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Strong Convergence Theorems for Common Fixed Points of Multistep Iterations with Errors in Banach Spaces
Journal of Inequalities and Applications volume 2009, Article number: 819036 (2009)
Abstract
We establish strong convergence theorem for multi-step iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in Banach spaces. Our results extend and improve the recent ones announced by Plubtieng and Wangkeeree (2006), and many others.
1. Introduction
Let be a subset of real normal linear space . A mapping is said to be asymptotically nonexpansive on if there exists a sequence in with such that for each ,
If , then is known as a nonexpansive mapping. is called asymptotically nonexpansive in the intermediate sense [1] provided is uniformly continuous and
From the above definitions, it follows that asymptotically nonexpansive mapping must be asymptotically nonexpansive in the intermediate sense.
Let be a nonempty subset of normed space , and Let be mappings. For a given and a fixed ( denotes the set of all positive integers), compute the iterative sequences defined by
where , are bounded sequences in and , , , are appropriate real sequences in such that for each .
The purpose of this paper is to establish a strong convergence theorem for common fixed points of the multistep iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in a uniformly convex Banach space. The results presented in this paper extend and improve the corresponding ones announced by Plubtieng and Wangkeeree [2], and many others.
2. Preliminaries
Definition 2.1 (see [1]).
A Banach space is said to be a uniformly convex if the modulus of convexity of is
Lemma 2.2 (see [3]).
Let , and be three nonnegative real sequences satisfying the following condition:
where and . Then
(1) exists;
(2)If , then .
Lemma 2.3 (see [4]).
Let be a uniformly convex Banach space and for all . Suppose that and are two sequences of such that
for some . Then
3. Main Results
Lemma 3.1.
Let be a uniformly convex Banach space, , are two sequences of , and be a real sequence. If there exists such that
(i) for all ;
(ii);
(iii);
(iv),
then .
Proof.
The proof is clear by Lemma 2.3.
Lemma 3.2.
Let be a uniformly convex Banach space, let be a nonempty closed bounded convex subset of , and let be asymptotically nonexpansive mappings in the intermediate sense such that . Put
so that . Let , , and be real sequences in satisfying the following condition:
(i) for all and ;
(ii) for all .
If is the iterative sequence defined by (1.3), then, for each , the limit exists.
Proof.
For each , we note that
where . Since
we see that
It follows from (3.2) that
where . Since
we see that
It follows from (3.5) that
where , and so
By continuing the above method, there are nonnegative real sequences such that
This together with Lemma 2.2 gives that exists. This completes the proof.
Lemma 3.3.
Let be a uniformly convex Banach space, let be a nonempty closed bounded convex subset of , and let be asymptotically nonexpansive mappings in the intermediate sense such that . Put
so that . Let the sequence be defined by (1.3) whenever , , satisfy the same assumptions as in Lemma 3.2 for each and the additional assumption that there exists such that for all . Then we have the following:
(1);
(2).
Proof.
Taking each , it follows from Lemma 3.2 that exists. Let
for some . We note that
where is a nonnegative real sequence such that
It follows that
which implies that
Next, we observe that
Thus we have
Also,
gives that
Note that
This together with (3.18), (3.20), and Lemma 3.1, gives
This completes the proof of .
For each ,
Since
we obtain
It follows that
which implies that
On the other hand, we note that
where is a nonnegative real sequence such that
Thus we have
and hence
Next, we observe that
Thus we have
Also,
gives that
Note that
Therefore, it follows from (3.33), (3.35), and Lemma 3.1 that
This completes the proof.
Theorem 3.4.
Let be a uniformly convex Banach space and let be a nonempty closed bounded convex subset of . Let be asymptotically nonexpansive mappings in the intermediate sense such that and there exists one member in which is completely continuous. Put
so that . Let the sequence be defined by (1.3) whenever , , satisfy the same assumptions as in Lemma 3.2 for each and the additional assumption that there exists such that for all . Then converges strongly to a common fixed point of the mappings .
Proof.
From Lemma 3.3, it follows that
which implies that
and so
It follows from (3.22), (3.37) that
Let for all . Then we have
Notice that . Thus and the above inequality becomes
and so
Since
we have
and so
Since is bounded and one of is completely continuous, we may assume that is completely continuous, without loss of generality. Then there exists a subsequence of such that as . Moreover, by (3.48), we have
which implies that as . By (3.48) again, we have
It follows that . Since exists, we have
that is,
Moreover, we observe that
for all and
Therefore,
for all . This completes the proof.
Remark 3.5.
Theorem 3.4 improves and extends the corresponding results of Plubtieng and Wangkeeree [2] in the following ways.
The iterative process defined by (1.3) in [2] is replaced by the new iterative process defined by (1.3) in this paper.
Theorem 3.4 generalizes Theorem of Plubtieng and Wangkeeree [2] from a asymptotically nonexpansive mappings in the intermediate sense to a finite family of asymptotically nonexpansive mappings in the intermediate sense.
Remark 3.6.
If and in Theorem 3.4, we obtain strong convergence theorem for Noor iteration scheme with error for asymptotically nonexpansive mapping in the intermediate sense in Banach space, we omit it here.
References
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972,35(1):171–174. 10.1090/S0002-9939-1972-0298500-3
Plubtieng S, Wangkeeree R: Strong convergence theorems for multi-step Noor iterations with errors in Banach spaces. Journal of Mathematical Analysis and Applications 2006,321(1):10–23. 10.1016/j.jmaa.2005.08.029
Liu Q: Iterative sequences for asymptotically quasi-nonexpansive mappings with error member. Journal of Mathematical Analysis and Applications 2001,259(1):18–24. 10.1006/jmaa.2000.7353
Schu J: Iterative construction of fixed points of strictly pseudocontractive mappings. Applicable Analysis 1991,40(2–3):67–72. 10.1080/00036819108839994
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Gu, F., Fu, Q. Strong Convergence Theorems for Common Fixed Points of Multistep Iterations with Errors in Banach Spaces. J Inequal Appl 2009, 819036 (2009). https://doi.org/10.1155/2009/819036
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DOI: https://doi.org/10.1155/2009/819036