Quasi-I-Nonexpansive Mapping in Banach Space

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Volume 2010, Article ID 719631,13pages doi:10.1155/2010/719631

Research Article

Weak and Strong Convergence of an Implicit Iteration Process for an Asymptotically

Quasi-I-Nonexpansive Mapping in Banach Space

Farrukh Mukhamedov and Mansoor Saburov

Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box 141, 25710 Kuantan, Malaysia

Correspondence should be addressed to Farrukh Mukhamedov,[email protected] Received 31 August 2009; Accepted 6 December 2009

Academic Editor: Mohamed A. Khamsi

Copyrightq2010 F. Mukhamedov and M. Saburov. 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.

We prove the weak and strong convergence of the implicit iterative process to a common fixed point of an asymptotically quasi-I-nonexpansive mappingT and an asymptotically quasi- nonexpansive mappingI, defined on a nonempty closed convex subset of a Banach space.

1. Introduction

LetKbe a nonempty subset of a real normed linear spaceXand letT :K → Kbe a mapping.

Denote byFTthe set of fixed points ofT, that is,FT {x∈K:Txx}. Throughout this paper, we always assume thatFT/∅. Now let us recall some known definitions.

Definition 1.1. A mappingT :K → Kis said to be

inonexpansive, ifTx−Ty ≤ x−yfor allx, y∈K;

iiasymptotically nonexpansive, if there exists a sequence {λn} ⊂ 1,∞ with lim_n_{→ ∞}λn1 such thatTⁿx−Tⁿy ≤λnx−yfor allx, y∈Kandn∈N;

iiiquasi-nonexpansive, ifTx−p ≤ x−pfor allx∈K, p∈FT;

ivasymptotically quasi-nonexpansive, if there exists a sequence {μn} ⊂ 1,∞with lim_n_{→ ∞}μn1 such thatTⁿx−p ≤μnx−pfor allx∈K, p∈FTandn∈N.

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Note that from the above definitions, it follows that a nonexpansive mapping must be asymptotically nonexpansive, and an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive, but the converse does not holdsee1.

IfKis a closed nonempty subset of a Banach space andT :K → Kis nonexpansive, then it is known thatTmay not have a fixed pointunlike the case ifTis a strict contraction, and even when it has, the sequence{xn}defined byx_n1Txnthe so-called Picard sequence may fail to converge to such a fixed point.

In2,3Browder studied the iterative construction for fixed points of nonexpansive mappings on closed and convex subsets of a Hilbert space. Note that for the past 30 years or so, the studies of the iterative processes for the approximation of fixed points of nonexpansive mappings and fixed points of some of their generalizations have been flourishing areas of research for many mathematicianssee for more details1,4.

In 5 Diaz and Metcalf studied quasi-nonexpansive mappings in Banach spaces.

Ghosh and Debnath6established a necessary and suﬃcient condition for convergence of the Ishikawa iterates of a quasi-nonexpansive mapping on a closed convex subset of a Banach space. The iterative approximation problems for nonexpansive mapping, asymptotically nonexpansive mapping and asymptotically quasi-nonexpansive mapping were studied extensively by Goebel and Kirk7, Liu8, Wittmann9, Reich10, Gornicki11, Schu 12Shioji and Takahashi 13, and Tan and Xu14 in the settings of Hilbert spaces and uniformly convex Banach spaces.

There are many methods for approximating fixed points of a nonexpansive mapping.

Xu and Ori15introduced implicit iteration process to approximate a common fixed point of a finite family of nonexpansive mappings in a Hilbert space. Recently, Sun16has extended an implicit iteration process for a finite family of nonexpansive mappings, due to Xu and Ori, to the case of asymptotically quasi-nonexpansive mappings in a setting of Banach spaces.

In17 it has been studied the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces, which extends and improves the mentioned paperssee also18,19for applications and other methods of implicit iteration processes.

There are many concepts which generalize a notion of nonexpansive mapping. One of such concepts isI-nonexpansivity of a mappingT20. Let us recall some notions.

Definition 1.2. LetT :K → K,I:K → Kbe two mappings of a nonempty subsetKof a real normed linear spaceX. ThenTis said to be

iI-nonexpansive, ifTx−Ty ≤ Ix−Iyfor allx, y∈K;

iiasymptotically I-nonexpansive, if there exists a sequence {λn} ⊂ 1,∞ with lim_n_{→ ∞}λn1 such thatTⁿx−Tⁿy ≤λnIⁿx−Iⁿyfor allx, y∈Kandn≥1;

iiiasymptotically quasi I-nonexpansive mapping, if there exists a sequence{μn} ⊂ 1,∞with limn→ ∞μn 1 such thatTⁿx−p ≤ μnIⁿx−pfor all x ∈ K, p ∈ FT∩FIandn≥1.

Remark 1.3. IfFT∩FI/∅then an asymptoticallyI-nonexpansive mapping is asymptotically quasi-I-nonexpansive. But, there exists a nonlinear continuous asymptotically quasi I-nonexpansive mappings which is asymptoticallyI-nonexpansive.

In 21 a weakly convergence theorem for I-asymptotically quasi-nonexpansive mapping defined in Hilbert space was proved. In 22 strong convergence of Mann iterations of I-nonexpansive mapping has been proved. Best approximation properties of

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I-nonexpansive mappings were investigated in20. In23the weak convergence of three- step Noor iterative scheme for an I-nonexpansive mapping in a Banach space has been established. Recently, in24the weak and strong convergence of implicit iteration process to a common fixed point of a finite family ofI-asymptotically nonexpansive mappings were studied. Assume that the family consists of oneI-asymptotically nonexpansive mappingT. Now let us consider an iteration method used in24, forT, which is defined by

x1∈K,

xn1 1−αnxnαnIⁿyn, yn

1−βn

xnβnTⁿxn.

n≥1, 1.1

where{αn}and{βn}are two sequences in0,1. From this formula one can easily see that the employed method, indeed, is not implicit iterative processes. The used process is some kind of modified Ishikawa iteration.

Therefore, in this paper we will extend of the implicit iterative process, defined in16, to I-asymptotically quasi-nonexpansive mapping defined on a uniformly convex Banach space. Namely, letKbe a nonempty convex subset of a real Banach spaceXandT :K → K be an asymptotically quasiI-nonexpansive mapping, and letI:K → Kbe an asymptotically quasi-nonexpansive mapping. Then for given two sequences{αn}and{βn}in0,1we will consider the following iteration scheme:

x0∈K,

xn 1−αnx_n−1αnTⁿyn, yn

1−βn

xnβnIⁿxn.

n≥1, 1.2

In this paper we will prove the weak and strong convergences of the implicit iterative process1.2to a common fixed point ofT andI. All results presented here generalize and extend the corresponding main results of15–17in a case of one mapping.

2. Preliminaries

Throughout this paper, we always assume thatXis a real Banach space. We denote byFT and DTthe set of fixed points and the domain of a mapping T, respectively. Recall that a Banach space X is said to satisfy Opial condition 25, if for each sequence{xn} in X, xn

converging weakly toximplies that lim inf

n→ ∞ xn−x<lim inf

n→ ∞ xn−y. 2.1

for ally∈Xwithy /x.It is well known thatsee26inequality2.1is equivalent to lim sup

n→ ∞ xn−x<lim sup

n→ ∞

xn−y. 2.2

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Definition 2.1. LetK be a closed subset of a real Banach space X and letT : K → K be a mapping.

iA mapping T is said to be semicloseddemiclosed at zero, if for each bounded sequence {xn} in K, the conditions xn converges weakly to x ∈ K and Txn

converges strongly to 0 implyTx0.

iiA mappingT is said to be semicompact, if for any bounded sequence{xn} inK such thatxn−Txn → 0, n → ∞,then there exists a subsequence{xnk} ⊂ {xn} such thatxnk → x^∗∈Kstrongly.

iiiTis called a uniformlyL-Lipschitzian mapping, if there exists a constantL >0 such thatTⁿx−Tⁿy ≤Lx−yfor allx, y∈Kandn≥1.

The following lemmas play an important role in proving our main results.

Lemma 2.2see12. LetXbe a uniformly convex Banach space and letb, cbe two constants with 0 < b < c <1.Suppose that{tn}is a sequence inb, cand{xn}and{yn}are two sequences inX such that

nlim→ ∞tnxn 1−tnynd, lim sup

n→ ∞ xn ≤d, lim sup

n→ ∞

yn≤d, 2.3

holds somed≤0.Then lim_{n→ ∞}xn−yn0.

Lemma 2.3 see 14. Let {an} and {bn} be two sequences of nonnegative real numbers with _∞

n1bn<∞.If one of the following conditions is satisfied:

ian1≤anbn, n≥1, iia_n1≤1bnan, n≥1, then the limit lim_n_{→ ∞}anexists.

3. Main Results

In this section we will prove our main results. To formulate one, we need some auxiliary results.

Lemma 3.1. LetX be a real Banach space and letKbe a nonempty closed convex subset ofX.Let T :K → Kbe an asymptotically quasiI-nonexpansive mapping with a sequence{λn} ⊂1,∞and I :K → Kbe an asymptotically quasi-nonexpansive mapping with a sequence{μn} ⊂1,∞such thatFFT∩FI/∅.SupposeA^∗sup_nαn, Λ sup_nλn≥1, Msup_nμn≥1 and{αn}and {βn}are two sequences in0,1which satisfy the following conditions:

i_∞

n1λnμn−1αn<∞, iiA^∗<1/Λ²M².

If{xn}is the implicit iterative sequence defined by1.2, then for eachp∈FFT∩FIthe limit lim_n_{→ ∞}xn−pexists.

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Proof. SinceFFT∩FI/∅,for any givenp∈F,it follows from1.2that xn−p1−αn

xn−1−p αn

Tⁿyn−p

≤1−αnxn−1−pαnTⁿyn−p

≤1−αnxn−1−pαnλnIⁿyn−p

≤1−αnxn−1−pαnλnμnyn−p.

3.1

Again from1.2we derive that

yn−p1−βn

xn−p βn

Iⁿxn−p

≤

1−βnxn−pβnμnxn−p

≤ 1−βn

μnxn−pβnμnIⁿxn−p

≤μnxn−p,

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which means

yn−p≤μnxn−p≤λnμnxn−p. 3.3 Then from3.3one finds

xn−p≤1−αnxn−1−pαnλ²_nμ²_nxn−p, 3.4 and so

1−αnλ²_nμ²_nxn−p≤1−αnx_n−1−p. 3.5

By conditioniiwe haveαnλ²_nμ²_n≤A^∗Λ²M²<1,and therefore

1−αnλ²_nμ²_n≥1−A^∗Λ²M²>0. 3.6 Hence from3.5we obtain

xn−p≤ 1−αn

1−αnλ²nμ²n

x_n−1−p

1

λ²_nμ²_n−1 αn

1−αnλ²nμ²n

xn−1−p

≤

1

1−A^∗Λ²M² xn−1−p.

3.7

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By puttingbn λ²_nμ²_n−1αn/1−A^∗Λ²M²the last inequality can be rewritten as follows:

xn−p≤1bnxn−1−p. 3.8

From conditioniwe find ∞ n1

bn 1

1−A^∗Λ²M² ∞ n1

1 1−A^∗Λ²M²

∞ n1

λnμn−1

λnμn1 αn

≤ ΛM1 1−A^∗Λ²M²

∞ n1

λnμn−1

αn<∞.

3.9

Denotinganx_n−1−pin3.8one gets

an1≤1bnan, 3.10

andLemma 2.3implies the existence of the limit lim_n_{→ ∞}an. This means the limit

nlim→ ∞xn−pd 3.11

exists, whered≥0 is a constant. This completes the proof.

Now we prove the following result.

Theorem 3.2. LetXbe a real Banach space and letKbe a nonempty closed convex subset ofX.Let T : K → K be a uniformlyL1-Lipschitzian asymptotically quasi-I-nonexpansive mapping with a sequence{λn} ⊂ 1,∞and let I : K → K be a uniformlyL2-Lipschitzian asymptotically quasi- nonexpansive mapping with a sequence {μn} ⊂ 1,∞such that F FT∩FI/∅. Suppose A^∗ sup_nαn, Λ sup_nλn ≥1, M sup_nμn ≥1,and{αn}and{βn}are two sequences in0,1 which satisfy the following conditions:

i_∞

n1λnμn−1αn<∞, iiA^∗<1/Λ²M².

Then the implicitly iterative sequence {xn}defined by1.2converges strongly to a common fixed point inF FT∩FI/∅if and only if

lim inf

n→ ∞ dxn, F 0. 3.12

Proof. The necessity of condition 3.12 is obvious. Let us proof the suﬃciency part of theorem.

SinceT, I:K → Kare uniformlyL-Lipschitzian mappings, soTandIare continuous mappings. Therefore the setsFTandFIare closed. HenceF FT∩FIis a nonempty closed set.

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For any givenp∈F,we havesee3.8

xn−p≤1bnx_n−1−p, 3.13 here as beforebn λ²_nμ²_n−1αn/1−A^∗Λ²M²with_∞

n1bn<∞.Hence, one finds

dxn, F≤1bndxn−1, F. 3.14

From 3.14 due to Lemma 2.3 we obtain the existence of the limit limn→ ∞dxn, F. By condition3.12, one gets

nlim→ ∞dxn, F lim inf

n→ ∞ dxn, F 0. 3.15

Let us prove that the sequence{xn}converges to a common fixed point ofTandI.In fact, due to 1t≤exptfor allt >0,and from3.13, we obtain

xn−p≤expbnx_n−1−p. 3.16 Hence, for any positive integersm, n,from3.16with_∞

n1bn<∞we find xnm−p≤expbnmxnm−1−p

≤expbnmbnm−1xnm−2−p

≤ · · ·

≤exp _nm

in1

bi xn−p

≤exp _∞

i1

bi xn−p,

3.17

which means that

x_nm−p≤Wxn−p 3.18 for allp∈F, whereW exp_∞

i1bi<∞.

Since limn→ ∞dxn, F 0, then for any givenε > 0, there exists a positive integer numbern0such that

dxn0, F< ε

W. 3.19

Therefore there existsp1∈Fsuch that

xn0−p1< ε

W. 3.20

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Consequently, for alln≥n0from3.18we derive

xn−p1≤Wxn0−p1

< W· ε W ε,

3.21

which means that the strong convergence of the sequence{xn}is a common fixed pointp1of TandI.This proves the required assertion.

We need one more auxiliary result.

Proposition 3.3. LetX be a real uniformly convex Banach space and letK be a nonempty closed convex subset of X. Let T : K → K be a uniformly L1-Lipschitzian asymptotically quasi-I- nonexpansive mapping with a sequence {λn} ⊂ 1,∞ and let I : K → K be a uniformly L2- Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence{μn} ⊂ 1,∞such that FFT∩FI/∅.SupposeA∗infnαn, A^∗sup_nαn, Λ sup_nλn≥1, Msup_nμn≥1 and {αn}and{βn}are two sequences in0,1which satisfy the following conditions:

i_∞

n1λnμn−1αn<∞, ii0< A∗≤A^∗<1/Λ²M²,

iii0< B∗inf_nβn≤sup_nβnB^∗<1.

Then the implicitly iterative sequence{xn}defined by1.2satisfies the following:

nlim→ ∞xn−Txn0, lim

n→ ∞xn−Ixn0. 3.22

Proof. First, we will prove that

nlim→ ∞xn−Tⁿxn0, lim

n→ ∞xn−Iⁿxn0. 3.23

According toLemma 3.1for anyp∈FFT∩FIwe have limn→ ∞xn−pd. It follows from1.2that

xn−p1−αn

xn−1−p αn

Tⁿyn−p−→d, n−→ ∞. 3.24 By means of asymptotically quasi-I-nonexpansivity of T and asymptotically quasi- nonexpansivity ofIfrom3.3we get

lim sup

n→ ∞

Tⁿyn−p≤lim sup

n→ ∞ λnμnyn−p≤lim sup

n→ ∞ λ²_nμ²_nxn−pd. 3.25 Now using

lim sup

n→ ∞

xn−1−pd 3.26

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with3.25and applyingLemma 2.2to3.24one finds

nlim→ ∞x_n−1−Tⁿyn0. 3.27

Now from1.2and3.27we infer that

nlim→ ∞xn−xn−1 lim

n→ ∞αn

Tⁿyn−xn−10. 3.28

On the other hand, we have

x_n−1−p≤x_n−1−TⁿynTⁿyn−p

≤xn−1−Tⁿynλnμnyn−p, 3.29

which implies

x_n−1−p−x_n−1−Tⁿyn≤λnμnyn−p. 3.30

The last inequality with3.3yields that

xn−1−p−xn−1−Tⁿyn≤λnμnyn−p≤λ²_nμ²xn−p. 3.31

Then3.27and3.24with the Squeeze theorem imply that

nlim→ ∞yn−pd. 3.32

Again from1.2we can see that yn−p1−βn

xn−p βn

Iⁿxn−p−→d, n−→ ∞. 3.33

From3.11one finds

lim sup

n→ ∞

Iⁿxn−p≤lim sup

n→ ∞ μnxn−pd. 3.34

Now applyingLemma 2.2to3.33we obtain

nlim→ ∞xn−Iⁿxn0. 3.35

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Consider

xn−Tⁿxn ≤ xn−xn−1xn−1−TⁿynTⁿyn−Tⁿxn

≤ xn−x_n−1x_n−1−TⁿynL1yn−xn xn−xn−1xn−1−TⁿynL1βnIⁿxn−xn xn−xn−1xn−1−TⁿynL1βnIⁿxn−xn.

3.36

Then from3.27,3.28, and3.35we get

nlim→ ∞xn−Tⁿxn0. 3.37

Finally, from

xn−Txn ≤ xn−TⁿxnTⁿxn−Txn

≤ xn−TⁿxnL1Tⁿ⁻¹xn−xn

≤ xn−TⁿxnL1Tⁿ⁻¹xn−Tⁿ⁻¹xn−1

Tⁿ⁻¹xn−1−xn−1xn−1−xn

≤ xn−TⁿxnL1

L1xn−xn−1

Tⁿ⁻¹xn−1−xn−1xn−1−xn

≤ xn−TⁿxnL1L11xn−x_n−1L1Tⁿ⁻¹x_n−1−x_n−1

3.38

with3.28and3.37we obtain

nlim→ ∞xn−Txn0. 3.39

Analogously, one has

xn−Ixn ≤ xn−IⁿxnL2L21xn−xn−1L2Iⁿ⁻¹xn−1−xn−1, 3.40

which with3.28and3.35implies

nlim→ ∞xn−Ixn0. 3.41

Now we are ready to formulate one of main results concerning weak convergence of the sequence{xn}.

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Theorem 3.4. Let X be a real uniformly convex Banach space satisfying Opial condition and let K be a nonempty closed convex subset of X. Let E : X → X be an identity mapping, let T : K → K be a uniformlyL1-Lipschitzian asymptotically quasi-I-nonexpansive mapping with a sequence{λn} ⊂ 1,∞, and,I : K → K be a uniformlyL2-Lipschitzian asymptotically quasi- nonexpansive mapping with a sequence {μn} ⊂ 1,∞such that F FT∩FI/∅. Suppose A_∗ inf_nαn, A^∗ sup_nαn, Λ sup_nλn ≥ 1, M sup_nμn ≥ 1,and {αn}and {βn}are two sequences in0,1satisfying the following conditions:

i_∞

n1λnμn−1αn<∞, ii0< A_∗≤A^∗<1/Λ²M².

iii0< B∗infnβn≤sup_nβnB^∗<1.

If the mappingsE−T andE−I are semiclosed at zero, then the implicitly iterative sequence{xn} defined by1.2converges weakly to a common fixed point ofT andI.

Proof. Let p ∈ F, then according to Lemma 3.1 the sequence {xn −p} converges. This provides that{xn}is a bounded sequence. SinceXis uniformly convex, then every bounded subset ofX is weakly compact. Since{xn}is a bounded sequence in K,then there exists a subsequence{xnk} ⊂ {xn}such that{xnk}converges weakly toq∈K.Hence from3.39and 3.41it follows that

nlimk→ ∞xnk−Txnk0, lim

nk→ ∞xnk−Ixnk0. 3.42 Since the mappingsE−T and E−I are semiclosed at zero, therefore, we find Tq qand Iqq,which meansq∈FFT∩FI.

Finally, let us prove that{xn}converges weakly toq.In fact, suppose the contrary, that is, there exists some subsequence{xnj} ⊂ {xn}such that{xnj}converges weakly toq1∈Kand q1/q. Then by the same method as given above, we can also prove thatq1∈FFT∩FI.

Takingpqandpq1and using the same argument given in the proof of3.11, we can prove that the limits lim_n_{→ ∞}xn−qand lim_n_{→ ∞}xn−q1exist, and we have

nlim→ ∞xn−qd, lim

n→ ∞xn−q1d1, 3.43

where dand d1 are two nonnegative numbers. By virtue of the Opial condition ofX, one finds

dlim sup

nk→ ∞

xnk−q<lim sup

nk→ ∞

xnk−q1d1

lim sup

nj→ ∞

xnj−q1<lim sup

nj→ ∞

xnj−qd.

3.44

This is a contradiction. Hence q1 q.This implies that {xn}converges weakly to q.This completes the proof ofTheorem 3.4.

Now we formulate next result concerning strong convergence of the sequence{xn}.

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Theorem 3.5. Let X be a real uniformly convex Banach space and let K be a nonempty closed convex subset of X. Let T : K → K be a uniformly L1-Lipschitzian asymptotically quasi-I- nonexpansive mapping with a sequence {λn} ⊂ 1,∞ and I : K → K be a uniformly L2- Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence{μn} ⊂ 1,∞such that FFT∩FI/∅.SupposeA∗infnαn, A^∗sup_nαn, Λ sup_nλn≥1, Msup_nμn≥1 and {αn}and{βn}are two sequences in0,1satisfying the following conditions:

i_∞

n1λnμn−1αn<∞, ii0< A∗≤A^∗<1/Λ²M².

iii0< B_∗inf_nβn≤sup_nβnB^∗<1

If at least one mapping of the mappingsT andIis semicompact, then the implicitly iterative sequence {xn}defined by1.2converges strongly to a common fixed point ofTandI.

Proof. Without any loss of generality, we may assume thatTis semicompact. This with3.39 means that there exists a subsequence{xnk} ⊂ {xn}such thatxnk → x^∗strongly andx^∗ ∈K.

SinceT, Iare continuous, then from3.39and3.41we find x^∗−Tx^∗ lim

nk→ ∞xnk −Txnk0, x^∗−Ix^∗ lim

nk→ ∞xnk−Ixnk0. 3.45 This shows thatx^∗ ∈ F FT∩FI.According toLemma 3.1the limit limn→ ∞xn−x^∗ exists. Then

nlim→ ∞xn−x^∗ lim

nk→ ∞xnk−x^∗0, 3.46

which means that{xn}converges tox^∗∈F.This completes the proof.

Note that all results presented here generalize and extend the corresponding main results of15–17in a case of one mapping.

Acknowledgment

The authors acknowledge the MOSTI Grant 01-01-08-SF0079.

References

1 K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.

2 F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965.

3 F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967.

4 Ch. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009.

5 J. B. Diaz and F. T. Metcalf, “On the structure of the set of subsequential limit points of successive approximations,” Bulletin of the American Mathematical Society, vol. 73, pp. 516–519, 1967.

6 M. K. Ghosh and L. Debnath, “Convergence of Ishikawa iterates of quasi-nonexpansive mappings,”

Journal of Mathematical Analysis and Applications, vol. 207, no. 1, pp. 96–103, 1997.

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7 K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”

Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.

8 Q. Liu, “Iterative sequences for asymptotically quasi-nonexpansive mappings,” Journal of Mathemati- cal Analysis and Applications, vol. 259, no. 1, pp. 1–7, 2001.

9 R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol.

58, no. 5, pp. 486–491, 1992.

10 S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,”

Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980.

11 J. Gornicki, “Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 30, no. 2, pp. 249–

252, 1989.

12 J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,”

Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991.

13 N. Shioji and W. Takahashi, “Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 1, pp.

87–99, 1998.

14 K.-K. Tan and H.-K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993.

15 H.-K. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 22, no. 5-6, pp. 767–773, 2001.

16 Z.-H. Sun, “Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp.

351–358, 2003.

17 F. Gu and J. Lu, “A new composite implicit iterative process for a finite family of nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2006, Article ID 82738, 11 pages, 2006.

18 H. Y. Li and H. Z. Li, “Strong convergence of an iterative method for equilibrium problems and variational inequality problems,” Fixed Point Theory and Applications, vol. 2009, Article ID 362191, 21 pages, 2009.

19 F. Zhang and Y. Su, “Strong convergence of modified implicit iteration processes for common fixed points of nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2007, Article ID 48174, 9 pages, 2007.

20 N. Shahzad, “Generalized I-nonexpansive maps and best approximations in Banach spaces,”

Demonstratio Mathematica, vol. 37, no. 3, pp. 597–600, 2004.

21 S. Temir and O. Gul, “Convergence theorem forI-asymptotically quasi-nonexpansive mapping in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 759–765, 2007.

22 B. H. Rhodes and S. Temir, “Convergebce thorems forI-nonexpansive mapping,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 63435, 4 pages, 2006.

23 P. Kumam, W. Kumethong, and N. Jewwaiworn, “Weak convergence theorems of three-step Noor iterative scheme forI-quasi-nonexpansive mappings in Banach spaces,” Applied Mathematical Sciences, vol. 2, no. 57–60, pp. 2915–2920, 2008.

24 S. Temir, “On the convergence theorems of implicit iteration process for a finite family of I- asymptotically nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 225, no. 2, pp. 398–405, 2009.

25 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.

26 E. Lami Dozo, “Multivalued nonexpansive mappings and Opial’s condition,” Proceedings of the American Mathematical Society, vol. 38, pp. 286–292, 1973.