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

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.

1. Introduction

Let 𝐶 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 {𝑘_𝑛} in [0,∞) such that 𝑘_𝑛→0 and

‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤(1+𝑘_𝑛)‖𝑥−𝑦‖ (1.1)

for all 𝑥,𝑦∈𝐶 and 𝑛≥1;(ii)asymptotically quasi-nonexpansive [2] if 𝐹(𝑇)={𝑝∈𝐶∶𝑇𝑝=𝑝}≠∅ and there exists a sequence {𝑘_𝑛} in [0,∞) such that 𝑘_𝑛→0 and

‖𝑇^𝑛𝑥−𝑝‖≤(1+𝑘_𝑛)‖𝑥−𝑝‖ (1.2)

for all 𝑥∈𝐶, 𝑝∈𝐹(𝑇) and 𝑛≥1;(iii)generalized asymptotically nonexpansive if there exist sequences {𝑘_𝑛},{𝑙_𝑛} in [0,∞) such that 𝑘_𝑛,𝑙_𝑛→0 and

‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤(1+𝑘_𝑛)‖𝑥−𝑦‖+𝑙_𝑛 (1.3)

for all 𝑥,𝑦∈𝐶 and 𝑛≥1;(iv)generalized asymptotically quasi-nonexpansive [3] if 𝐹(𝑇)≠∅ and there exist sequences {𝑘_𝑛},{𝑙_𝑛} in [0,∞) such that 𝑘_𝑛,𝑙_𝑛→0 and

‖𝑇^𝑛𝑥−𝑝‖≤(1+𝑘_𝑛)‖𝑥−𝑝‖+𝑙_𝑛 (1.4)

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 {𝑇_𝑖}^𝑁_𝑖=1 be a finite family of self-mappings of a closed convex subset 𝐶 of 𝑋. The sequence {𝑥_𝑛} is generated from 𝑥₁∈𝐶, and

𝑦_1𝑛=𝛼_1𝑛𝑇^𝑛₁𝑥_𝑛+𝛽_1𝑛𝑥_𝑛+𝛾_1𝑛𝑢_1𝑛,𝑦_2𝑛=𝛼_2𝑛𝑇^𝑛₂𝑦_1𝑛+𝛽_2𝑛𝑥_𝑛+𝛾_2𝑛𝑢_2𝑛,⋮𝑦_{(𝑁−1)𝑛}=𝛼_{(𝑁−1)𝑛}𝑇^𝑛_𝑁−1𝑦_{(𝑁−2)𝑛}+𝛽_{(𝑁−1)𝑛}𝑥_𝑛+𝛾_{(𝑁−1)𝑛}𝑢_{(𝑁−1)𝑛},𝑥_𝑛+1=𝛼_𝑁𝑛𝑇^𝑛_𝑁𝑦_{(𝑁−1)𝑛}+𝛽_𝑁𝑛𝑥_𝑛+𝛾_𝑁𝑛𝑢_𝑁𝑛, (1.5) where {𝑢_1𝑛},{𝑢_2𝑛},…,{𝑢_𝑁𝑛} are bounded sequences in 𝐶, and {𝛼_𝑖𝑛},{𝛽_𝑖𝑛}, and {𝛾_𝑖𝑛} are sequences in [0,1] such that 𝛼_𝑖𝑛+𝛽_𝑖𝑛+𝛾_𝑖𝑛=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 {𝑥_𝑛} be a sequence in a metric space (𝑋,𝑑) and 𝐹 a subset of 𝑋. We say that {𝑥_𝑛} is of (i)monotone type (1) with respect to 𝐹 [6] if there exist sequences {𝑟_𝑛} and {𝑠_𝑛} of nonnegative real numbers such that ∑^∞_𝑛=1𝑟_𝑛<∞, ∑^∞_𝑛=1𝑠_𝑛<∞ and𝑑(𝑥_𝑛+1,𝑝)≤(1+𝑟_𝑛)𝑑(𝑥_𝑛,𝑝)+𝑠_𝑛 (2.1)for all 𝑛≥1 and 𝑝∈𝐹; (ii)monotone type (2) with respect to 𝐹 if for each 𝑝∈𝐹 there exist sequences {𝑟_𝑛} and {𝑠_𝑛} of nonnegative real numbers such that ∑^∞_𝑛=1𝑟_𝑛<∞, ∑^∞_𝑛=1𝑠_𝑛<∞ and𝑑(𝑥_𝑛+1,𝑝)≤(1+𝑟_𝑛)𝑑(𝑥_𝑛,𝑝)+𝑠_𝑛 (2.2)for all 𝑛≥1.

Proposition 2.2. If {𝑥_𝑛} 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 {𝑎_𝑛}, {𝑏_𝑛}, and {𝛼_𝑛} be sequences of nonnegative real numbers such that 𝑎_𝑛+1≤(1+𝛼_𝑛)𝑎_𝑛+𝑏_𝑛,𝑛≥1. (2.3) If ∑^∞_𝑛=1𝛼_𝑛<∞ and ∑^∞_𝑛=1𝑏_𝑛<∞, then lim_𝑛→∞𝑎_𝑛 exists.

Theorem 2.4. Let (𝑋,𝑑) be a complete metric space, 𝐹⊂𝑋, and {𝑥_𝑛} a sequence in 𝑋. Then one has the following assertions. (a)If {𝑥_𝑛} is of monotone type (2) with respect to 𝐹, then lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exists for all 𝑝∈𝐹. (b)If {𝑥_𝑛} is of monotone type (1) with respect to 𝐹, then lim_𝑛→∞𝑑(𝑥_𝑛,𝐹) exists. (c)If {𝑥_𝑛} is of monotone type (1) with respect to 𝐹 and liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, then 𝑥_𝑛→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 {𝑟_𝑛} and {𝑠_𝑛} are independent of 𝑝∈𝐹. Taking infimum over all 𝑝∈𝐹 in (2.1) gives 𝑑(𝑥_𝑛+1,𝐹)≤(1+𝑟_𝑛)𝑑(𝑥_𝑛,𝐹)+𝑠_𝑛∀𝑛≥1. (2.4) Again, by Lemma 2.3, we get that lim_𝑛→∞𝑑(𝑥_𝑛,𝐹) exists.(c) It follows from (b) and liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0 that lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0. (2.5) To show that {𝑥_𝑛} is a Cauchy sequence, let 𝜀>0. Since lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, we may assume without loss of generality that there is a sequence {𝑝_𝑛} in 𝐹 such that 𝑑(𝑥_𝑛,𝑝_𝑛)≤𝜀/4 for all 𝑛≥1. As {𝑥_𝑛} is bounded, we put 𝑀=sup{𝑑(𝑥_𝑚,𝑝_𝑛)∶𝑚,𝑛≥1}. From (2.1), we have 𝑑(𝑥_𝑛+1,𝑝_𝑘)≤𝑑(𝑥_𝑛,𝑝_𝑘)+𝑡_𝑛∀𝑛,𝑘≥1, (2.6) where 𝑡_𝑛≡𝑟_𝑛𝑀+𝑠_𝑛. Consequently, 𝑑(𝑥_𝑛+𝑘,𝑝_𝑛)≤𝑑(𝑥_𝑛,𝑝_𝑛)+^{𝑛+𝑘−1}_𝑗=𝑛𝑡_𝑗≤𝜀4+^∞_𝑗=𝑛𝑡_𝑗∀𝑛,𝑘≥1. (2.7) Notice that ∑^∞_𝑛=1𝑡_𝑛<∞. So there exists 𝑁≥1 such that ∑^∞_𝑛=𝑁𝑡_𝑛<𝜀/2. Then for all 𝑛≥𝑁,𝑘≥1, we have 𝑑(𝑥_𝑛+𝑘,𝑥_𝑛)≤𝑑(𝑥_𝑛+𝑘,𝑝_𝑛)+𝑑(𝑥_𝑛,𝑝_𝑛)<𝜀. (2.8) Hence, {𝑥_𝑛} is a Cauchy sequence in 𝑋. By the completeness of 𝑋, we assume that 𝑥_𝑛→𝑝 for some 𝑝∈𝑋. Since |𝑑(𝑥_𝑛,𝐹)−𝑑(𝑝,𝐹)|≤𝑑(𝑥_𝑛,𝑝)→0, (2.9) 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,…,𝑇_𝑁∶𝐶→𝐶 be 𝑁 generalized asymptotically quasi-nonexpansive mappings with 𝐹∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)≠∅. Then there exist sequences {𝑘_𝑛},{𝑙_𝑛} in [0,∞) such that 𝑘_𝑛,𝑙_𝑛→0 and ‖𝑇^𝑛_𝑖𝑥−𝑝‖≤(1+𝑘_𝑛)‖𝑥−𝑝‖+𝑙_𝑛, (2.10) 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,…,𝑇_𝑁∶𝐶→𝐶 are equipped with the sequences {𝑘_𝑛},{𝑙_𝑛} 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,…,𝑇_𝑁} a finite family of generalized asymptotically quasi-nonexpansive self-mappings of 𝐶 with the sequence {(𝑘_𝑛,𝑙_𝑛)} such that ∑^∞_𝑛=1𝑘_𝑛<∞ and ∑^∞_𝑛=1𝑙_𝑛<∞. Assume that 𝐹∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)≠∅ is closed, and {𝑥_𝑛} is the sequence in 𝐶 defined by (1.5) such that ∑^∞_𝑛=1𝛾_𝑖𝑛<∞ for each 𝑖=1,2,…,𝑁. Then the sequence {𝑥_𝑛} converges strongly to a common fixed point of the family of mappings if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.

Remark 2.7. There is a small gap in [4, Theorem 3.2]. More precisely, the sequence {𝑥_𝑛} generated by (1.5) is shown in [4, Theorem 3.2] to be of monotone type (2) with respect to 𝐹, that is, ‖𝑥_𝑛+1−𝑝‖≤(1+𝑘_𝑛)^𝑁‖𝑥_𝑛−𝑝‖+𝑒_𝑘𝑛 where each 𝑒_𝑘𝑛 is a nonnegative real number depending on 𝑝. Then the expression 𝑑(𝑥_𝑛+1,𝐹)≤(1+𝑘_𝑛)^𝑁𝑑(𝑥_𝑛,𝐹)+𝑒_𝑘𝑛 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 {𝑥_𝑛} is of monotone type (2) with respect to 𝐹. Let 𝑝∈𝐹. We have that ‖𝑦_1𝑛−𝑝‖=‖𝛼_1𝑛𝑇^𝑛₁𝑥_𝑛+𝛽_1𝑛𝑥_𝑛+𝛾_1𝑛𝑢_1𝑛−𝑝‖≤𝛼_1𝑛‖𝑇^𝑛₁𝑥_𝑛−𝑝‖+𝛽_1𝑛‖𝑥_𝑛−𝑝‖+𝛾_1𝑛‖𝑢_1𝑛−𝑝‖≤(𝛼_1𝑛+𝛽_1𝑛)(1+𝑘_𝑛)‖𝑥_𝑛−𝑝‖+𝛼_1𝑛𝑙_𝑛+𝛾_1𝑛‖𝑢_1𝑛−𝑝‖≤(1+𝑘_𝑛)‖𝑥_𝑛−𝑝‖+𝑙̃_1𝑛, (2.11)(2.12) where 𝑙̃_1𝑛≡𝛼_1𝑛𝑙_𝑛+𝛾_1𝑛‖𝑢_1𝑛−𝑝‖. Notice that ∑^∞_𝑛=1𝑙_𝑛<∞ and {𝑢_1𝑛} is bounded. Then ∑^∞_𝑛=1𝑙̃_1𝑛<∞. It follows from (2.12) that ‖𝑦_2𝑛−𝑝‖≤𝛼_2𝑛‖𝑇^𝑛₂𝑦_1𝑛−𝑝‖+𝛽_2𝑛‖𝑥_𝑛−𝑝‖+𝛾_2𝑛‖𝑢_2𝑛−𝑝‖≤𝛼_2𝑛(1+𝑘_𝑛)‖𝑦_1𝑛−𝑝‖+𝛼_2𝑛𝑙_𝑛+𝛽_2𝑛‖𝑥_𝑛−𝑝‖+𝛾_2𝑛‖𝑢_2𝑛−𝑝‖≤(𝛼_2𝑛+𝛽_2𝑛)(1+𝑘_𝑛)²‖𝑥_𝑛−𝑝‖+𝛼_2𝑛((1+𝑘_𝑛)𝑙̃_1𝑛+𝑙_𝑛)+𝛾_2𝑛‖𝑢_2𝑛−𝑝‖≤(1+𝑘_𝑛)²‖𝑥_𝑛−𝑝‖+𝑙̃_2𝑛, (2.13) where 𝑙̃_2𝑛≡𝛼_2𝑛((1+𝑘_𝑛)𝑙̃_1𝑛+𝑙_𝑛)+𝛾_2𝑛‖𝑢_2𝑛−𝑝‖. Notice that ∑^∞_𝑛=1𝑘_𝑛<∞,∑^∞_𝑛=1𝑙_𝑛<∞,∑^∞_𝑛=1𝑙̃_1𝑛<∞ and {𝑢_2𝑛} is bounded. Then ∑^∞_𝑛=1𝑙̃_2𝑛<∞. By continuing this process, there is a sequence {𝑙̃_𝑘𝑛} of nonnegative real numbers such that ∑^∞_𝑛=1𝑙̃_𝑘𝑛<∞ and ‖𝑥_𝑛+1−𝑝‖≤(1+𝑘_𝑛)^𝑁‖𝑥_𝑛−𝑝‖+𝑙̃_𝑘𝑛. (2.14) Then {𝑥_𝑛} is of monotone type (2) with respect to 𝐹. By Theorem 2.4(a), we get that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists and {𝑥_𝑛} is bounded. Next, we show that {𝑥_𝑛} is of monotone type (1) with respect to 𝐹. It follows from (2.11) that ‖𝑦_1𝑛−𝑝‖≤(𝛼_1𝑛+𝛽_1𝑛)(1+𝑘_𝑛)‖𝑥_𝑛−𝑝‖+𝛼_1𝑛𝑙_𝑛+𝛾_1𝑛‖𝑢_1𝑛−𝑝‖≤(𝛼_1𝑛+𝛽_1𝑛)(1+𝑘_𝑛)‖𝑥_𝑛−𝑝‖+𝛼_1𝑛𝑙_𝑛+𝛾_1𝑛(‖𝑥_𝑛−𝑝‖+‖𝑥_𝑛−𝑢_1𝑛‖)≤(1+𝑘_𝑛)‖𝑥_𝑛−𝑝‖+𝑙̃_1𝑛, (2.15) where 𝑙̃_1𝑛≡𝛼_1𝑛𝑙_𝑛+𝛾_1𝑛‖𝑥_𝑛−𝑢_1𝑛‖. Notice that {𝑢_1𝑛}, {𝑥_𝑛} are bounded and ∑^∞_𝑛=1𝑙_𝑛<∞. Then ∑^∞_𝑛=1𝑙̃_1𝑛<∞ and {𝑙̃_1𝑛} is independent of 𝑝. Again, by continuing this process, we obtain a sequence {𝑙̃_𝑘𝑛} of nonnegative real numbers such that it is independent of 𝑝, ∑^∞_𝑛=1𝑙̃_𝑘𝑛<∞ and ‖𝑥_𝑛+1−𝑝‖≤(1+𝑘_𝑛)^𝑁‖𝑥_𝑛−𝑝‖+𝑙̃_𝑘𝑛 (2.16) for all 𝑛≥1 and 𝑝∈𝐹. Then {𝑥_𝑛} 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 1𝑇𝑥=𝑥,if𝑥∈[−21,0),4𝑥,if𝑥=0,₂1,if𝑥∈(0,2]. (2.17) Then 𝑇 is generalized asymptotically nonexpansive.

Proof. Notice that 𝐹(𝑇)=[−1/2,0) is not closed. We prove that |𝑇^𝑛𝑥−𝑇^𝑛1𝑦|≤|𝑥−𝑦|+2^2𝑛 (2.18) 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 |𝑇^𝑛𝑥−𝑇^𝑛𝑦|=|𝑥^2𝑛−𝑦^2𝑛1|≤2^2𝑛. (2.19)Case 2 (𝑥∈[−1/2,0) and 𝑦=0). Then |𝑇^𝑛𝑥−𝑇^𝑛1𝑦|=|𝑥−2^2𝑛1|≤|𝑥−𝑦|+2^2𝑛. (2.20)Case 3 (𝑥∈[−1/2,0) and 𝑦∈(0,1/2]). Then |𝑇^𝑛𝑥−𝑇^𝑛𝑦|=|𝑥−𝑦^2𝑛|≤|𝑥−𝑦|. (2.21)Case 4 (𝑥=0 and 𝑦∈(0,1/2]). Then |𝑇^𝑛𝑥−𝑇^𝑛1𝑦|=|2^2𝑛−𝑦^2𝑛1|≤|𝑥−𝑦|+2^2𝑛. (2.22)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 𝑥_𝑛+1=𝛼_𝑛𝑇^𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑥_𝑛, (2.23) where 0<𝑎≤𝛼_𝑛≤1 and 𝑛≥1. It is not hard to show that 𝑥_𝑛→0∉𝐹(𝑇) and 𝑑(𝑥_𝑛,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 {𝑝_𝑛} be a sequence in 𝐹(𝑇) such that 𝑝_𝑛→𝑝. Since 𝑇 is a generalized asymptotically quasi-nonexpansive mapping with the sequence {(𝑘_𝑛,𝑙_𝑛)}, we have ‖𝑇^𝑛𝑝−𝑝‖≤‖𝑇^𝑛𝑝−𝑝_𝑛‖+‖𝑝_𝑛−𝑝‖≤(1+𝑘_𝑛)‖𝑝−𝑝_𝑛‖+𝑙_𝑛+‖𝑝_𝑛−𝑝‖→0. (2.24) Then 𝑇^𝑛𝑝→𝑝, and so 𝑇(𝑇^𝑛𝑝)=𝑇^𝑛+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 {𝑘_𝑛}. Let {𝑝_𝑛} be a sequence in 𝐹(𝑇) such that 𝑝_𝑛→𝑝. We have ‖𝑇𝑝−𝑝‖≤‖𝑇𝑝−𝑝_𝑛‖+‖𝑝_𝑛−𝑝‖≤(1+𝑘₁)‖𝑝−𝑝_𝑛‖+‖𝑝_𝑛−𝑝‖→0. (2.25) 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 {𝑟_𝑛}⊂[0,∞) and {𝑠_𝑛}⊂[0,1) such that 𝑟_𝑛,𝑠_𝑛→0 and ‖𝑇^𝑛𝑥−𝑝‖≤(1+𝑟_𝑛)‖𝑥−𝑝‖+𝑠_𝑛‖𝑥−𝑇^𝑛𝑥‖ (3.1) 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 {𝑟_𝑛}⊂[0,∞) and {𝑠_𝑛}⊂[0,1) such that 𝑟_𝑛,𝑠_𝑛→0 and ‖𝑇^𝑛𝑥−𝑝‖≤(1+𝑟_𝑛)‖𝑥−𝑝‖+𝑠_𝑛‖𝑥−𝑇^𝑛𝑥‖ (3.2) for all 𝑥∈𝐶, 𝑝∈𝐹(𝑇), and 𝑛∈ℕ. Consequently, ‖𝑇^𝑛𝑥−𝑝‖≤(1+𝑟_𝑛)‖𝑥−𝑝‖+𝑠_𝑛(‖𝑥−𝑝‖+‖𝑇^𝑛𝑥−𝑝‖). (3.3) This implies ‖𝑇^𝑛𝑥−𝑝‖≤1+𝑟_𝑛+𝑠_𝑛1−𝑠_𝑛𝑟‖𝑥−𝑝‖=(1+^𝑛+2𝑠_𝑛1−𝑠_𝑛)‖𝑥−𝑝‖. (3.4) It is also clear that (𝑟_𝑛+2𝑠_𝑛)/(1−𝑠_𝑛)→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.