Strong Convergence to Common Fixed Points of Countable Relatively Quasi-Nonexpansive Mappings

Volume 2008, Article ID 312454,19pages doi:10.1155/2008/312454

Research Article

Strong Convergence to Common Fixed Points of Countable Relatively Quasi-Nonexpansive Mappings

Weerayuth Nilsrakoo¹and Satit Saejung²

1Department of Mathematics, Statistics and Computer, Ubon Rajathanee University, Ubon Ratchathani 34190, Thailand

2Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

Correspondence should be addressed to Satit Saejung,[email protected] Received 30 August 2007; Accepted 24 December 2007

Recommended by Simeon Reich

We prove that a sequence generated by the monotone CQ-method converges strongly to a common fixed point of a countable family of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. Our result is applicable to a wide class of mappings.

Copyrightq2008 W. Nilsrakoo and S. Saejung. 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

LetEbe a real Banach space, letCbe a nonempty closed convex subset ofE, and letT:C→E be a mapping. Recall thatT is nonexpansive if

Tx−Ty ≤ x−y ∀x, y∈C. 1.1

We denote by FTthe set of fixed points ofT, that is, FT {x∈C:xTx}. A mappingTis said to be quasi-nonexpansive if FT/∅and

Tx−y ≤ x−y ∀x∈C, y∈FT. 1.2

It is easy to see that ifT is nonexpansive with FT/∅, then it is quasi-nonexpansive. There are many methods for approximating fixed points of a quasi-nonexpansive mapping. In 1953, Mann1introduced the iteration as follows: a sequence{xn}is defined by

x_n1α_nx_n 1−α_n

Tx_n, 1.3

where the initial guess element x0 ∈ C is arbitrary and {αn} is a real sequence in 0,1.

Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich2. In an infinite-dimensional Hilbert space, Mann iteration can yield only weak convergencesee3,4. Attempts to modify the Mann iteration method 1.3 so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi 5 proposed the following modification of Mann iteration method1.3for a nonexpansive mappingTfromCinto itself in a Hilbert space:

x0∈Cis arbitrary, y_nα_nx_n

1−α_n Tx_n, Cn

z∈C:yn−z≤xn−z, Q_n

z∈C:

x_n−z, x₀−x_n

≥0 , xn1PCn∩Qnx0, n0,1,2, . . . ,

1.4

whereP_Kdenotes the metric projection from a Hilbert spaceH onto a closed convex subset K of H and prove that the sequence {xn} converges strongly to P_FTx₀.A projection onto intersection of two halfspaces is computed by solving a linear system of two equations with two unknownssee6, Section 3.

Recently, Su and Qin7modified iteration1.4, so-called the monotone CQ method for nonexpansive mapping, as follows:

x0∈Cis arbitrary, y_nα_nx_n

1−α_n Tx_n, C0

z∈C:y0−z≤x0−z, Q₀C,

C_n

z∈C_n−1∩Q_n−1:y_n−z≤x_n−z, Qn

z∈Cn−1∩Qn−1:

xn−z, x0−xn

≥0 , x_n1P_Cn∩Qnx₀, n0,1,2, . . . ,

1.5

and prove that the sequence{xn}converges strongly toP_FTx₀.

We now recall some definitions concerning relatively quasi-nonexpansive mappings and what have been proved until now. LetEbe a real smooth Banach space with norm · and letE^∗be the dual ofE. Denote by·,·the pairing betweenEandE^∗. The normalized duality mappingJfromEtoE^∗is defined by

Jx

f ∈E^∗:x, fx²f²

, where x∈E. 1.6

The reader is directed to8 and its review9, where the properties on the duality mapping and several related topics are presented. The functionφ:E×E→Ris defined by

φx, y x²−2x, Jyy² ∀x, y∈E. 1.7

LetT be a mapping fromCintoE. A pointpinCis said to be an asymptotic fixed point ofT 10ifCcontains a sequence{xn}which converges weakly topand limn→∞xn−Txn 0. The set of asymptotic fixed points ofT is denoted byFT. We say that the mappingT is relatively nonexpansive if the following conditions are satisfied:

R1FT/∅;

R2φp, Tx≤φp, xfor eachx∈C, p∈FT; R3FT FT.

IfT satisfiesR1andR2, thenTis called relatively quasi-nonexpansive.

Several articles have appeared providing method for approximating fixed points of relatively quasi-nonexpansive mappings11–16. Matsushita and Takahashi12introduced the following iteration: a sequence{xn}defined by

xn1

C

J⁻¹

αnJxn 1−αn

JTxn

, 1.8

where the initial guess elementx₀ ∈ Cis arbitrary,{αn}is a real sequence in 0,1,T is a relatively nonexpansive mapping, andΠC denotes the generalized projection fromEonto a closed convex subsetCofE. They prove that the sequence{xn}converges weakly to a fixed point ofT. Moreover, Matsushita and Takahashi13proposed the following modification of iteration1.8:

x₀∈Cis arbitrary, ynJ⁻¹

αnJxn

1−αnJTxn , C_n

z∈C:φ

z, y_n≤φ z, x_n

, Qn

z∈C:

xn−z, x0−xn

≥0 , xn1

Cn∩Qn

x₀, n0,1,2, . . . ,

1.9

and prove that the sequence{xn}converges strongly toΠFTx₀.

Recently, Kohsaka and Takahashi 11 extended iteration 1.8 to obtain a weak convergence theorem for common fixed points of a finite family of relatively nonexpansive mapping{Ti}^m_i1by the following iteration:

xn1

C

J⁻¹ _m

i1

wn,i

αn,iJxn

1−αn,i

JTixn

, n1,2, . . . , 1.10 whereαn,i⊂0,1andwn,i⊂0,1with_m

i1wn,i1 for alln∈N.

Employing the ideas of Su and Qin7, and of Aoyama et al.17, we modify iterations 1.5,1.8–1.10to obtain strong convergence theorems for common fixed points of countable relatively quasi-nonexpansive mappings in a Banach space. Consequently, we obtain strong convergence theorems for quasi-nonexpansive mappings in a Hilbert space without using demiclosedness principle. Moreover, we introduce a new certain condition for an infinite family of mappings which is inspired by Aoyama et al.17, and we also show how to generate a corresponding sequence of mappings satisfying our condition.

2. Preliminaries

Throughout the paper, let E be a real Banach space. We say that E is strictly convex if the following implication holds forx, y∈E:

xy1, x /yimply xy

2

<1. 2.1

It is also said to be uniformly convex if for anyε >0, there existsδ >0 such that xy1, x−y ≥εimply

xy 2

≤1−δ. 2.2 It is known that ifEis uniformly convex Banach space, thenEis reflexive and strictly convex.

A Banach spaceEis said to be smooth if limt→0

xty − x

t 2.3

exists for each x, y ∈ SE : {x ∈ E : x 1}. In this case, the norm of Eis said to be Gˆateaux differentiable. The spaceEis said to have uniformly Gˆateaux differentiable norm if for each y∈SE, the limit2.3is attained uniformly forx∈SE.The norm ofEis said to be Fr´echet differentiable if for eachx∈SE, the limit2.3is attained uniformly fory ∈SE.The norm ofEis said to be uniformly Fr´echet differentiableandEis said to be uniformly smoothif the limit 2.3is attained uniformly forx, y∈SE.

We also know the following propertiessee, e.g.,18for details.

aEE^∗, resp.is uniformly convex if and only ifE^∗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 E is uniformly smooth, then J is uniformly norm to norm continuous on each bounded subset ofE.

hIfEis a Hilbert space, thenJis the identity operator.

LetEbe a smooth Banach space. The functionφ:E×E→Ris defined by

φx, y x²−2x, Jyy² ∀x, y∈E. 2.4 It is obvious from the definition of the functionφthat

x − y₂

≤φx, y≤

xy₂

∀x, y∈E. 2.5

Moreover, we know the following results.

Lemma 2.1 see 13, Remark 2.1. Let E be a strictly convex and smooth Banach space, then φx, y 0 if and only ifxy.

Lemma 2.2see11, Lemma 2.5. Let Ebe a uniformly convex and smooth Banach space and let r > 0.Then there exists a continuous, strictly increasing, and convex functiong :0,2r → 0,∞ such thatg0 0 and

g

x−y

≤φx, y 2.6

for allx, y∈B_r{z∈E:z ≤r}.

Let Cbe a nonempty closed convex subset of E. Suppose that E is reflexive, strictly convex, and smooth. It is known that19for anyx ∈ E, there exists a unique pointx^∗ ∈ C such that

φ x^∗, x

min

y∈C φy, x. 2.7

Following Alber20, we denote such anx^∗byΠCx. The mappingΠC is called the generalized projection fromEontoC. It is easy to see that in a Hilbert space, the mappingΠCcoincides with the metric projectionPC. Concerning the generalized projection, the following are well known.

Lemma 2.3see19, Proposition 4. LetCbe a nonempty closed convex subset of a smooth Banach spaceEand letx∈E. Then

x^∗

C

x⇐⇒

x^∗−y, Jx−Jx^∗

≥0 for eachy∈C. 2.8

Lemma 2.4see19, Proposition 5. LetEbe a reflexive, strictly convex, and smooth Banach space, letCbe a nonempty closed convex subset ofE, and letx∈E. Then

φ

y,

C

x

φ

C

x, x

≤φy, x for eachy∈C. 2.9

Dealing with the generalized projection fromEonto the fixed point set of a relatively quasi-nonexpansive mapping, we get the following result.

Lemma 2.5. LetEbe a strictly convex and smooth Banach space, letCbe a nonempty closed convex subset ofE, and letTbe a relatively quasi-nonexpansive mapping fromCintoE. Then FTis closed and convex.

Proof. The proof of13, Proposition 2.4does not invoke conditionR3at all. So the conclusion holds for relatively quasi-nonexpansive mappings as well.

LetCbe a subset of a Banach spaceEand let{Tn}be a family of mappings fromCinto E. For a subsetBofC, we say that

i {Tn}, Bsatisfies condition AKTT if ∞

n1

supT_n1z−T_nz:z∈B

<∞; 2.10

ii {Tn}, Bsatisfies condition^∗AKTT if ∞

n1

supJTn1z−JTnz:z∈B

<∞. 2.11

Aoyama et al.17, Lemma 3.2prove the following result which is very useful in our main result.

Lemma 2.6. LetCbe a nonempty subset of a Banach spaceEand let{Tn}be a sequence of mappings fromCintoE. LetB be a subset ofCwith{Tn}, Bsatisfying condition AKTT, then there exists a mappingT:B→Esuch that

Tx lim

n→ ∞Tnx ∀x∈B 2.12

and lim_{n→ ∞}sup{Tz −T_nz:z∈B}0.

Inspired by the preceding lemma, we have the following result.

Lemma 2.7. LetEbe a reflexive and strictly convex Banach space whose norm is Fr´echet differentiable, letCbe a nonempty subset ofE, and 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 2.13

and limn→ ∞sup{JTz−JTnz:z∈B}0.

Proof. Forx∈B, we show that{JTnx}is a Cauchy sequence inE^∗. Letε >0. By the condition

∗AKTT of{Tn}, B, there existsl₀∈Nsuch that ∞

nl0

supJT_n1z−JT_nz:z∈B

< ε. 2.14

In particular, ifk > l≥l0, then

JT_kx−JT_lx≤^k−1

nl

supJT_n1z−JT_nz:z∈B

≤^∞

nl0

supJT_n1z−JT_nz:z∈B

< ε.

2.15

Hence,{JTnx}is a Cauchy sequence inE^∗. It follows then that limn→ ∞JT_nxexists for allx∈B.

Moreover, it is noted that the convergence is uniform onB. Since Eis reflexive and strictly convex,Jis bijective and we can define a mappingT fromBintoEsuch that

TxJ⁻¹

nlim→ ∞JTnx

∀x∈B. 2.16

SinceEhas a Fr´echet differentiable norm,Jis norm-to-norm continuous and hence TxJ⁻¹J

n→ ∞limTnx lim

n→ ∞Tnx ∀x∈B. 2.17

This completes the proof.

Combining Lemmas2.6and2.7, we obtain a crucial tool for our main result.

Lemma 2.8. LetEbe a reflexive and strictly convex Banach space whose norm is Fr´echet differentiable, letCbe a nonempty subset ofE, 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 :C→Esuch that Tx lim

n→ ∞T_nx ∀x∈C. 2.18

Proof. To see that T is well defined, we suppose that {Tn},{x} satisfies condition AKTT and condition ^∗AKTT. Then, by Lemmas 2.6 and 2.7, there exist T and T such that Tx limn→ ∞TnxTx.

Lemma 2.9see11, Lemma 3.2. LetEbe a reflexive, strictly convex, and smooth Banach space, let z∈E, and let{ti}^m_i1⊂0,1with_m

i1ti1. If{xi}^m_i1is a finite sequence inEsuch that φ

z, J⁻¹

_m

i1

tiJxi

^m

i1

tiφ z, xi

, 2.19

thenx1x2· · ·xm.

Lemma 2.10. LetEbe a strictly convex Banach space and let{tn} ⊂0,1with_∞

n1tn1. If{xn}is a sequence inEsuch that_∞

n1tnxnand_∞

n1tnxn²converge, and

∞ n1

t_nx_n

2

^∞

n1

t_nx_n², 2.20

then{xn}is a constant sequence.

Proof. Suppose thatxi/xjfor somei, j∈N. Then, by the strict convexity ofE, t_i

t_it_jx_i tj

t_it_jx_j ²< t_i

t_it_jx_i² tj

t_it_jx_j². 2.21 It follows that

∞ n1

t_nx_n

2

t_it_j t_i

t_it_jx_i tj

t_it_jx_j

n /i,j

t_nx_n

2

≤

titj ti

titjxi t_j titjxj

²

n /i,j

tnxn²

<

titj ti

titj

xi² tj

titj

xj²

n /i,j

tnxn²

^∞

n1

t_nx_n².

2.22

This is a contradiction.

3. Main results

In this section, we establish strong convergence theorem for finding common fixed points of a countable family of relatively quasi-nonexpansive mappings in a Banach space.

This theorem generalizes a recent theorem by Su et al.21, Theorem 3.1. It is noted that relative quasi-nonexpansiveness considered in the paper and hemirelative nonexpansiveness of21are the same. We do prefer the former name because in a Hilbert space setting, relatively quasi-nonexpansive mappings are just quasi-nonexpansive.

Recall that an operatorTin a Banach space is closed ifxn→xandTxn→y, thenTxy.

Theorem 3.1. Let E be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset ofE. Let{Tn}be a sequence of relatively quasi-nonexpansive mappings fromCintoEsuch that_∞

n0FTnis nonempty and let{xn}be a sequence inCdefined as follows:

x₀∈C, C−1Q−1C, y_nJ⁻¹

α_nJx_n 1−α_n

JT_nx_n , Cn

z∈Cn−1∩Qn−1:φ z, yn

≤φ z, xn

, Qn

z∈Cn−1∩Qn−1:

xn−z, Jx0−Jxn

≥0 , x_n1

Cn∩Qn

x₀, n0,1,2, . . . ,

3.1

where{αn}is a sequence in 0,1 with lim sup_n→∞αn < 1. Suppose that for each bounded subset B ofC, the ordered pair{Tn}, Bsatisfies either condition AKTT or condition^∗AKTT. LetT be the mapping fromCintoEdefined byTz lim_n→∞T_nzfor allz ∈ Cand suppose thatT is closed and

FT _∞

n0FTn. Then{xn}converges strongly toΠFTx₀.

Proof. We first note that eachC_nandQ_n are closed and convex. This follows sinceφz, yn ≤ φz, xnis equivalent to

2

z, Jxn−Jyn

≤xn²−yn². 3.2 It is clear that_∞

n0FTn⊂CC₋₁∩Q₋₁. Next, we show that ∞

n0

F T_n

⊂C_n∩Q_n ∀n∈N∪ {0}. 3.3

Suppose that_∞

n0FTn⊂Ck−1∩Qk−1for somek∈N∪ {0}. Letp∈_∞

n0FTn. Then φ

p, yk φ

p, J⁻¹

αkJxk 1−αk

JTkxk p²−2

p, αkJxk 1−αk

JTkxk

αkJxk 1−αk

JTkxk²

≤ p²−2αk p, Jxk

−2

1−αk

p, JTkxk

αkxk²

1−αkTkxk² α_k

p²−2 p, Jx_k

x_k²

1−α_k

p²−2

p, JT_kx_k

T_kx_k² αkφ

p, xk

1−αk φ

p, Tkxk

≤α_kφ p, x_k

1−α_k

φ p, x_k φ

p, x_k .

3.4

This implies that_∞

n0FTn⊂Ck.Fromxk ΠC_k−1∩Q_k−1x0and byLemma 2.3, we have x_k−z, Jx₀−Jx_k

≥0 for eachz∈C_k−1∩Q_k−1. 3.5 In particular,

xk−p, Jx0−Jxk

≥0 for everyp∈^∞

n0

F Tn

3.6 and hence_∞

n0FTn⊂Q_k. It follows that ∞ n0

F Tn

⊂Ck∩Qk. 3.7

By induction,3.3holds. This implies that{xn}is well defined. It follows from the definition ofQnandLemma 2.3thatxn ΠQnx0. Sincexn1 ΠCn∩Qnx0∈Qn, we have

φ x_n, x₀

≤φ

x_n1, x₀

∀n∈N∪ {0}. 3.8

Therefore,φxn, x₀is nondecreasing. Usingxn ΠQnx₀andLemma 2.4, we have φ

xn, x0

φ

ΠQnx0, x0

≤φ p, x0

−φ p, xn

≤φ p, x0

3.9 for allp∈_∞

n0FTnfor alln∈N∪ {0}.Therefore,φxn, x₀is bounded. So

nlim→ ∞φ xn, x0

exists. 3.10

In particular, by 2.5, the sequence {xn − x0²} is bounded. This implies that {xn} is bounded. Noticing again that xn ΠQnx₀, and for any positive integer k, we havexnk ∈ Qnk−1⊂Qn. ByLemma 2.4,

φ

xnk, xn φ

xnk,

Qn

x0

≤φ

xnk, x0

−φ

Qn

x0, x0

φ

xnk, x0

−φ xn, x0

. 3.11 UsingLemma 2.2, we have, form, nwithm > n,

gxm−xn≤φ xm, xn

≤φ xm, x0

−φ xn, x0

, 3.12

whereg:0,∞→0,∞is a continuous, strictly increasing, and convex function withg0 0. Then the properties of the functiong yield that {xn}is a Cauchy sequence in C,so there existsw ∈Csuch thatxn →w.In view ofxn1 ΠCn∩Qnx0 ∈Cnand the definition ofCn, we also have

φ

x_n1, y_n

≤φ

x_n1, x_n

∀n∈N∪ {0}. 3.13

It follows that

n→ ∞limφ

x_n1, y_n lim

n→ ∞φ

x_n1, x_n

0. 3.14 By usingLemma 2.2, we obtain

nlim→ ∞xn1−yn lim

n→ ∞xn1−xn0. 3.15

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

nlim→ ∞Jxn1−Jyn lim

n→ ∞Jxn1−Jxn0. 3.16

On the other hand, we have, for eachn∈N∪ {0}, Jx_n1−Jy_nJx_n1−

α_nJx_n 1−α_n

JT_nx_n 1−α_n

Jx_n1−JT_nx_n

−α_n

Jx_n−Jx_n1

≥

1−αnJxn1−JTnxn−αnJxn−Jxn1,

3.17

and hence

Jxn1−JTnxn≤ 1

1−αnJxn1−Jyn αn

1−αnJxn−Jxn1. 3.18 From3.16and lim sup_{n→ ∞}α_n<1, we obtain

nlim→ ∞Jxn1−JTnxn0. 3.19

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

nlim→ ∞x_n1−T_nx_n lim

n→ ∞J⁻¹ Jx_n1

−J⁻¹

JT_nx_n0. 3.20 It follows from3.15that

xn−Tnxn≤xn−xn1xn1−Tnxn−→0 3.21

and so

nlim→ ∞Jxn−JTnxn0. 3.22

Case 1.{Tn},{xn}satisfies condition AKTT. We applyLemma 2.6to get x_n−Tx_n≤x_n−T_nx_nT_nx_n−Tx_n

≤x_n−T_nx_nsupT_nz−Tz:z∈ x_n

−→0. 3.23

Case 2.{Tn},{xn}satisfies condition^∗AKTT. It follows fromLemma 2.7that Jx_n−JTx_n≤Jx_n−JT_nx_nJT_nx_n−JTx_n

≤Jx_n−JT_nx_nsupJT_nz−JTz:z∈ x_n

−→0. 3.24 Hence,

nlim→ ∞xn−Txn lim

n→ ∞J⁻¹ Jxn

−J⁻¹

JTxn0. 3.25

From both cases, we obtain

nlim→ ∞xn−Txn0. 3.26

SinceTis closed andxn→w, we havew∈FT. Furthermore, by3.9, φ

w, x₀ lim

n→ ∞φ x_n, x₀

≤φ p, x₀

∀p∈FT. 3.27

Hence,w ΠFTx₀.

Corollary 3.2see21, Theorem 3.1. LetEbe a uniformly convex and uniformly smooth Banach space and letCbe a nonempty closed convex subset ofE. LetTbe a closed relatively quasi-nonexpansive mapping fromCintoEsuch that FTis nonempty and let{xn}be a sequence inCdefined as follows:

x₀∈C, C₋₁Q₋₁C, y_nJ⁻¹

α_nJx_n 1−α_n

JTx_n , C_n

z∈C_n−1∩Q_n−1:φ z, y_n

≤φ z, x_n

, Qn

z∈Cn−1∩Qn−1:

xn−z, Jx₀−Jxn

≥0 , x_n1

Cn∩Qn

x₀, n0,1,2, . . . ,

3.28

where{αn}is a sequence in0,1with lim sup_{n→ ∞}αn<1. Then{xn}converges strongly toΠFTx₀. Remark 3.3. If, in Theorem 3.1, Tn is continuous for each n ∈ N, then the mapping T is continuous and closed.

In our main theorem, we assume that for each bounded subsetBofC, the ordered pair {Tn}, B satisfies either condition AKTT or condition ^∗AKTT. As in 17, we can generate a sequence {Tn}of relatively quasi-nonexpansive mappings satisfying such an assumption by using convex combination of a given sequence {Sk} of relatively quasi-nonexpansive mappings with a nonempty common fixed point set.

Let{β^k_n}be a family of positive real numbers with indicesn,k∈N∪ {0}withk≤nsuch that

i_n

k0β^k_n1 for everyn∈N∪ {0};

iilimn→ ∞β^k_nβ^k>0 for everyk∈N∪ {0}; and iii_∞

n0 _n

k0|β^k_n1−β^k_n|<∞.

LetEbe a uniformly convex and uniformly smooth Banach space and letCbe a nonempty closed convex subset ofE. For a sequence{Sk}^∞_k1of continuous relatively quasi-nonexpansive mappings with a common fixed point andS0 is the identity mapping, we define a sequence {Tn}of mappings fromCintoEby

TnxJ⁻¹ _n

k0

β^k_nJSkx

3.29

forx∈Candn∈N∪ {0}. We note that ∞

k0

F Sk

⊂ⁿ

k0

F Sk

⊂F Tn

∀n∈N∪ {0}. 3.30

Forn∈N∪ {0}, letp∈_n

k0FSk. Then

φ p, Tnx

φ

p, J⁻¹ _n

k0

β^k_nJSkx

p²−2

p,ⁿ

k0

β^k_nJS_kx

n k0

β_n^kJS_kx

2

≤ p²−2 n k0

β_n^k

p, JSkx ⁿ

k0

β^k_nSkx²

ⁿ

k0

β_n^kφ p, S_kx

≤φp, x

3.31

for allx∈C. Then, for allz∈FTnand fixq∈_∞

k0FSk,

φq, z φ q, T_nz

φ

q, J⁻¹ _n

k0

β^k_nJS_kz

≤ⁿ

k0

β^k_nφ q, S_kz

≤φq, z, 3.32

that is,

φ

q, J⁻¹ _n

k0

β^k_nJS_kz

ⁿ

k0

β^k_nφ q, S_kz

φq, z. 3.33

ByLemma 2.9, we havezS₀zS₁z· · ·S_nz. So F

Tn

⊂ⁿ

k0

F Sk

∀n∈N∪ {0}. 3.34

This implies that

F T_n

ⁿ

k0

F S_k

∀n∈N∪ {0}, 3.35

and so

∞ n0

F T_n

^∞

k0

F S_k

/∅. 3.36

Then, by3.31, we have that{Tn}is a sequence of relatively quasi-nonexpansive mappings.

LetBbe a bounded subset ofCand letp∈_∞

k0FSk. By2.5, we have S_kx− p₂

≤φ p, S_kx

≤φp, x≤

xp₂

, 3.37

and hence

S_kx≤2psup

z:z∈B

3.38 for allx ∈ B andk ∈ N∪ {0}. LetM sup{Skx : x ∈ B, k ∈ N∪ {0}}. Forx ∈ B and n∈N∪ {0}, we have

JTn1x−JTnx

n1

k0

β_n1^k JSkx−ⁿ

k0

β_n^kJSkx

≤ⁿ

k0

β^k_n1−β^k_nJSkxβⁿ¹_n1JSkx

ⁿ

k0

β^k_n1−β^k_nSkx

1−ⁿ

k0

β^k_n1

Skx

≤ⁿ

k0

β^k_n1−β^k_nM _n

k0

β_n^k−ⁿ

k0

β_n1^k

M

≤2Mⁿ

k0

β_n1^k −β^k_n.

3.39

Therefore,

supJT_n1x−JT_nx:x∈B

≤2Mⁿ

k0

β^k_n1−β_n^k. 3.40

It follows fromiiithat ∞ n0

supJT_n1x−JT_nx:x∈B

≤2M^∞

n0

n k0

β^k_n1−β^k_n<∞. 3.41

ByLemma 2.7, we can define a mappingTby Tx lim

n→ ∞Tnx, ∀x∈C. 3.42

Using the same argument presented in the proof of17, pages 2357-2358, we have

nlim→ ∞

n k0

β_n^k−β^k0, ^∞

k0

β^k1. 3.43

For eachx∈C, the series_∞

k0β^kJSkxconverges absolutely and

JTx−^∞

k0

β^kJSkx lim

n→ ∞

JTnx−^∞

k0

β^kJSkx lim

n→ ∞

n k0

β^k_nJSkx−^∞

k0

β^kJSkx

≤ lim

n→ ∞

_n

k0

β^k_n−β^kJSkx ^∞

kn1

β^kJSkx

≤ lim

n→ ∞

n k0

β^k_n−β^kS_kx lim

n→ ∞

∞ kn1

β^kS_kx

≤ lim

n→ ∞

n k0

β^k_n−β^kM lim

n→ ∞

∞ kn1

β^kM0.

3.44

This implies that

TxJ⁻¹ _∞

k0

β^kJS_kx

∀x∈C. 3.45

It is obvious that

∞ k0

F Sk

⊂FT. 3.46

Letz∈FTand fixp∈_∞

k0FSk. Then φp, z φ

p, Tz φ

p, J⁻¹

_∞

k0

β^kJSkz

lim

n→ ∞φ

p, J⁻¹ _n

k0

β^kJSkz

lim

n→ ∞

p²−2

p,ⁿ

k0

β^kJSkz

n k0

β^kJSkz

2

≤ lim

n→ ∞

p²−2

p,ⁿ

k0

β^kJSkz

ⁿ

k0

β^kJSkz² lim

n→ ∞

_∞

k0

β^kp²−2 n k0

β^k

p, JSkz ⁿ

k0

β^kSkz² lim

n→ ∞

_n

k0

β^kφ p, S_kz

^∞

kn1

β^kp²

lim

n→ ∞

n k0

β^kφ p, S_kz

^∞

k0

β^kφ p, S_kz

≤^∞

k0

β^kφp, z

φp, z.

3.47

It follows that

∞ k0

β^kJS_kz

2

^∞

k0

β^kJS_kz². 3.48

By the strict convexity ofE^∗andLemma 2.10,

JSkzJS₀zJz ∀k∈N. 3.49

SinceJis one to one,

S_kzS₀zz ∀k∈N. 3.50

Soz∈_∞

k0FSk.Therefore,

FT⊂^∞

k0

F Sk

. 3.51

This together with3.36and3.46gives

FT ^∞

n0

F Tn

^∞

k0

F Sk

. 3.52

Hence, we obtain that{Tn}satisfies all the conditions of our main theorem. Now, we have the following result.

Theorem 3.4. Let E be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let {βn^k} be a family of positive real numbers with indicesn, k∈N∪ {0}withk≤nsuch that

i_n

k0β^k_n1 for everyn∈N∪ {0};

iilimn→∞β_n^kβ^k>0 for everyk∈N∪ {0};

iii_∞

n0_n

k0|β_n1^k −β^k_n|<∞.

Let{Sk} be a sequence of continuous relatively quasi-nonexpansive mappings with a common fixed point and letS0 be the identity operator, one defines a sequence{Tn}of relatively quasi-nonexpansive mappings fromCintoEby

TnxJ⁻¹ _n

k0

β^k_nJSkx

3.53 for allx ∈ Candn ∈ N∪ {0}. Then the sequence{xn}inCdefined by3.1converges strongly to Π^∞_k0FSkx0.

4. Deduced theorems

In Hilbert spaces, relatively quasi-nonexpansive mappings and quasi-nonexpansive mappings are the same. We obtain the following result.

Theorem 4.1. LetCbe a nonempty closed convex subset of a Hilbert spaceH. Let{Tn}be a sequence of quasi-nonexpansive mappings fromCintoEsuch that _∞

n0FTnis nonempty and let{xn} be a sequence inCdefined as follows:

x0∈C, C−1Q−1C, ynαnxn

1−αn Tnxn, Cn

z∈Cn−1∩Qn−1:yn−z≤xn−z, Qn

z∈Cn−1 ∩Qn−1:

xn−z, x₀−xn

≥0 , x_n1P_Cn∩Qnx₀, n0,1,2, . . . ,

4.1

where{αn}is a sequence in0,1with lim sup_n→∞α_n < 1. Suppose that for each bounded subsetB ofC, the ordered pair{Tn}, Bsatisfies condition AKTT. LetT be the mapping fromCintoEdefined byTz limn→∞Tnzfor allz ∈ Cand suppose thatT is closed and FT _∞

n0FTn. Then{xn} converges strongly toP_FTx₀.

Proof. SinceJis an identity operator, we have

φx, y x−y², 4.2

for everyx, y∈H. Therefore,

T_nx−p≤ x−p ⇐⇒φ p, T_nx

≤φp, x 4.3

for everyx∈Candp ∈FTn. Hence,T_nis quasi-nonexpansive if and only ifT_nis relatively quasi-nonexpansive. Then, byTheorem 3.1, we obtain the result.

Corollary 4.2see22, Theorem 2.1. LetCbe a nonempty closed convex subset of a Hilbert space H. LetT be a closed quasi-nonexpansive mapping fromCintoEsuch that FTis nonempty and let {xn}be a sequence inCdefined as follows:

x₀∈C, C−1Q−1C, y_nα_nx_n

1−α_n Tx_n, C_n

z∈C_n−1∩Q_n−1:y_n−z≤x_n−z, Q_n

z∈C_n−1∩Q_n−1:

x_n−z, x₀−x_n

≥0 , x_n1P_Cn∩Qnx₀, n0,1,2, . . . ,

4.4

where{αn}is a sequence in0,1with lim sup_n→∞α_n<1. Then{xn}converges strongly toP_FTx₀. We give an example of a countable family of quasi-nonexpansive mappings which are not nonexpansive but satisfy all the requirements of our main theorem.

Example 4.3. LetERwith the usual norm. Forn∈N, we define a mappingT_nonRby

Tnx

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

0 ifx≤ 1 n², 1

n² ifx > 1

n², 4.5

for allx∈R. Then_∞

n1FTn FTn {0}and

Tnx−0≤ |x−0| ∀x∈R. 4.6

So{Tn}is a sequence of quasi-nonexpansive mappings. Letz∈R, then

T_n1z−T_nz

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

0 ifz≤ 1

n1², 1

n² if 1

n1² < z≤ 1 n², 1

n² − 1

n1² ifz > 1 n²,

4.7