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Convergence of Three-Step Iterations Scheme for Nonself Asymptotically Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 783178 (2010)
Abstract
Weak and strong convergence theorems of three-step iterations are established for nonself asymptotically nonexpansive mappings in uniformly convex Banach space. The results obtained in this paper extend and improve the recent ones announced by Suantai (2005), Khan and Hussain (2008), Nilsrakoo and Saejung (2006), and many others.
1. Introduction
Suppose that is a real uniformly convex Banach space,
is a nonempty closed convex subset of
. Let
be a self-mapping of
.
A mapping is called nonexpansive provided
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ1_HTML.gif)
for all .
is called asymptotically nonexpansive mapping if there exists a sequence
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ2_HTML.gif)
for all and
.
The class of asymptotically nonexpansive maps which is an important generalization of the class nonexpansive maps was introduced by Goebel and Kirk [1]. They proved that every asymptotically nonexpansive self-mapping of a nonempty closed convex bounded subset of a uniformly convex Banach space has a fixed point.
is called uniformly
-Lipschitzian if there exists a constant
such that
, the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ3_HTML.gif)
for all .
Asymptotically nonexpansive self-mappings using Ishikawa iterative and the Mann iterative processes have been studied extensively by various authors to approximate fixed points of asymptotically nonexpansive mappings (see [2, 12]). Noor [3] introduced a three-step iterative scheme and studied the approximate solutions of variational inclusion in Hilbert spaces. Glowinski and Le Tallec [4] applied a three-step iterative process for finding the approximate solutions of liquid crystal theory, and eigenvalue computation. It has been shown in [1] that the three-step iterative scheme gives better numerical results than the two-step and one-step approximate iterations. Xu and Noor [5] introduced and studied a three-step scheme to approximate fixed point of asymptotically nonexpansive mappings in a Banach space. Very recently, Nilsrakoo and Saejung [6] and Suantai [7] defined new three-step iterations which are extensions of Noor iterations and gave some weak and strong convergence theorems of the modified Noor iterations for asymptotically nonexpansive mappings in Banach space. It is clear that the modified Noor iterations include Mann iterations [8], Ishikawa iterations [9], and original Noor iterations [3] as special cases. Consequently, results obtained in this paper can be considered as a refinement and improvement of the previously known results
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ4_HTML.gif)
where ,
,
,
,
,
,
, and
in
satisfy certain conditions.
If , then (1.4) reduces to the modified Noor iterations defined by Suantai [7] as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ5_HTML.gif)
where ,
,
,
,
,
and
in
satisfy certain conditions.
If , then (1.4) reduces to Noor iterations defined by Xu and Noor [5] as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ6_HTML.gif)
If , then (1.4) reduces to modified Ishikawa iterations as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ7_HTML.gif)
If , then (1.4) reduces to Mann iterative process as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ8_HTML.gif)
Let be a real normed space and
be a nonempty subset of
. A subset
of
is called a retract of
if there exists a continuous map
such that
for all
. Every closed convex subset of a uniformly convex Banach space is a rectract. A map
is called a retraction if
. In particular, a subset
is called a nonexpansive retract of
if there exists a nonexpansive retraction
such that
for all
.
Iterative techniques for converging fixed points of nonexpansive nonself-mappings have been studied by many authors (see, e.g., Khan and Hussain [10], Wang [11]). Evidently, we can obtain the corresponding nonself-versions of (1.5)(1.7). We will obtain the weak and strong convergence theorems using (1.12) for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space. Very recently, Suantai [7] introduced iterative process and used it for the weak and strong convergence of fixed points of self-mappings in a uniformly convex Banach space. As remarked earlier, Suantai [7] has established weak and strong convergence criteria for asymptotically nonexpansive self-mappings, while Chidume et al. [12] studied the Mann iterative process for the case of nonself-mappings. Our results will thus improve and generalize corresponding results of Suantai [7] and others for nonself-mappings and those of Chidume et al. [12] in the sense that our iterative process contains the one used by them. The concept of nonself asymptotically nonexpansive mappings was introduced by Chidume et al. [12] as the generalization of asymptotically nonexpansive self-mappings and obtained some strong and weak convergence theorems for such mappings given (1.9) as follows: for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ9_HTML.gif)
where and
for some
.
A nonself-mapping is called asymptotically nonexpansive if there exists a sequence
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ10_HTML.gif)
for all , and
.
is called uniformly
-Lipschitzian if there exists constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ11_HTML.gif)
for all , and
. From the above definition, it is obvious that nonself asymptotically nonexpansive mappings are uniformly
-Lipschitzian.
Now, we give the following nonself-version of (1.4):
for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ12_HTML.gif)
, where
,
,
,
,
,
,
, and
in
satisfy certain conditions.
The aim of this paper is to prove the weak and strong convergence of the three-step iterative sequence for nonself asymptotically nonexpansive mappings in a real uniformly convex Banach space. The results presented in this paper improve and generalize some recent papers by Suantai [7], Khan and Hussain [10], Nilsrakoo and Saejung [6], and many others.
2. Preliminaries
Throughout this paper, we assume that is a real Banach space,
is a nonempty closed convex subset of
, and
is the set of fixed points of mapping
. A Banach space
is said to be uniformly convex if the modulus of convexity of
is as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ13_HTML.gif)
for all (i.e.,
is a function
Recall that a Banach space is said to satisfy Opial's condition [13] if, for each sequence
in
, the condition
weakly as
and for all
with
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ14_HTML.gif)
Lemma 2.1 (see [12]).
Let be a uniformly convex Banach space,
a nonempty closed convex subset of
and
a nonself asymptotically nonexpansive mapping with a sequence
and
, then
is demiclosed at zero.
Lemma 2.2 (see [12]).
Let be a real uniformly convex Banach space,
a nonempty closed subset of
with
as a sunny nonexpansive retraction and
a mapping satisfying weakly inward condition, then
.
Lemma 2.3 (see [14]).
Let ,
, and
be sequences of nonnegative real sequences satisfying the following conditions:
,
, where
and
, then
exists.
Lemma 2.4 (see [6]).
Let be a uniformly convex Banach space and
, then there exists a continuous strictly increasing convex function
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ15_HTML.gif)
for all , and
with
.
Lemma 2.5 (See [7], Lemma ).
Let be a Banach space which satisfies Opial's condition and let
be a sequence in
. Let
be such that
and
. If
,
are the subsequences of
which converge weakly to
, respectively, then
.
3. Main Results
In this section, we prove theorems of weak and strong of the three-step iterative scheme given in (1.12) to a fixed point for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space. In order to prove our main results the followings lemmas are needed.
Lemma 3.1.
If and
are sequences in
such that
and
is sequence of real numbers with
for all
and
, then there exists a positive integer
and
such that
for all
.
Proof.
By , there exists a positive integer
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ16_HTML.gif)
Let with
. From
, then there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ17_HTML.gif)
from which we have . Put
, then we have
for all
.
Lemma 3.2.
Let be a real Banach space and
a nonempty closed and convex subset of
. Let
be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set
and a sequence
of real numbers such that
and
. Let
,
,
,
,
, and
be real sequences in
, such that
and
in
for all
. Let
be a sequence in
defined by (1.12), then we have, for any
,
exists.
Proof.
Consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ18_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ19_HTML.gif)
Since and from Lemma 2.3, it follows that
exits.
Lemma 3.3.
Let be a real uniformly convex Banach space and
a nonempty closed and convex subset of
. Let
be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set
and a sequence
of real numbers such that
and
. Let
,
,
,
,
, and
be real sequences in
, such that
and
in
for all
. Let
be a sequence in
defined by (1.12), then one has the following conclusions.
If , then
If either or
and
, then
If the following conditions
,
either and
or
and
are satisfied, then
Proof.
Let . By Lemma 3.2, we know that
exits for any
. Then the sequence
is bounded. It follows that the sequences
and
are also bounded. Since
is a nonself asymptotically nonexpansive mapping, then the sequences
,
, and
are also bounded. Therefore, there exists
such that
,
,
,
,
,
. By Lemma 2.4 and (1.12), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ20_HTML.gif)
Let .
Therefore, the assumption implies that
.
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ21_HTML.gif)
From the last inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ23_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ24_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ25_HTML.gif)
By condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ26_HTML.gif)
there exists a positive integer and
such that
and
for all
then it follows from (3.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ27_HTML.gif)
for all . Thus, for
, we write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ28_HTML.gif)
Letting , we have
, so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ29_HTML.gif)
From is continuous strictly increasing with
and (1), then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ30_HTML.gif)
By using a similar method for inequalities (3.8) and (3.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ31_HTML.gif)
Next, to prove
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ32_HTML.gif)
we assume that and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ33_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ34_HTML.gif)
By Lemma 3.1, there exists a positive integer and
such that
for all
. This together with (3.18) implies that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ35_HTML.gif)
It follows from (3.15) and (3.16) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ36_HTML.gif)
This completes the proof.
Next, we show that .
Lemma 3.4.
Let be a real uniformly convex Banach space and
a nonempty closed convex subset of
. Let
be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set
and a sequence
of real numbers such that
and
. Let
,
,
,
,
, and
be real sequences in
, such that
and
in
for all
. Let
be a sequence in
defined by (1.12) with the following restrictions:
and
,
,
then .
Proof.
We first consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ37_HTML.gif)
We note that every asymptotically nonexpansive mapping is uniformly -Lipschitzian. Also note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ38_HTML.gif)
In addition,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ39_HTML.gif)
We denote as the identity maps from
into itself. Thus, by above inequality, we write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ40_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ41_HTML.gif)
In the next result, we prove our first strong convergence theorem as follows.
Theorem 3.5.
Let be a real uniformly convex Banach space and
a nonempty closed convex subset of
. Let
be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set
and a sequence
of real numbers such that
and
. Let
,
,
,
,
, and
be real sequences in
, such that
and
in
for all
. Let
be a sequence in
defined by (1.12) with the following restrictions:
and
,
.
If, in addition, is either completely continuous or demicompact, then
converges strongly to a fixed point of
.
Proof.
By Lemma 3.2, is bounded. It follows by our assumption that
is completely continuous, there exists a subsequence
of
such that
as
. Therefore, by Lemma 3.4, we have
which implies that
as
. Again by Lemma 3.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F783178/MediaObjects/13663_2010_Article_1342_Equ42_HTML.gif)
It folows that . Moreover, since
exists, then
, that is,
converges strongly to a fixed point
of
.
We assume that is demicompact. Then, using the same ideas and argument, we also prove that
converges strongly to a fixed point of
.
Finally, we prove the weak convergence of the iterative scheme (1.12) for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space satisfying Opial's condition.
Theorem 3.6.
Let be a real uniformly convex Banach space satisfying Opial's condition and
a nonempty closed convex subset of
. Let
be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set
and a sequence
of real numbers such that
and
. Let
,
,
,
,
, and
be real sequences in
, such that
and
in
for all
. Let
be a sequence in
defined by (1.12) with the following restrictions:
and
,
,
then converges weakly to a fixed point of
.
Proof.
Let . Then as in Lemma 3.2,
exists. We prove that
has a unique weak subsequential limit in
. We assume that
and
are weak limits of the subsequences
,
, or
, respectively. By Lemma 3.4,
and
is demiclosed by Lemma 2.1,
and in the same way,
. Therefore, we have
. It follows from Lemma 2.5 that
. Thus,
converges weakly to an element of
This completes the proof.
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Temir, S. Convergence of Three-Step Iterations Scheme for Nonself Asymptotically Nonexpansive Mappings. Fixed Point Theory Appl 2010, 783178 (2010). https://doi.org/10.1155/2010/783178
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DOI: https://doi.org/10.1155/2010/783178