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.
1. Introduction
Let C be a nonempty subset of a Banach space X. A mapping T:C→C is said to be
asymptotically nonexpansive [1] if there exists a sequence {kn} in [0,∞) such that kn→0 and
∥Tnx-Tny∥≤(1+kn)∥x-y∥
for all x,y∈C and n≥1;
asymptotically quasi-nonexpansive [2] if F(T)={p∈C:Tp=p}≠∅ and there exists a sequence {kn} in [0,∞) such that kn→0 and
∥Tnx-p∥≤(1+kn)∥x-p∥
for all x∈C, p∈F(T) and n≥1;
generalized asymptotically nonexpansive if there exist sequences {kn},{ln} in [0,∞) such that kn,ln→0 and
∥Tnx-Tny∥≤(1+kn)∥x-y∥+ln
for all x,y∈C and n≥1;
generalized asymptotically quasi-nonexpansive [3] if F(T)≠∅ and there exist sequences {kn},{ln} in [0,∞) such that kn,ln→0 and
∥Tnx-p∥≤(1+kn)∥x-p∥+ln
for all x∈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
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 Results2.1. 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
monotone type (1) with respect to F [6] if there exist sequences {rn} and {sn} of nonnegative real numbers such that ∑n=1∞rn<∞, ∑n=1∞sn<∞ and
d(xn+1,p)≤(1+rn)d(xn,p)+sn
for all n≥1 and p∈F;
monotone type (2) with respect to F if for each p∈F there exist sequences {rn} and {sn} of nonnegative real numbers such that ∑n=1∞rn<∞, ∑n=1∞sn<∞ and
d(xn+1,p)≤(1+rn)d(xn,p)+sn
for all n≥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
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.
If {xn} is of monotone type (2) with respect to F, then limn→∞d(xn,p) exists for all p∈F.
If {xn} is of monotone type (1) with respect to F, then limn→∞d(xn,F) exists.
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.
It is easy to see that the result follows from (2.2) and Lemma 2.3.
Note that {rn} and {sn} are independent of p∈F. Taking infimum over all p∈F in (2.1) gives
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.
It follows from (b) and lim infn→∞d(xn,F)=0 that
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
d(xn+1,pk)≤d(xn,pk)+tn∀n,k≥1,
where tn≡rnM+sn. Consequently,
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
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
|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.
Let C 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
∥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 that N 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.
Let C 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 Theorem 2.6.
Necessity is obvious. Conversely, we show first that {xn} is of monotone type (2) with respect to F. Let p∈F. We have that
∥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∥≤(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
∥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
∥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
∥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
∥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.
Theorem 2.4 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).
Let T:[-1/2,1/2]→[-1/2,1/2] be a mapping defined by
Tx={x,ifx∈[-12,0),14,ifx=0,x2,ifx∈(0,12].
Then T is generalized asymptotically nonexpansive.
Proof.
Notice that F(T)=[-1/2,0) is not closed. We prove that
|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
|Tnx-Tny|=|x2n-y2n|≤122n.
Case 2 (x∈[-1/2,0) and y=0).
Then
|Tnx-Tny|=|x-122n|≤|x-y|+122n.
Case 3 (x∈[-1/2,0) and y∈(0,1/2]).
Then
|Tnx-Tny|=|x-y2n|≤|x-y|.
Case 4 (x=0 and y∈(0,1/2]).
Then
|Tnx-Tny|=|122n-y2n|≤|x-y|+122n.
Hence, (2.18) holds. This completes the proof.
Remark 2.11.
For T which is defined in Example 2.10 and x1∈(0,1/2], we define
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.
Let C 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
∥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 of G(T) can be dropped if T is asymptotically quasi-nonexpansive.
Theorem 2.14.
Let C 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 that T is an asymptotically quasi-nonexpansive mapping with the sequence {kn}. Let {pn} be a sequence in F(T) such that pn→p. We have
∥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 mapping T 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)]).
Let C 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
∥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.
If T: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
∥Tnx-p∥≤(1+rn)∥x-p∥+sn∥x-Tnx∥
for all x∈C, p∈F(T), and n∈ℕ. Consequently,
∥Tnx-p∥≤(1+rn)∥x-p∥+sn(∥x-p∥+∥Tnx-p∥).
This implies
∥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.
Acknowledgments
The research of the first and second authors is partially supported by the Centre of Excellence in Mathematics, the Commission on Higher Education of Thailand. The third author was supported by the Human Resource Development in Science Project.
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