Strong Convergence to Common Fixed Points of Countable Relatively Quasi-Nonexpansive Mappings

We prove that a sequence generated by the monotone CQ-method converges strongly to a common fixed point of a countable family of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. Our result is applicable to a wide class of mappings.

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 quasi-nonexpansive if and

It is easy to see that if is nonexpansive with , then it is quasi-nonexpansive. There are many methods for approximating fixed points of a quasi-nonexpansive 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 infinite-dimensional 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), so-called the monotone CQ method for nonexpansive mapping, as follows:

and prove that the sequence converges strongly to

We now recall some definitions concerning relatively quasi-nonexpansive 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 quasi-nonexpansive.

Several articles have appeared providing method for approximating fixed points of relatively quasi-nonexpansive 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 quasi-nonexpansive mappings in a Banach space. Consequently, we obtain strong convergence theorems for quasi-nonexpansive 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.

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 quasi-nonexpansive 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 quasi-nonexpansive 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 quasi-nonexpansive 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 norm-to-norm 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.

In this section, we establish strong convergence theorem for finding common fixed points of a countable family of relatively quasi-nonexpansive mappings in a Banach space.

This theorem generalizes a recent theorem by Su et al. [21, Theorem 3.1]. It is noted that relative quasi-nonexpansiveness 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 quasi-nonexpansive mappings are just quasi-nonexpansive.

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 quasi-nonexpansive 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 norm-to-norm 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 norm-to-norm 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 quasi-nonexpansive 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 quasi-nonexpansive mappings satisfying such an assumption by using convex combination of a given sequence of relatively quasi-nonexpansive 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 quasi-nonexpansive 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 quasi-nonexpansive 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 2357-2358], 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 quasi-nonexpansive mappings with a common fixed point and let be the identity operator, one defines a sequence of relatively quasi-nonexpansive mappings from into by

for all and . Then the sequence in defined by (3.1) converges strongly to .

In Hilbert spaces, relatively quasi-nonexpansive mappings and quasi-nonexpansive 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 quasi-nonexpansive 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 quasi-nonexpansive if and only if is relatively quasi-nonexpansive. 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 quasi-nonexpansive 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 quasi-nonexpansive 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 quasi-nonexpansive 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 .

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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