for

y∈U. The norm ofEis said to beuniformly Fr´echet diﬀerentiableandEis said to be uniformly smoothif the limit2.3is attained uniformly forx, y∈U.

In our work, the concept duality mapping is very important. Here, we list some known facts, related to the duality mappingJ, as follows.

aEE∗_{, resp.}_{is uniformly convex if and only if}_E∗_E_{, resp.}_{is uniformly smooth.}

bJx/∅for eachx∈E.

cIfEis reflexive, thenJis a mapping ofEontoE∗. dIfEis strictly convex, thenJx∩Jy/∅for allx /y.

eIfEis smooth, thenJis single valued.

fIfEhas a Fr´echet diﬀerentiable norm, thenJis norm to norm continuous.

gIf Eis uniformly smooth, thenJ is uniformly norm to norm continuous on each bounded subset ofE.

hIfEis a Hilbert space, thenJis the identity operator.

For more information, the readers may consult20,21.

IfCis a nonempty closed convex subset of a real Hilbert spaceHandPC :H → Cis

themetric projection, thenPCis nonexpansive. Alber22has recently introduced ageneralized

projectionoperatorΠCin a Banach spaceEwhich is an analogue representation of the metric

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The generalized projectionΠC : E → Cis a map that assigns to an arbitrary point

x∈Ethe minimum point of the functionalφy, x, that is,ΠCxx∗, wherex∗is the solution to the minimization problem

φx∗_{, x} _min

y∈C φ

y, x. _2.4

Notice that the existence and uniqueness of the operatorΠCis followed from the properties of the functionalφy, xand strict monotonicity of the mappingJ, and moreover, in the Hilbert spaces setting we haveΠC PC. It is obvious from the definition of the functionφthat

_y₋x2_≤_φ_{y, x}_≤_y_x2_, _∀_{x, y}_∈_E. _2.5

Remark 2.1. IfEis a strictly convex and a smooth Banach space, then for allx, y∈E,φy, x

0 if and only ifxy, see Matsushita and Takahashi14.

To obtain our results, following lemmas are important.

Lemma 2.2 Kamimura and Takahashi23. LetEbe a uniformly convex and smooth Banach

space and let r > 0. Then there exists a continuous strictly increasing and convex function g :

0,2r → 0,∞such thatg0 0and

gx−y≤φx, y, 2.6

for allx, y∈Br {z∈E:z ≤r}.

Lemma 2.3Kamimura and Takahashi23. LetEbe a uniformly convex and smooth real Banach space and let{xn},{yn}be two sequences ofE. Ifφxn, yn → 0and either{xn}or{yn}is bounded,

thenxn−yn → 0.

Lemma 2.4Alber22. LetCbe a nonempty closed convex subset of a smooth real Banach space E

andx∈E. Then,x0 ΠCxif and only if

x0−y, Jx−Jx0 ≥0, ∀y∈C. 2.7

Lemma 2.5Alber22. LetEbe a reflexive strict convex and smooth real Banach space, letCbe a

nonempty closed convex subset of E and letx∈E. Then

φy,ΠCxφΠCx, x≤φy, x, ∀y∈C. 2.8

Lemma 2.6Matsushita and Takahashi14. LetEbe a strictly convex and smooth real Banach

space, letCbe a closed convex subset of E, and let T be a hemi-relatively nonexpansive mapping from

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LetCbe a subset of a Banach spaceEand let{Tn}be a family of mappings fromCintoE. For

a subsetBofC, one says that

a {Tn}, Bsatisfies condition AKTT if

∞

n1

sup{Tn1z−Tnz:z∈B}<∞, 2.9

b {Tn}, Bsatisfies condition∗AKTT if

∞

n1

sup{JTn1z−JTnz:z∈B}<∞. 2.10

For more information, see Aoyama et al. [10].

Lemma 2.7Aoyama et al.10. LetCbe a nonempty subset of a Banach spaceEand let{Tn}be a

sequence of mappings fromCintoE. LetBbe a subset ofCwith{Tn}, Bsatisfying condition AKTT,

then there exists a mappingT:B → Esuch that

Tx lim

n→ ∞Tnx, ∀x∈B 2.11

andlim sup_n_{→ ∞}{Tz −Tnz:z∈B}0.

Inspired byLemma 2.7, Nilsrakoo and Saejung19prove the following results.

Lemma 2.8Nilsrakoo and Saejung19. LetEbe a reflexive and strictly convex Banach space

whose norm is Fr´echet diﬀerentiable, letCbe a nonempty subset of a Banach spaceE, and let{Tn}be a

sequence of mappings fromCintoE. LetBbe a subset ofCwith{Tn}, Bsatisfies condition∗AKTT,

then there exists a mappingT:B → Esuch that

Tx lim

n→ ∞Tnx, ∀x∈B 2.12

andlim sup_n_{→ ∞}{JTz −JTnz:z∈B}0.

Lemma 2.9Nilsrakoo and Saejung19. LetEbe a reflexive and strictly convex Banach space

whose norm is Fr´echet diﬀerentiable, letCbe a nonempty subset of a Banach spaceE,and let{Tn}be

a sequence of mappings fromCintoE. Suppose that for each bounded subsetBofC, the ordered pair

{Tn}, Bsatisfies either condition AKTT or condition∗AKTT. Then there exists a mappingT :B →

Esuch that

Tx lim

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3. Modified Ishikawa Iterative Scheme

In this section, we establish the strong convergence theorems for finding common fixed points of a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. It is worth mentioning that our main theorem generalizes recent theorems by Su et al.12 from relatively nonexpansive mappings to a more general concept. Moreover, our results also improve and extend the corresponding results of Nilsrakoo and Saejung19. In order to prove the main result, we recall a concept as follows. An operator T in a Banach space is closed if xn → x and Txn → y, then

Txy.

Theorem 3.1. Let E be a uniformly convex and uniformly smooth Banach space and let C be a

nonempty bounded closed convex subset ofE. Let{Tn}be a sequence of hemi-relatively nonexpansive

mappings fromCinto itself such that∞_n₀FTnis nonempty. Assume that{an}∞n0and{βn}∞n0are

sequences in0,1such thatlim sup_n_{→ ∞}αn<1andlimn→ ∞βn1and let a sequence{xn}inCby

the following algorithm be:

x0∈C, chosen arbitrarity, C0C,

ynJ−1αnJxn 1−αnJTnzn,

znJ−1βnJxn1−βnJTnxn,

Cn1

v∈Cn:φv, yn≤φv, xn,

xn1 ΠCn1x0,

3.1

for n ∈ N ∪ {0}, where J is the single-valued duality mapping on E. Suppose that for each

bounded subset B of C, the ordered pair {Tn}, B satisfies either condition AKTT or condition

∗_{AKTT. Let}_T _{be the mapping from}_C _{into itself defined by}_Tv _lim_n

→ ∞Tnvfor allv ∈ Cand

suppose that T is closed and FT ∞_n₀FTn. If Tn is uniformly continuous for all n ∈ N,

then {xn} converges strongly to ΠFTx0, where ΠFT is the generalized projection from C onto

FT.

Proof. We first show that Cn1 is closed and convex for each n ≥ 0. Obviously, from the

definition ofCn1, we see thatCn1is closed for eachn≥0. Now we show thatCn1is convex

for anyn≥0. Since

φv, yn≤φv, xn⇐⇒2v, Jxn−Jyn yn2− xn2≤0, 3.2

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letp∈∞_n₀FTn, we have

φp, ynφ

p, J−1_α

nJxn 1−αnJTnzn

p2₋

2p, αnJxn 1−αnJTnzn αnJxn 1−αnJTnzn2

≤p2₋

2αnp, Jxn −21−αnp, JTnzn αnxn2 1−αnTnzn2

αnp2−2p, Jxn xn2

1−αnp2−2p, JTnzn Tnzn2

≤αnφp, xn 1−αnφp, Tnzn

≤αnφp, xn 1−αnφp, zn,

3.3

φp, znφ

p, J−1_β

nJxn1−βnJTnxn

p2₋

2p, βnJxn1−βnJTnxn βnJxn1−βnJTnxn2

p2₋

2βnp, Jxn −21−βnp, JTnxn βnxn21−βnTnxn2

βnp2−2p, Jxn xn2

1−βnp2−2p, JTnxn Tnxn2

≤βnφp, xn1−βnφp, Tnxn

≤βnφp, xn1−βnφp, xn

≤φp, xn.

3.4

Substituting3.4into3.3, we have

φp, yn ≤ φp, xn. 3.5

This means that,p ∈ Cn1 for alln ≥ 0. Consequently, the sequence{xn} is well defined. Moreover, sincexn ΠCnx0andxn1∈Cn1⊂Cn, we get

φxn, x0≤φxn1, x0, 3.6

for alln≥0. Therefore,{φxn, x0}is nondecreasing.

By the definition ofxnandLemma 2.5, we have

φxn, x0 φΠCnx0, x0≤φ

p, x0−φp,ΠCnx0

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for allp∈∞_n₀FTn⊂Cn. Thus,{φxn, x0}is a bounded sequence. Moreover, by2.5, we know that{xn}is bounded. So, limn→ ∞φxn, x0exists. Again, byLemma 2.5, we have

φxn1, xn φxn1,ΠCnx0

≤φxn1, x0−φΠCnx0, x0

φxn1, x0−φxn, x0,

3.8

for alln≥0. Thus,φxn1, xn → 0 asn → ∞.

Next, we show that{xn}is a Cauchy sequence. UsingLemma 2.2, form, nsuch that

m > n, we have

gxm−xn≤φxm, xn≤φxm, x0−φxn, x0, 3.9

whereg:0,∞ → 0,∞is a continuous stricly increasing and convex function withg0 0. Then the properties of the functiong yield that{xn}is a Cauchy sequence. Thus, we can say that{xn}converges strongly topfor some pointpinC. However, since limn→ ∞βn 1 and{xn}is bounded, we obtain

φxn1, zn φ

xn1, J−1βnJxn1−βnJTnxn

xn12−2

xn1, βnJxn1−βnJTnxn βnJxn1−βnJTnxn2

≤ xn12−2βn xn1, Jxn −21−βn xn1, JTnxnβnxn21−βnTnxn2 βnφxn1, xn

1−βnφxn1, Tnxn.

3.10

Thereforeφxn1, zn → 0 asn → ∞.

Sincexn1 ΠCn1x0∈Cn1, from the definition ofCn, we have

φxn1, yn≤φxn1, xn, 3.11

for alln≥0. Thus

φxn1, yn

−→0, asn−→ ∞. 3.12

By usingLemma 2.3, we also have

lim

n→ ∞xn1−ynnlim→ ∞xn1−xnnlim→ ∞xn1−zn0. 3.13

SinceJis uniformly norm-to-norm continuous on bounded sets, we have

lim

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For eachn∈N∪ {0},we observe that

Jxn1−JynJxn1−αnJxn 1−αnJTnzn

αnJxn1−Jxn 1−αnJxn1−JTnzn 1−αnJxn1−JTnzn−αnJxn−Jxn1

≥1−αnJxn1−JTnzn −αnJxn−Jxn1.

3.15

It follows that

Jxn1−JTnzn ≤ 1 1−αn

Jxn1−JynαnJxn−Jxn1

. 3.16

By3.14and lim sup_n_{→ ∞}αn <1, we obtain

lim

n→ ∞Jxn1−JTnzn0. 3.17

SinceJ−1_{is uniformly norm-to-norm continuous on bounded sets, we have}

lim

n→ ∞xn1−Tnzn0. 3.18

By3.13, we have

zn−xn ≤ zn−xn1xn1−xn −→0, asn−→ ∞. 3.19

SinceTnis uniformly continuous, by3.13and3.18, we obtain

xn−Tnxn ≤ xn−xn1xn1−TnznTnzn−Tnxn −→0, 3.20

asn → ∞, and so

lim

n→ ∞Jxn−JTnxn0. 3.21

Based on the hypothesis, we now consider the following two cases.

Case 1. {Tn},{xn}satisfies condition∗AKTT. ApplyingLemma 2.8to get

Jxn−JTxn ≤ Jxn−JTnxnJTnxn−JTxn

≤ Jxn−JTnxnsup{JTnz−JTz:z∈ {xn}} −→0.

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Case 2. {Tn},{xn}satisfies condition AKTT. ApplyLemma 2.7to get

xn−Txn ≤ xn−TnxnTnxn−Txn

≤ xn−Tnxnsup{Tnz−Tz:z∈ {xn}} −→0.

3.23

Hence

lim

n→ ∞xn−Txnnlim→ ∞

J−1_Jx_n₋_J−1_JTx_n₀_. _3.24

Therefore, from the both two cases, we have

lim

n→ ∞xn−Txn0. 3.25

SinceT is closed andxn → p, we havep∈FT.Moreover, by3.7, we obtain

φp, x0

lim

n→ ∞φxn, x0≤φ

p, x0

, _3.26

for allp∈FT. Therefore,p ΠFTx0.This completes the proof.

Since every relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, we obtain the following result for a countable family of relatively nonexpansive mappings of modified Ishikawa iterative process.

Corollary 3.2. Let E be a uniformly convex and uniformly smooth Banach space and let Cbe a

nonempty bounded closed convex subset of E. Let {Tn} be a sequence of relatively nonexpansive

mappings from Cinto itself such that∞_n₀FTnis nonempty. Assume that{αn}∞n0 and {βn}∞n0

are sequences in0,1such thatlim sup_n_{→ ∞}αn <1andlimn→ ∞βn 1and let a sequence{xn}in

Cbe defined by the following algorithm:

Cn1

xn1 ΠCn1x0,

3.27

forn∈ N∪ {0}, whereJ is the single-valued duality mapping onE. Suppose that for each bounded

subsetBofC, the ordered pair{Tn}, Bsatisfies either condition AKTT or condition∗AKTT. LetTbe

the mapping fromCinto itself defined byTvlimn→ ∞Tnvfor allv∈Cand suppose thatTis closed

andFT ∞_n₀FTn. IfTnis uniformly continuous for alln∈N, then{xn}converges strongly to

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mappings fromCinto itself such that∞_n₀FTnis nonempty. Assume that{αn}∞n0is a sequence in

0,1such thatlim sup_n_{→ ∞}αn<1and let a sequence{xn}inCbe defined by the following algorithm:

Cn1

v∈Cn:φv, yn≤φv,xn,

xn1 ΠCn1x0,

3.28

subsetBofC, the ordered pair{Tn}, Bsatisfies either condition AKTT or condition∗AKTT. LetT

be the mapping fromCinto itself defined byTv limn→ ∞Tnvfor allv∈ Cand suppose thatT is

closed andFT ∞_n₀FTn. Then{xn}converges strongly toΠFTx0.

Proof. InTheorem 3.1, ifβn1 for alln∈N∪ {0}then3.1reduced to3.28.

mappings fromCinto itself such that∞_n₀FTnis nonempty. Assume that{αn}∞n0 is a sequence

in 0,1 such that lim sup_n_{→ ∞}αn < 1 and let a sequence {xn} in Cbe defined by the following

algorithm:

x0∈C, chosen arbitrarity, C0C,

Cn1

xn1 ΠCn1x0,

3.29

Notice that every uniformly continuous mapping must be a continuous and closed mapping. Then settingTn ≡T for alln∈N, in Theorems3.1and3.3, we immediately obtain the following results.

nonempty bounded closed convex subset ofE. LetT :C → Cbe a closed hemi-relatively nonexpansive

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lim sup_n_{→ ∞}αn < 1and limn→ ∞βn 1 and let a sequence{xn}in Cbe defined by the following

algorithm:

Cn1

xn1 ΠCn1x0,

3.30

forn∈N∪{0}, whereJis the single-valued duality mapping onE. IfTis uniformly continuous, then

{xn}converges strongly toΠFTx0.

nonempty bounded closed convex subset ofE. LetT : C → Cbe a closed relatively nonexpansive

mapping such that FT/∅. Assume that {αn}∞n0 and {βn}∞n0 are sequences in 0,1 such that

lim sup_n_{→ ∞}αn < 1and limn→ ∞βn 1 and let a sequence{xn}in Cbe defined by the following

algorithm:

x0∈C, chosen arbitrarity, C0C,

Cn1

xn1 ΠCn1x0,

3.31

Proof. Since a closed relatively nonexpansive mapping is a closed hemi-relatively one,

Corollary 3.6is implied byCorollary 3.5.

nonempty bounded closed convex subset ofE. LetT :C → Cbe a closed hemi-relatively nonexpansive

mapping fromCinto itself such thatFT/∅. Assume that{αn}∞n0is a sequence in0,1such that

lim sup_n_{→ ∞}αn <1and let a sequence{xn}inCbe defined by the following algorithm:

Cn1

xn1 ΠCn1x0,

3.32

forn∈N∪ {0}, whereJis the single-valued duality mapping onE. Then{xn}converges strongly to

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nonempty bounded closed convex subset ofE. LetT : C → Cbe a closed relatively nonexpansive

mapping fromCinto itself such thatFT/∅. Assume that{αn}∞n0is a sequence in0,1such that

Cn1

xn1 ΠCn1x0,

3.33

ΠFTx0.

Similarly, as in the proof ofTheorem 3.1, we obtain the following results.

mappings fromCinto itself such that∞_n₀FTnis nonempty. Assume that{αn}∞n0and{βn}∞n0are

sequences in0,1such thatlim sup_n_{→ ∞}αn <1andlimn→ ∞βn <1and let a sequence{xn}inCbe

defined by the following algorithm:

Cnv∈C:φv, yn≤φv, xn,

Qn{v∈C: v−xn, Jxn−Jx0 ≥0}, xn1 ΠCn∩Qnx0,

3.34

subsetBofC, the ordered pair{Tn}, Bsatisfies either condition AKTT or condition∗AKTT. LetTbe

the mapping fromCinto itself defined byTvlimn→ ∞Tnv for allv∈Cand suppose thatTis closed

andFT ∞_n₀FTn. IfTnis uniformly continuous for alln∈N, then{xn}converges strongly to

ΠFTx0, whereΠFTis the generalized projection fromContoFT.

Corollary 3.10. LetE be a uniformly convex and uniformly smooth Banach space and letC be a

nonempty bounded closed convex subset ofE. LetT :C → Cbe closed hemi-relatively nonexpansive

mappings fromCinto itself such thatFT/∅. Assume that{αn}∞n0 and{βn}∞n0are sequences in

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by the following algorithm:

Qn{v∈C: v−xn, Jxn−Jx0 ≥0},

xn1 ΠCn∩Qnx0,

3.35

Theorem 3.11. LetE be a uniformly convex and uniformly smooth Banach space and let C be a

mappings fromCinto itself such that∞_n₀FTnis a nonempty. Assume that{αn}∞n0is a sequence in

0,1such thatlim sup_n_{→ ∞}αn<1and let a sequence{xn}inCbe defined by the following algorithm:

xn1 ΠCn∩Qnx0,

3.36

Proof. Puttingβn1, for alln∈N∪{0}, inTheorem 3.9we immediately obtainTheorem 3.11.

Corollary 3.12. LetE be a uniformly convex and uniformly smooth Banach space and letC be a

nonempty bounded closed convex subset ofE. LetT :C → Cbe closed hemi-relatively nonexpansive

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Qn{v∈C: v−xn, Jxn−Jx0 ≥0},

xn1 ΠCn∩Qnx0,

3.37

ΠFTx0.

Remark 3.13. Our results extend and improve the corresponding results in the following

senses.

iCorollary 3.10improves Theorem 2.1 of Qin and Su15from relatively nonexpan-sive mappings to more general hemi-relatively nonexpannonexpan-sive mappings.

iiTheorem 3.11 improves the algorithm in Theorem 3.1 of Nilsakoo and Saejung 19 from the Mann iteration process to modify Ishikawa iteration process and from countable relatively nonexpansive mappings to more general countable hemi-relatively nonexpansive mappings; that is, we relax the strong restrictionFT

FT. Fromiandii, it means that we relax the strongly restriction asFT FT from the assumption.

iiiCorollary 3.12 improves Theorem 3.1 of Matsushita and Takahashi 14 from relatively nonexpansive mappings to more general hemi-relatively nonexpansive mappings in a Banach space.

4. Halpern Iterative Scheme

In this section, we prove the strong convergence theorems for finding common fixed points of a countable family of hemi-relatively nonexpansive mappings, which can be viewed as a generalization of the recently result of15, Theorem 2.2.

mappings fromCinto itself such that∞_n₀FTnis nonempty. Assume that{αn}∞n0is a sequence in

0,1such thatlimn→ ∞αn0and let a sequence{xn}inCbe defined by the following algorithm:

x0 ∈C, chosen arbitrarity, C0C,

ynJ−1αnJx0 1−αnJTnxn,

Cn1

v∈Cn:φv, yn≤αnφv, x0 1−αnφv, xn

,

xn1 ΠCn1x0,

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Proof. As in the proof ofTheorem 3.1, we have thatCn1is closed and convex for eachn≥0.

Next, we show that∞_n₀FTn⊂Cnfor alln≥0. Indeed, letp∈∞n0FTn, we have

φp, ynφ

p, J−1_α

nJx0 1−αnJTnxn

p2₋₂_{p, α}

nJx0 1−αnJTnxn αnJx0 1−αnJTnxn2

≤p2₋

2αnp, Jx0 −21−αnp, JTnxn αnx02 1−αnTnxn2

αnp2−2p, Jx0 x02

1−αnp2₋

2p, JTnxn Tnxn2

≤αnφp, x0

1−αnφp, Tnxn

≤αnφp, x0 1−αnφp, xn.

4.2

This means that,p∈Cn1for alln≥0.FromTheorem 3.1, we obtain limn→ ∞φxn1, xn 0

and limn→ ∞φxn, x0exists. Sincexn1 ΠCn1x0and hencexn1∈Cn1⊂Cn, we also get

φxn1, yn≤αnφxn1, x0 1−αnφxn1, xn, 4.3

for alln≥0. Since limn→ ∞αn0, thus,φxn1, yn → 0 asn → ∞.

By using the same argument as in Theorem 3.1, we obtain that {xn} is a Cauchy sequence, thus {xn} converges strongly to p for some point p in C. By usingLemma 2.3, we also have

lim

n→ ∞xn1−ynnlim→ ∞xn1−xn0. 4.4

SinceJis uniformly norm-to-norm continuous on bounded sets, we have

lim

n→ ∞Jxn1−Jynnlim→ ∞Jxn1−Jxn0. 4.5

Observe that

Jxn1−JynJxn1−αnJx0 1−αnJTnxn

αnJxn1−Jx0 1−αnJxn1−JTnxn 1−αnJxn1−JTnxn−αnJx0−Jxn1

≥1−αnJxn1−JTnxn −αnJx0−Jxn1,

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this gives

Jxn1−JTnxn ≤ 1 1−αn

Jxn1−JynαnJx0−Jxn1

. 4.7

By4.5and limn→ ∞αn 0, we obtain limn→ ∞Jxn1−JTnxn 0.SinceJ−1 is uniformly norm-to-norm continuous on bounded sets, we have

lim

n→ ∞xn1−Tnxn0. 4.8

It follows from4.4thatxn−Tnxn ≤ xn−xn1xn1−Tnxn → 0,asn → ∞, and since

J−1_{is uniformly norm-to-norm continuous on bounded sets, we get lim}

n→ ∞Jxn−JTnxn0. From the conditions∗AKTT, AKTT, Lemmas2.7and2.8, by using the same line as in the proof ofTheorem 3.1, the both two cases, we know that

lim

n→ ∞xn−Txn0. 4.9

Finally, we prove thatxn → p, wherep ΠFTx0. Let{xni}be a subsequence of{xn}such

that{xni} q ∈C.Replacingq ΠFTx0, fromxn1 ΠCn1x0 andq∈F ⊂Cn1, we have

φxn1, x0≤φq, x0. On the other hand, from weakly lower semicontinuity of the norm, we

have

φq, x0

q2₋

2q, Jx0 x02

≤ lim i→ ∞inf

xni

2₋_x

ni, Jx0x0

2

≤ lim

i→ ∞infφxni, x0≤ilim→ ∞sup φxni, x0 ≤φq_{, x0}_.

4.10

From the definition of ΠFTx0, since q ΠFTx0, we have limn→ ∞φxni, x0 φq, x0.

This implies limn→ ∞xni q. Using the Kadec-Klee property24of the spaceE, we

obtain that{xni}converges strongly toΠFTx0. Since{xni}is an arbitrary weakly convergent

sequence of{xn}, we can conclude that{xn}convergence strongly toΠFTx0.

nonempty bounded closed convex subset ofE. LetTbe a closed hemi-relatively nonexpansive mapping

fromCinto itself such thatFTis nonempty. Assume that{αn}∞n0is a sequence in0,1such that

limn→ ∞αn0and let a sequence{xn}inCbe defined by the following algorithm:

Cn1

,

xn1 ΠCn1x0,

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forn∈N∪ {0}, whereJis the single-valued duality mapping onE. Then{xn}converges strongly to ΠFTx0.

Proof. By settingTn ≡Tfor alln∈N∪ {0}, we immediately obtain the result.

Since every relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, we immediately obtain the following corollaries.

mappings fromCinto itself such that∞_n₀FTnis nonempty. Assume that{αn}∞n0 is a sequence

in0,1such thatlimn→ ∞αn0and let a sequence{xn}inCbe defined by the following algorithm:

Cn1

,

xn1 ΠCn1x0,

4.12

nonempty bounded closed convex subset of E. Let T be a closed relatively nonexpansive mapping

fromCinto itself such thatFTis nonempty. Assume that{αn}∞n0is a sequence in0,1such that

limn→ ∞αn0and let a sequence{xn}inCbe defined by the following algorithm:

ynJ−1αnJx0 1−αnJTxn, Cn1

v∈Cn:φv, yn≤αnφv, x0 1−αnφv, xn,

xn1 ΠCn1x0,

4.13

ΠFTx0.

Similarly, as in the proof ofTheorem 4.1, we obtain the following result.

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in0,1such thatlimn→ ∞αn0and let a sequence{xn}inCbe defined by the following algorithm:

,

xn1 ΠCn∩Qnx0,

4.14

be the mapping fromCinto itself defined byTv limn→ ∞Tnv for allv∈Cand suppose thatT is

IfTnT, thenTheorem 4.5reduces to the following corollary.

Corollary 4.6see15, Theorem 2.2. LetEbe a uniformly convex and uniformly smooth Banach

space and letCbe a nonempty bounded closed convex subset ofE. LetT:C → Cbe a closed relatively

nonexpansive mapping fromCinto itself such thatFT/∅. Assume that{αn}nn0 is a sequence in

0,1such thatlimn→ ∞αn0and let a sequence{xn}inCbe defined by the following algorithm:

,

xn1 ΠCn∩Qnx0,

4.15

ΠFTx0.

5. Some Applications to Hilbert Spaces

It is well known that, in the Hilbert space setting, the concepts of hemi-relatively nonexpansive mappings and quasi-nonexpansive mappings are the equivalent. Thus, the following results can be obtained.

Theorem 5.1. LetHbe a Hilbert space and let Cbe a nonempty bounded closed convex subset of

H. Let{Tn}be a sequence of quasi-nonexpansive mappings fromCinto itself such that∞n0FTn

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andlimn→ ∞βn1and let a sequence{xn}inCbe defined by the following algorithm:

ynαnxn 1−αnTnzn,

znβnxn1−βnTnxn,

Cn1

v∈Cn:yn−v≤ xn−v,

xn1PCn1x0,

5.1

forn ∈ N∪ {0}. Suppose that for each bounded subsetBofC, the ordered pair{Tn}, Bsatisfies

condition AKTT. LetT be the mapping fromCinto itself defined byTv limn→ ∞Tnvfor allv∈C

and suppose thatT is closed andFT ∞_n₀FTn. IfTnis uniformly continuous for alln∈N, then

{xn}converges strongly toPFTx0.

Proof. SinceJis an identity operator, we have

φx, yx−y2_, _5.2

for everyx, y∈H.Therefore

_T_n_x₋_p_≤_x₋_p_⇐⇒_φ

p, Tnx≤φp, x, 5.3

for everyx ∈ Cand p ∈ FTn.Hence,Tn is quasi-nonexpansive if and only if Tn is hemi-relatively nonexpansive. Then, byTheorem 3.1, we obtain the result.

H. Let{Tn}be a sequence of quasi-nonexpansive mappings fromCinto itself such that∞n0FTnis

nonempty. Assume that{αn}∞n0is sequence in0,1such thatlim supn→ ∞αn<1and let a sequence

{xn}inCbe defined by the following algorithm:

ynαnxn 1−αnTnxn,

Cn1

v∈Cn:yn−v≤ xn−v,

xn1PCn1x0,

5.4

and suppose thatTis closed andFT ∞_n₀FTn. Then{xn}converges strongly toPFTx0.

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H. Let{Tn}be a sequence of quasi-nonexpansive mappings fromCinto itself such that∞n0FTn

is nonempty. Assume that{αn}∞n0 is a sequence in0,1such that lim supn→ ∞αn < 1 and let a

sequence{xn}inCbe defined by the following algorithm:

x0 ∈C, chosen arbitrarity, C0C,

yn αnx0 1−αnTnxn,

Cn1

v∈Cn:yn−v≤αnx0−v 1−αnxn−v

,

xn1 PCn1x0,

5.5

and suppose thatTis closed andFT ∞_n₀FTn. Then{xn}converges strongly toPFTx0.

Acknowledgments

The authors would like to thank the referees for the valuable suggestions which helped to improve this manuscript. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

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