- Research Article
- Open access
- Published:
Construction of Fixed Points by Some Iterative Schemes
Fixed Point Theory and Applications volume 2009, Article number: 612491 (2009)
Abstract
We obtain strong convergence theorems of two modifications of Mann iteration processes with errors in the doubly sequence setting. Furthermore, we establish some weakly convergence theorems for doubly sequence Mann's iteration scheme with errors in a uniformly convex Banach space by a Frechét differentiable norm.
1. Introduction
Let be a real Banach space and let
be a nonempty closed convex subset of
. A self-mapping
is said to be nonexpansive if
for all
A point
is a fixed point of
provided
. Denote by
the set of fixed points of
that is,
It is assumed throughout this paper that
is a nonexpansive mapping such that
Construction of fixed points of nonexpansive mappings is an important subject in the theory of nonexpansive mappings and its applications in a number of applied areas, in particular, in image recovery and signal processing (see [1–3]). One way to overcome this difficulty is to use Mann's iteration method that produces a sequence
via the recursive sequence manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ1_HTML.gif)
Reich [4] proved that if is a uniformly convex Banach space with a Frechét differentiable norm and if
is chosen such that
then the sequence
defined by (1.1) converges weakly to a fixed point of
However, this scheme has only weak convergence even in a Hilbert space (see [5]). Some attempts to modify Mann's iteration method (1.1) so that strong convergence is guaranteed have recently been made.
The following modification of Mann's iteration method (1.1) in a Hilbert space is given by Nakajo and Takahashi [6]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ2_HTML.gif)
where denotes the metric projection from
onto a closed convex subset
of
. They proved that if the sequence
is bounded from one, then
defined by (1.2) converges strongly to
Their argument does not work outside the Hilbert space setting. Also, at each iteration step, an additional projection is needed to calculate.
Let be a closed convex subset of a Banach space and
is a nonexpansive mapping such that
Define
in the following way:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ3_HTML.gif)
where is an arbitrary (but fixed) element in
, and
and
are two sequences in
It is proved, under certain appropriate assumptions on the sequences
and
that
defined by (1.3) converges to a fixed point of
(see [7]).
The second modification of Mann's iteration method (1.1) is an adaption to (1.3) for finding a zero of an -accretive operator
, for which we assume that the zero set
The iteration process is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ4_HTML.gif)
where for each is the resolvent of
. In [7], it is proved, in a uniformly smooth Banach space and under certain appropriate assumptions on the sequences
and
, that
defined by (1.4) converges strongly to a zero of
2. Preliminaries
Let be a real Banach space. Recall that the (normalized) duality map
from
into
the dual space of
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ5_HTML.gif)
Now, we define Opial's condition in the sense of doubly sequence.
Definition 2.1.
A Banach space is said to satisfy Opial's condition if for any sequence
in
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ6_HTML.gif)
where denotes that
converges weakly to
We are going to work in uniformly smooth Banach spaces that can be characterized by duality mappings as follows (see [8] for more details).
Lemma 2.2 (see [8]).
A Banach space is uniformly smooth if and only if the duality map
is single-valued and norm-to-norm uniformly continuous on bounded sets of
Lemma 2.3 (see [8]).
In a Banach space there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ7_HTML.gif)
where
If and
are nonempty subsets of a Banach space
such that
is a nonempty closed convex subset and
then the map
is called a retraction from
onto
provided
for all
A retraction
is sunny [1, 4] provided
for all
and
whenever
A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. A sunny nonexpansive retraction plays an important role in our argument.
If is a smooth Banach space, then
is a sunny nonexpansive retraction if and only if there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ8_HTML.gif)
Lemma 2.4 (see [9]).
Let be a uniformly smooth Banach space and let
be a nonexpansive mapping with a fixed point. For each fixed
and every
, the unique fixed point
of the contraction
converges strongly as
to a fixed point of
. Define
by
Then,
is the unique sunny nonexpansive retract from
onto
that is,
satisfies the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ9_HTML.gif)
Let be a sequence of nonnegative real numbers satisfying the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ10_HTML.gif)
where and
are such that
(i)
(ii)
Then, converges to zero.
Lemma 2.6 (see [8]).
Assume that has a weakly continuous duality map
with gauge
. Then,
is demiclosed in the sense that
is closed in the product space
, where
is equipped with the norm topology and
with the weak topology. That is, if
then
Lemma 2.7 (see [12]).
Let be a Banach space and
Then,
(i) is uniformly convex if and only if, for any positive number r, there is a strictly increasing continuous function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ11_HTML.gif)
where the closed ball of
centered at the origin with radius r, and
(ii) is
-uniformly convex if and only if there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ12_HTML.gif)
where is a constant.
Lemma 2.8 (see [4]).
Let be a closed convex subset of a uniformly convex Banach space with a Fréchet differentiable norm, and let
be a sequence of nonexpansive self mapping of
with a nonempty common fixed point set
If
and
for
then
exists for all
In particular,
where
and
are weak limit points of
Lemma 2.9 (the demiclosedness principle of nonexpansive mappings [13]).
Let be a nonexpansive selfmapping of a closed convex subset of
of a uniformly convex Banach space. Suppose that
has a fixed point. Then
is demiclosed. This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ13_HTML.gif)
In 2005, Kim and Xu [7], proved the following theorem.
Theorem 2.
Let be a closed convex subset of a uniformly smooth Banach space
, and let
be a nonexpansive mapping such that
Given a point
and given sequences
and
in
the following conditions are satisfied.
(i),
(ii),
(iii)
Define a sequence in
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ14_HTML.gif)
Then is strongly converges to a fixed point of
.
Recently, the study of fixed points by doubly Mann iteration process began by Moore (see [14]). In [15, 16], we introduced the concept of Mann-type doubly sequence iteration with errors, then we obtained some fixed point theorems for some different classes of mappings. In this paper, we will continue our study in the doubly sequence setting. We propose two modifications of the doubly Mann iteration process with errors in uniformly smooth Banach spaces: one for nonexpansive mappings and the other for the resolvent of accretive operators. The two modified doubly Mann iterations are proved to have strong convergence. Also, we append this paper by obtaining weak convergence theorems for Mann's doubly sequence iteration with errors in a uniformly convex Banach space by a Fréchet differentiable norm. Our results in this paper extend, generalize, and improve a lot of known results (see, e.g., [4, 7, 8, 17]). Our generalizations and improvements are in the use of doubly sequence settings as well as by adding the error part in the iteration processes.
3. A Fixed Point of Nonexpansive Mappings
In this section, we propose a modification of doubly Mann's iteration method with errors to have strong convergence. Modified doubly Mann's iteration process is a convex combination of a fixed point in , and doubly Mann's iteration process with errors can be defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ15_HTML.gif)
The advantage of this modification is that not only strong convergence is guaranteed, but also computations of iteration processes are not substantially increased.
Now, we will generalize and extend Theorem A by using scheme (3.1).
Theorem 3.1.
Let be a closed convex subset of a uniformly smooth Banach space
and let
be a nonexpansive mapping such that
Given a point
and given sequences
and
in
the following conditions are satisfied.
(i),
(ii)
Define a sequence in
by (3.1). Then,
converges strongly to a fixed point of
Proof.
First, we observe that is bounded. Indeed, if we take a fixed point
of
noting that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ16_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ17_HTML.gif)
Now, an induction yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ18_HTML.gif)
Hence, is bounded, so is
. As a result, we obtain by condition (i)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ19_HTML.gif)
We next show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ20_HTML.gif)
It suffices to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ21_HTML.gif)
Indeed, if (3.7) holds, in view of (3.5), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ22_HTML.gif)
Hence, (3.6) holds. In order to prove (3.7), we calculate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ23_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ24_HTML.gif)
Hence, by assumptions (i)-(ii), we obtain
Next, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ25_HTML.gif)
where with
being the fixed point of the contraction
In order to prove (3.11), we need some more information on
, which is obtained from that of
(cf. [18]). Indeed,
solves the fixed point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ26_HTML.gif)
Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ27_HTML.gif)
We apply Lemma 2.3 to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ29_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ30_HTML.gif)
Letting in (3.16) and noting (3.15) yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ31_HTML.gif)
where is a constant such that
for all
and
. Since the set
is bounded, the duality map
is norm-to-norm uniformly continuous on bounded sets of
(Lemma 2.2), and
strongly converges to
By letting
in (3.17), thus (3.11) is therefore proved. Finally, we show that
strongly and this concludes the proof. Indeed, using Lemma 2.3 again, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ32_HTML.gif)
Now we apply Lemma 2.5, and using (3.11) we obtain that
We support our results by giving the following examples.
Example 3.2.
Let be given by
Then, the modified doubly Mann's iteration process with errors converges to the fixed point
, and both Picard and Mann iteration processes converge to the same point too.
Proof.
-
(I)
Doubly Picards iteration converges.
For every point in is a fixed point of
Let
be a point in
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ33_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ34_HTML.gif)
Let for all
Take
and
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ35_HTML.gif)
-
(II)
Doubly Mann's iteration converges.
Let be a point in
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ36_HTML.gif)
Since doubly Mann's iteration is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ37_HTML.gif)
Take to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ38_HTML.gif)
-
(III)
Modified doubly Mann's iteration process with errors converges because the sequence
as we can see and by using (3.1), we obtain
(3.25)
In (3.1), we suppose that ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ40_HTML.gif)
Let and using Theorem 3.1 (
we obtain
Example 3.3.
Let be given by
Then the doubly Mann's iteration converges to the fixed point of
but modified doubly Mann's iteration process with errors does not converge.
Proof.
-
(I)
Doubly Mann's iteration converges because the sequence
as we can see,
(3.27)
The last inequality is true because for all
and
(II)The origin is the unique fixed point of
(III)Note that, modified doubly Mann's iteration process with errors does not converge to the fixed point of because the sequence
as we can see and by using (3.1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ42_HTML.gif)
Putting ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ43_HTML.gif)
Letting we deduce that
4. Convergence to a Zero of Accretive Operator
In this section, we prove a convergence theorem for -accretive operator in Banach spaces. Let
be a real Banach space. Recall that, the (possibly multivalued) operator
with domain
and range
in
is accretive if, for each
and
there exists a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ44_HTML.gif)
An accretive operator is m-accretive if
for each
. Throughout this section, we always assume that
is
-accretive and has a zero. The set of zeros of
is denoted by
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ45_HTML.gif)
For each , we denote by
the resolvent of
that is,
Note that if
is
-accretive, then
is nonexpansive and
for all
We need the resolvent identity (see [19, 20] for more information).
Lemma 4.1 ([7] (the resolvent identity)).
For ,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ46_HTML.gif)
Theorem 4.2.
Assume that is a uniformly smooth Banach space, and
is an
-accretive operator in
such that
Let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ47_HTML.gif)
Suppose and
satisfy the conditions,
(i),
(ii),
(iii) for some
and for all
Also assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ48_HTML.gif)
Then, converges strongly to a zero of
Proof.
First of all we show that is bounded. Take
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ49_HTML.gif)
By induction, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ50_HTML.gif)
This implies that is bounded. Then, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ51_HTML.gif)
A simple calculation shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ52_HTML.gif)
The resolvent identity (4.3) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ53_HTML.gif)
which in turn implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ54_HTML.gif)
Combining (4.9) and (4.11), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ55_HTML.gif)
where is a constant such that
for all
and
By assumptions (i)–(iii) in the theorem, we have that
and
Hence, Lemma 2.5 is applicable to (4.12), and we conclude that
Take a fixed number such that
Again from the resolvent identity (4.3), we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ56_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ57_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ58_HTML.gif)
Since in a uniformly smooth Banach space the sunny nonexpansive retract from
onto the fixed point set
of
is unique, it must be obtained from Reich's theorem (Lemma 2.4). Namely,
where
and
solve the fixed point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ59_HTML.gif)
Applying Lemma 2.3, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ60_HTML.gif)
where by (4.15). It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ61_HTML.gif)
Therefore, letting in (4.18), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ62_HTML.gif)
where is a constant such that
for all
and
. Since
strongly and the duality map
is norm-to-norm uniformly continuous on bounded sets of
it follows that (by letting
in (4.19))
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ63_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ64_HTML.gif)
Now we apply Lemma 2.5 and using (4.20), we obtain that
5. Weakly Convergence Theorems
We next introduce the following iterative scheme. Given an initial , we define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ65_HTML.gif)
Theorem 5.1.
Let be a uniformly convex Banach space with a Frechét differentiable norm. Assume that
has a weakly continuous duality map
with gauge
. Assume also that
(i),
(ii)
Then, the scheme (5.1) converges weakly to a point in
Proof.
First, we observe that for any , the sequence
is nonincreasing.
Indeed, we have by nonexpansivity of ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ66_HTML.gif)
In particular, is bounded, so is
. Let
be the set of weak limit point of the sequence
Note that we can rewrite the scheme (5.1) in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ67_HTML.gif)
where is the nonexpansive mapping given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ68_HTML.gif)
Then, we have for
Hence, by Lemma 2.7, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ69_HTML.gif)
Therefore, will converge weakly to a point in
if we can show that
To show this, we take a point
in
Then we have a subsequence
of
such that
. Noting that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ70_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ71_HTML.gif)
By Lemma 2.6, we conclude that that is,
.
Theorem 5.2.
Let be a uniformly convex Banach space which either has a Frechét differentiable norm or satisfies Opial's property. Assume for some
(i) for
(ii) for
Then, the scheme (5.1) converges weakly to a point in
Proof.
We have shown that exists for all
Applying Lemma 2.7(i), we have a strictly increasing continuous function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ72_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ73_HTML.gif)
Since we obtain by (5.9) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ74_HTML.gif)
For any fixed by Lemma 4.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ75_HTML.gif)
We deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ76_HTML.gif)
Therefore we obtain by (5.9) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ77_HTML.gif)
Apply Lemma 2.9 to find out that It remains to show that
is a singleton set. Towards this end, we take
and distinguish the two cases.
In case has a Frechét differentiable norm, we apply Lemma 2.8 to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ78_HTML.gif)
hence, In case
satisfies Opial's condition, we can find two subsequences
such that
If
, Opial's property creates the contradiction,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F612491/MediaObjects/13663_2008_Article_1164_Equ79_HTML.gif)
In either case, we have shown that consists of exact one point, which is clearly the weak limit of
Remark 5.3.
The schemes (3.1), (4.4), and (5.1) generalize and extend several iteration processes from literature (see [7, 8, 17, 21–25] and others).
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El-Sayed Ahmed, A., Kamal, A. Construction of Fixed Points by Some Iterative Schemes. Fixed Point Theory Appl 2009, 612491 (2009). https://doi.org/10.1155/2009/612491
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DOI: https://doi.org/10.1155/2009/612491