Convergence Theorems of Modified Ishikawa Iterative Scheme for Two Nonexpansive Semigroups

Abstract

We prove convergence theorems of modified Ishikawa iterative sequence for two nonexpansive semigroups in Hilbert spaces by the two hybrid methods. Our results improve and extend the corresponding results announced by Saejung (2008) and some others.

1. Introduction

Let be a subset of real Hilbert spaces with the inner product and the norm . is called a nonexpansive mapping if

(1.1)

We denote by the set of fixed points of , that is, .

Let be a family of mappings from a subset of into itself. We call it a nonexpansive semigroup on if the following conditions are satisfied:

(i)

(ii) for all

(iii)for each the mapping is continuous;

(iv) for all and

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

(1.2)

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

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

(1.3)

see also Browder [3]. He pointed out that the conditions and are necessary in the sence that, if the iteration (1.3) converges to a fixed point of , then these conditions must be satisfied.

On the other hand, in 2002, Suzuki [4] was the first to introduce the following implicit iteration process in Hilbert spaces:

(1.4)

for the nonexpansive semigroup. In 2005, Xu [5] established a Banach space version of the sequence (1.4) of Suzuki [4].

In 2007, Chen and He [6] studied the viscosity approximation process for a nonexpansive semigroup and prove another strong convergence theorem for a nonexpansive semigroup in Banach spaces, which is defined by

(1.5)

where is a fixed contractive mapping.

Recently He and Chen [7] is proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. Very recently, Saejung [8] proved a convergence theorem by the new iterative method introduced by Takahashi et al. [9] without Bochner integrals for a nonexpansive semigroup with in Hilbert spaces:

(1.6)

where denotes the metric projection from onto a closed convex subset of .

In 1974, Ishikawa [10] introduced a new iterative scheme, which is defined recursively by

(1.7)

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

In this paper, motivated by the iterative sequences (1.6) given by Saejung in [8] and Ishikawa [10], we introduce the modified Ishikawa iterative scheme for two nonexpansive semigroups in Hilbert spaces. Further, we obtain strong convergence theorems by using the hybrid methods. This result extends and improves the result of Saejung [8] and some others.

2. Preliminaries

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

It is known that every Hilbert space satisfies the Opial's condition [11], that is,

(2.1)

Recall that the metric (nearest point) projection from a Hilbert space to a closed convex subset of is defined as follows. Given is the only point in with the property

(2.2)

is characterized as follows.

Lemma 2.1.

Let be a real Hilbert space, a closed convex subset of . Given and . Then if and only if there holds the inequality

(2.3)

Lemma 2.2.

There holds the identity in a Hilbert space

(2.4)

for all and

Lemma 2.3 (see [12, Lemma ]).

Let be a real sequence and let be a real number such that . Suppose that either of the following holds:

(i) or

(ii),

then is a cluster point of . Moreover, for , there exists such that for every integer with

3. Main Results

3.1. The Shrinking Projection Method

In this section, we prove strong convergence of an iterative sequence generated by the shrinking hybrid projection method in mathematical programming.

Theorem 3.1.

Let be a closed convex subset of a real Hilbert space . Let and be nonexpansive semigroups on with a nonempty common fixed point set , that is, . Let , and be the sequences such that , and . Suppose that is a sequence generated by the following iterative scheme:

(3.1)

then converges strongly to

Proof.

We first show that is closed and convex for each . From the definition of it is obvious that is closed for each . We show that is convex for any . Since

(3.2)

and hence is convex. Next we show that for all . Let , then we have

(3.3)

(3.4)

Substituting (3.3) into (3.4), we have

(3.5)

This means that for all . Thus, is well defined. Since and , we get

(3.6)

Consequently,

(3.7)

for . This implies that

(3.8)

Therefore, is nondecreasing. From , we also have , for all

Since , we get

(3.9)

Thus, for , we obtain

(3.10)

Thus, , for all and . Then exists and is bounded.

Next, we show that as . From (3.6) we have

(3.11)

Since exists, then

(3.12)

Further, as in the proof of [8, page 3], we have which is a Cauchy sequence. So, we have By definition of , we have

(3.13)

Since and (3.12), we obtain

(3.14)

We now show that .

For , we have This implies that and hence Moreover, since

(3.15)

we have

(3.16)

And since is a nonexpansive mapping, we obtain

(3.17)

Since and , we obtain

(3.18)

As in the proof of [12, Theorem ], by Lemma 2.3, we can choose a sequence of positive real numbers such that

(3.19)

In similar way, we also have

(3.20)

Next, we show that . To see this, we fix

(3.21)

As and (3.19), we obtain and so Similarly, we have Thus .

Finally, we show that Since and

(3.22)

But as , we have

(3.23)

Hence as required. This completes the proof.

Corollary 3.2.

Let be a closed convex subset of a real Hilbert space . Let be nonexpansive semigroups on with a nonempty common fixed point set , that is, . Let , and be the sequences such that , and . Suppose that is a sequence iteratively generated by the following iterative scheme:

(3.24)

then converges strongly to

Proof.

Putting , in Theorem 3.1, we obtain the conclusion immediately.

Corollary 3.3 (see [8, Theorem ]).

Let be a closed convex subset of a real Hilbert space . Let be a nonexpansive semigroups on with a nonempty common fixed point set , that is, . Let and be the sequences such that , and . Suppose that is a sequence iteratively generated by the following iterative scheme:

(3.25)

then

Proof.

If for all and for every in Theorem 3.1 then (3.1) reduced to (3.25). By using Theorem 3.1, we get the following conclusion.

3.2. The CQ Hybrid Method

In this section, we consider the modified Ishikawa iterative scheme computing by the CQ hybrid method [13–15]. We use the same idea as Saejung's Theorem in [8] and our Theorem 3.1 to obtain the following result and the proof is omitted.

Theorem 3.4.

Let be a closed convex subset of a real Hilbert space . Let and be nonexpansive semigroups on with a nonempty common fixed point set , that is, . Let , and be the sequences such that , and . Suppose that is a sequence generated by the following iterative scheme:

(3.26)

then converges strongly to

Proof.

First, we show that both and are closed and convex, and for all . It follows easily from the definition that and are just intersection of and the half-spaces see also [9]. As in the proof of the preceding theorem, we have for all . Clearly, . Suppose that for some , we have . In particular, that is, . It follows from the induction that for all . This proves the claim.

Next, we show that and

We first claim that Indeed, as and ,

(3.27)

For fixed . It follows from for all that

(3.28)

This implies that sequence is bounded and

(3.29)

Notice that

(3.30)

This implies that

(3.31)

By using the same argument of Saejung [8, Theorem , page 6] and in the proof of Theorem 3.1, we have and . And we can choose a subsequence of such that , , and as .

From (3.21), we obtain

(3.32)

By the Opial's condition of , we have and for all , that is, .

We note that

(3.33)

This implies that

(3.34)

Therefore,

(3.35)

Hence the whole sequence must converge to , as required. This completes the proof.

Corollary 3.5.

Let be a closed convex subset of a real Hilbert space . Let be nonexpansive semigroups on with a nonempty common fixed point set , that is, . Let , and be the sequences such that , and . Suppose that is a sequence iteratively generated by the following iterative scheme:

(3.36)

then converges strongly to

Proof.

If for all , in Theorem 3.4 then (3.26) reduced to (3.36). So, we obtain the result immediately.

We also deduce the following corollary.

Corollary 3.6 (see [8, Theorem ]).

Let be a closed convex subset of a real Hilbert space . Let be a nonexpansive semigroups on with a nonempty common fixed point set , that is, . Let and be the sequences such that , and . Suppose that is a sequence iteratively generated by the following iterative scheme:

(3.37)

then