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Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 618767 (2010)
Abstract
Let be a nonempty compact convex subset of a uniformly convex Banach space
, and let
and
be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that
and
for all
. We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary
by
,
, where
and
,
are sequences of positive numbers satisfying
,
, converges strongly to a common fixed point of
and
; that is, there exists
such that
.
1. Introduction
Let be a Banach space, and let
be a nonempty subset of
. We will denote by
the family of nonempty bounded closed subsets of
and by
the family of nonempty compact convex subsets of
. Let
be the Hausdorff distance on
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ1_HTML.gif)
where is the distance from the point
to the subset
.
A mapping is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ2_HTML.gif)
A point is called a fixed point of
if
.
A multivalued mapping is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ3_HTML.gif)
A point is called a fixed point for a multivalued mapping
if
.
We use the notation standing for the set of fixed points of a mapping
and
standing for the set of common fixed points of
and
. Precisely, a point
is called a common fixed point of
and
if
.
In 2006, S. Dhompongsa et al. [1] proved a common fixed point theorem for two nonexpansive commuting mappings.
Theorem 1.1 (see [1, Theorem 4.2]).
Let be a nonempty bounded closed convex subset of a uniformly Banach space
, and let
, and
be a nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume that
and
are commuting; that is, if for every
such that
and
, there holds
. Then,
and
have a common fixed point.
In this paper, we introduce an iterative process in a new sense, called the modified Ishikawa iteration method with respect to a pair of single-valued and multivalued nonexpansive mappings. We also establish the strong convergence theorem of a sequence from such process in a nonempty compact convex subset of a uniformly convex Banach space.
2. Preliminaries
The important property of the uniformly convex Banach space we use is the following lemma proved by Schu [2] in 1991.
Lemma 2.1 (see [2]).
Let be a uniformly convex Banach space, let
be a sequence of real numbers such that
for all
, and let
and
be sequences of
such that
,
, and
for some
.  Then,
.
The following observation will be used in proving our results, and the proof is straightforward.
Lemma 2.2.
Let be a Banach space, and let
be a nonempty closed convex subset of
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ4_HTML.gif)
where and
is a multivalued nonexpansive mapping from
into
.
A fundamental principle which plays a key role in ergodic theory is the demiclosedness principle. A mapping defined on a subset
of a Banach space
is said to be demiclosed if any sequence
in
the following implication holds:
and
implies
.
Theorem 2.3 (see [3]).
Let be a nonempty closed convex subset of a uniformly convex Banach space
, and let
be a nonexpansive mapping. If a sequence
in
converges weakly to
and
converges to 0 as
, then
.
In 1974, Ishikawa introduced the following well-known iteration.
Definition 2.4 (see [4]).
Let be a Banach space, let
be a closed convex subset of
, and let
be a selfmap on
. For
, the sequence
of Ishikawa iterates of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ5_HTML.gif)
where and
are real sequences.
A nonempty subset of
is said to be proximinal if, for any
, there exists an element
such that
= dist
. We will denote
by the family of nonempty proximinal bounded subsets of
.
In 2005, Sastry and Babu [5] defined the Ishikawa iterative scheme for multivalued mappings as follows.
Let be a compact convex subset of a Hilbert space
, and let
be a multivalued mapping, and fix
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ6_HTML.gif)
where ,
are sequences in [
] with
such that
and
.
They also proved the strong convergence of the above Ishikawa iterative scheme for a multivalued nonexpansive mapping with a fixed point
under some certain conditions in a Hilbert space.
Recently, Panyanak [6] extended the results of Sastry and Babu [5] to a uniformly convex Banach space and also modified the above Ishikawa iterative scheme as follows.
Let be a nonempty convex subset of a uniformly convex Banach space
, and let
be a multivalued mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ7_HTML.gif)
where ,
are sequences in [
] with
and
such that
and
, respectively. Moreover,
and
such that
and
, respectively.
Very recently, Song and Wang [7, 8] improved the results of [5, 6] by means of the following Ishikawa iterative scheme.
Let be a multivalued mapping, where
. The Ishikawa iterative scheme
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ8_HTML.gif)
where and
such that
and
, respectively. Moreover,
such that
.
At the same period, Shahzad and Zegeye [9] modified the Ishikawa iterative scheme and extended the result of [7, Theorem 2] to a multivalued quasinonexpansive mapping as follows.
Let be a nonempty convex subset of a Banach space
, and let
be a multivalued mapping, where
. The Ishikawa iterative scheme
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ9_HTML.gif)
where and
.
In this paper, we introduce a new iteration method modifying the above ones and call it the modified Ishikawa iteration method.
Definition 2.5.
Let be a nonempty closed bounded convex subset of a Banach space
, let
be a single-valued nonexpansive mapping, and let
be a multivalued nonexpansive mapping. The sequence
of the modified Ishikawa iteration is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ10_HTML.gif)
where ,
, and
,
.
3. Main Results
We first prove the following lemmas, which play very important roles in this section.
Lemma 3.1.
Let be a nonempty compact convex subset of a uniformly convex Banach space
, and let
and
be a single-valued and a multivalued nonexpansive mapping, respectively, and
satisfying
for all
. Let
be the sequence of the modified Ishikawa iteration defined by (2.7). Then,
exists for all
.
Proof.
Letting and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ11_HTML.gif)
Since is a decreasing and bounded sequence, we can conclude that the limit of
exists.
We can see how Lemma 2.1 is useful via the following lemma.
Lemma 3.2.
Let be a nonempty compact convex subset of a uniformly convex Banach space
, and let
and
be a single-valued and a multivalued nonexpansive mapping, respectively, and
satisfying
for all
. Let
be the sequence of the modified Ishikawa iteration defined by (2.7). If
for some
, then,
.
Proof.
Let . By Lemma 3.1, we put
and consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ12_HTML.gif)
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ13_HTML.gif)
Further, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ14_HTML.gif)
By Lemma 2.1, we can conclude that .
Lemma 3.3.
Let be a nonempty compact convex subset of a uniformly convex Banach space
, and let
and
be a single-valued and a multivalued nonexpansive mapping, respectively, and
satisfying
for all
. Let
be the sequence of the modified Ishikawa iteration defined by (2.7). If
,
for some
, then
.
Proof.
Let . We put, as in Lemma 3.2,
. For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ15_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ16_HTML.gif)
Therefore, since ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ17_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ18_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ19_HTML.gif)
Since, from (3.3), , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ20_HTML.gif)
Recall that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ21_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ22_HTML.gif)
Using the fact that and by (3.10), we can conclude that
.
The following lemma allows us to go on.
Lemma 3.4.
Let be a nonempty compact convex subset of a uniformly convex Banach space
, and let
and
be a single-valued and a multivalued nonexpansive mapping, respectively, and
satisfying
for all
. Let
be the sequence of the modified Ishikawa iteration defined by (2.7). If
,
, then
.
Proof.
Consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ23_HTML.gif)
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ24_HTML.gif)
Hence, by Lemmas 3.2 and 3.3, .
We give the sufficient conditions which imply the existence of common fixed points for single-valued mappings and multivalued nonexpansive mappings, respectively, as follows
Theorem 3.5.
Let be a nonempty compact convex subset of a uniformly convex Banach space
, and let
and
be a single-valued and a multivalued nonexpansive mapping, respectively, and
satisfying
for all
. Let
be the sequence of the modified Ishikawa iteration defined by (2.7). If
,
, then
for some subsequence
of
implies
.
Proof.
Assume that . From Lemma 3.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ25_HTML.gif)
Since is demiclosed at 0, we have
, and hence
, that is,
. By Lemma 2.2 and by Lemma 3.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F618767/MediaObjects/13663_2010_Article_1314_Equ26_HTML.gif)
It follows that . Therefore
as desired.
Hereafter, we arrive at the convergence theorem of the sequence of the modified Ishikawa iteration. We conclude this paper with the following theorem.
Theorem 3.6.
Let be a nonempty compact convex subset of a uniformly convex Banach space
, and let
and
be a single-valued and a multivalued nonexpansive mapping, respectively, and
satisfying
for all
. Let
be the sequence of the modified Ishikawa iteration defined by (2.7) with
,
. Then
converges strongly to a common fixed point of
and
.
Proof.
Since is contained in
which is compact, there exists a subsequence
of
such that
converges strongly to some point
, that is,
. By Theorem 3.5, we have
, and by Lemma 3.1, we have that
exists. It must be the case in which
. Therefore,
converges strongly to a common fixed point
of
and
.
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Acknowledgments
The first author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree for this research. The authors would like to express their deep gratitude to Prof. Dr. Sompong Dhompongsa whose guidance and support were valuable for the completion of the paper. This work was completed with the support of the Commission on Higher Education and The Thailand Research Fund under Grant MRG5180213.
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Sokhuma, K., Kaewkhao, A. Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces. Fixed Point Theory Appl 2010, 618767 (2010). https://doi.org/10.1155/2010/618767
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DOI: https://doi.org/10.1155/2010/618767