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Mann Type Implicit Iteration Approximation for Multivalued Mappings in Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 140530 (2010)
Abstract
Let be a nonempty compact convex subset of a uniformly convex Banach space
and let
be a multivalued nonexpansive mapping. For the implicit iterates
,
,
,
. We proved that
converges strongly to a fixed point of
under some suitable conditions. Our results extended corresponding ones and revised a gap in the work of Panyanak (2007).
1. Introduction
Let be a nonempty subset of a Banach space
. We will denote
by the family of all subsets of
the family of nonempty closed and bounded subsets of
the family of nonempty compact subsets of
. Let
symbolize the family of nonempty compact convex subsets of
. Let
be Hausdorff metric on
; that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ1_HTML.gif)
where . A multivalued mapping
is called nonexpansive (resp., contractive), if for any
, there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ2_HTML.gif)
A point is called a fixed point of
if
. In this paper,
stands for the fixed point set of a mapping
.
The fixed point theory of multivalued nonexpansive mappings is much more complicated and difficult than the corresponding theory of single-valued nonexpansive mappings. However, some classical fixed point theorems for single-valued nonexpansive mappings have already been extended to multivalued mappings.
In 1968, Markin [1] firstly established the nonexpansive multivalued convergence results in Hilbert space. Banach's Contraction Principle was extended to a multivalued contraction in 1969. (Below is stated in a Banach space setting.)
Theorem 1.1 (see [2]).
Let be a nonempty closed subset of a Banach space
and
a multivalued contraction. Then
has a fixed point.
In 1974, one breakthrough was achieved by Lim using Edelstein's method of asymptotic centers [3].
Theorem 1.2 (see Lim [3]).
Let be a nonempty closed bounded convex subset of a uniformly convex Banach space
and
a multivalued nonexpansive mapping. Then
has a fixed point.
In 1990, Kirk and Massa [4] obtained another important result for multivalued nonexpansive mappings.
Theorem 1.3 (see Kirk and Massa [4]).
Let be a nonempty closed bounded convex subset of a Banach space
and
a multivalued nonexpansive mapping. Suppose that the asymptotic center in E of each bounded sequence of X is nonempty and compact. Then T has a fixed point.
In 1999, Sahu [5] obtained the strong convergence theorems of the nonexpansive type and nonself multivalued mappings for the following (1.3) iteration process:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ3_HTML.gif)
where and
. He proved that
converges strongly to some fixed points of
. Xu [6] extended Theorem 1.3 to a multivalued nonexpansive nonself mapping and obtained the fixed theorem in 2001. The recent fixed point results for nonexpansive mappings can be found in [7–12] and references therein.
Recently, Panyanak [13] studied the Mann iteration (1.4) and Ishikawa iterative processes (1.5) for as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ4_HTML.gif)
where , and fixed
are such that
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ5_HTML.gif)
where , and fixed
are such that
and
and proved the strong convergence theorems for multivalued nonexpansive mappings
in Banach spaces.
In this paper, motivated by Panyanak [13] and the previous results, we will study the following iteration process (1.6). Let be a nonempty convex subset of
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ6_HTML.gif)
and we prove some strong convergence theorems of the sequence defined by (1.6) for nonexpansive multivalued mappings in Banach spaces. The results presented in this paper establish a new type iteration convergence theorems for multivalued nonexpansive mappings in Banach spaces and extend the corresponding results of Panyanak [13].
2. Preliminaries
Let be a real Banach space and let
denote the normalized duality mapping from
to
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ7_HTML.gif)
where denotes the dual space of
and
denotes the generalized duality pair. It is well known that if
is strictly convex, then
is single valued. And if Banach space
admits sequentially continuous duality mapping
from weak topology to weak star topology, then, by [14, Lemma  1], we know that the duality mapping
is also single valued. In this case, the duality mapping
is also said to be weakly sequentially continuous; that is, if
is a subject of
with
, then
. By Theorem  1 of [14], we know that if
admits a weakly sequentially continuous duality mapping, then
satisfies Opial's condition, and
is smooth; for the details, see [14]. In the sequel, we will denote the single-valued duality mapping by
.
Throughout this paper, we write (resp.,
) to indicate that the sequence
weakly (resp., weak *) converges to
, as usual
will symbolize strong convergence. In order to show our main results, the following concepts and lemmas are needed.
A Banach space is called uniformly convex if for each
there is a
such that for
with
and
holds. The modulus of convexity of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ8_HTML.gif)
for all .
is said to be uniformly convex if
, and
for all
.
Lemma 2.1 (see [10]).
In Banach space , the following result (subdifferential inequality) is well known: for all
, for all
, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ9_HTML.gif)
Definition 2.2.
A Banach space is said to satisfy Opial's condition if for any sequence
in
(
) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ10_HTML.gif)
We know that Hilbert spaces, , and Banach space with weakly sequentially continuous duality mappings satisfy Opial's condition; for the details, see [14, 15].
Definition 2.3.
A multivalued mapping is said to satisfy Condition 
if there is a nondecreasing function
with
for
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ11_HTML.gif)
Example of mappings that satisfy Condition   can be founded in [13].
3. Main Results
Now, we prove our results.
Theorem 3.1.
Let be a nonempty compact convex subset of a uniformly convex Banach space
and let
be a multivalued nonexpansive mapping, where
and
, the sequence
is generated by (1.6).
Then,
(i)by the compactness of , there exists a subsequence
of
such that
for some
. In addition if
then
(ii) is a fixed point of
and the sequence
converges strongly to
.
Proof.
Part (i) is trivial. And part (ii) remains to be proved.
Due to the compactness of and boundness of
, there exists a real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ12_HTML.gif)
It follows from (1.6), that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ13_HTML.gif)
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ14_HTML.gif)
therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ15_HTML.gif)
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ16_HTML.gif)
Hence, is a fixed point of
.
Next we show that exists.
For all , there exist 
, using Lemma 2.1, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ17_HTML.gif)
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ18_HTML.gif)
If , then
apparently holds.
Let , from (3.7), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ19_HTML.gif)
We get that is a decreasing sequence, so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ20_HTML.gif)
So the desired conclusion follows.
The proof is completed.
Remark 3.2.
The above result modified the gap in the proof of Theorem  3.1 in [13] by a new method; the gap discovered by Song and Wang [16] is as follows.
Panyanak [13] introduced the Ishikawa iterates (1.5) of a multivalued mapping . It is obvious that
depends on
and
. For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ21_HTML.gif)
Clearly, if and
, then the above inequalities cannot be assured. Indeed, from the monotone decreasing sequence of
in the proof of (Theorem  3.1 [13]), we cannot obtain that
is a decreasing sequence. Hence, the conclusion of Theorem  3.1 in [13] cannot be achieved.
Theorem 3.3.
Let be a Banach space satisfying Opial's condition and let
be a nonempty weakly compact convex subset of
. Suppose that
is a multivalued nonexpansive mapping, where
and
, the sequence
is generated by (1.6).
Then,
(i)by the weak compactness of , there exists a subsequence
of
such that
for some
. In addition if,
then
(ii) is a fixed point of
and the sequence
converges weakly to
.
Proof.
Part (i) is trivial. Now we prove part (ii).
It follows from (3.3) of Theorem 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ22_HTML.gif)
Since is weakly compact, from part (i), there exists a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ23_HTML.gif)
Suppose that does not belong to
. By the compactness of
, for any given
, there exist
such that
and
.
Thus , from
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ24_HTML.gif)
This is a contradiction by satisfying Opial's condition.
Hence, is a fixed point of
.
It follows from (3.7) of Theorem 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ25_HTML.gif)
Next we show . Suppose not. There exists another subsequence
of
such that
.
Then, we also obtain . From Opial's condition, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ26_HTML.gif)
Which is a contradiction, so the conclusion of the theorem follows.
The proof is completed.
Corollary 3.4.
Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping
from
to
, and let
be a nonempty weakly compact convex subset of
. Suppose that
is a multivalued nonexpansive mapping, where
and
, the sequence
is generated by (1.6).
Then,
(i)by the weak compactness of , there exists a subsequence
of
such that
for some
. In addition if,
then
(ii) is a fixed point of
and the sequence
converges weakly to
.
Proposition 3.5.
Let be a nonempty compact convex subset of a uniformly convex Banach space
and let
be a multivalued nonexpansive mapping. Then
is a closed subset of
.
Proof.
Suppose , such that
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ27_HTML.gif)
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ28_HTML.gif)
Hence, is a fixed point of
.
Thus, is a closed subset of
.
The proof is completed.
Theorem 3.6.
Let be a nonempty compact convex subset of a uniformly convex Banach space
and let
be a multivalued nonexpansive mapping satisfying Condition  
, where
and
, then the sequence
generated by (1.6) converges strongly to a fixed point.
Proof.
It follows from (3.3) of Theorem 3.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F140530/MediaObjects/13663_2010_Article_1204_Equ29_HTML.gif)
The proof of remained part is omitted because it is similar to the proof of Theorem  3.8 in [13].
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Acknowledgment
The work was supported by the Fundamental Research Funds for the Central Universities, No. JY10000970006, and National Nature Science Foundation, No. 60974082.
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He, H., Liu, S. & Chen, R. Mann Type Implicit Iteration Approximation for Multivalued Mappings in Banach Spaces. Fixed Point Theory Appl 2010, 140530 (2010). https://doi.org/10.1155/2010/140530
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DOI: https://doi.org/10.1155/2010/140530