Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space

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,

Kriengsak Wattanawitoon,

and Poom Kumam

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

2Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand

3Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

4Department 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,poom.kum@kmutt.ac.th 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 be nonexpansive ifTx−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 be quasi-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

x_n1αnxn 1−αnTxn, n≥0, 1.1

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,

x_n1αnu 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

y_nβ_nx_n 1−β_n

Tx_n,

x_n1αnxn 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}^N_i1in a real Hilbert space:

x₀∈C is arbitrary, y_nα_nx_n 1−α_nTnz_n, znβnyn

1−βn Tnyn, C_n

z∈C:y_n−z≤ x_n−z , Qn{z∈C: xn−z, x₀−xn ≥0}, x_n1PCn∩Qnx0, n0,1,2, . . . ,

1.4

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

In 2008, Nakprasit et al.8 established weak and strong convergence theorems for finding common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. In the same year, Cho et al.9introduced the normal Mann’s iterative process and proved some strong convergence theorems for a finite family nonexpansive mapping in the framework Banach spaces.

To find a common fixed point of a family of nonexpansive mappings, Aoyama et al.

10introduced the following iterative sequence. Letx₁x∈Cand

x_n1αnx 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. LetH 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.

ForC₁Candx₁P_C1x₀, define a sequence{xn}ofCas follows:

yn αnxn 1−αnTnxn, C_n1

z∈C_n:y_n−z≤ x_n−z , x_n1 PC_n1x₀, n∈N,

1.6

where 0 ≤αn < 1 for alln∈Nand{Tn}is said to satisfy the NST-conditionIwithTif for each bounded sequence{zn} ⊂C, limn→ ∞zn−Tnzn 0 implies that limn→ ∞zn−Tzn0 for all T ∈ T. Then,{xn}converges strongly toP_FTx₀.

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 2^E∗is defined byJx{f ∈E^∗: x, fx² f²}, for allx∈E. The functionφ:E×E → Ris defined by

φ x, y

x²−2

x, Jy y², ∀x, y∈E. 1.7

A mappingTis said to be hemi-relatively nonexpansivesee12ifFT/∅and φ

p, Tx

≤φ p, x

, ∀x∈C, p∈FT. 1.8

A pointpinCis said to be an asymptotic fixed point ofT13ifCcontains a sequence {xn} which converges weakly to p such that the strong lim_{n→ ∞}xn − Tx_n 0. The set of asymptotic fixed points of T will be denoted by FT . A hemi-relatively nonexpansive mappingT fromCinto itself is called relatively nonexpansive ifFT FT; see14–16for more details.

On the other hand, Matsushita and Takahashi17introduced the following iteration.

A sequence{xn},defined by

x_n1 ΠCJ⁻¹αnJx_n 1−α_nJTxn, 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}^m_i1by the following iteration:

x_n1 ΠCJ⁻¹ _m

i1

wn,iαn,iJxn 1−αn,iJTixn

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

whereαn,i⊂0,1andwn,i⊂0,1with_m

i1wn,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, C_n

z∈C:φ z, y_n

≤φz, xn , Qn{z∈C: xn−z, Jx−Jxn ≥0}, x_n1 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, z_nJ⁻¹

β_nJx_n 1−β_n

JTx_n , Cn

v∈C:φ v, yn

≤αnφv, xn 1−αnφv, zn , Q_n{v∈C: Jx0−Jx_n, x_n−v ≥0},

x_n1 ΠCn∩Qnx₀

1.12

converges strongly toΠ_FTx₀.

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

x₀∈C, chosen arbitrarily, ynJ⁻¹αnJx0 1−αnJTxn, C_n

v∈C:φ v, y_n

≤α_nφv, x0 1−α_nφv, xn , Qn{v∈C: Jx0−Jxn, xn−v ≥0},

x_n1 ΠCn∩Qnx₀,

1.13

converges strongly toΠ_FTx₀.

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

x₀∈C is arbitrary, C₋₁Q₋₁C,

y_nJ⁻¹αnJx_n 1−α_nJTnx_n, Cn

v∈Cn:φ v, yn

≤φv, xn , Q_n{v∈C: Jx0−Jx_n, x_n−v ≥0}, x_n1 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.

In this paper, motivated by Qin and Su15, Nilsrakoo and Saejung19, we consider the modified Ishikawa iterative 1.12 and Halpern iterative processes 1.13, which is different from those of1.12–1.14, for countable hemi-relatively nonexpansive mappings.

By using the shrinking projection method, some strong convergence theorems in a uniformly convex and uniformly smooth Banach space are provided. Our results extend and improve the recent results by Nilsrakoo and Saejung’s result 19, Qin and Su 15, Su et al. 12, Takahashi et al.’s theorem11, and many others.

2. Preliminaries

In this section, we will recall some basic concepts and useful well-known results.

A Banach spaceEis said to be strictly convex if xy

2

<1, 2.1

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

nlim→ ∞x_ny_n2, 2.2

lim_{n→ ∞}xn−y_n0 holds.

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

limt→0

xty− x

t 2.3

exists for eachx, y∈U.It is said to be uniformly smooth if the limit is attained uniformly for x, y ∈E. In this case, the norm ofEis said to be Gˆateaux differentiable. The spaceEis said to have uniformly Gˆateaux differentiable if for eachy∈U, the limit2.3is attained uniformly for y∈U. The norm ofEis said to be uniformly 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 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 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 spaceHandP_C :H → Cis the metric projection, thenPCis nonexpansive. Alber22has recently introduced a generalized projection operatorΠCin a Banach spaceEwhich is an analogue representation of the metric projection in Hilbert spaces.

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 P_C. It is obvious from the definition of the functionφthat

y− x₂

≤φ y, x

≤yx₂

, ∀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 functiong : 0,2r → 0,∞such thatg0 0 and

gx−y≤φ x, y

, 2.6

for allx, y∈B_r {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 → 0 and either{xn}or{yn}is bounded, thenxn−y_n → 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

φ

y,ΠCx

φΠCx, x≤φ y, 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 C into itself. Then F(T) is closed and convex.

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{T_n1z−T_nz: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

and lim 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

and lim 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

n→ ∞Tnx, ∀x∈C. 2.13

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 that lim sup_{n→ ∞}α_n<1 and lim_{n→ ∞}β_n1 and let a sequence{xn}inCby the following algorithm be:

x₀∈C, chosen arbitrarity, C₀C, ynJ⁻¹αnJxn 1−αnJTnzn, z_nJ⁻¹

β_nJx_n 1−β_n

JT_nx_n , C_n1

v∈Cn:φ v, yn

≤φv, xn , x_n1 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. LetT be the mapping fromC into itself defined byTv lim_{n→ ∞}T_nvfor allv ∈ Cand suppose that T is closed and FT _∞

n0FTn. If Tn is uniformly continuous for all n ∈ N, then {xn} converges strongly to Π_FTx₀, where Π_FT is the generalized projection from C onto FT.

Proof. We first show that C_n1 is closed and convex for each n ≥ 0. Obviously, from the definition ofC_n1, we see thatC_n1is closed for eachn≥0. Now we show thatC_n1is convex for anyn≥0. Since

φ v, yn

≤φv, xn⇐⇒2

v, Jxn−Jyn yn²− xn²≤0, 3.2 this implies thatC_n1is a convex set. Next, we show that_∞

n0FTn⊂C_nfor alln≥0. Indeed,

letp∈_∞

n0FTn, we have φ

p, y_n φ

p, J⁻¹αnJx_n 1−α_nJTnz_n p²−2

p, αnJxn 1−αnJTnzn αnJxn 1−αnJTnzn²

≤p²−2α_n

p, Jx_n −21−α_n

p, JT_nz_n α_nx_n² 1−α_nT_nz_n² αnp²−2

p, Jxn xn²

1−αnp²−2

p, JTnzn Tnzn²

≤αnφ p, xn

1−αnφ

p, Tnzn

≤αnφ p, xn

1−αnφ p, zn

,

3.3

φ p, z_n

φ p, J⁻¹

β_nJx_n 1−β_n

JT_nx_n p²−2

p, βnJxn 1−βn

JTnxn βnJxn 1−βn

JTnxn² p²−2βn

p, Jxn −2

1−βn

p, JTnxn βnxn² 1−βn

Tnxn² βnp²−2

p, Jxn xn²

1−βnp²−2

p, JTnxn Tnxn²

≤βnφ p, xn

1−βn

φ

p, Tnxn

≤βnφ p, xn

1−βn

φ p, xn

≤φ p, x_n

.

3.4

Substituting3.4into3.3, we have φ

p, yn

≤ φ p, xn

. 3.5

This means that,p ∈ C_n1 for alln ≥ 0. Consequently, the sequence{xn} is well defined.

Moreover, sincexn ΠCnx₀andx_n1∈C_n1⊂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

−φ

p,ΠCnx0

≤φ p, x0

, 3.7

for allp∈_∞

n0FTn⊂Cn. Thus,{φxn, x0}is a bounded sequence. Moreover, by2.5, we know that{xn}is bounded. So, lim_{n→ ∞}φxn, x₀exists. Again, byLemma 2.5, we have

φxn1, xn φxn1,ΠCnx0

≤φx_n1, 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−x_n≤φxm, x_n≤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 lim_{n→ ∞}β_n 1 and{xn}is bounded, we obtain

φxn1, zn φ

x_n1, J⁻¹

βnJxn 1−βn

JTnxn xn1²−2

x_n1, βnJxn 1−βn

JTnxn βnJxn 1−βn

JTnxn²

≤ x_n1²−2βn xn1, Jxn −2 1−βn

xn1, JTnxnβnxn² 1−βn

Tnxn² β_nφx_n1, x_n

1−β_n

φx_n1, T_nx_n.

3.10 Thereforeφx_n1, z_n → 0 asn → ∞.

Sincex_n1 ΠCn1x₀∈C_n1, from the definition ofC_n, we have φ

x_n1, y_n

≤φx_n1, x_n, 3.11

for alln≥0. Thus

φ

x_n1, yn

−→0, asn−→ ∞. 3.12

By usingLemma 2.3, we also have

nlim→ ∞x_n1−yn lim

n→ ∞xn1−xn lim

n→ ∞xn1−zn0. 3.13

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

nlim→ ∞Jx_n1−Jyn lim

n→ ∞Jx_n1−Jx_n lim

n→ ∞Jx_n1−Jz_n0. 3.14

For eachn∈N∪ {0},we observe that

Jx_n1−Jy_nJx_n1−α_nJx_n 1−α_nJT_nz_n

αnJx_n1−Jx_n 1−α_nJx_n1−JT_nz_n 1−αnJxn1−JTnzn−αnJxn−Jx_n1

≥1−α_nJx_n1−JT_nz_n −α_nJxn−Jx_n1.

3.15

It follows that

Jx_n1−JT_nz_n ≤ 1

1−α_nJx_n1−Jy_nα_nJxn−Jx_n1

. 3.16

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

nlim→ ∞Jx_n1−JT_nz_n0. 3.17

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

nlim→ ∞x_n1−T_nz_n0. 3.18

By3.13, we have

zn−xn ≤ zn−x_n1xn1−xn −→0, asn−→ ∞. 3.19

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

xn−T_nx_n ≤ xn−x_n1x_n1−T_nz_nTnz_n−T_nx_n −→0, 3.20

asn → ∞, and so

n→ ∞limJxn−JT_nx_n0. 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−JTx_n ≤ Jxn−JT_nx_nJTnx_n−JTx_n

≤ Jxn−JTnxnsup{JTnz−JTz:z∈ {xn}} −→0. 3.22

Case 2. {Tn},{xn}satisfies condition AKTT. ApplyLemma 2.7to get xn−Tx_n ≤ xn−T_nx_nTnx_n−Tx_n

≤ xn−Tnxnsup{Tnz−Tz:z∈ {xn}} −→0. 3.23

Hence

nlim→ ∞xn−Txn lim

n→ ∞

J⁻¹Jxn−J⁻¹JTxn0. 3.24

Therefore, from the both two cases, we have

nlim→ ∞xn−Txn0. 3.25

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

p, x₀ lim

n→ ∞φxn, x₀≤φ p, x₀

, 3.26

for allp∈FT. Therefore,p Π_FTx₀.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 that lim sup_{n→ ∞}α_n <1 and lim_{n→ ∞}β_n 1 and let a sequence{xn}in Cbe defined by the following algorithm:

x₀∈C, chosen arbitrarity, C₀C, y_nJ⁻¹αnJx_n 1−α_nJTnz_n, z_nJ⁻¹

β_nJx_n 1−β_n

JT_nx_n , C_n1

v∈C_n:φ v, y_n

≤φv, xn , x_n1 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 Π_FTx₀, whereΠ_FTis the generalized projection fromContoFT.

Theorem 3.3. 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{αn}^∞_n0is a sequence in 0,1such that lim sup_{n→ ∞}αn<1 and let a sequence{xn}inCbe defined by the following algorithm:

x₀∈C, chosen arbitrarity, C₀C, y_nJ⁻¹αnJx_n 1−α_nJTnx_n, C_n1

v∈C_n:φ v, y_n

≤φv,x_n , x_n1 ΠCn1x₀,

3.28

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. LetT be the mapping fromCinto itself defined byTv lim_{n→ ∞}T_nvfor 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.

Corollary 3.4. 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 fromCinto itself such that_∞

n0FTnis nonempty. Assume that{αn}^∞_n0 is a sequence in 0,1 such that lim sup_{n→ ∞}α_n < 1 and let a sequence {xn} in Cbe defined by the following algorithm:

x0∈C, chosen arbitrarity, C0C, ynJ⁻¹αnJxn 1−αnJTnxn, C_n1

v∈Cn:φ v, yn

≤φv, xn , x_n1 ΠC_n1x₀,

3.29

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. LetT be the mapping fromCinto itself defined byTv limn→ ∞Tnvfor allv∈ Cand suppose thatT is closed andFT _∞

n0FTn. Then{xn}converges strongly toΠ_FTx₀.

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.

Corollary 3.5. Let E be a uniformly convex and uniformly smooth Banach space and let Cbe a nonempty bounded closed convex subset ofE. LetT :C → Cbe a closed hemi-relatively nonexpansive mapping such that FT/∅. Assume that {αn}^∞_n0 and {βn}^∞_n0 are sequences in 0,1 such that

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

x₀∈C, chosen arbitrarity, C₀C, ynJ⁻¹αnJxn 1−αnJTzn, znJ⁻¹

βnJxn 1−βn

JTxn , C_n1

v∈Cn:φ v, yn

≤φv, xn , x_n1 ΠCn1x₀,

3.30

forn∈N∪{0}, whereJis the single-valued duality mapping onE. IfTis uniformly continuous, then {xn}converges strongly toΠ_FTx₀.

Corollary 3.6. Let E be a uniformly convex and uniformly smooth Banach space and let Cbe a 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 < 1 and limn→ ∞βn 1 and let a sequence{xn}in Cbe defined by the following algorithm:

x₀∈C, chosen arbitrarity, C₀C, ynJ⁻¹αnJxn 1−αnJTzn, z_nJ⁻¹

β_nJx_n 1−β_n

JTx_n , C_n1

v∈C_n:φ v, y_n

≤φv, xn , x_n1 ΠCn1x₀,

3.31

forn∈N∪{0}, whereJis the single-valued duality mapping onE. IfTis uniformly continuous, then {xn}converges strongly toΠ_FTx₀.

Proof. Since a closed relatively nonexpansive mapping is a closed hemi-relatively one, Corollary 3.6is implied byCorollary 3.5.

Corollary 3.7. Let E be a uniformly convex and uniformly smooth Banach space and let Cbe a 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→ ∞}α_n <1 and let a sequence{xn}inCbe defined by the following algorithm:

x₀∈C, chosen arbitrarity, C₀C, y_nJ⁻¹αnJx_n 1−α_nJTxn, C_n1

v∈C_n:φ v, y_n

≤φv, xn , x_n1 ΠCn1x₀,

3.32

forn∈N∪ {0}, whereJis the single-valued duality mapping onE. Then{xn}converges strongly to Π_FTx₀.

Corollary 3.8. Let E be a uniformly convex and uniformly smooth Banach space and let Cbe a 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 lim sup_{n→ ∞}αn <1 and let a sequence{xn}inCbe defined by the following algorithm:

x₀∈C, chosen arbitrarity, C₀C, ynJ⁻¹αnJxn 1−αnJTxn, C_n1

v∈C_n:φ v, y_n

≤φv, xn , x_n1 ΠC_n1x₀,

3.33

forn∈N∪ {0}, whereJis the single-valued duality mapping onE. Then{xn}converges strongly to Π_FTx₀.

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

Theorem 3.9. 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{αn}^∞_n0and{βn}^∞_n0are sequences in0,1such that lim sup_{n→ ∞}α_n <1 and lim_{n→ ∞}β_n <1 and let a sequence{xn}inCbe defined by the following algorithm:

x0∈C, chosen arbitrarity, C0C, y_nJ⁻¹αnJx_n 1−α_nJTnz_n, znJ⁻¹

βnJxn 1−βn

JTnxn , C_n

v∈C:φ v, y_n

≤φv, xn , Qn{v∈C: v−xn, Jxn−Jx₀ ≥0}, x_n1 ΠCn∩Qnx0,

3.34

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→ ∞T_nv for allv∈Cand suppose thatTis closed

andFT _∞

n0FTn. IfT_nis 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 0,1such that lim sup_{n→ ∞}α_n <1 and lim sup_{n→ ∞}β_n1 and let a sequence{xn}inCbe defined

by the following algorithm:

x₀∈C, chosen arbitrarity, C₀C, ynJ⁻¹αnJxn 1−αnJTzn, z_nJ⁻¹

β_nJx_n 1−β_n

JTx_n , Cn

v∈C:φ v, yn

≤φv, xn , Q_n{v∈C: v−x_n, Jx_n−Jx₀ ≥0}, x_n1 ΠCn∩Qnx0,

3.35

forn∈N∪{0}, whereJis the single-valued duality mapping onE. IfTis uniformly continuous, then {xn}converges strongly toΠ_FTx₀.

Theorem 3.11. LetE be a uniformly convex and uniformly smooth Banach space and let C be a nonempty bounded closed convex subset of E. Let {Tn} be a sequence of relatively nonexpansive mappings fromCinto itself such that_∞

n0FTnis a nonempty. Assume that{αn}^∞_n0is a sequence in 0,1such that lim sup_{n→ ∞}αn<1 and let a sequence{xn}inCbe defined by the following algorithm:

x₀∈C, chosen arbitrarity, C₀C, ynJ⁻¹αnJxn 1−αnJTnxn, C_n

v∈C:φ v, y_n

≤φv, xn , Qn{v∈C: v−xn, Jxn−Jx₀ ≥0}, x_n1 ΠCn∩Qnx0,

3.36

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. LetT be the mapping fromCinto itself defined byTv limn→ ∞T_nvfor allv∈ Cand suppose thatT is closed andFT _∞

n0FTn. Then{xn}converges strongly toΠ_FTx₀.

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 mappings fromCinto itself such thatFT/∅. Assume that{αn}^∞_n0is a sequence in0,1such that