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Viscosity Approximation to Common Fixed Points of Families of Nonexpansive Mappings with Weakly Contractive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 476913 (2010)
Abstract
Let X be a reflexive Banach space which has a weakly sequentially continuous duality mapping. In this paper, we consider the following viscosity approximation sequence , where
(0, 1),
is a uniformly asymptotically regular sequence, and f is a weakly contractive mapping. Strong convergence of the sequence
is proved.
1. Introduction
Let be a nonempty closed convex subset of a Banach space
. Recall that a self-mapping
is nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ1_HTML.gif)
Alber and Guerre-Delabriere [1] defined the weakly contractive maps in Hilbert spaces, and Rhoades [2] showed that the result of [1] is also valid in the complete metric spaces as follows.
Definition 1.1.
Let be a complete metric space. A mapping
is called weakly contractive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ2_HTML.gif)
where and
is a continuous and nondecreasing function such that
if and only if
and
.
Theorem 1.2.
Let be a weakly contractive mapping, where
is a complete metric space, then
has a unique fixed point.
In 2007, Song and Chen [3] considered the iterative sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ3_HTML.gif)
They proved the strong convergence of the iterative sequence , where
is a contraction mapping and
is a uniformly asymptotically regular sequence of nonexpansive mappings in a reflexive Banach space
, as follows.
Theorem 1.3 (see [3, Theorem ]).
Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping
from
to
. Suppose that
is a nonempty closed convex subset of
and
is a uniformly asymptotically regular sequence of nonexpansive mappings from
into itself such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ4_HTML.gif)
where . Let
be defined by (1.3) and
, such that
. Then as
, the sequence
converges strongly to
, such that
is the unique solution, in
, to the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ5_HTML.gif)
In this paper, inspired by the above results, strong convergence of sequence (1.3) is proved, where is a weakly contractive mapping.
2. Preliminaries
A Banach space is called strictly convex if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ6_HTML.gif)
A Banach space is called uniformly convex, if for all
there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ7_HTML.gif)
The following results are well known which can be founded in [4].
(1)A uniformly convex Banach space is reflexive and strictly convex.
(2)If is a nonempty convex subset of a strictly convex Banach space
and
is a nonexpansive mapping, then the fixed point set
of
is a closed convex subset of
.
By a gauge function we mean a continuous strictly increasing function defined on
such that
and
. The mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ8_HTML.gif)
is called the duality mapping with gauge function . In the case where
then
which is the normalized duality mapping.
Proposition 2.1 (see [5]).
-
(1)
if and only if
is a Hilbert space.
-
(2)
is surjective if and only if
is reflexive.
-
(3)
for all
; in particular
, for all
.
We say that a Banach space has a weakly sequentially continuous duality mapping if there exists a gauge function
such that the duality mapping
is single-valued and continuous from the weak topology to the wea
topology of
.
We recall [6] that a Banach space is said to satisfy Opial's condition, if for any sequence
in
, which converges weakly to
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ9_HTML.gif)
It is known [7] that any separable Banach space can be equivalently renormed such that it satisfies Opial's condition. A space with a weakly sequentially continuous duality mapping is easily seen to satisfy Opial's condition [8].
Lemma 2.2 (see [9, Lemma ]).
Let be a Banach space satisfying Opial's condition and
a nonempty, closed, and convex subset of
. Suppose that
is a nonexpansive mapping. Then
is demiclosed at zero, that is, if
is a sequence in
which converges weakly to
and if the sequence
converges strongly to zero, then
.
Definition 2.3 (see [3]).
Let be a nonempty closed convex subset of a Banach space
and
, where
. Then the mapping sequence
is called uniformly asymptotically regular on
, if for all
and any bounded subset
of
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ10_HTML.gif)
3. Main Result
In this section, we prove a new version of Theorem 1.3.
Theorem 3.1.
Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping
from
to
. Suppose that
is a nonempty closed convex subset of
and
is a uniformly asymptotically regular sequence of nonexpansive mappings such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ11_HTML.gif)
Let be a weakly contractive mapping. Suppose that
is a sequence of positive numbers in
satisfying
. Assume that
is defined by the following iterative process:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ12_HTML.gif)
Then the above sequence converges strongly to a common fixed point
of
such that
is the unique solution, in
, to the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ13_HTML.gif)
Proof.
Step 1.
We prove the uniqueness of the solution to the variational inequality (3.3). Suppose that are distinct solutions to (3.3). Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ14_HTML.gif)
By adding up the above relations, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ15_HTML.gif)
Thus , hence
. We denote by
the unique solution, in
, to (3.3).
Step 2.
We show that the sequence is bounded. Let
; from (3.2) we get then that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ16_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ17_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ18_HTML.gif)
Therefore is bounded.
Step 3.
We prove that , for all
. Since the sequence
is bounded, so
and
are bounded. Hence
, thus
. Let
be a bounded subset of
which contains
. Since the sequence
is uniformly asymptotically regular, we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ19_HTML.gif)
Let , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ20_HTML.gif)
Hence , for all
.
Step 4.
We show that the sequence is sequentially compact. Since
is reflexive and
is bounded, there exists a subsequence
of
such that
is weakly convergent to
as
. Since
for all
, by Lemma 2.2, we have
for all
. Thus
.
Step 2 implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ21_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ22_HTML.gif)
Since is single valued and weakly sequentially continuous from
to
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ23_HTML.gif)
Thus . Hence the sequence
is sequentially compact.
Step 5.
We now prove that is a solution to the variational inequality (3.3). Suppose that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ24_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ25_HTML.gif)
Since as
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ26_HTML.gif)
as . Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ27_HTML.gif)
Thus is a solution to the variational inequality (3.3). By uniqueness,
. Since the sequence
is sequentially compact and each cluster point of it is equal to
, then
as
. The proof is completed.
It is known that [10, Example ] in a uniformly convex Banach space
, the Cesà ro means
for nonexpansive mapping
is uniformly asymptotically regular. So we have the following corollary, which is a new version of [10, Theorem
].
Corollary 3.2.
Let be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping
from
to
and
a nonempty closed convex subset of
. Suppose that
is a nonexpansive mapping,
and
is a weakly contractive mapping. Let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ28_HTML.gif)
where and
. Then as
,
converges strongly to a fixed point
of
, where
is the unique solution in
to the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476913/MediaObjects/13663_2010_Article_1291_Equ29_HTML.gif)
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Acknowledgment
A. Razani would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Teheran, Iran for supporting this paper (Grant no.89470126).
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Razani, A., Homaeipour, S. Viscosity Approximation to Common Fixed Points of Families of Nonexpansive Mappings with Weakly Contractive Mappings. Fixed Point Theory Appl 2010, 476913 (2010). https://doi.org/10.1155/2010/476913
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DOI: https://doi.org/10.1155/2010/476913