Approximation of Common Fixed Points of a Countable Family of Relatively Nonexpansive Mappings

Abstract

We introduce two general iterative schemes for finding a common fixed point of a countable family of relatively nonexpansive mappings in a Banach space. Under suitable setting, we not only obtain several convergence theorems announced by many authors but also prove them under weaker assumptions. Applications to the problem of finding a common element of the fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem are also discussed.

1. Introduction and Preliminaries

Let be a nonempty subset of a Banach space , and let be a mapping from into itself. When is a sequence in , we denote strong convergence of to by and weak convergence by . We also denote the weak convergence of a sequence to in the dual by . A point is an asymptotic fixed point of if there exists in such that and . We denote and by the set of fixed points and of asymptotic fixed points of , respectively. A Banach space is said to be strictly convex if for and . It is also said to be uniformly convex if for each , there exists such that for and . The space is said to be smooth if the limit

(1.1)

exists for all . It is also said to be uniformly smooth if the limit exists uniformly in .

Many problems in nonlinear analysis can be formulated as a problem of finding a fixed point of a certain mapping or a common fixed point of a family of mappings. This paper deals with a class of nonlinear mappings, so-called relatively nonexpansive mappings introduced by Matsushita and Takahashi [1]. This type of mappings is closely related to the resolvent of maximal monotone operators (see [2–4]).

Let be a smooth, strictly convex and reflexive Banach space and let be a nonempty closed convex subset of . Throughout this paper, we denote by the function defined by

(1.2)

where is the normalized duality mapping from to the dual space given by the following relation:

(1.3)

We know that if is smooth, strictly convex, and reflexive, then the duality mapping is single-valued, one-to-one, and onto. The duality mapping is said to be weakly sequentially continuous if implies that (see [5] for more details).

Following Matsushita and Takahashi [6], a mapping is said to be relatively nonexpansive if the following conditions are satisfied:

(R1) is nonempty;

(R2) for all , ;

(R3).

If satisfies (R1) and (R2), then is called relatively quasi-nonexpansive [7]. Obviously, relative nonexpansiveness implies relative quasi-nonexpansiveness but the converse is not true. Relatively quasi-nonexpansive mappings are sometimes called hemirelatively nonexpansive mappings. But we do prefer the former name because in a Hilbert space setting, relatively quasi-nonexpansive mappings are nothing but quasi-nonexpansive.

In [2], Alber introduced the generalized projection from onto as follows:

(1.4)

If is a Hilbert space, then and becomes the metric projection of onto . Alber's generalized projection is an example of relatively nonexpansive mappings. For more example, see [1, 8].

In 2004, Masushita and Takahashi [1, 6] also proved weak and strong convergence theorems for finding a fixed point of a single relatively nonexpansive mapping. Several iterative methods, as a generalization of [1, 6], for finding a common fixed point of the family of relatively nonexpansive mappings have been further studied in [7, 9–14].

Recently, a problem of finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping is studied by Takahashi and Zembayashi in [15, 16]. The purpose of this paper is to introduce a new iterative scheme which unifies several ones studied by many authors and to deduce the corresponding convergence theorems under the weaker assumptions. More precisely, many restrictions as were the case in other papers are dropped away.

First, we start with some preliminaries which will be used throughout the paper.

Lemma 1.1 (see [7, Lemma ]).

Let be a nonempty closed convex subset of a strictly convex and smooth Banach space and let be a relatively quasi-nonexpansive mapping from into itself. Then is closed and convex.

Lemma 1.2 (see [17, Proposition ]).

Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space . Then

(1.5)

for all and .

Lemma 1.3 (see [17]).

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

(1.6)

for all .

Lemma 1.4 (see [17, Proposition ]).

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

Lemma 1.5 (see [2]).

Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space , let and let . Then

(1.7)

Lemma 1.6 (see [18]).

Let be a uniformly convex Banach space and let . Then there exists a strictly increasing, continuous, and convex function such that and

(1.8)

for all and .

We next prove the following three lemmas which are very useful for our main results.

Lemma 1.7.

Let Let be a closed convex subset of a smooth Banach space . Let be a relatively quasi-nonexpansive mapping from into and let be a family of relatively quasi-nonexpansive mappings from into itself such that . The mapping is defined by

(1.9)

for all and , such that . If and , then

(1.10)

Proof.

The proof of this lemma can be extracted from that of Lemma 1.8; so it is omitted.

If has a stronger assumption, we have the following lemma.

Lemma 1.8.

Let be a closed convex subset of a uniformly smooth Banach space . Let . Then, there exists a strictly increasing, continuous, and convex function such that and for each relatively quasi-nonexpansive mapping and each finite family of relatively quasi-nonexpansive mappings such that ,

(1.11)

for all and , where

(1.12)

and , such that .

Proof.

Let . From Lemma 1.6 and is uniformly convex, then there exists a strictly increasing, continuous, and convex function such that and

(1.13)

for all and . Let and be relatively quasi-nonexpansive for all such that . For and . It follows that

(1.14)

and hence . Consequently, for ,

(1.15)

Then

(1.16)

Thus

(1.17)

Lemma 1.9.

Let be a closed convex subset of a uniformly smooth and strictly convex Banach space . Let be a relatively quasi-nonexpansive mapping from into and let be a family of relatively quasi-nonexpansive mappings from into itself such that . The mapping is defined by

(1.18)

for all and , such that . Then, the following hold:

(1),

(2) is relatively quasi-nonexpansive.

Proof.

() Clearly, . We want to show the reverse inclusion. Let and . Choose

(1.19)

From Lemma 1.8, we have

(1.20)

From for all and by the properties of , we have

(1.21)

for all . From is one to one, we have

(1.22)

for all . Consider

(1.23)

Thus .

• (2) It follows directly from the above discussion.

2. Weak Convergence Theorem

Theorem 2.1.

Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space . Let be a family of relatively quasi-nonexpansive mappings and let be a family of relatively quasi-nonexpansive mappings such that . Let the sequence be generated by ,

(2.1)

for any , for all , such that for all . Then converges strongly to , where is the generalized projection of onto .

Proof.

Let . Put

(2.2)

From Lemma 1.7, we have

(2.3)

Therefore exists. This implies that and are bounded for all .

Let . From (2.3) and , we have

(2.4)

Consequently,

(2.5)

In particular,

(2.6)

This implies that exists. This together with the boundedness of gives . Using Lemma 1.3, there exists a strictly increasing, continuous, and convex function such that and

(2.7)

Since is a convergent sequence, it follows from the properties of that is a Cauchy sequence. Since is closed, there exists such that .

We first establish weak convergence theorem for finding a common fixed point of a countable family of relatively quasi-nonexpansive mappings. Recall that, for a family of mappings with , we say that satisfies the NST-condition [19] if for each bounded sequence in ,

(2.8)

where denotes the set of all weak subsequential limits of a sequence .

Theorem 2.2.

Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space . Let be a family of relatively quasi-nonexpansive mappings satisfying NST-condition and let be a family of relatively nonexpansive mappings such that and suppose that

(2.9)

for all , and . Let the sequence be generated by ,

(2.10)

for any , for all , such that for all , for all . If is weakly sequentially continuous, then converges weakly to , where .

Proof.

Let . From Theorem 2.1, exists and hence and are bounded for all . Let

(2.11)

By Lemma 1.8, there exists a strictly increasing, continuous, and convex function such that and

(2.12)

In particular, for all ,

(2.13)

Hence,

(2.14)

for all . Since for all and the properties of , we have

(2.15)

for all . Since is uniformly norm-to-norm continuous on bounded sets, we have

(2.16)

for all . Since is bounded, there exists a subsequence of such that . Since is relatively nonexpansive, for all .

We show that . Let

(2.17)

We note from (2.15) that

(2.18)

Since is uniformly norm-to-norm continuous on bounded sets, it follows that

(2.19)

Moreover, by (2.9) and the existence of , we have

(2.20)

It follows from Lemma 1.4 that . From (2.19) and , we have . Since satisfies NST-condition, we have . Hence .

Let . From Lemma 1.5 and , we have

(2.21)

From Theorem 2.1, we know that . Since is weakly sequentially continuous, we have

(2.22)

Moreover, since is monotone,

(2.23)

Then

(2.24)

Since is strictly convex, . This implies that and hence .

We next apply our result for finding a common element of a fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem. This problem is extensively studied in [11, 14–16]. Let be a subset of a Banach space and let be a bifunction. The equilibrium problem for a bifunction is to find such that for all . The set of solutions above is denoted by , that is

(2.25)

To solve the equilibrium problem, we usually assume that a bifunction satisfies the following conditions ( is closed and convex):

(A1) for all ;

(A2) is monotone, that is, ;

(A3)for all , ;

(A4)for all , is convex and lower semicontinuous.

The following lemma gives a characterization of a solution of an equilibrium problem.

Lemma 2.3.

Let be a nonempty closed convex subset of a Banach space . Let be a bifunction from satisfying (A1)–(A4). Suppose that . Then if and only if for all .

Proof.

Let , then for all . From (A2), we get that for all .

Conversely, assume that for all . For any , let

(2.26)

Then and hence

(2.27)

So for all . From (A3), we have

(2.28)

Hence .

Takahashi and Zembayashi proved the following important result.

Lemma 2.4 (see [15, Lemma ]).

Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space . Let be a bifunction from satisfying (A1)–(A4). For and , define a mapping as follows:

(2.29)

for all . Then, the following hold:

(1) is single-valued;

(2) is a firmly nonexpansive-type mapping [20], that is, for all

(2.30)

(3);

(4) is closed and convex.

We now deduce Takahashi and Zembayashi's recent result from Theorem 2.2.

Corollary 2.5 (see [15, Theorem ]).

Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space . Let be a bifunction from to satisfying (A1)–(A4) and let be a relatively nonexpansive mapping from into itself such that . Let the sequence be generated by ,

(2.31)

for every , satisfying and for some . If is weakly sequentially continuous, then converges weakly to , where .

Proof.

Put where is defined by Lemma 2.4. Then . By reindexing the sequences and of this iteration, we can apply Theorem 2.2 by showing that the family satisfies the condition (2.9) and NST-condition. It is proved in [15, Lemma ] that

(2.32)

To see that satisfies NST-condition, let be a bounded sequence in such that and . Suppose that there exists a subsequence of such that . Then . Since is uniformly continuous on bounded sets and , we have

(2.33)

From the definition of , we have

(2.34)

Since

(2.35)

and is lower semicontinuous and convex in the second variable, we have

(2.36)

Thus for all . From Lemma 2.3, we have . Then satisfies the NST-condition. From Theorem 2.2 where , converges weakly to , where .

Using the same proof as above, we have the following result.

Corollary 2.6 (see [11, Theorem ]).

Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfies (A1)–(A4) and let be two relatively nonexpansive mappings such that . Let the sequence be generated by the following manner:

(2.37)

Assume that , and are three sequences in satisfying the following restrictions:

(a);

(b), ;

(c) for some .

If is weakly sequentially continuous, then converges weakly to , where .

The following result also follows from Theorem 2.2.

Corollary 2.7 (see [9, Theorem ]).

Let be a uniformly smooth and uniformly convex Banach space and let be a nonempty closed convex subset of . Let be a finite family of relatively nonexpansive mappings from into itself such that is a nonempty and let and be sequences such that and for all and for all . Let be a sequence of mappings defined by

(2.38)

for all and let the sequence be generated by and

(2.39)

Then the following hold:

(1)the sequence is bounded and each weak subsequential limit of belongs to ;

(2)if the duality mapping from into is weakly sequentially continuous, then converges weakly to the strong limit of .

Proof.

Since is relatively nonexpansive, the family satisfies the NST-condition. Moreover, and

(2.40)

Thus the conclusions of this corollary follow.

3. Strong Convergence Theorem

In this section, we prove strong convergence of an iterative sequence generated by the hybrid method in mathematical programming. We start with the following useful common tools.

Lemma 3.1.

Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space . Let and be families of relatively quasi-nonexpansive mappings such that , and

(3.1)

for all , and . Let be such that and are bounded for all , and

(3.2)

where for all and satisfy for all , for all and . Then the following statements hold:

(1) for all ,

(2),

(3),

(4)if , then for all ,

(5)if , then and .

Proof.

() Since and is uniformly norm-to-norm continuous on bounded sets,

(3.3)

We note here that is also bounded. For any , we have

(3.4)

 () Let . Using (3.1) and the relative quasi-nonexpansiveness of each , we have

(3.5)

By Lemma 1.4 and the boundedness of , we have

(3.6)

  () Since

(3.7)

we have .

() Assume that . From , we get that . Since is uniformly norm-to-norm continuous on bounded sets, we have

(3.8)

So,

(3.9)

From (3.8), we have

(3.10)

It follows from for all that

(3.11)

for all . Since is uniformly norm-to-norm continuous on bounded sets and , we have

(3.12)

for all , as desired.

() Assume that . From the assumption and (), we have

(3.13)

Hence and .

Lemma 3.2 (see [21, Lemma ]).

Let be a closed convex subset of a strictly convex, smooth and reflexive Banach space satisfying Kadec-Klee property. Let and be a sequence in such that and for all . Then .

Recall that a Banach space satisfies Kadec–Klee property if whenever is a sequence in with and , it follows that .

3.1. The CQ-Method

Theorem 3.3.

Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space . Let be a family of relatively quasi-nonexpansive mappings satisfying NST-condition and let be a family of relatively nonexpansive mappings such that , and

(3.14)

for all , and . Let the sequence be generated by

(3.15)

for every , for all and satisfying for all , for all . Then converges strongly to .

Proof.

The proof is broken into 4 steps.

Step 1 ( is well defined).

First, we show that is closed and convex. Clearly, is closed and convex. Since

(3.16)

then is closed and convex. Thus is closed and convex.

We next show that . Let . Then, from Lemma 1.7,

(3.17)

Thus . Hence for all .

Next, we show by induction that for all . Since , we have

(3.18)

Suppose that for some . From and the definition of the generalized projection, we have

(3.19)

for all . From ,

(3.20)

for all . Hence , and we also have . So, we have for all and hence the sequence is well defined.

Step 2 ().

From the definition of , we have . Using Lemma 1.2, we get

(3.21)

for all . In particular, since and ,

(3.22)

(3.23)

for all . This implies that exists and is bounded. Moreover, from (3.21) and ,

(3.24)

Hence

(3.25)

It follows from that

(3.26)

From (3.25), (3.26), and Lemma 1.4, we have

(3.27)

So . Using Lemma 3.1(4), we get that

(3.28)

for all . Since each is relatively nonexpansive,

(3.29)

Step 3 ().

Let . From Lemma 3.1(2), we have

(3.30)

and . It follows from NST-condition that .

Step 4 ().

From Steps 2 and 3, we have . The conclusion follows by Lemma 3.2 and (3.23).

We apply Theorem 3.3 and the proof of Corollary 2.5 and then obtain the following result.

Corollary 3.4.

Let , , , be as in Corollary 2.5. Let the sequence be generated by

(3.31)

for every , satisfying and for some . Then, converges strongly to , where is the generalized projection of onto .

Remark 3.5.

Corollary 3.4 improves the restriction on of [15, Theorem ]. In fact, it is assumed in [15, Theorem ] that .

3.2. The Monotone CQ-Method

Let be a closed subset of a Banach space . Recall that a mapping is closed if for each in , if and , then . A family of mappings with is said to satisfy the -condition if for each bounded sequence in ,

(3.32)

Remark 3.6.

• (1) If satisfies NST-condition, then satisfies -condition.

• (2) If and is closed, then satisfies -condition.

Theorem 3.7.

Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space. Let be a family of relatively quasi-nonexpansive mappings satisfying -condition and let be a family of closed relatively quasi-nonexpansive mappings such that , and

(3.33)

for all , and . Let the sequence be generated by

(3.34)

for every , satisfying and for all . Then converges strongly to .

Proof.

Step 1 ( is well defined).

This step is almost the same as Step 1 of the proof of Theorem 3.3, so it is omitted.

Step 2 ( is a Cauchy sequence in ).

We can follow the proof of Theorem 3.3 and conclude that

(3.35)

exists. Moreover, as for all and ,

(3.36)

Since is bounded, it follows from Lemma 1.3 that there exists a strictly increasing, continuous, and convex function such that and

(3.37)

Since exists, we have that is a Cauchy sequence. Therefore, for some .

Step 3 ().

Since , we have

(3.38)

By Lemma 1.4 and the boundedness of , we have

(3.39)

So, we have . Using Lemma 3.1(4), we get that

(3.40)

for all . Since each is closed, .

Step 4 ().

Let . From Lemma 3.1(2), we have and . It follows from -condition that .

Step 5 ().

From Steps 3 and 4, we have . The conclusion follows by Lemma 3.2 and (3.23).

Letting identity and yield the following result.

Corollary 3.8 (see [12, Theorem ]).

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth real Banach space . Let be a closed relatively quasi-nonexpansive mapping such that . Assume that is a sequence in such that . Define a sequence in by the following algorithm:

(3.41)

Then converges strongly to .

Letting identity and yield the following result.

Corollary 3.9 (see [13, Theorem ]).

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth real Banach space . Let be two closed relatively quasi-nonexpansive mappings from into itself such that . Define a sequence in be the following algorithm:

(3.42)

with the conditions: with and

(1);

(2);

(3) for some .

Then converges strongly to .

Remark 3.10.

Using Theorem 3.7, we can show that the conclusion of Corollary 3.9 remains true under the more general restrictions on and :

(1) are arbitrary;

(2) and .

3.3. The Shrinking Projection Method

Theorem 3.11.

Let be as in Theorem 3.7. Let the sequence be generated by

(3.43)

for every , for all and satisfies for all , for all . Then converges strongly to .

Proof.

The proof is almost the same as the proofs of Theorems 3.3 and 3.7; so it is omitted.

In particular, applying Theorem 3.11 gives the following result.

Corollary 3.12.

Let be as in Corollary 2.5. Let the sequence be generated by , and

(3.44)

for every , where is the duality mapping on . Assume that satisfies and for some . Then converges strongly to , where is the generalized projection of onto .

Remark 3.13.

Corollary 3.12 improves the restriction on of [16, Theorem ]. In fact, it is assumed in [16, Theorem ] that .

Corollary 3.14 (see [11, Theorem ]).

Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying (A1)–(A4) and let be two closed relatively quasi-nonexpansive mappings such that . Let the sequence be generated by the following manner:

(3.45)

Assume that , and are three sequences in satisfying the restrictions:

(a);

(b), ;

(c) for some .

Then converges strongly to .

Remark 3.15.

The conclusion of Corollary 3.14 remains true under the more general assumption; that is, we can replace (b) by the following one:

(b′) and .

We also deduce the following result.

Corollary 3.16 (see [14, Theorem ]).

Let be as in Corollary 3.14. Let the sequences , and be generated by the following:

(3.46)

Assume that , and are three sequences in satisfying the following restrictions:

(a);

(b) for all and ;

(c), ;

(d) for some .

Then and converge strongly to .

Remark 3.17.

The conclusion of Corollary 3.16 remains true under the more general restrictions; that is, we replace (b) and (c) by the following one:

(b′) and .

Corollary 3.18 (see [10, Theorem ]).

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space . Let be a family of relatively nonexpansive mappings such that and let . For and , define a sequence of as follows:

(3.47)

where satisfies the following restrictions:

(i) for all and ;

(ii) for all , for all . If

(a)either for all or

(b) and for all .

then the sequence converges strongly to .

Remark 3.19.

The conclusion of Corollary 3.18 remains true under the more general restrictions on :

(1) are arbitrary.

(2) for all .