- Research Article
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Convergence Comparison of Several Iteration Algorithms for the Common Fixed Point Problems
Fixed Point Theory and Applications volume 2009, Article number: 824374 (2009)
Abstract
We discuss the following viscosity approximations with the weak contraction for a non-expansive mapping sequence
,
,
. We prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern's type iteration and Mouda's viscosity approximations with the weak contraction.
1. Introduction
The following famous theorem is referred to as the Banach Contraction Principle.
Theorem 1.1 (Banach [1]).
Let be a complete metric space and let
be a contraction on
, that is, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ1_HTML.gif)
Then has a unique fixed point.
In 2001, Rhoades [2] proved the following very interesting fixed point theorem which is one of generalizations of Theorem 1.1 because the weakly contractions contains contractions as the special cases .
Theorem 1.2 (Rhoades[2], Theorem  2).
Let be a complete metric space, and let
be a weak contraction on
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ2_HTML.gif)
for some is a continuous and nondecreasing function such that
is positive on
and
. Then
has a unique fixed point.
The concept of the weak contraction is defined by Alber and Guerre-Delabriere [3] in 1997. The natural generalization of the contraction as well as the weak contraction is nonexpansive. Let be a nonempty subset of Banach space
,
is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ3_HTML.gif)
One classical way to study nonexpansive mappings is to use a contraction to approximate a nonexpansive mapping. More precisely, take and define a contraction
by
where
is a fixed point. Banach Contraction Principle guarantees that
has a unique fixed point
in
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ4_HTML.gif)
Halpern [4] also firstly introduced the following explicit iteration scheme in Hilbert spaces: for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ5_HTML.gif)
In the case of having a fixed point, Browder [5] (resp. Halpern [4]) proved that if
is a Hilbert space, then
(resp.
) converges strongly to the fixed point of
, that is, nearest to
. Reich [6] extended Halpern's and Browder's result to the setting of Banach spaces and proved that if
is a uniformly smooth Banach space, then
and
converge strongly to a same fixed point of
, respectively, and the limit of
defines the (unique) sunny nonexpansive retraction from
onto
. In 1984, Takahashi and Ueda [7] obtained the same conclusion as Reich's in uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Recently, Xu [8] showed that the above result holds in a reflexive Banach space which has a weakly continuous duality mapping
. In 1992, Wittmann [9] studied the iterative scheme (1.5) in Hilbert space, and obtained convergence of the iterations. In particular, he proved a strong convergence result [9, Theorem  2] under the control conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ6_HTML.gif)
In 2002, Xu [10, 11] extended wittmann's result to a uniformly smooth Banach space, and gained the strong convergence of under the control conditions
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ7_HTML.gif)
Actually, Xu [10, 11] and Wittmann [9] proved the following approximate fixed points theorem. Also see [12, 13].
Theorem 1.3.
Let be a nonempty closed convex subset of a Banach space
. provided that
is nonexpansive with
, and
is given by (1.5) and
satisfies the condition
,
and
(or
). Then
is bounded and
In 2000, for a nonexpansive selfmapping with
and a fixed contractive selfmapping
, Moudafi [14] introduced the following viscosity approximation method for
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ8_HTML.gif)
and proved that converges to a fixed point
of
in a Hilbert space. They are very important because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential equations. Xu [15] extended Moudafi's results to a uniformly smooth Banach space. Recently, Song and Chen [12, 13, 16–18] obtained a number of strong convergence results about viscosity approximations (1.8). Very recently, Petrusel and Yao [19], Wong, et al. [20] also studied the convergence of viscosity approximations, respectively.
In this paper, we naturally introduce viscosity approximations (1.9) and (1.10) with the weak contraction for a nonexpansive mapping sequence
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ10_HTML.gif)
We will prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern's type iteration and Moudafi's viscosity approximations with the weak contraction.
2. Preliminaries and Basic Results
Throughout this paper, a Banach space will always be over the real scalar field. We denote its norm by
and its dual space by
. The value of
at
is denoted by
and the normalized duality mapping
from
into
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ11_HTML.gif)
Let denote the set of all fixed points for a mapping
, that is,
, and let
denote the set of all positive integers. We write
(resp.
) to indicate that the sequence
weakly (resp. wea
) converges to
; as usual
will symbolize strong convergence.
In the proof of our main results, we need the following definitions and results. Let denote the unit sphere of a Banach space
.
is said to have (i) a Gâteaux differentiable norm (we also say that
is smooth), if the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ12_HTML.gif)
exists for each ; (ii) a uniformly Gâteaux differentiable norm, if for each
in
, the limit (2.2) is uniformly attained for
; (iii) a Fréchet differentiable norm, if for each
, the limit (2.2) is attained uniformly for
; (iv) a uniformly Fréchet differentiable norm (we also say that
is uniformly smooth), if the limit (2.2) is attained uniformly for
. A Banach space
is said to be (v) strictly convex if
(vi) uniformly convex if for all
,
such that
For more details on geometry of Banach spaces, see [21, 22].
If is a nonempty convex subset of a Banach space
and
is a nonempty subset of
, then a mapping
is called a retraction if
is continuous with
. A mapping
is called sunny if
whenever
and
. A subset
of
is said to be a sunny nonexpansive retract of
if there exists a sunny nonexpansive retraction of
onto
. We note that if
is closed and convex of a Hilbert space
, then the metric projection coincides with the sunny nonexpansive retraction from
onto
. The following lemma is well known which is given in [22, 23].
Lemma 2.1 (see [22, Lemma  5.1.6]).
Let be nonempty convex subset of a smooth Banach space
,
,
the normalized duality mapping of
, and
a retraction. Then
is both sunny and nonexpansive if and only if there holds the inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ13_HTML.gif)
Hence, there is at most one sunny nonexpansive retraction from onto
.
In order to showing our main outcomes, we also need the following results. For completeness, we give a proof.
Proposition 2.2.
Let be a convex subset of a smooth Banach space
. Let
be a subset of
and let
be the unique sunny nonexpansive retraction from
onto
. Suppose
is a weak contraction with a function
on
and
is a nonexpansive mapping. Then
(i)the composite mapping is a weak contraction on
;
(ii)For each , a mapping
is a weak contraction on
. Moreover,
defined by (2.4) is well definition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ14_HTML.gif)
(iii) if and only if
is a unique solution of the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ15_HTML.gif)
Proof.
For any , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ16_HTML.gif)
So, is a weakly contractive mapping with a function
. For each fixed
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ17_HTML.gif)
Namely, is a weakly contractive mapping with a function
. Thus, Theorem 1.2 guarantees that
has a unique fixed point
in
, that is,
satisfying (2.4) is uniquely defined for each
. (i) and (ii) are proved.
Subsequently, we show (iii). Indeed, by Theorem 1.2, there exists a unique element such that
. Such a
fulfils (2.5) by Lemma 2.1. Next we show that the variational inequality (2.5) has a unique solution
. In fact, suppose
is another solution of (2.5). That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ18_HTML.gif)
Adding up gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ19_HTML.gif)
Hence by the property of
. This completes the proof.
Let be a sequence of nonexpansive mappings with
on a closed convex subset
of a Banach space
and let
be a sequence in
with (C1).
is said to have Browder's property if for each
, a sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ20_HTML.gif)
for , converges strongly. Let
be a sequence in
with (C1) and (C2). Then
is said to have Halpern's property if for each
, a sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ21_HTML.gif)
for , converges strongly.
We know that if is a uniformly smooth Banach space or a uniformly convex Banach space with a uniformly Gâteaux differentiable norm,
is bounded,
is a constant sequence
, then
has both Browder's and Halpern's property (see [7, 10, 11, 23], resp.).
Lemma 2.3 (see [24, Proposition   4]).
Let have Browder's property. For each
, put
, where
is a sequence in
defined by (2.10). Then
is a nonexpansive mapping on
.
Lemma 2.4 (see [24, Proposition  5]).
Let have Halpern's property. For each
, put
, where
is a sequence in
defined by (2.11). Then the following hold: (i)
does not depend on the initial point
. (ii)
is a nonexpansive mapping on
.
Proposition 2.5.
Let be a smooth Banach space, and
have Browder's property. Then
is a sunny nonexpansive retract of
, and moreover,
define a sunny nonexpansive retraction from
to
.
Proof.
For each , it is easy to see from (2.10) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ23_HTML.gif)
This implies for any and some
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ24_HTML.gif)
The smoothness of implies the norm weak
continuity of
[22, Theorems   4.3.1,  4.3.2], so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ25_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ26_HTML.gif)
By Lemma 2.1, is a sunny nonexpansive retraction from
to
.
We will use the following facts concerning numerical recursive inequalities (see [25–27]).
Lemma 2.6.
Let and
be two sequences of nonnegative real numbers, and
a sequence of positive numbers satisfying the conditions
and
. Let the recursive inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ27_HTML.gif)
be given where is a continuous and strict increasing function for all
with
. Then ( 1)
converges to zero, as
; ( 2) there exists a subsequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ28_HTML.gif)
3. Main Results
We first discuss Browder's type convergence.
Theorem 3.1.
Let have Browder's property. For each
, put
, where
is a sequence in
defined by (2.10). Let
be a weak contraction with a function
. Define a sequence
in
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ29_HTML.gif)
Then converges strongly to the unique point
satisfying
.
Proof.
We note that Proposition 2.2(ii) assures the existence and uniqueness of . It follows from Proposition 2.2(i) and Lemma 2.3 that
is a weak contraction on
, then by Theorem 1.2, there exists the unique element
such that
. Define a sequence
in
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ30_HTML.gif)
Then by the assumption, converges strongly to
. For every
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ31_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ32_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ33_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ34_HTML.gif)
Consequently, converges strongly to
. This completes the proof.
We next discuss Halpern's type convergence.
Theorem 3.2.
Let have Halpern's property. For each
, put
, where
is a sequence in
defined by (2.11). Let
be a weak contraction with a function
. Define a sequence
in
by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ35_HTML.gif)
Then converges strongly to the unique point
satisfying
. Moreover, there exist a subsequence
, and
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ36_HTML.gif)
Proof.
It follows from Proposition 2.2(i) and Lemma 2.4 that is a weak contraction on
, then by Theorem 1.2, there exists a unique element
such that
. Thus we may define a sequence
in
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ37_HTML.gif)
Then by the assumption, as
. For every
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ38_HTML.gif)
Thus, we get for the following recursive inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ39_HTML.gif)
where , and
. Thus by Lemma 2.6,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ40_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ41_HTML.gif)
Consequently, we obtain the strong convergence of to
, and the remainder estimates now follow from Lemma 2.6.
Theorem 3.3.
Let be a Banach space
whose norm is uniformly Gâteaux differentiable, and
satisfies the condition (C2). Assume that
have Browder's property and
for every
, where
is a bounded sequence in
defined by (2.10). then
have Halpern's property.
Proof.
Define a sequence in
by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ42_HTML.gif)
It follows from Proposition 2.5 and the assumption that is the unique sunny nonexpansive retraction from
to
Subsequently, we approved that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ43_HTML.gif)
In fact, since , then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ44_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ45_HTML.gif)
where is a constant such that
by the boundedness of
and
Therefore, using
, and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ46_HTML.gif)
On the other hand, since the duality map is norm topology to weak
topology uniformly continuous in a Banach space
with uniformly Gâteaux differentiable norm, we get that as
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ47_HTML.gif)
Therefore for any ,
such that for all
and all
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ48_HTML.gif)
Hence noting (3.18), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ49_HTML.gif)
(3.15) is proved. From (2.10) and
, we have for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ50_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ51_HTML.gif)
Consequently, we get for the following recursive inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ52_HTML.gif)
where and
The strong convergence of
to
follows from Lemma 2.6. Namely,
have Halpern's property.
4. Deduced Theorems
Using Theorems 3.1, 3.2, and 3.3, we can obtain many convergence theorems. We state some of them.
We now discuss convergence theorems for families of nonexpansive mappings. Let be a nonempty closed convex subset of a Banach space
. A (one parameter) nonexpansive semigroups is a family
of selfmappings of
such that
(i) for
(ii) for
and
;
(iii) for
;
(iv)for each is nonexpansive, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ53_HTML.gif)
We will denote by the common fixed point set of
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ54_HTML.gif)
A continuous operator semigroup is said to be uniformly asymptotically regular (in short, u.a.r.) (see [28–31]) on
if for all
and any bounded subset
of
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ55_HTML.gif)
Recently, Song and Xu [31] showed that have both Browder's and Halpern's property in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm whenever
. As a direct consequence of Theorems 3.1, 3.2, and 3.3, we obtain the following.
Theorem 4.1.
Let be a real reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm, and
a nonempty closed convex subset of
, and
a u.a.r. nonexpansive semigroup from
into itself such that
, and
a weak contraction. Suppose that
and
satisfies the condition
, and
satisfies the conditions
and
. If
and
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ56_HTML.gif)
Then as , both
and
strongly converge to
, where
is a sunny nonexpansive retraction from
to
.
Let a sequence of positive real numbers divergent to
, and for each
and
,
is the average given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ57_HTML.gif)
Recently, Chen and Song [32] showed that have both Browder's and Halpern's property in a uniformly convex Banach space with a uniformly Gâeaux differentiable norm whenever
. Then we also have the following.
Theorem 4.2.
Let be a uniformly convex Banach space with uniformly Gâteaux differentiable norm, and let
be as in Theorem 4.1. Suppose that
a nonexpansive semigroups from
into itself such that
,
and
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F824374/MediaObjects/13663_2009_Article_1184_Equ58_HTML.gif)
where , and
satisfies the condition
, and
satisfies the conditions
and
. Then as
, both
and
strongly converge to
, where
is a sunny nonexpansive retraction from
to
.
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Song, Y., Liu, X. Convergence Comparison of Several Iteration Algorithms for the Common Fixed Point Problems. Fixed Point Theory Appl 2009, 824374 (2009). https://doi.org/10.1155/2009/824374
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DOI: https://doi.org/10.1155/2009/824374