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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 170540, 11 pages
Strong Convergence of Hybrid Algorithm for Asymptotically Nonexpansive Mappings in Hilbert Spaces
1Department of Basic Courses, Jiangxi Vocational and Technical College, Jiangxi Nanchang 330013, China
2Department of Mathematics, Nanchang University, Nanchang 330031, China
Received 24 January 2012; Accepted 5 March 2012
Academic Editor: Rudong Chen
Copyright © 2012 Juguo Su et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The hybrid algorithms for constructing fixed points of nonlinear mappings have been studied extensively in recent years. The advantage of this methods is that one can prove strong convergence theorems while the traditional iteration methods just have weak convergence. In this paper, we propose two types of hybrid algorithm to find a common fixed point of a finite family of asymptotically nonexpansive mappings in Hilbert spaces. One is cyclic Mann's iteration scheme, and the other is cyclic Halpern's iteration scheme. We prove the strong convergence theorems for both iteration schemes.
Let be a real Hilbert space and be a nonempty closed convex subset of , and denote the inner product and norm in , respectively. Let be a self-mapping of . Then, is said to be a Lipschitzian mapping if for each there exists an nonnegative real number such that for all . A Lipschitzian mapping is said to be nonexpansive mapping if for all and asymptotically nonexpansive mapping  if , respectively. We use to denote the set of fixed points of (i.e., ). It is well known that if is asymptotically nonexpansive mapping with , then is closed and convex.
Iterative methods for finding fixed points of nonexpansive mappings are an important topic in the theory of nonexpansive mappings and have wide applications in a number of applied areas, such as the convex feasibility problem [2–4], the split feasibility problem [5–7] and image recovery and signal processing [8–10]. The Mann's iteration is defined by the following: where is chosen arbitrarily and . Reich  proved that if is a uniformly convex Banach space with a Fréchet differentiable norm and if is chosen such that , then the sequence defined by (1.2) converges weakly to a fixed point of nonexpansive mapping . However, we highlight that the Mann's iterations have only weak convergence even in a Hilbert space (see e.g., ).
In order to obtain the strong convergence theorem for the Mann iteration method (1.2) to nonexpansive mappings, in 2003, Nakajo and Takahashi  proved the following theorem in a Hilbert space by using an idea of the hybrid method in mathematical programming.
Theorem 1.1 (see ). Let be a closed convex subset of a Hilbert space and let be a nonexpansive mapping of into itself such that is nonempty. Let be the metric projection of onto . Let and where satisfies and is the metric projection of onto . Then converges strongly to .
The iterative algorithm (1.3) is often referred to as hybrid algorithm or CQ algorithm in the literature. We call it hybrid algorithm. Since then, the hybrid algorithm has been studied extensively by many authors (see, e.g., [14–18]). Specifically, Kim and Xu  extended the results of Nakajo and Takahashi  from nonexpansive mapping to asymptotically nonexpansive mapping; they proposed the following hybrid algorithm: where . Zhang and Chen in , studied the following hybrid algorithm of Halpern's type for asymptotically nonexpansive mappings: where . Some other related works can be found in [21–25].
The hybrid algorithm of (1.3)–(1.5) just considered a single nonexpansive and asymptotically nonexpansive mapping. In order to extend them to a finite family of mappings. Recall that in 1996, Bauschke  investigated the following cyclic Halpern's type algorithm for a finite family of nonexpansive mappings : or, more compactly, where , and the function takes values in .
If and each nonexpansive mapping is a projection onto a closed convex set, then (1.7) reduces to the famous Algebraic Reconstruction Technique (ART), which has numerous applications from computer tomograph to image reconstruction.
For the cyclic Mann's type algorithm, a finite family of asymptotically nonexpansive mappings was introduced by Qin et al.  and Osilike and Shehu , independently. Let be a finite family of asymptotically nonexpansive self-mappings of . For a given , and a real sequence , the sequence is generated as follows: The algorithm can be expressed in a compact form as where , , positive integer and . Similarly, we can define the cyclic Halpern's type algorithm for asymptotically nonexpansive mappings as follows:
The purpose of this paper is to extend the hybrid algorithms (1.4) and (1.5) to the cyclic Mann's type (1.9) and the cyclic Halpern's type (1.10). Our results generalize the corresponding results of Kim and Xu  and Zhang and Chen  from a single asymptotically nonexpansive mapping to a finite family of asymptotically nonexpansive mappings, respectively.
In this section, we collect some useful results which will be used in the following section.
We use the following notations: (i) for weak convergence and for strong convergence; (ii) denotes the weak -limit set of .
It is well known that a Hilbert space satisfies the Opial's condition ; that is, for each sequence in which converges weakly to a point , we have for all .
Recall that given a closed convex subset of of a real Hilbert space , the nearest point projection form onto assigns to each its nearest point denoted in from to ; that is, is the unique point in with the property The following Lemmas 2.1 and 2.2 are well known.
Lemma 2.1. Let be a closed convex subset of a real Hilbert space . Given and , then if and only if there holds the relation
Lemma 2.2. Let be a real Hilbert space, then for all
Lemma 2.3 (see ). Let be a uniformly convex Banach space, a nonempty closed convex subset of , and an asymptotically nonexpansive mapping. Then is demiclosed at zero, that is, if and , then .
Lemma 2.4 (see ). Let be a closed convex subset of a real Hilbert space . Let be sequences in and . Let . If is such that and satisfies the condition then converges strongly to .
Lemma 2.5 (see ). Let be a closed convex subset of a real Hilbert space . For any and real number , the set is convex and closed.
3. Main Results
In this section, we consider a finite family of asymptotically nonexpansive mappings ; that is, there exists , with , for all such that for all and . Let , then , and for all , and for all and .
We prove the following theorems.
Theorem 3.1. Let be a bounded closed convex subset of a Hilbert space , and let be a finite family of asymptotically nonexpansive mappings with . Assume that such that . Suppose the sequence generated by where , as . Then converges strongly to .
Proof. By Lemma 2.5, we conclude that is closed and convex. It is obvious that and are closed and convex. Then, the projection mappings and are well defined. We divide the proof into several steps.
Step 1. We show that , for all . Let . By the hybrid algorithm (3.3) and that is convex, we have where . Hence, , that is, , for all .
Next, we prove that , for all . Indeed, for , , then . Assuming that , we show that . Since is the projection of onto , it follows from Lemma 2.1 that As , in particular, we have Thus, . Therefore, , for all .
Step 2. We prove that
Since the definition of implies that , we have By Step 1, , we have In particular, Since , we have and . The second inequality shows that the sequence is nondecreasing. Since is bounded, we obtain that the exists.
With the help of Lemma 2.2, we obtain Consequently,
Step 3. We now claim that , as , for all . Notice that for all , , since , we obtain . So that . Hence and .
By the hybrid algorithm (3.3) and the condition , we get It follows from the fact that we have Putting , we deduce that Hence, Consequently, for all , we have Thus, , as , for all .
Step 4. Since is bounded, then has a weakly convergent subsequence . Suppose converges weakly to . Since C is weakly closed and , we have . By Lemma 2.3, is demiclosed at 0 for all , and we get , that is . Suppose does not converge weakly to , then there exists another subsequence of which converges weakly to some . Similarly we can prove that . It follows from the proof of above that we know that and exist. Since every Hilbert space satisfies Opial’s condition, we have This is a contradiction. Hence, . Then by virtue of (3.10) and Lemma 2.4, we conclude that as , where .
Recall that a mapping is said to be asymptotically strictly pseudocontractive , if there exist and a sequence with such that for all and .
Theorem 3.2. Let be a bounded closed convex subset of a Hilbert space , and let be a finite family of asymptotically nonexpansive mappings with . Assume that , for some . Suppose the sequence generated by where . Then converges strongly to .
Proof. Since is asymptotically nonexpansive if and only if is asymptotically strictly pseudocontractive mapping with . Then, the rest of proof follows from Theorem 3.2 of Osilike and Shehu  and Theorem 2.2 of Qin et al.  directly by letting .
The authors are deeply grateful to Professor Rudong Chen (Editor) for managing the review process. This work was supported by the Natural Science Foundations of Jiangxi Province (2009GZS0021, CA201107114) and the Youth Science Funds of the Education Department of Jiangxi Province (GJJ12141).
- K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.
- H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996.
- D. Butnariu, Y. Censor, P. Gurfil, and E. Hadar, “On the behavior of subgradient projections methods for convex feasibility problems in Euclidean spaces,” SIAM Journal on Optimization, vol. 19, no. 2, pp. 786–807, 2008.
- S. Maruster and C. Popirlan, “On the Mann-type iteration and the convex feasibility problem,” Journal of Computational and Applied Mathematics, vol. 212, no. 2, pp. 390–396, 2008.
- C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
- Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071–2084, 2005.
- H.-K. Xu, “A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problem,” Inverse Problems, vol. 22, no. 6, pp. 2021–2034, 2006.
- E. T. Hale, W. Yin, and Y. Zhang, “Fixed-point continuation applied to compressed sensing: implementation and numerical experiments,” Journal of Computational Mathematics, vol. 28, no. 2, pp. 170–194, 2010.
- C. I. Podilchuk and R. J. Mammone, “Image recovery by convex projections using a least-squares constraint,” Journal of the Optical Society of America A, vol. 7, pp. 517–521, 1990.
- D. Youla, “Mathematical theory of image restoration by the method of convex projections,” in Image Recovery Theory and Applications, H. Stark, Ed., pp. 29–77, Academic Press, Orlando, Fla, USA, 1987.
- S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.
- A. Genel and J. Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics, vol. 22, no. 1, pp. 81–86, 1975.
- K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003.
- M. O. Osilike and Y. Shehu, “Explicit averaging cyclic algorithm for common fixed points of asymptotically strictly pseudocontractive maps,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 548–553, 2009.
- S. Plubtieng and K. Ungchittrakool, “Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis A, vol. 67, no. 7, pp. 2306–2315, 2007.
- X. Qin, H. Zhou, and S. M. Kang, “Strong convergence of Mann type implicit iterative process for demi-continuous pseudo-contractions,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 217–228, 2009.
- X. Qin, Y. J. Cho, S. M. Kang, and M. Shang, “A hybrid iterative scheme for asymptotically k-strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis A, vol. 70, no. 5, pp. 1902–1911, 2009.
- B. S. Thakur, “Convergence of strictly asymptotically pseudo-contractions,” Thai Journal of Mathematics, vol. 5, no. 1, pp. 41–52, 2007.
- T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis A, vol. 64, no. 5, pp. 1140–1152, 2006.
- L. J. Zhang and J. M. Chen, “Strong convergence of modified Ishikawa iteration for asymptotically nonexpansive mappings,” Acta Mathematica Sinica, vol. 51, no. 1, pp. 123–128, 2008.
- X. L. Qin, C. Q. Wu, and M. J. Shang, “Strong convergence theorems for asymptotically strictly pseudocontractive maps in Hilbert spaces,” Journal of Mathematical Research and Exposition, vol. 29, no. 1, pp. 43–51, 2009.
- C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis A, vol. 64, no. 11, pp. 2400–2411, 2006.
- H. Zhou, “Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 546–556, 2008.
- Y. Yao, Y.-C. Liou, and G. Marino, “A hybrid algorithm for pseudo-contractive mappings,” Nonlinear Analysis A, vol. 71, no. 10, pp. 4997–5002, 2009.
- Q.-b. Zhang and C.-z. Cheng, “Strong convergence theorem for a family of Lipschitz pseudocontractive mappings in a Hilbert space,” Mathematical and Computer Modelling, vol. 48, no. 3-4, pp. 480–485, 2008.
- H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 202, no. 1, pp. 150–159, 1996.
- Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
- Y. J. Cho, H. Zhou, and G. Guo, “Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings,” Computers & Mathematics with Applications, vol. 47, no. 4-5, pp. 707–717, 2004.
- Q. Liu, “Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings,” Nonlinear Analysis A, vol. 26, no. 11, pp. 1835–1842, 1996.