Convergence Theorems of Modified Ishikawa Iterative Scheme for Two Nonexpansive Semigroups
© K. Wattanawitoon and P. Kumam. 2010
Received: 26 September 2009
Accepted: 24 November 2009
Published: 25 November 2009
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.
Let be a subset of real Hilbert spaces with the inner product and the norm . is called a nonexpansive mapping if
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:
(ii) for all
(iii)for each the mapping is continuous;
(iv) for all and
where the initial guess is taken in arbitrarily and the sequence is in the interval .
In 1967, Halpern  first introduced the following iterative scheme:
see also Browder . 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  was the first to introduce the following implicit iteration process in Hilbert spaces:
In 2007, Chen and He  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
where is a fixed contractive mapping.
Recently He and Chen  is proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. Very recently, Saejung  proved a convergence theorem by the new iterative method introduced by Takahashi et al.  without Bochner integrals for a nonexpansive semigroup with in Hilbert spaces:
where denotes the metric projection from onto a closed convex subset of .
In 1974, Ishikawa  introduced a new iterative scheme, which is defined recursively by
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  and Ishikawa , 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  and some others.
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 , that is,
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
is characterized as follows.
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:
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.
then converges strongly to
Therefore, is nondecreasing. From , we also have , for all
Since , we get
Thus, , for all and . Then exists and is bounded.
Next, we show that as . From (3.6) we have
We now show that .
For , we have This implies that and hence Moreover, since
As and (3.19), we obtain and so Similarly, we have Thus .
Finally, we show that Since and
Hence as required. This completes the proof.
then converges strongly to
Putting , in Theorem 3.1, we obtain the conclusion immediately.
Corollary 3.3 (see [8, Theorem ]).
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  and our Theorem 3.1 to obtain the following result and the proof is omitted.
then converges strongly to
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 . 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 ,
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
By the Opial's condition of , we have and for all , that is, .
We note that
Hence the whole sequence must converge to , as required. This completes the proof.
then converges strongly to
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 ]).
The authors would like to thank the editors and the anonymous referees for their valuable suggestions which help to improve this paper. This research was supported by the Computational Science and Engineering Research Cluster, King Mongkut's University of Technology Thonburi (KMUTT) (National Research Universities under CSEC Project no. E01008).
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