• Research Article
• Open Access

# Strong Convergence Theorems of Modified Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space

Fixed Point Theory and Applications20092009:483497

https://doi.org/10.1155/2009/483497

• Accepted: 12 September 2009
• Published:

## Abstract

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 result (2007), Nilsrakoo and Saejung's result (2008), Su et al.'s result (2008), and some known corresponding results in the literatures.

## Keywords

• Hilbert Space
• Banach Space
• Nonexpansive Mapping
• Real Banach Space
• Nonempty Closed Convex Subset

## 1. Introduction

Let be a nonempty closed convex subset of a real Banach space . A mapping is said to be nonexpansive if for all We denote by the set of fixed points of , that is . A mapping is said to be quasi-nonexpansive if and for all and . It is easy to see that if is nonexpansive with , 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

(1.1)

where the initial guess is taken in arbitrarily and the sequence is in the interval .

In 1976, Halpern [2] first introduced the following iterative scheme:

(1.2)

see also Browder [3]. He pointed out that the conditions and are necessary in the sence that, if the iteration (1.2) converges to a fixed point of , then these conditions must be satisfied.

In 1974, Ishikawa [4] introduced a new iterative scheme, which is defined recursively by

(1.3)

where the initial guess is taken in arbitrarily and the sequences and are in the interval .

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 importance (see [6]).

Zhang and Su [7] introduced the following implicit hybrid method for a finite family of nonexpansive mappings in a real Hilbert space:

(1.4)

where , and are sequences in and for some and for some .

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. [9] introduced 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. [10] introduced the following iterative sequence. Let and

(1.5)

for all , where is a nonempty closed convex subset of a Banach space, is a sequence of and is a sequence of nonexpansive mappings. Then they proved that, under some suitable conditions, the sequence defined by (1.5) converges strongly to a common fixed point of .

In 2008, by using a (new) hybrid method, Takahashi et al. [11] proved the following theorem.

Theorem 1.1 (Takahashi et al. [11]).

Let be a Hilbert space and let be a nonempty closed convex subset of . Let and be families of nonexpansive mappings of into itself such that and let . Suppose that satisfies the NST-condition with . For and , define a sequence of as follows:
(1.6)

where for all and is said to satisfy the NST-condition with if for each bounded sequence , implies that for all . Then, converges strongly to .

Note that, recently, many authors try to extend the above result from Hilbert spaces to a Banach space setting.

Let be a real Banach space with dual . Denote by the duality product. The normalized duality mapping from to is defined by for all . The function is defined by

(1.7)

A mapping is said to be hemi-relatively nonexpansive (see [12]) if and

(1.8)

A point in is said to be an asymptotic fixed point of [13] if contains a sequence which converges weakly to such that the strong . The set of asymptotic fixed points of will be denoted by . A hemi-relatively nonexpansive mapping from into itself is called relatively nonexpansive if ; see [1416]) for more details.

On the other hand, Matsushita and Takahashi [17] introduced the following iteration. A sequence defined by

(1.9)

where the initial guess element is arbitrary, is a real sequence in , is a relatively nonexpansive mapping, and denotes the generalized projection from onto a closed convex subset of . Under some suitable conditions, they proved that the sequence converges weakly to a fixed point of .

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 by the following iteration:

(1.10)

where and with , for all . Moreover, Matsushita and Takahashi [14] proposed the following modification of iteration (1.9) in a Banach space :

(1.11)

and proved that the sequence converges strongly to .

Qin and Su [15] showed that the sequence , which is generated by relatively nonexpansive mappings in a Banach space , as follows:

(1.12)

converges strongly to

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

(1.13)

converges strongly to

In 2008, Nilsrakoo and Saejung [19] used the following Mann's iterative process:

(1.14)

and showed that the sequence 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. [12] showed their results by using the method as a monotone (CQ) hybrid method.

In this paper, motivated by Qin and Su [15], Nilsrakoo and Saejung [19], we consider the modified Ishikawa iterative (1.12) and Halpern iterative processes (1.13), which is different from those of (1.12)–(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 theorem [11], and many others.

## 2. Preliminaries

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

A Banach space is said to be strictly convex if

(2.1)

for all with and . It is said to be uniformly convex if for any two sequences in such that and

(2.2)

holds.

Let be the unit sphere of . Then the Banach space is said to be smooth if

(2.3)

exists for each It is said to be uniformly smooth if the limit is attained uniformly for . In this case, the norm of is said to be Gâteaux differentiable. The space is said to have uniformly Gâteaux differentiable if for each , the limit (2.3) is attained uniformly for . The norm of is said to be uniformly Fréchet differentiable (and is said to be uniformly smooth) if the limit (2.3) is attained uniformly for .

In our work, the concept duality mapping is very important. Here, we list some known facts, related to the duality mapping , as follows.

(a) ( , resp.) is uniformly convex if and only if ( , resp.) is uniformly smooth.

(b) for each .
1. (c)

If is reflexive, then is a mapping of onto .

2. (d)

If is strictly convex, then for all .

3. (e)

If is smooth, then is single valued.

4. (f)

If has a Fr chet differentiable norm, then is norm to norm continuous.

5. (g)

If is uniformly smooth, then is uniformly norm to norm continuous on each bounded subset of .

6. (h)

If is a Hilbert space, then is the identity operator.

If is a nonempty closed convex subset of a real Hilbert space and is the metric projection, then is nonexpansive. Alber [22] has recently introduced a generalized projection operator in a Banach space which is an analogue representation of the metric projection in Hilbert spaces.

The generalized projection is a map that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem

(2.4)

Notice that the existence and uniqueness of the operator is followed from the properties of the functional and strict monotonicity of the mapping , and moreover, in the Hilbert spaces setting we have . It is obvious from the definition of the function that

(2.5)

Remark 2.1.

If is a strictly convex and a smooth Banach space, then for all , if and only if , see Matsushita and Takahashi [14].

To obtain our results, following lemmas are important.

Lemma 2.2 (Kamimura and Takahashi [23]).

Let be a uniformly convex and smooth Banach space and let . Then there exists a continuous strictly increasing and convex function such that and
(2.6)

for all

Lemma 2.3 (Kamimura and Takahashi [23]).

Let be a uniformly convex and smooth real Banach space and let be two sequences of . If and either or is bounded, then .

Lemma 2.4 (Alber [22]).

Let be a nonempty closed convex subset of a smooth real Banach space E and . Then, if and only if
(2.7)

Lemma 2.5 (Alber [22]).

Let be a reflexive strict convex and smooth real Banach space, let be a nonempty closed convex subset of E and let . Then
(2.8)

Lemma 2.6 (Matsushita and Takahashi [14]).

Let be a strictly convex and smooth real Banach space, let be 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.

Let be a subset of a Banach space and let be a family of mappings from into . For a subset of , one says that

(a) satisfies condition AKTT if

(2.9)

(b) satisfies condition AKTT if

(2.10)

Lemma 2.7 (Aoyama et al. [10]).

Let be a nonempty subset of a Banach space and let be a sequence of mappings from into . Let be a subset of with satisfying condition AKTT, then there exists a mapping such that
(2.11)

and

Inspired by Lemma 2.7, Nilsrakoo and Saejung [19] prove the following results.

Lemma 2.8 (Nilsrakoo and Saejung [19]).

Let be a reflexive and strictly convex Banach space whose norm is Fr chet differentiable, let be a nonempty subset of a Banach space , and let be a sequence of mappings from into . Let be a subset of with satisfies condition , then there exists a mapping such that
(2.12)

and

Lemma 2.9 (Nilsrakoo and Saejung [19]).

Let be a reflexive and strictly convex Banach space whose norm is Fr chet differentiable, let be a nonempty subset of a Banach space and let be a sequence of mappings from into . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Then there exists a mapping such that
(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 Saejung [19]. In order to prove the main result, we recall a concept as follows. An operator in a Banach space is closed if and , then .

Theorem 3.1.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a sequence of hemi-relatively nonexpansive mappings from into itself such that is nonempty. Assume that and are sequences in such that and and let a sequence in by the following algorithm be:
(3.1)

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . If is uniformly continuous for all , then converges strongly to , where is the generalized projection from onto .

Proof.

We first show that is closed and convex for each . Obviously, from the definition of , we see that is closed for each . Now we show that is convex for any . Since
(3.2)
this implies that is a convex set. Next, we show that for all . Indeed, let , we have
(3.3)
(3.4)
Substituting (3.4) into (3.3), we have
(3.5)
This means that, for all . Consequently, the sequence is well defined. Moreover, since and , we get
(3.6)

for all . Therefore, is nondecreasing.

By the definition of and Lemma 2.5, we have
(3.7)
for all . Thus, is a bounded sequence. Moreover, by (2.5), we know that is bounded. So, exists. Again, by Lemma 2.5, we have
(3.8)

for all . Thus, as .

Next, we show that is a Cauchy sequence. Using Lemma 2.2, for such that , we have
(3.9)
where is a continuous stricly increasing and convex function with . Then the properties of the function yield that is a Cauchy sequence. Thus, we can say that converges strongly to for some point in . However, since and is bounded, we obtain
(3.10)

Therefore as .

Since , from the definition of , we have
(3.11)
for all . Thus
(3.12)
By using Lemma 2.3, we also have
(3.13)
Since is uniformly norm-to-norm continuous on bounded sets, we have
(3.14)
For each we observe that
(3.15)
It follows that
(3.16)
By (3.14) and , we obtain
(3.17)
Since is uniformly norm-to-norm continuous on bounded sets, we have
(3.18)
By (3.13), we have
(3.19)
Since is uniformly continuous, by (3.13) and (3.18), we obtain
(3.20)
as , and so
(3.21)

Based on the hypothesis, we now consider the following two cases.

Case 1.

satisfies condition . Applying Lemma 2.8 to get
(3.22)

Case 2.

satisfies condition AKTT. Apply Lemma 2.7 to get
(3.23)
Hence
(3.24)
Therefore, from the both two cases, we have
(3.25)
Since is closed and , we have Moreover, by (3.7), we obtain
(3.26)

for all . Therefore, 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 be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a sequence of relatively nonexpansive mappings from into itself such that is nonempty. Assume that and are sequences in such that and and let a sequence in be defined by the following algorithm:
(3.27)

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . If is uniformly continuous for all , then converges strongly to , where is the generalized projection from onto .

Theorem 3.3.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a sequence of hemi-relatively nonexpansive mappings from into itself such that is nonempty. Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(3.28)

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

Proof.

In Theorem 3.1, if for all then (3.1) reduced to (3.28).

Corollary 3.4.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a sequence of relatively nonexpansive mappings from into itself such that is nonempty. Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(3.29)

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

Notice that every uniformly continuous mapping must be a continuous and closed mapping. Then setting for all , in Theorems 3.1 and 3.3, we immediately obtain the following results.

Corollary 3.5.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a closed hemi-relatively nonexpansive mapping such that . Assume that and are sequences in such that and and let a sequence in be defined by the following algorithm:
(3.30)

for , where is the single-valued duality mapping on . If is uniformly continuous, then converges strongly to .

Corollary 3.6.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a closed relatively nonexpansive mapping such that . Assume that and are sequences in such that and and let a sequence in be defined by the following algorithm:
(3.31)

for , where is the single-valued duality mapping on . If is uniformly continuous, then converges strongly to .

Proof.

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

Corollary 3.7.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a closed hemi-relatively nonexpansive mapping from into itself such that . Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(3.32)

for , where is the single-valued duality mapping on . Then converges strongly to .

Corollary 3.8.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a closed relatively nonexpansive mapping from into itself such that . Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(3.33)

for , where is the single-valued duality mapping on . Then converges strongly to .

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

Theorem 3.9.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a sequence of hemi-relatively nonexpansive mappings from into itself such that is nonempty. Assume that and are sequences in such that and and let a sequence in be defined by the following algorithm:
(3.34)

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . If is uniformly continuous for all , then converges strongly to , where is the generalized projection from onto .

Corollary 3.10.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be closed hemi-relatively nonexpansive mappings from into itself such that . Assume that and are sequences in such that and and let a sequence in be defined by the following algorithm:
(3.35)

for , where is the single-valued duality mapping on . If is uniformly continuous, then converges strongly to .

Theorem 3.11.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a sequence of relatively nonexpansive mappings from into itself such that is a nonempty. Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(3.36)

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

Proof.

Putting , for all , in Theorem 3.9 we immediately obtain Theorem 3.11.

Corollary 3.12.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be closed hemi-relatively nonexpansive mappings from into itself such that . Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(3.37)

for , where is the single-valued duality mapping on . Then converges strongly to .

Remark 3.13.

Our results extend and improve the corresponding results in the following senses.

1. (i)

Corollary 3.10 improves Theorem 2.1 of Qin and Su [15] from relatively nonexpansive mappings to more general hemi-relatively nonexpansive mappings.

2. (ii)

Theorem 3.11 improves the algorithm in Theorem 3.1 of Nilsakoo and Saejung [19] from the Mann iteration process to modify Ishikawa iteration process and from countable relatively nonexpansive mappings to more general countable hemi-relatively nonexpansive mappings; that is, we relax the strong restriction . From (i) and (ii), it means that we relax the strongly restriction as from the assumption.

1. (iii)

Corollary 3.12 improves Theorem 3.1 of Matsushita and Takahashi [14] from relatively nonexpansive mappings to more general hemi-relatively nonexpansive mappings in a Banach space.

## 4. Halpern Iterative Scheme

In this section, we prove the strong convergence theorems for finding common fixed points of a countable family of hemi-relatively nonexpansive mappings, which can be viewed as a generalization of the recently result of [15, Theorem 2.2].

Theorem 4.1.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a sequence of hemi-relatively nonexpansive mappings from into itself such that is nonempty. Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(4.1)

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

Proof.

As in the proof of Theorem 3.1, we have that is closed and convex for each .

Next, we show that for all . Indeed, let , we have
(4.2)
This means that, for all From Theorem 3.1, we obtain and exists. Since and hence , we also get
(4.3)

for all . Since , thus, as .

By using the same argument as in Theorem 3.1, we obtain that is a Cauchy sequence, thus converges strongly to for some point in . By using Lemma 2.3, we also have
(4.4)
Since is uniformly norm-to-norm continuous on bounded sets, we have
(4.5)
Observe that
(4.6)
this gives
(4.7)
By (4.5) and , we obtain Since is uniformly norm-to-norm continuous on bounded sets, we have
(4.8)
It follows from (4.4) that as , and since is uniformly norm-to-norm continuous on bounded sets, we get From the conditions , AKTT, Lemmas 2.7 and 2.8, by using the same line as in the proof of Theorem 3.1, the both two cases, we know that
(4.9)
Finally, we prove that , where . Let be a subsequence of such that Replacing , from and , we have . On the other hand, from weakly lower semicontinuity of the norm, we have
(4.10)

From the definition of , since , we have This implies . Using the Kadec-Klee property ([24]) of the space , we obtain that converges strongly to . Since is an arbitrary weakly convergent sequence of , we can conclude that convergence strongly to

Corollary 4.2.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a closed hemi-relatively nonexpansive mapping from into itself such that is nonempty. Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(4.11)

for , where is the single-valued duality mapping on . Then converges strongly to .

Proof.

By setting for all , we immediately obtain the result.

Since every relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, we immediately obtain the following corollaries.

Corollary 4.3.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a sequence of relatively nonexpansive mappings from into itself such that is nonempty. Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(4.12)

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

Corollary 4.4.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a closed relatively nonexpansive mapping from into itself such that is nonempty. Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(4.13)

for , where is the single-valued duality mapping on . Then converges strongly to .

Similarly, as in the proof of Theorem 4.1, we obtain the following result.

Theorem 4.5.

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a sequence of hemi-relatively nonexpansive mappings from into itself such that is nonempty. Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(4.14)

for , where is the single-valued duality mapping on . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition . Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

If , then Theorem 4.5 reduces to the following corollary.

Corollary 4.6 (see [15, Theorem 2.2]).

Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty bounded closed convex subset of . Let be a closed relatively nonexpansive mapping from into itself such that . Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(4.15)

for , where is the single-valued duality mapping on . Then converges strongly to .

## 5. Some Applications to Hilbert Spaces

It is well known that, in the Hilbert space setting, the concepts of hemi-relatively nonexpansive mappings and quasi-nonexpansive mappings are the equivalent. Thus, the following results can be obtained.

Theorem 5.1.

Let be a Hilbert space and let be a nonempty bounded closed convex subset of . Let be a sequence of quasi-nonexpansive mappings from into itself such that is nonempty. Assume that and are sequences in such that and and let a sequence in be defined by the following algorithm:
(5.1)

for . Suppose that for each bounded subset of , the ordered pair satisfies condition AKTT. Let be the mapping from into itself defined by for all and suppose that is closed and . If is uniformly continuous for all , then converges strongly to .

Proof.

Since is an identity operator, we have
(5.2)
for every Therefore
(5.3)

for every and Hence, is quasi-nonexpansive if and only if is hemi-relatively nonexpansive. Then, by Theorem 3.1, we obtain the result.

Theorem 5.2.

Let be a Hilbert space and let be a nonempty bounded closed convex subset of . Let be a sequence of quasi-nonexpansive mappings from into itself such that is nonempty. Assume that is sequence in such that and let a sequence in be defined by the following algorithm:
(5.4)

for . Suppose that for each bounded subset of , the ordered pair satisfies condition AKTT. Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

Proof.

In Theorem 5.1 setting for all , then (5.1) reduces to (5.4).

Theorem 5.3.

Let be a Hilbert space and let be a nonempty bounded closed convex subset of . Let be a sequence of quasi-nonexpansive mappings from into itself such that is nonempty. Assume that is a sequence in such that and let a sequence in be defined by the following algorithm:
(5.5)

for . Suppose that for each bounded subset of , the ordered pair satisfies condition AKTT. Let be the mapping from into itself defined by for all and suppose that is closed and . Then converges strongly to .

## Declarations

### Acknowledgments

The authors would like to thank the referees for the valuable suggestions which helped to improve this manuscript. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand
(2)
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok, 10140, Thailand
(3)
Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak, 63000, Thailand
(4)
Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok, 10400, Thailand

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