Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups

Abstract

We prove strong convergence theorems for countable families of asymptotically nonexpansive mappings and semigroups in Hilbert spaces. Our results extend and improve the recent results of Nakajo and Takahashi (2003) and of Zegeye and Shahzad (2008) from the class of nonexpansive mappings to asymptotically nonexpansive mappings.

1. Introduction

Throughout this paper, Let be a real Hilbert space with inner product and norm , and we write to indicate that the sequence converges strongly to . Let be a nonempty closed convex subset of , and let be a mapping. Recall that is nonexpansive if , for all . We denote the set of fixed points of by , that is, . A mapping is said to be asymptotically nonexpansive if there exists a sequence with for all , , and

(1.1)

Mann's iterative algorithm was introduced by Mann [1] in 1953. This iteration process is now known as Mann's iteration process, which is defined as

(1.2)

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

In 1967, Halpern [2] first introduced the following iteration scheme:

(1.3)

for all , where and is a sequence in . This iteration process is called a Halpern-type iteration.

Recall also that a one-parameter family of self-mappings of a nonempty closed convex subset of a Hilbert space is said to be a (continuous) Lipschitzian semigroup on if the following conditions are satisfied:

(a),

(b), for all , ;

(c)for each , the map is continuous on

(d)there exists a bounded measurable function such that, for each , , for all

A Lipschitzian semigroup is called nonexpansive if for all , and asymptotically nonexpansive if . We denote by the set of fixed points of the semigroup , that is, .

In 2003, Nakajo and Takahashi [3] proposed the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space :

(1.4)

where denotes the metric projection from onto a closed convex subset of . They proved that the sequence converges weakly to a fixed point of . Moreover, they introduced and studied an iteration process of a nonexpansive semigroup in a Hilbert space :

(1.5)

In 2006, Kim and Xu [4] adapted iteration (1.4) to an asymptotically nonexpansive mapping in a Hilbert space :

(1.6)

where as . They also proved that if for all and for some , then the sequence converges weakly to a fixed point of . Moreover, they modified an iterative method (1.5) to the case of an asymptotically nonexpansive semigroup in a Hilbert space :

(1.7)

where as .

In 2007, Zegeye and Shahzad [5] developed the iteration process for a finite family of asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups with a closed convex bounded subset of a Hilbert space :

(1.8)

where as and

(1.9)

where as , with , for each

Recently, Su and Qin [6] modified the hybrid iteration method of Nakajo and Takahashi through the monotone hybrid method, and to prove strong convergence theorems.

In 2008, Takahashi et al. [7] proved strong convergence theorems by the new hybrid methods for a family of nonexpansive mappings and nonexpansive semigroups in Hilbert spaces:

(1.10)

where , and

(1.11)

where , and .

In this paper, motivated and inspired by the above results, we modify iteration process (1.4)–(1.11) by the new hybrid methods for countable families of asymptotically nonexpansive mappings and semigroups in a Hilbert space, and to prove strong convergence theorems. Our results presented are improvement and extension of the corresponding results in [3, 5–8] and many authors.

2. Preliminaries

This section collects some lemmas which will be used in the proofs for the main results in the next section.

Lemma 2.1.

Here holds the identity in a Hilbert space :

(2.1)

for all and .

Using this Lemma 2.1, we can prove that the set of fixed points of is closed and convex. Let be a nonempty closed convex subset of . Then, for any , there exists a unique nearest point in , denoted by , such that for all , where is called the metric projection of onto . We know that for and , is equivalent to for all . We know that a Hilbert space satisfies Opial's condition, that is, for any sequence with , the inequality

(2.2)

hold for every with . We also know that has the Kadec-Klee property, that is, and imply . In fact, from

(2.3)

we get that a Hilbert space has the Kadec-Klee property.

Let be a nonempty closed convex subset of a Hilbert space . Motivated by Nakajo et al. [9], we give the following definitions: Let and be families of nonexpansive mappings of into itself such that , where is the set of all fixed points of and is the set of all common fixed points of . We consider the following conditions of and (see [9]):

(i)NST-condition (I). For each bounded sequence , implies that for all .

(ii)NST-condition (II). For each bounded sequence , implies that for all .

(iii)NST-condition (III). There exists with such that for every bounded subset of , there exists such that holds for all and

Lemma 2.2.

Let be a nonempty closed convex subset of and let be a nonexpansive mapping of into itself with . Then, the following hold:

(i) with and satisfy the condition (I) with .

(ii) with and satisfy the condition (I) with

Lemma 2.3 (Opial [10]).

Let be a closed convex subset of a real Hilbert space and let be a nonexpansive mapping such that . If is a sequence in such that and , then .

Lemma 2.4 (Lin et al. [11]).

Let be an asymptotically nonexpansive mapping defined on a bounded closed convex subset of a bounded closed convex subset of a Hilbert space . If is a sequence in such that and , then .

Lemma 2.5 (Nakajo and Takahashi [3]).

Let be a real Hilbert space. Given a closed convex subset and points . Given also a real number . The set is convex and closed.

Lemma 2.6 (Kim and Xu [4]).

Let be a nonempty bounded closed convex subset of and be an asymptotically nonexpansive semigroup on . If is a sequence in satisfying the properties

(a);

(b),

then .

Lemma 2.7 (Kim and Xu [4]).

Let be a nonempty bounded closed convex subset of and be an asymtotically nonexpansive semigroup on . Then it holds that

(2.4)

3. Strong Convergence for a Family of Asymptotically Nonexpansive Mappings

Theorem 3.1.

Let be a nonempty bounded closed convex subset of a Hilbert space and let for be a countable family of asymptotically nonexpansive mapping with sequence for , respectively. Assume such that for all and as . Let . Further, suppose that satisfies NST-condition (I) and (III) with T. Define a sequence in by the following algorithm:

(3.1)

where as . Then converges in norm to .

Proof.

We first show that is closed and convex for all . From the Lemma 2.5, it is observed that is closed and convex for each .

Next, we show that for all . Indeed, let , we have

(3.2)

Thus and hence for all . Thus is well defined.

From and , we have

(3.3)

So, for , we have

(3.4)

for all . This implies that

(3.5)

hence

(3.6)

for all Therefore is nondecreasing.

From , we have

(3.7)

Using , we also have

(3.8)

So, for , we have

(3.9)

This implies that

(3.10)

Thus, is bounded. So, exists.

Next, we show that . From (3.3), we have

(3.11)

Since exists, we conclude that .

Since , we have which implies that . Now we claim that as for all . We first show that as . Indeed, by the definition of , we have

(3.12)

for all and it follows that

(3.13)

Since as we obtain

(3.14)

for all .

Let . Now, for we get

(3.15)

from (3.14) and as , yields

(3.16)

for each Let and take with . By NST-condition (III), there exists such that

(3.17)

By (3.16) and , we get

(3.18)

By the assumption of and NST-condition (I), we have

(3.19)

Put . Since for all , is bounded. Let be a subsequence of such that . Since is closed and convex, is weakly closed and hence . From (3.19), we have that . If not, since satisfies Opial's condition, we have

(3.20)

This is a contradiction. So, we have that . Then, we have

(3.21)

and hence . From , we have . This implies that converges weakly to , and we have

(3.22)

and hence . From , we also have . Since satisfies the Kadec-Klee property, it follows that . So, we have

(3.23)

and hence . This completes the proof.

Corollary 3.2.

Let be a nonempty bounded closed convex subset of a Hilbert space and let be an asymptotically nonexpansive mapping with sequence . Assume such that for all and as . Let . Define a sequence in by the following algorithm:

(3.24)

where as . Then converges in norm to .

Proof.

Setting for all from Lemma 2.2(i) and Theorem 3.1, we immediately obtain the corollary.

Since every family's nonexpansive mapping is family's asymptotically nonexpansive mapping we obtain the following result.

Corollary 3.3.

Let be a nonempty bounded closed convex subset of a Hilbert space and let be a family of nonexpansive mappings with sequence . Assume such that for all and as . Let . Further, suppose that satisfies NST-condition (I) with T. Define a sequence in by the following algorithm:

(3.25)

Assume that if for each bounded sequence , for all implies that . Then converges in norm to .

We have the following corollary for nonexpansive mappings by Lemma 2.2(i) and Theorem 3.1.

Corollary 3.4 (Takahashi et al. [7, Theorem ]).

Let be a bounded closed convex subset of a Hilbert space and let be a nonexpansive mapping such that . Assume that for all . Then the sequence generated by

(3.26)

converges in norm to

4. Strong Convergence for a Family of Asymptotically Nonexpansive Semigroups

Theorem 4.1.

Let be a nonempty bounded closed convex subset of a Hilbert space and let , be a countable family of asymptotically nonexpansive semigroups. Assume such that for all and as . Let be a countable positive and divergent real sequence. Let . Further, suppose that satisfies NST-condition (I) with T. Define a sequence in by the following algorithm:

(4.1)

where as with . Then converges in norm to .

Proof.

First observe that for all . Indeed, we have for all

(4.2)

So, . Hence for all . By the same argument as in the proof of Theorem 3.1, is closed and convex, is well defined. Also, similar to the proof of Theorem 3.1

(4.3)

We next claim that Indeed, by definition of and we have

(4.4)

and then

(4.5)

Since , we have

(4.6)

which in turn implies that

(4.7)

It follows from (4.5) that

(4.8)

Let and for each , we get that

(4.9)

By (4.8) and Lemma 2.7, we obtain that

(4.10)

Furthermore, from (4.9) and Lemma 2.6 and the boundedness of we obtain that . By the fact that for any , where and the weak lower semi-continuity of the norm, we have for all . However, since , we must have for all . Thus and then converges weakly to . Moreover, following the method of Theorem 3.1, . This completes the proof.

Corollary 4.2.

Let be a bounded closed convex subset of a Hilbert space and be an asymptotically nonexpansive semigroup on . Assume also that for all and is a positive real divergent sequence. Then, the sequence generated by

(4.11)

converges in norm to , where as .

Proof.

By Theorem 4.1, if the semigroup , then for all and for all . Hence for all and then, (4.1) reduces to (4.11).

Corollary 4.3 (Takahashi et al. [7, Theorem ]).

Let be a nonempty closed convex subset of a Hilbert space and be a nonexpansive semigroup on . Assume that for all and is a positive real divergent sequence. If , then the sequence generated by

(4.12)

converges in norm to