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Convergence Theorems of Fixed Points for a Finite Family of Nonexpansive Mappings in Banach Spaces
Fixed Point Theory and Applications volume 2008, Article number: 856145 (2007)
Abstract
We modify the normal Mann iterative process to have strong convergence for a finite family nonexpansive mappings in the framework of Banach spaces without any commutative assumption. Our results improve the results announced by many others.
1. Introduction and Preliminaries
Throughout this paper, we assume that is a real Banach space with the normalized duality mapping
from
into
give by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ1_HTML.gif)
where denotes the dual space of
and
denotes the generalized duality pairing. We assume that
is a nonempty closed convex subset of
and
a mapping. A point
is a fixed point of
provided
. Denote by
the set of fixed points of
, that is,
. Recall that
is nonexpansive if
for all
One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping (see [1, 2]). More precisely, take and define a contraction
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ2_HTML.gif)
where is a fixed point. Banach's contraction mapping principle guarantees that
has a unique fixed point
in
. It is unclear, in general, what is the behavior of
as
even if
has a fixed point. However, in the case of
having a fixed point, Browder [1] proved that if
is a Hilbert space, then
converges strongly to a fixed point of
that is nearest to
. Reich [2] extended Broweder's result to the setting of Banach spaces and proved that if
is a uniformly smooth Banach space, then
converges strongly to a fixed point of
and the limit defines the (unique) sunny nonexpansive retraction from
onto
.
Recall that the normal Mann iterative process was introduced by Mann [3] in 1953. The normal Mann iterative process generates a sequence in the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ3_HTML.gif)
where the sequence is in the interval (0,1). If
is a nonexpansive mapping with a fixed point and the control sequence
is chosen so that
then the sequence
generated by normal Mann's iterative process (1.3) converges weakly to a fixed point of
(this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm [4]). In an infinite-dimensional Hilbert space, the normal Mann iteration algorithm has only weak convergence, in general, even for nonexpansive mappings. Therefore, many authors try to modify normal Mann's iteration process to have strong convergence for nonexpansive mappings (see, e.g., [5–8] and the references therein).
Recently, Kim and Xu [5] introduced the following iteration process:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ4_HTML.gif)
where is a nonexpansive mapping of
into itself and
is a given point. They proved that the sequence
defined by (1.4) converges strongly to a fixed point of
provided the control sequences
and
satisfy appropriate conditions.
Concerning a family of nonexpansive mappings it has been considered by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings; see, for example, [9]. The problem of finding an optimal point that minimizes a given cost function over common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see, e.g., [10]).
In this paper, we consider the mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ5_HTML.gif)
where are sequences in
. Such a mapping
is called the
-mapping generated by
and
. Nonexpansivity of each
ensures the nonexpansivity of
. Moreover, in [11], it is shown that
Motivated by Atsushiba and Takahashi [11], Kim and Xu [5], and Shang et al. [7], we study the following iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ6_HTML.gif)
where is defined by (1.5) and
is given point. We prove, under certain appropriate assumptions on the sequences
and
, that
defined by (1.6) converges to a common fixed point of the finite family nonexpansive mappings without any commutative assumptions.
In order to prove our main results, we need the following definitions and lemmas.
Recall that if and
are nonempty subsets of a Banach space
such that
is nonempty closed convex and
, then a map
is sunny (see [12, 13]) provided
for all
and
whenever
A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. Sunny nonexpansive retractions play an important role in our argument. They are characterized as follows [12, 13]: if
is a smooth Banach space, then
is a sunny nonexpansive retraction if and only if there holds the inequality
for all
and
Reich [2] showed that if is uniformly smooth and
is the fixed point set of a nonexpansive mapping from
into itself, then there is a sunny nonexpansive retraction from
onto
and it can be constructed as follows.
Lemma 1.1.
Let be a uniformly smooth Banach space and let
be a nonexpansive mapping with a fixed point. For each fixed
and
, 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
for all
and
Lemma 1.2 (See [14]).
Let and
be bounded sequences in a Banach space
and let
be a sequence in [0,1] with
. Suppose
for all integers
and
Then
Lemma 1.3.
In a Banach space , there holds the inequality
for all
where
.
Lemma 1.4 (See [15]).
Assume that is a sequence of nonnegative real numbers such that
where
is a sequence in (0,1) and
is a sequence such that
and
or
Then
2. Main Results
Theorem 2.1.
Let be a closed convex subset of a uniformly smooth and strictly convex Banach space
. Let
be a nonexpansive mapping from
into itself for
. Assume that
. Given a point
and given sequences
and
in (0,1), the following conditions are satisfied:
(i)
(ii)
(iii)
Let be the composite process defined by (1.6). Then
converges strongly to
, where
and
is the unique sunny nonexpansive retraction from
onto
.
Proof.
We divide the proof into four parts.
Step 1.
First we observe that sequences and
are bounded.
Indeed, take a point and notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ7_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ8_HTML.gif)
By simple inductions, we have which gives that the sequence
is bounded, so is
.
Step 2.
In this part, we will claim that as
Put . Now, we compute
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ9_HTML.gif)
Observing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ10_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ11_HTML.gif)
From the proof of Yao [8], we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ12_HTML.gif)
where is an appropriate constant. Substituting (2.6) into (2.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ13_HTML.gif)
Observing the conditions (i)–(iii), we get We can obtain
easily by Lemma 1.2. Observe that (2.3) yields
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ14_HTML.gif)
Step 3.
We will prove .
Observing that and the condition (i), we can easily get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ15_HTML.gif)
On the other hand, we have Combining (2.8) with (2.9), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ16_HTML.gif)
Notice that This implies
From the condition (iii) and (2.10), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ17_HTML.gif)
Step 4.
Finally, we will show as
.
First, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ18_HTML.gif)
where with
being the fixed point of the contraction
Then
solves the fixed point equation
Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ19_HTML.gif)
It follows from Lemma 1.3 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ20_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ21_HTML.gif)
It follows from (2.14) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ22_HTML.gif)
Letting in (2.16) and noting (2.15) yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ23_HTML.gif)
where is an appropriate constant. Taking
in (2.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ24_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ25_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ26_HTML.gif)
Noticing that is norm-to-norm uniformly continuous on bounded subsets of
and from (2.18), we have
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ27_HTML.gif)
Hence, (2.12) holds. Now, from Lemma 1.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F856145/MediaObjects/13663_2007_Article_1111_Equ28_HTML.gif)
Applying Lemma 1.4 to (2.22) we have as
Remark 2.2.
Theorem 2.1 improves the results of Kim and Xu [5] from a single nonexpansive mapping to a finite family of nonexpansive mappings.
Remark 2.3.
If is a contraction map and we replace
by
in the recursion formula (1.6), we obtain what some authors now call viscosity iteration method. We note that our theorem in this paper carries over trivially to the so-called viscosity process. Therefore, our results also include Yao et al. [16] as a special case.
Remark 2.4.
Our results partially improve Shang et al. [7] from a Hilbert space to a Banach space.
Remark 2.5.
If is a single nonexpansive mapping, then the strict convexity of
may not be needed.
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Acknowledgment
This paper was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2007-313-C00040).
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Cho, Y.J.e., Kang, S.M. & Qin, X. Convergence Theorems of Fixed Points for a Finite Family of Nonexpansive Mappings in Banach Spaces. Fixed Point Theory Appl 2008, 856145 (2007). https://doi.org/10.1155/2008/856145
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DOI: https://doi.org/10.1155/2008/856145