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.
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
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:
In 1974, Ishikawa  introduced a new iterative scheme, which is defined recursively by
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,
Lemma 2.3 (see [12, Lemma ]).
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.
Corollary 3.3 (see [8, Theorem ]).
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.
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.
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
We note that
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).
- Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar
- Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar
- Browder FE: Fixed-point theorems for noncompact mappings in Hilbert space. Proceedings of the National Academy of Sciences of the United States of America 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleMATHGoogle Scholar
- Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proceedings of the American Mathematical Society 2002,131(7):2133–2136.View ArticleMathSciNetMATHGoogle Scholar
- Xu H-K: A strong convergence theorem for contraction semigroups in Banach spaces. Bulletin of the Australian Mathematical Society 2005,72(3):371–379. 10.1017/S000497270003519XMathSciNetView ArticleMATHGoogle Scholar
- Chen R, He H: Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space. Applied Mathematics Letters 2007,20(7):751–757. 10.1016/j.aml.2006.09.003MathSciNetView ArticleMATHGoogle Scholar
- He H, Chen R: Strong convergence theorems of the CQ method for nonexpansive semigroups. Fixed Point Theory and Applications 2007, 2007:-8.Google Scholar
- Saejung S: Strong convergence theorems for nonexpansive semigroups without Bochner integrals. Fixed Point Theory and Applications 2008, 2008:-7.Google Scholar
- Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062MathSciNetView ArticleMATHGoogle Scholar
- Ishikawa S: Fixed points by a new iteration method. Proceedings of the American Mathematical Society 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar
- Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar
- Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar
- Haugazeau Y: Sur les Inéquations variationnelles et la minimisation de fonctionnelles convexes, Ph.D. thesis. Université Paris, Paris, France; 1968.Google Scholar
- Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar
- Nakajo K, Shimoji K, Takahashi W: Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces. Taiwanese Journal of Mathematics 2006,10(2):339–360.MathSciNetMATHGoogle Scholar
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