We prove that a sequence generated by the monotone CQmethod converges strongly to a common fixed point of a countable family of relatively quasinonexpansive mappings in a uniformly convex and uniformly smooth Banach space. Our result is applicable to a wide class of mappings.
1. Introduction
Let be a real Banach space, let be a nonempty closed convex subset of , and let be a mapping. Recall that is nonexpansive if
We denote by the set of fixed points of , that is, . A mapping is said to be quasinonexpansive if and
It is easy to see that if is nonexpansive with , then it is quasinonexpansive. There are many methods for approximating fixed points of a quasinonexpansive mapping. In 1953, Mann [1] introduced the iteration as follows: a sequence is defined by
where the initial guess element is arbitrary and is a real sequence in . Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [2]. In an infinitedimensional Hilbert space, Mann iteration can yield only weak convergence (see [3, 4]). Attempts to modify the Mann iteration method (1.3) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.3) for a nonexpansive mapping from into itself in a Hilbert space:
where denotes the metric projection from a Hilbert space onto a closed convex subset of and prove that the sequence converges strongly to A projection onto intersection of two halfspaces is computed by solving a linear system of two equations with two unknowns (see [6, Section 3]).
Recently, Su and Qin [7] modified iteration (1.4), socalled the monotone CQ method for nonexpansive mapping, as follows:
and prove that the sequence converges strongly to
We now recall some definitions concerning relatively quasinonexpansive mappings and what have been proved until now. Let be a real smooth Banach space with norm and let be the dual of . Denote by the pairing between and . The normalized duality mapping from to is defined by
The reader is directed to [8] (and its review [9]), where the properties on the duality mapping and several related topics are presented. The function is defined by
Let be a mapping from into . A point in is said to be an asymptotic fixed point of [10] if contains a sequence which converges weakly to and . The set of asymptotic fixed points of is denoted by . We say that the mapping is relatively nonexpansive if the following conditions are satisfied:
(R1)
(R2) for each
(R3)
If satisfies (R1) and (R2), then is called relatively quasinonexpansive.
Several articles have appeared providing method for approximating fixed points of relatively quasinonexpansive mappings [11–16]. Matsushita and Takahashi [12] introduced the following iteration: a sequence defined by
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 . They prove that the sequence converges weakly to a fixed point of . Moreover, Matsushita and Takahashi [13] proposed the following modification of iteration (1.8):
and prove that the sequence converges strongly to
Recently, Kohsaka and Takahashi [11] extended iteration (1.8) to obtain a weak convergence theorem for common fixed points of a finite family of relatively nonexpansive mapping by the following iteration:
where and with for all .
Employing the ideas of Su and Qin [7], and of Aoyama et al. [17], we modify iterations (1.5), (1.8)–(1.10) to obtain strong convergence theorems for common fixed points of countable relatively quasinonexpansive mappings in a Banach space. Consequently, we obtain strong convergence theorems for quasinonexpansive mappings in a Hilbert space without using demiclosedness principle. Moreover, we introduce a new certain condition for an infinite family of mappings which is inspired by Aoyama et al. [17], and we also show how to generate a corresponding sequence of mappings satisfying our condition.
2.preliminaries
Throughout the paper, let be a real Banach space. We say that is strictly convex if the following implication holds for :
It is also said to be uniformly convex if for any , there exists such that
It is known that if is uniformly convex Banach space, then is reflexive and strictly convex. A Banach space is said to be smooth if
exists for each . In this case, the norm of is said to be Gâteaux differentiable. The space is said to have uniformly Gâteaux differentiable norm if for each , the limit (2.3) is attained uniformly for The norm of is said to be Fréchet 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 .
We also know the following properties (see, e.g., [18] for details).
(a) (, resp.) is uniformly convex if and only if (, resp.) is uniformly smooth.
(b) for each
(c) If is reflexive, then is a mapping of onto
(d) If is strictly convex, then for all .
(e) If is smooth, then is single valued.
(f)If has a Fréchet differentiable norm, then is norm to norm continuous.
(g)If is uniformly smooth, then is uniformly norm to norm continuous on each bounded subset of .
(h) If is a Hilbert space, then is the identity operator.
Let be a smooth Banach space. The function is defined by
It is obvious from the definition of the function that
Moreover, we know the following results.
Lemma(see 2.1 (see [13, Remark 2.1]).
Let be a strictly convex and smooth Banach space, then if and only if .
Lemma(see 2.2 (see [11, Lemma 2.5]).
Let be a uniformly convex and smooth Banach space and let Then there exists a continuous, strictly increasing, and convex function such that and
for all .
Let be a nonempty closed convex subset of . Suppose that is reflexive, strictly convex, and smooth. It is known that [19] for any , there exists a unique point such that
Following Alber [20], we denote such an by . The mapping is called the generalized projection from onto . It is easy to see that in a Hilbert space, the mapping coincides with the metric projection . Concerning the generalized projection, the following are well known.
Lemma(see 2.3 (see [19, Proposition 4]).
Let be a nonempty closed convex subset of a smooth Banach space and let . Then
Lemma(see 2.4 (see [19, Proposition 5]).
Let be a reflexive, strictly convex, and smooth Banach space, let be a nonempty closed convex subset of , and let . Then
Dealing with the generalized projection from onto the fixed point set of a relatively quasinonexpansive mapping, we get the following result.
Lemma 2.5.
Let be a strictly convex and smooth Banach space, let be a nonempty closed convex subset of , and let be a relatively quasinonexpansive mapping from into . Then is closed and convex.
Proof.
The proof of [13, Proposition 2.4] does not invoke condition (R3) at all. So the conclusion holds for relatively quasinonexpansive mappings as well.
Let be a subset of a Banach space and let be a family of mappings from into . For a subset of , we say that
(i) satisfies condition AKTT if
(ii) satisfies condition ^{*}AKTT if
Aoyama et al. [17, Lemma 3.2] prove the following result which is very useful in our main result.
Lemma 2.6.
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
and .
Inspired by the preceding lemma, we have the following result.
Lemma 2.7.
Let be a reflexive and strictly convex Banach space whose norm is Fréchet differentiable, let be a nonempty subset of , 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
and .
Proof.
For , we show that is a Cauchy sequence in . Let . By the condition ^{*}AKTT of , there exists such that
In particular, if , then
Hence, is a Cauchy sequence in . It follows then that exists for all . Moreover, it is noted that the convergence is uniform on . Since is reflexive and strictly convex, is bijective and we can define a mapping from into such that
Since has a Fréchet differentiable norm, is normtonorm continuous and hence
This completes the proof.
Combining Lemmas 2.6 and 2.7, we obtain a crucial tool for our main result.
Lemma 2.8.
Let be a reflexive and strictly convex Banach space whose norm is Fréchet differentiable, let be a nonempty subset of , 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 ^{*}AKTT. Then there exists a mapping such that
Proof.
To see that is well defined, we suppose that satisfies condition AKTT and condition ^{*}AKTT. Then, by Lemmas 2.6 and 2.7, there exist and such that
Lemma(see 2.9 (see [11, Lemma 3.2]).
Let be a reflexive, strictly convex, and smooth Banach space, let , and let with . If is a finite sequence in such that
then
Lemma 2.10.
Let be a strictly convex Banach space and let with . If is a sequence in such that and converge, and
then is a constant sequence.
Proof.
Suppose that for some . Then, by the strict convexity of ,
It follows that
This is a contradiction.
3. Main Results
In this section, we establish strong convergence theorem for finding common fixed points of a countable family of relatively quasinonexpansive mappings in a Banach space.
This theorem generalizes a recent theorem by Su et al. [21, Theorem 3.1]. It is noted that relative quasinonexpansiveness considered in the paper and hemirelative nonexpansiveness of [21] are the same. We do prefer the former name because in a Hilbert space setting, relatively quasinonexpansive mappings are just quasinonexpansive.
Recall that 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 closed convex subset of . Let be a sequence of relatively quasinonexpansive mappings from into such that is nonempty and let be a sequence in defined as follows:
where is a sequence in with . Suppose that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition ^{*}AKTT. Let be the mapping from into defined by for all and suppose that is closed and . Then converges strongly to .
Proof.
We first note that each and are closed and convex. This follows since is equivalent to
It is clear that . Next, we show that
Suppose that for some . Let . Then
This implies that From and by Lemma 2.3, we have
In particular,
and hence . It follows that
By induction, (3.3) holds. This implies that is well defined. It follows from the definition of and Lemma 2.3 that . Since , we have
Therefore, is nondecreasing. Using and Lemma 2.4, we have
for all for all Therefore, is bounded. So
In particular, by (2.5), the sequence is bounded. This implies that is bounded. Noticing again that , and for any positive integer , we have . By Lemma 2.4,
Using Lemma 2.2, we have, for with ,
where is a continuous, strictly increasing, and convex function with . Then the properties of the function yield that is a Cauchy sequence in so there exists such that In view of and the definition of , we also have
It follows that
By using Lemma 2.2, we obtain
Since is uniformly normtonorm continuous on bounded sets, we have
On the other hand, we have, for each ,
and hence
From (3.16) and , we obtain
Since is uniformly normtonorm continuous on bounded sets, we have
It follows from (3.15) that
and so
Case 1.
satisfies condition AKTT. We apply Lemma 2.6 to get
Case 2.
satisfies condition ^{*}AKTT. It follows from Lemma 2.7 that
Hence,
From both cases, we obtain
Since is closed and , we have . Furthermore, by (3.9),
Hence, .
Corollary(see 3.2 (see [21, Theorem 3.1]).
Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a closed relatively quasinonexpansive mapping from into such that is nonempty and let be a sequence in defined as follows:
where is a sequence in with . Then converges strongly to
Remark 3.3.
If, in Theorem 3.1, is continuous for each , then the mapping is continuous and closed.
In our main theorem, we assume that for each bounded subset of , the ordered pair satisfies either condition AKTT or condition ^{*}AKTT. As in [17], we can generate a sequence of relatively quasinonexpansive mappings satisfying such an assumption by using convex combination of a given sequence of relatively quasinonexpansive mappings with a nonempty common fixed point set.
Let be a family of positive real numbers with indices , with such that
(i) for every ;
(ii) for every ; and
(iii).
Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of . For a sequence of continuous relatively quasinonexpansive mappings with a common fixed point and is the identity mapping, we define a sequence of mappings from into by
for and . We note that
For , let . Then
for all . Then, for all and fix ,
that is,
By Lemma 2.9, we have . So
This implies that
and so
Then, by (3.31), we have that is a sequence of relatively quasinonexpansive mappings. Let be a bounded subset of and let . By (2.5), we have
and hence
for all and . Let . For and , we have
Therefore,
It follows from (iii) that
By Lemma 2.7, we can define a mapping by
Using the same argument presented in the proof of [17, pages 23572358], we have
For each , the series converges absolutely and
This implies that
It is obvious that
Let and fix . Then
It follows that
By the strict convexity of and Lemma 2.10,
Since is one to one,
So Therefore,
This together with (3.36) and (3.46) gives
Hence, we obtain that satisfies all the conditions of our main theorem. Now, we have the following result.
Theorem 3.4.
Let be a uniformly convex and uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a family of positive real numbers with indices , with such that
(i) for every ;
(ii) for every ;
(iii).
Let be a sequence of continuous relatively quasinonexpansive mappings with a common fixed point and let be the identity operator, one defines a sequence of relatively quasinonexpansive mappings from into by
for all and . Then the sequence in defined by (3.1) converges strongly to .
4. Deduced Theorems
In Hilbert spaces, relatively quasinonexpansive mappings and quasinonexpansive mappings are the same. We obtain the following result.
Theorem 4.1.
Let be a nonempty closed convex subset of a Hilbert space . Let be a sequence of quasinonexpansive mappings from into such that is nonempty and let be a sequence in defined as follows:
where is a sequence in with . Suppose that for each bounded subset of , the ordered pair satisfies condition AKTT. Let be the mapping from into defined by for all and suppose that is closed and . Then converges strongly to .
Proof.
Since is an identity operator, we have
for every . Therefore,
for every and . Hence, is quasinonexpansive if and only if is relatively quasinonexpansive. Then, by Theorem 3.1, we obtain the result.
Corollary 4.2 (see [22, Theorem 2.1]).
Let be a nonempty closed convex subset of a Hilbert space . Let be a closed quasinonexpansive mapping from into such that is nonempty and let be a sequence in defined as follows:
where is a sequence in with . Then converges strongly to .
We give an example of a countable family of quasinonexpansive mappings which are not nonexpansive but satisfy all the requirements of our main theorem.
Example 4.3.
Let with the usual norm. For , we define a mapping on by
for all . Then and
So is a sequence of quasinonexpansive mappings. Let , then
for all . It follows that
We now define a mapping on by
Hence, the sequence satisfies all conditions in our main result. We also note that each is neither nonexpansive nor relatively nonexpansive. Actually, above fails to have the condition (R3). Let be a sequence define by . Then
This implies that and .
Acknowledgments
The authors would like to thank Professor Simeon Reich and the referee for the valuable suggestions on the manuscript. Satit Saejung was supported by the Commission on Higher Education and the Thailand Research Fund (MRG4980022).
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