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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 205875, 6 pages
Strong Convergence on Iterative Methods of Cesàro Means for Nonexpansive Mapping in Banach Space
1Faculty of Statistics, Jilin University of Finance and Economics, Changchun, Jilin 130117, China
2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Received 23 March 2014; Accepted 4 May 2014; Published 12 May 2014
Academic Editor: Jen-Chih Yao
Copyright © 2014 Zhichuan Zhu and Rudong Chen. 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.
Two new iterations with Cesàro's means for nonexpansive mappings are proposed and the strong convergence is obtained as . Our main results extend and improve the corresponding results of Xu (2004), Song and Chen (2007), and Yao et al. (2009).
Let be a nonempty closed convex subset of a real Banach space and let be nonexpansive mapping from into itself (recall that a mapping is nonexpansive if , ). We denote fixed points of as ; that is, .
Recall that a mapping is contractive if there exists a constant such that , .
In 1975, Baillon  proved the first nonlinear ergodic theorem.
Theorem 1. Suppose that is a nonempty closed convex subset of Hilbert space and mapping such that ; then , and the Cesàro means weakly converges to a fixed point of .
In 1979, Bruck  showed the nonlinear ergodic theorem for nonexpansive mapping in uniformly convex Banach space with Fréchet differentiable norms.
In 2004, Xu  introduced the following viscosity iterative scheme given by where parameter , satisfying(X1);(X2);(X3)either , or .He proved that the explicit iterative scheme converges strongly to a fixed point of in uniformly smooth Banach space.
In 2007, Song and Chen  defined the following viscosity iteration of Cesàro means for nonexpansive mapping : and they proved that the sequence converges strongly to some point in in a uniformly convex Banach space with weakly sequentially continuous duality mapping.
In 2009, Yao et al.  introduced the following process : They proved that the sequence converges strongly to a fixed point of under the following control conditions of parameters:(YLZ1), for all ;(YLZ2) and ;(YLZ3).
Motivated by the above results, we propose the following new iterations with Cesàro’s means for nonexpansive mappings: and viscosity iteration:
Some examples are given to show the generation of our new iterations with Cesàro’s means as follows.
Example 2. If , iteration (5) with Cesàro’s means for nonexpansive mappings is , which is reduced as the same iteration of Yao et al. . If , iteration (5) can be written as follows: which is a generation of Yao et al. .
Example 3. Let with the usual metric, nonexpansive mapping defined by , the fixed contractive mapping , and the parameters are defined as , , and . The new iterations with Cesro’s means which is related to iterative step can be written as follows:
Throughout the paper, let be a real Banach space with norm . The normalized duality mapping is defined by where denotes the dual space of and denotes the generalized duality pairing. We will denote the single-valued normalized duality mapping by .
Let be the unit sphere of a Banach space. The space is said to have a Gâteaux differentiable norm (or is said to be smooth), if the limit exists for every , and is said to have a uniformly Gâteaux differentiable norm if for each the limit (10) is attained uniformly for . Further, is said to be uniformly smooth if the limit (10) exists uniformly for .
The following two results can be found in .
If is smooth the duality mapping is single-valued and continuous.
If is Banach space with uniformly Gâteaux differentiable norm, then duality mapping is single-valued and norm to weak star uniformly continuous on bounded sets of .
In order to prove our main results, the following lemmas will be used.
Lemma 4 (see ). Let be a nonempty closed convex subset of a uniformly smooth Banach space . Let be a nonexpansive mapping with . For each fixed and every , the unique fixed point of the contraction as converges strongly to which is the nearest to .
Lemma 5 (see ). Let be a uniformly smooth Banach space, a closed convex subset of , a nonexpansive mapping with , and a fixed contraction. Then defined by converges strongly to a unique fixed point in as .
Lemma 6 (see ). Let be a real Banach space and let be the normalized duality mapping. Then for any given , one has
Lemma 7 (see ). Assume is a sequence of nonnegative real numbers such that where is a sequence in (0,1) and is a sequence such that(1);(2). Then .
3. Main Results
Let be a nonempty closed convex subset of a uniformly smooth Banach space. Let be a nonexpansive mapping such that . Let , , and be three real sequences in (0,1) satisfying(i), for all ;(ii) and ;(iii).
In the following, we will present the first main result. For each and , let be the unique fixed point of the contractive mapping given by That is, From Lemma 4, for fixed , we have which is the unique fixed point.
Theorem 8. Let be a nonempty closed convex subset of a uniformly smooth Banach space . Let be a nonexpansive mapping such that . Let , , and be three real sequences in (0,1) satisfying conditions (i)–(iii). Then, for given arbitrarily, let the sequence be generated iteratively by (5). Then the sequence defined by (5) converges strongly to a fixed point of .
Proof. Taking a fixed point , we have
By induction, we get that is bounded. We observe that (14) can be rewritten as follows:
By (14), we have
Applying Lemma 6 to (18), we have
where is some constant such that
Hence, we get
where is a constant such that
It follows that
By the fact that the order of and is changeable, we have
Finally, we prove . Indeed, applying Lemma 6 to (5), we obtain
Hence, by Lemma 7, we have that as .
The proof is complete.
Now we will give the second main result. In order to prove the strong convergence of viscosity iterative (6), we assume that is the unique fixed point of the following contractive mapping given by That is, From Lemma 5, for fixed , we have which is the unique fixed point.
Theorem 9. Let be a nonempty closed convex subset of a uniformly smooth Banach space . Let be a nonexpansive mapping such that . Let , , and be three real sequences in (0,1) satisfying the following control conditions (i–iii). Then, for given arbitrarily, the sequence defined by (6) converges strongly to a fixed point of .
Proof. Taking a fixed point , we have
and is bounded so are and .
We observe that (27) can be rewritten as and , for all .
From (31), we have Applying Lemma 6 to (32), we get where is some constant such that Hence, we have where is also a constant such that It follows that Since the order of and is exchangeable, hence
Finally, we prove that . Indeed, applying Lemma 6 to (6), we obtain Therefore, we have Put It follows that It is easily seen from (ii) and (38) that Hence, applying Lemma 7 to (42), we have that as .
The proof is complete.
Remark 10. Our main result extends the main result of Yao et al. to Cesàro means and viscosity iteration method. Our results are new and the proofs are simple and different from many others.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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