# A Note on “Common Fixed Point of Multistep Noor Iteration with Errors for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings”

Authors: Satit Saejung; Suthep Suantai; Pongsakorn Yotkaew
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

The purpose of this paper is to give a general and short principle
for proving some convergence results of certain types of iterative sequences.
A small gap in the paper by Imnang and Suantai (2009) is discussed and corrected. Finally, we prove that the generalized asymptotically
quasi-nonexpansive mappings in the sense of Lan (2006) are nothing but asymptotically quasi-nonexpansive. Hence several results concerning these mappings become a special case of the known ones.


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## Body

## 1. Introduction

LetC be a nonempty subset of a Banach space X. A mapping T:C→C is said to be(i)
asymptotically nonexpansive [1] if there exists a sequence {kn} in [0,∞) such that kn→0 and(1.1)∥Tnx-Tny∥≤(1+kn)∥x-y∥for allx,y∈C and n≥1;(ii)
asymptotically quasi-nonexpansive [2] if F(T)={p∈C:Tp=p}≠∅ and there exists a sequence {kn} in [0,∞) such that kn→0 and(1.2)∥Tnx-p∥≤(1+kn)∥x-p∥for allx∈C, p∈F(T) and n≥1;(iii)
generalized asymptotically nonexpansive if there exist sequences {kn},{ln} in [0,∞) such that kn,ln→0 and(1.3)∥Tnx-Tny∥≤(1+kn)∥x-y∥+lnfor allx,y∈C and n≥1;(iv)
generalized asymptotically quasi-nonexpansive [3] if F(T)≠∅ and there exist sequences {kn},{ln} in [0,∞) such that kn,ln→0 and(1.4)∥Tnx-p∥≤(1+kn)∥x-p∥+lnfor allx∈C, p∈F(T) and n≥1.Many researchers have paid their attention on the approximation of a fixed point of a single mapping or a common fixed point of a family of mappings. One effective way is to use a sequence generated by an appropriate iteration. In this paper, we propose a general and short principle for proving some convergence results of certain types of iterative sequences. We also discuss and correct a small gap in the recent paper by Imnang and Suantai [4]. In the last section, we give a remark on the generalized asymptotically quasi-nonexpansive mapping in the sense of Lan [5].Let{Ti}i=1N be a finite family of self-mappings of a closed convex subset C of X. The sequence {xn} is generated from x1∈C, and(1.5)y1n=α1nT1nxn+β1nxn+γ1nu1n,y2n=α2nT2ny1n+β2nxn+γ2nu2n,⋮y(N-1)n=α(N-1)nTN-1ny(N-2)n+β(N-1)nxn+γ(N-1)nu(N-1)n,xn+1=αNnTNny(N-1)n+βNnxn+γNnuNn,
where {u1n},{u2n},…,{uNn} are bounded sequences in C, and {αin},{βin}, and {γin} are sequences in [0,1] such that αin+βin+γin=1 for all i=1,2,…,N and n≥1.


## 2. Main Results

### 2.1. Sequences of Monotone Types (1) and (2)

Definition 2.1.
Let{xn} be a sequence in a metric space (X,d) and F a subset of X. We say that {xn} is of  (i)
monotone type (1) with respect toF [6] if there exist sequences {rn} and {sn} of nonnegative real numbers such that ∑n=1∞rn<∞, ∑n=1∞sn<∞ and(2.1)d(xn+1,p)≤(1+rn)d(xn,p)+snfor alln≥1 and p∈F;(ii)
monotone type (2) with respect toF if for each p∈F there exist sequences {rn} and {sn} of nonnegative real numbers such that ∑n=1∞rn<∞, ∑n=1∞sn<∞ and(2.2)d(xn+1,p)≤(1+rn)d(xn,p)+snfor alln≥1.Proposition 2.2.
If{xn} is of monotone type (1) with respect to F, then it is of monotone type (2) with respect to F.Lemma 2.3 ([7, Lemma 1]).
Let{an}, {bn}, and {αn} be sequences of nonnegative real numbers such that
(2.3)an+1≤(1+αn)an+bn,n≥1.
If ∑n=1∞αn<∞ and ∑n=1∞bn<∞, then limn→∞an exists.Theorem 2.4.
Let(X,d) be a complete metric space, F⊂X, and {xn} a sequence in X. Then one has the following assertions.  (a)
If{xn} is of monotone type (2) with respect to F, then limn→∞d(xn,p) exists for all p∈F.(b)
If{xn} is of monotone type (1) with respect to F, then limn→∞d(xn,F) exists.(c)
If{xn} is of monotone type (1) with respect to F and lim infn→∞d(xn,F)=0, then xn→p for some p∈X satisfying d(p,F)=0. In particular, if F is closed, then p∈F.Proof.
(a)
It is easy to see that the result follows from (2.2) and Lemma 2.3.(b)
Note that{rn} and {sn} are independent of p∈F. Taking infimum over all p∈F in (2.1) gives(2.4)d(xn+1,F)≤(1+rn)d(xn,F)+sn∀n≥1.
Again, by Lemma 2.3, we get that limn→∞d(xn,F) exists.(c)
It follows from (b) andlim infn→∞d(xn,F)=0 that(2.5)limn→∞d(xn,F)=0.
To show that {xn} is a Cauchy sequence, let ε>0. Since limn→∞d(xn,F)=0, we may assume without loss of generality that there is a sequence {pn} in F such that d(xn,pn)≤ε/4 for all n≥1. As {xn} is bounded, we put M=sup{d(xm,pn):m,n≥1}. From (2.1), we have
(2.6)d(xn+1,pk)≤d(xn,pk)+tn∀n,k≥1,
where tn≡rnM+sn. Consequently,
(2.7)d(xn+k,pn)≤d(xn,pn)+∑j=nn+k-1tj≤ε4+∑j=n∞tj∀n,k≥1.
Notice that ∑n=1∞tn<∞. So there exists N≥1 such that ∑n=N∞tn<ε/2. Then for all n≥N,k≥1, we have
(2.8)d(xn+k,xn)≤d(xn+k,pn)+d(xn,pn)<ε.
Hence, {xn} is a Cauchy sequence in X. By the completeness of X, we assume that xn→p for some p∈X. Since
(2.9)|d(xn,F)-d(p,F)|≤d(xn,p)→0,
we obtain d(p,F)=0. This completes the proof.


### 2.2. A Correction of Recent Results of Imnang and Suantai

The following observation is an auxiliary result.Proposition 2.5.
LetC be a nonempty subset of a Banach space X, and let T1,T2,…,TN:C→C be N generalized asymptotically quasi-nonexpansive mappings with F:=⋂i=1NF(Ti)≠∅. Then there exist sequences {kn},{ln} in [0,∞) such that kn,ln→0 and
(2.10)∥Tinx-p∥≤(1+kn)∥x-p∥+ln,
for all x∈C,p∈F,n≥1, and i=1,2,…,N.From now on, we assume thatN generalized asymptotically quasi-nonexpansive mappings T1,T2,…,TN:C→C are equipped with the sequences {kn},{ln} in [0,∞) as mentioned in the preceding proposition.Theorem 2.6.
LetC be a nonempty closed convex subset of a Banach space X, and {T1,T2,…,TN} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of C with the sequence {(kn,ln)} such that ∑n=1∞kn<∞ and ∑n=1∞ln<∞. Assume that F:=⋂i=1NF(Ti)≠∅ is closed, and {xn} is the sequence in C defined by (1.5) such that ∑n=1∞γin<∞ for each i=1,2,…,N. Then the sequence {xn} converges strongly to a common fixed point of the family of mappings if and only if lim infn→∞d(xn,F)=0.Remark 2.7.
There is a small gap in [4, Theorem 3.2]. More precisely, the sequence {xn} generated by (1.5) is shown in [4, Theorem 3.2]  to be of monotone type (2) with respect to F, that is, ∥xn+1-p∥≤(1+kn)N∥xn-p∥+ekn where each ekn is a nonnegative real number depending on p. Then the expression d(xn+1,F)≤(1+kn)Nd(xn,F)+ekn cannot warrant.Remark 2.8.
The same gap also appears in [8,Lemma 2.3] and [6, Theorem 3.2].Proof of Theorem2.6.
Necessity is obvious. Conversely, we show first that{xn} is of monotone type (2) with respect to F. Let p∈F. We have that
(2.11)∥y1n-p∥=∥α1nT1nxn+β1nxn+γ1nu1n-p∥≤α1n∥T1nxn-p∥+β1n∥xn-p∥+γ1n∥u1n-p∥≤(α1n+β1n)(1+kn)∥xn-p∥+α1nln+γ1n∥u1n-p∥(2.12)≤(1+kn)∥xn-p∥+l̃1n,
where l̃1n≡α1nln+γ1n∥u1n-p∥. Notice that ∑n=1∞ln<∞ and {u1n} is bounded. Then ∑n=1∞l̃1n<∞. It follows from (2.12) that
(2.13)∥y2n-p∥≤α2n∥T2ny1n-p∥+β2n∥xn-p∥+γ2n∥u2n-p∥≤α2n(1+kn)∥y1n-p∥+α2nln+β2n∥xn-p∥+γ2n∥u2n-p∥≤(α2n+β2n)(1+kn)2∥xn-p∥+α2n((1+kn)l̃1n+ln)+γ2n∥u2n-p∥≤(1+kn)2∥xn-p∥+l̃2n,
where l̃2n≡α2n((1+kn)l̃1n+ln)+γ2n∥u2n-p∥. Notice that ∑n=1∞kn<∞,∑n=1∞ln<∞,∑n=1∞l̃1n<∞ and {u2n} is bounded. Then ∑n=1∞l̃2n<∞. By continuing this process, there is a sequence {l̃kn} of nonnegative real numbers such that ∑n=1∞l̃kn<∞ and
(2.14)∥xn+1-p∥≤(1+kn)N∥xn-p∥+l̃kn.
Then {xn} is of monotone type (2) with respect to F. By Theorem 2.4(a), we get that limn→∞∥xn-p∥ exists and {xn} is bounded. Next, we show that {xn} is of monotone type (1) with respect to F. It follows from (2.11) that
(2.15)∥y1n-p∥≤(α1n+β1n)(1+kn)∥xn-p∥+α1nln+γ1n∥u1n-p∥≤(α1n+β1n)(1+kn)∥xn-p∥+α1nln+γ1n(∥xn-p∥+∥xn-u1n∥)≤(1+kn)∥xn-p∥+l̃1n,
where l̃1n≡α1nln+γ1n∥xn-u1n∥. Notice that {u1n}, {xn} are bounded and ∑n=1∞ln<∞. Then ∑n=1∞l̃1n<∞ and {l̃1n} is independent of p. Again, by continuing this process, we obtain a sequence {l̃kn} of nonnegative real numbers such that it is independent of p, ∑n=1∞l̃kn<∞ and
(2.16)∥xn+1-p∥≤(1+kn)N∥xn-p∥+l̃kn
for all n≥1 and p∈F. Then {xn} is of monotone type (1) with respect to F. Hence the result follows from (2.16) and Theorem 2.4(c). This completes the proof.Remark 2.9.
Theorem2.4 is a correction of [4,Theorem 3.2]. In fact, the closedness of F is not assumed there (this defect is now corrected after the submission of this article). Moreover, it is shown in the following example that the fixed point set of a generalized asymptotically nonexpansive mapping is not necessarily closed even in a Hilbert space.Example 2.10 (A generalized asymptotically nonexpansive mapping whose fixed point set is not closed).
LetT:[-1/2,1/2]→[-1/2,1/2] be a mapping defined by
(2.17)Tx={x,ifx∈[-12,0),14,ifx=0,x2,ifx∈(0,12].
Then T is generalized asymptotically nonexpansive.Proof.
Notice thatF(T)=[-1/2,0) is not closed. We prove that
(2.18)|Tnx-Tny|≤|x-y|+122n
for all x,y∈[-1/2,1/2] and n≥1. The inequality above holds trivially if x=y=0 or x,y∈[-1/2,0). Then it suffices to consider the following cases.Case 1 (x,y∈(0,1/2]).
Then(2.19)|Tnx-Tny|=|x2n-y2n|≤122n.Case 2 (x∈[-1/2,0) and y=0).
Then(2.20)|Tnx-Tny|=|x-122n|≤|x-y|+122n.Case 3 (x∈[-1/2,0) and y∈(0,1/2]).
Then(2.21)|Tnx-Tny|=|x-y2n|≤|x-y|.Case 4 (x=0 and y∈(0,1/2]).
Then(2.22)|Tnx-Tny|=|122n-y2n|≤|x-y|+122n.Hence, (2.18) holds. This completes the proof.Remark 2.11.
ForT which is defined in Example 2.10 and x1∈(0,1/2], we define
(2.23)xn+1=αnTnxn+(1-αn)xn,
where 0<a≤αn≤1 and n≥1. It is not hard to show that xn→0∉F(T) and d(xn,F(T))→0. Hence [4, Theorems 3.2 and 3.6] do not hold even for a single mapping if the closedness of the fixed point set is not assumed.We present a sufficient condition guaranteeing the closedness of the fixed point set of a generalized asymptotically quasi-nonexpansive mapping.Theorem 2.12.
LetC be a nonempty subset of a Banach space X and T:C→C a generalized asymptotically quasi-nonexpansive mapping. If G(T):={(x,Tx):x∈C} is closed, then F(T) is closed.Proof.
Let{pn} be a sequence in F(T) such that pn→p. Since T is a generalized asymptotically quasi-nonexpansive mapping with the sequence {(kn,ln)}, we have
(2.24)∥Tnp-p∥≤∥Tnp-pn∥+∥pn-p∥≤(1+kn)∥p-pn∥+ln+∥pn-p∥→0.
Then Tnp→p, and so T(Tnp)=Tn+1p→p. Hence, by the closedness of G(T), Tp=p. This completes the proof.Remark 2.13.
It is also worth mentioning that the(L-γ) uniform Lipschitz condition of mappings in [4, Theorems 4.2 and 4.3] implies the closedness of their graphs.The following result shows that the closedness ofG(T) can be dropped if T is asymptotically quasi-nonexpansive.Theorem 2.14.
LetC be a nonempty subset of a Banach space X, and T:C→C an asymptotically quasi-nonexpansive mapping. Then F(T) is closed.Proof.
Suppose thatT is an asymptotically quasi-nonexpansive mapping with the sequence {kn}. Let {pn} be a sequence in F(T) such that pn→p. We have
(2.25)∥Tp-p∥≤∥Tp-pn∥+∥pn-p∥≤(1+k1)∥p-pn∥+∥pn-p∥→0.
Then Tp=p. This completes the proof.Remark 2.15.
Not every generalized asymptotically quasi-nonexpansive mapping is asymptotically quasi-nonexpansive. In fact, the mappingT in Example 2.10 is not asymptotically quasi-nonexpansive since F(T) is not closed.


## 2.1. Sequences of Monotone Types (1) and (2)

Definition 2.1.
Let{xn} be a sequence in a metric space (X,d) and F a subset of X. We say that {xn} is of  (i)
monotone type (1) with respect toF [6] if there exist sequences {rn} and {sn} of nonnegative real numbers such that ∑n=1∞rn<∞, ∑n=1∞sn<∞ and(2.1)d(xn+1,p)≤(1+rn)d(xn,p)+snfor alln≥1 and p∈F;(ii)
monotone type (2) with respect toF if for each p∈F there exist sequences {rn} and {sn} of nonnegative real numbers such that ∑n=1∞rn<∞, ∑n=1∞sn<∞ and(2.2)d(xn+1,p)≤(1+rn)d(xn,p)+snfor alln≥1.Proposition 2.2.
If{xn} is of monotone type (1) with respect to F, then it is of monotone type (2) with respect to F.Lemma 2.3 ([7, Lemma 1]).
Let{an}, {bn}, and {αn} be sequences of nonnegative real numbers such that
(2.3)an+1≤(1+αn)an+bn,n≥1.
If ∑n=1∞αn<∞ and ∑n=1∞bn<∞, then limn→∞an exists.Theorem 2.4.
Let(X,d) be a complete metric space, F⊂X, and {xn} a sequence in X. Then one has the following assertions.  (a)
If{xn} is of monotone type (2) with respect to F, then limn→∞d(xn,p) exists for all p∈F.(b)
If{xn} is of monotone type (1) with respect to F, then limn→∞d(xn,F) exists.(c)
If{xn} is of monotone type (1) with respect to F and lim infn→∞d(xn,F)=0, then xn→p for some p∈X satisfying d(p,F)=0. In particular, if F is closed, then p∈F.Proof.
(a)
It is easy to see that the result follows from (2.2) and Lemma 2.3.(b)
Note that{rn} and {sn} are independent of p∈F. Taking infimum over all p∈F in (2.1) gives(2.4)d(xn+1,F)≤(1+rn)d(xn,F)+sn∀n≥1.
Again, by Lemma 2.3, we get that limn→∞d(xn,F) exists.(c)
It follows from (b) andlim infn→∞d(xn,F)=0 that(2.5)limn→∞d(xn,F)=0.
To show that {xn} is a Cauchy sequence, let ε>0. Since limn→∞d(xn,F)=0, we may assume without loss of generality that there is a sequence {pn} in F such that d(xn,pn)≤ε/4 for all n≥1. As {xn} is bounded, we put M=sup{d(xm,pn):m,n≥1}. From (2.1), we have
(2.6)d(xn+1,pk)≤d(xn,pk)+tn∀n,k≥1,
where tn≡rnM+sn. Consequently,
(2.7)d(xn+k,pn)≤d(xn,pn)+∑j=nn+k-1tj≤ε4+∑j=n∞tj∀n,k≥1.
Notice that ∑n=1∞tn<∞. So there exists N≥1 such that ∑n=N∞tn<ε/2. Then for all n≥N,k≥1, we have
(2.8)d(xn+k,xn)≤d(xn+k,pn)+d(xn,pn)<ε.
Hence, {xn} is a Cauchy sequence in X. By the completeness of X, we assume that xn→p for some p∈X. Since
(2.9)|d(xn,F)-d(p,F)|≤d(xn,p)→0,
we obtain d(p,F)=0. This completes the proof.


## 2.2. A Correction of Recent Results of Imnang and Suantai

The following observation is an auxiliary result.Proposition 2.5.
LetC be a nonempty subset of a Banach space X, and let T1,T2,…,TN:C→C be N generalized asymptotically quasi-nonexpansive mappings with F:=⋂i=1NF(Ti)≠∅. Then there exist sequences {kn},{ln} in [0,∞) such that kn,ln→0 and
(2.10)∥Tinx-p∥≤(1+kn)∥x-p∥+ln,
for all x∈C,p∈F,n≥1, and i=1,2,…,N.From now on, we assume thatN generalized asymptotically quasi-nonexpansive mappings T1,T2,…,TN:C→C are equipped with the sequences {kn},{ln} in [0,∞) as mentioned in the preceding proposition.Theorem 2.6.
LetC be a nonempty closed convex subset of a Banach space X, and {T1,T2,…,TN} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of C with the sequence {(kn,ln)} such that ∑n=1∞kn<∞ and ∑n=1∞ln<∞. Assume that F:=⋂i=1NF(Ti)≠∅ is closed, and {xn} is the sequence in C defined by (1.5) such that ∑n=1∞γin<∞ for each i=1,2,…,N. Then the sequence {xn} converges strongly to a common fixed point of the family of mappings if and only if lim infn→∞d(xn,F)=0.Remark 2.7.
There is a small gap in [4, Theorem 3.2]. More precisely, the sequence {xn} generated by (1.5) is shown in [4, Theorem 3.2]  to be of monotone type (2) with respect to F, that is, ∥xn+1-p∥≤(1+kn)N∥xn-p∥+ekn where each ekn is a nonnegative real number depending on p. Then the expression d(xn+1,F)≤(1+kn)Nd(xn,F)+ekn cannot warrant.Remark 2.8.
The same gap also appears in [8,Lemma 2.3] and [6, Theorem 3.2].Proof of Theorem2.6.
Necessity is obvious. Conversely, we show first that{xn} is of monotone type (2) with respect to F. Let p∈F. We have that
(2.11)∥y1n-p∥=∥α1nT1nxn+β1nxn+γ1nu1n-p∥≤α1n∥T1nxn-p∥+β1n∥xn-p∥+γ1n∥u1n-p∥≤(α1n+β1n)(1+kn)∥xn-p∥+α1nln+γ1n∥u1n-p∥(2.12)≤(1+kn)∥xn-p∥+l̃1n,
where l̃1n≡α1nln+γ1n∥u1n-p∥. Notice that ∑n=1∞ln<∞ and {u1n} is bounded. Then ∑n=1∞l̃1n<∞. It follows from (2.12) that
(2.13)∥y2n-p∥≤α2n∥T2ny1n-p∥+β2n∥xn-p∥+γ2n∥u2n-p∥≤α2n(1+kn)∥y1n-p∥+α2nln+β2n∥xn-p∥+γ2n∥u2n-p∥≤(α2n+β2n)(1+kn)2∥xn-p∥+α2n((1+kn)l̃1n+ln)+γ2n∥u2n-p∥≤(1+kn)2∥xn-p∥+l̃2n,
where l̃2n≡α2n((1+kn)l̃1n+ln)+γ2n∥u2n-p∥. Notice that ∑n=1∞kn<∞,∑n=1∞ln<∞,∑n=1∞l̃1n<∞ and {u2n} is bounded. Then ∑n=1∞l̃2n<∞. By continuing this process, there is a sequence {l̃kn} of nonnegative real numbers such that ∑n=1∞l̃kn<∞ and
(2.14)∥xn+1-p∥≤(1+kn)N∥xn-p∥+l̃kn.
Then {xn} is of monotone type (2) with respect to F. By Theorem 2.4(a), we get that limn→∞∥xn-p∥ exists and {xn} is bounded. Next, we show that {xn} is of monotone type (1) with respect to F. It follows from (2.11) that
(2.15)∥y1n-p∥≤(α1n+β1n)(1+kn)∥xn-p∥+α1nln+γ1n∥u1n-p∥≤(α1n+β1n)(1+kn)∥xn-p∥+α1nln+γ1n(∥xn-p∥+∥xn-u1n∥)≤(1+kn)∥xn-p∥+l̃1n,
where l̃1n≡α1nln+γ1n∥xn-u1n∥. Notice that {u1n}, {xn} are bounded and ∑n=1∞ln<∞. Then ∑n=1∞l̃1n<∞ and {l̃1n} is independent of p. Again, by continuing this process, we obtain a sequence {l̃kn} of nonnegative real numbers such that it is independent of p, ∑n=1∞l̃kn<∞ and
(2.16)∥xn+1-p∥≤(1+kn)N∥xn-p∥+l̃kn
for all n≥1 and p∈F. Then {xn} is of monotone type (1) with respect to F. Hence the result follows from (2.16) and Theorem 2.4(c). This completes the proof.Remark 2.9.
Theorem2.4 is a correction of [4,Theorem 3.2]. In fact, the closedness of F is not assumed there (this defect is now corrected after the submission of this article). Moreover, it is shown in the following example that the fixed point set of a generalized asymptotically nonexpansive mapping is not necessarily closed even in a Hilbert space.Example 2.10 (A generalized asymptotically nonexpansive mapping whose fixed point set is not closed).
LetT:[-1/2,1/2]→[-1/2,1/2] be a mapping defined by
(2.17)Tx={x,ifx∈[-12,0),14,ifx=0,x2,ifx∈(0,12].
Then T is generalized asymptotically nonexpansive.Proof.
Notice thatF(T)=[-1/2,0) is not closed. We prove that
(2.18)|Tnx-Tny|≤|x-y|+122n
for all x,y∈[-1/2,1/2] and n≥1. The inequality above holds trivially if x=y=0 or x,y∈[-1/2,0). Then it suffices to consider the following cases.Case 1 (x,y∈(0,1/2]).
Then(2.19)|Tnx-Tny|=|x2n-y2n|≤122n.Case 2 (x∈[-1/2,0) and y=0).
Then(2.20)|Tnx-Tny|=|x-122n|≤|x-y|+122n.Case 3 (x∈[-1/2,0) and y∈(0,1/2]).
Then(2.21)|Tnx-Tny|=|x-y2n|≤|x-y|.Case 4 (x=0 and y∈(0,1/2]).
Then(2.22)|Tnx-Tny|=|122n-y2n|≤|x-y|+122n.Hence, (2.18) holds. This completes the proof.Remark 2.11.
ForT which is defined in Example 2.10 and x1∈(0,1/2], we define
(2.23)xn+1=αnTnxn+(1-αn)xn,
where 0<a≤αn≤1 and n≥1. It is not hard to show that xn→0∉F(T) and d(xn,F(T))→0. Hence [4, Theorems 3.2 and 3.6] do not hold even for a single mapping if the closedness of the fixed point set is not assumed.We present a sufficient condition guaranteeing the closedness of the fixed point set of a generalized asymptotically quasi-nonexpansive mapping.Theorem 2.12.
LetC be a nonempty subset of a Banach space X and T:C→C a generalized asymptotically quasi-nonexpansive mapping. If G(T):={(x,Tx):x∈C} is closed, then F(T) is closed.Proof.
Let{pn} be a sequence in F(T) such that pn→p. Since T is a generalized asymptotically quasi-nonexpansive mapping with the sequence {(kn,ln)}, we have
(2.24)∥Tnp-p∥≤∥Tnp-pn∥+∥pn-p∥≤(1+kn)∥p-pn∥+ln+∥pn-p∥→0.
Then Tnp→p, and so T(Tnp)=Tn+1p→p. Hence, by the closedness of G(T), Tp=p. This completes the proof.Remark 2.13.
It is also worth mentioning that the(L-γ) uniform Lipschitz condition of mappings in [4, Theorems 4.2 and 4.3] implies the closedness of their graphs.The following result shows that the closedness ofG(T) can be dropped if T is asymptotically quasi-nonexpansive.Theorem 2.14.
LetC be a nonempty subset of a Banach space X, and T:C→C an asymptotically quasi-nonexpansive mapping. Then F(T) is closed.Proof.
Suppose thatT is an asymptotically quasi-nonexpansive mapping with the sequence {kn}. Let {pn} be a sequence in F(T) such that pn→p. We have
(2.25)∥Tp-p∥≤∥Tp-pn∥+∥pn-p∥≤(1+k1)∥p-pn∥+∥pn-p∥→0.
Then Tp=p. This completes the proof.Remark 2.15.
Not every generalized asymptotically quasi-nonexpansive mapping is asymptotically quasi-nonexpansive. In fact, the mappingT in Example 2.10 is not asymptotically quasi-nonexpansive since F(T) is not closed.


## 3. Remark on Lan’s Generalized Asymptotically Quasi-Nonexpansive Mappings

The following mapping introduced by Lan [5] also bears the name generalized asymptotically quasi-nonexpansive mappings. We recall his definition here.Definition 3.1 (see [5, Definition 2.1(4)]).
LetC be a subset of a Banach space X. A mapping T:C→C is called generalized asymptotically quasi-nonexpansive in the sense of Lan if there exists two sequences {rn}⊂[0,∞) and {sn}⊂[0,1) such that rn,sn→0 and
(3.1)∥Tnx-p∥≤(1+rn)∥x-p∥+sn∥x-Tnx∥
for all x∈C, p∈F(T), and n≥1.Lan [5] and many authors (e.g., [8–11]) have investigated convergence theorems for such mappings without awareness that Lan's mappings are not new ones.Proposition 3.2.
IfT:C→C is generalized asymptotically quasi-nonexpansive in the sense of Lan, then it is asymptotically quasi-nonexpansive.Proof.
By Lan's definition, there exist two sequences{rn}⊂[0,∞) and {sn}⊂[0,1) such that rn,sn→0 and
(3.2)∥Tnx-p∥≤(1+rn)∥x-p∥+sn∥x-Tnx∥
for all x∈C, p∈F(T), and n∈ℕ. Consequently,
(3.3)∥Tnx-p∥≤(1+rn)∥x-p∥+sn(∥x-p∥+∥Tnx-p∥).
This implies
(3.4)∥Tnx-p∥≤1+rn+sn1-sn∥x-p∥=(1+rn+2sn1-sn)∥x-p∥.
It is also clear that (rn+2sn)/(1-sn)→0 and this completes the proof.



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*Source: 283461-2010-02-14.xml*