- Research Article
- Open access
- Published:
Strong Convergence of an Implicit Iteration Algorithm for a Finite Family of Pseudocontractive Mappings
Journal of Inequalities and Applications volume 2008, Article number: 280908 (2007)
Abstract
Strong convergence theorems for approximation of common fixed points of a finite family of pseudocontractive mappings are proven in Banach spaces using an implicit iteration scheme. The results presented in this paper improve and extend the corresponding results of Osilike, Xu and Ori, Chidume and Shahzad, and others.
1. Introduction
Let be a real Banach space and let
denote the normalized duality mapping from
into
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ1_HTML.gif)
where denotes the dual space of
and
denotes the generalized duality pairing. If
is strictly convex, then
is single valued. In the sequel, we will denote the single-value duality mapping by
.
Let be a nonempty closed convex subset of
. Recall that a self-mapping
is said to be a contraction if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ2_HTML.gif)
We use to denote the collection of all contractions on
. That is,
.
A mapping with domain
and
in
is called pseudocontractive if, for all
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ3_HTML.gif)
We use to denote the fixed point set of
, that is,
.
Recently, Xu and Ori [1] have introduced an implicit iteration process below for a finite family of nonexpansive mappings. Let be
self-mappings of
and suppose that
, the set of common fixed points of
An implicit iteration process for a finite family of nonexpansive mappings is defined as follows with
a real sequence in
,
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ4_HTML.gif)
which can be written in the following compact form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ5_HTML.gif)
where .
Xu and Ori proved the weak convergence of the above iterative process (1.5) to a common fixed point of a finite family of nonexpansive mappings in a Hilbert space. They further remarked that it is yet unclear what assumptions on the mapping and/or the parameters
are sufficient to guarantee the strong convergence of the sequence
.
Very recently, Osilike [2] first extended Xu and Ori [1] from the class of nonexpansive mappings to the more general class of strictly pseudocontractive mappings in a Hilbert space. He proved the following two convergence theorems.
Theorem 1.
Let be a real Hilbert space and let
be a nonempty closed convex subset of
. Let
be
strictly pseudocontractive self-mappings of
such that
. Let
and let
be a sequence in
such that
. Then the sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ6_HTML.gif)
where , converges weakly to a common fixed point of the mappings
.
Theorem 1.
Let be a real Banach space and let
be a nonempty closed convex subset of
. Let
be
strictly pseudocontractive self-mappings of
such that
, and let
be a real sequence satisfying the conditions
,
and
. Let
and let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ7_HTML.gif)
where . Then
converges strongly to a common fixed point of the mappings
if and only if
.
Remark 1.1.
We note that Theorem O1 has only weak convergence even in a Hilbert space and Theorem O2 has strong convergence, but imposed condition .
In 2005, Chidume and Shahzad [3] also proved the strong convergence of the implicit iteration process (1.5) to a common fixed point for a finite family of nonexpansive mappings. They gave the following theorem.
Theorem 1 CS.
Let be a uniformly convex Banach space, let
be a nonempty closed convex subset of
. Let
be
nonexpansive self-mappings of
with
. Suppose that one of the mappings in
is semicompact. Let
for some
. From arbitrary
, define the sequence
by (1.5). Then
converges strongly to a common fixed point of the mappings
.
Remark 1.2.
Chidume and Shahzad gave an affirmative response to the question raised by Xu and Ori [1], but they imposed compactness condition on some mapping of .
In this paper, we will consider a process for a finite family of pseudocontractive mappings which include the nonexpansive mappings as special cases. Let be a contraction. Let
,
, and
be three real sequences in
and an initial point
. Let the sequence
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ8_HTML.gif)
which can be written in the following compact form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ9_HTML.gif)
where .
Motivated by the works in [1–6], our purpose in this paper is to study the implicit iteration process (1.9) in the general setting of a uniformly smooth Banach space and prove the strong convergence of the iterative process (1.9) to a common fixed point of a finite family of pseudocontractive mappings . The results presented in this paper generalize and extend the corresponding results of Chidume and Shahzad [3], Osilike [2], Xu and Ori [1], and others.
2. Preliminaries
Let be a Banach space. Recall the norm of
is said to be Gateaux differentiable (and
is said to be smooth) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ10_HTML.gif)
exists for each in its unit sphere
. It is said to be uniformly Frechet differentiable (and
is said to be uniformly smooth) if the limit in (2.1) is attained uniformly for
It is well known that a Banach space
is uniformly smooth if and only if the duality map
is single valued and norm-to-norm uniformly continuous on bounded sets of
.
Recall that if and
are nonempty subsets of a Banach space
such that
is nonempty closed convex and
, then a map
is called a retraction from
onto
provided
for all
. A retraction
is sunny provided
for all
and
whenever
. A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive.
We need the following lemmas for proof of our main results.
Lemma 2.1 (see [7]).
Let be a uniformly smooth Banach space,
a closed convex subset of
,
a nonexpansive with
. For each
and every
, then
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ11_HTML.gif)
converges strongly as to a fixed point of
.
In particular, if is a constant, then (2.2) is reduced to the sunny nonexpansive retraction of Reich from
onto
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ12_HTML.gif)
Lemma 2.2 (see [8]).
Let be a real uniformly smooth Banach space, then there exists a nondecreasing continuous function
satisfying
(i) for all
;
(ii);
(iii), for all
.
The inequality (iii) is called Reich's inequality.
Lemma 2.3 (see [9]).
Let be a sequences of nonegative real numbers satisfying the property
where
and
are such that
(i);
(ii)either or
.
Then converges to
.
3. Main Results
Theorem 3.1.
Let be a uniformly smooth Banach space and let
be a nonempty closed convex subset of
. Let
be
pseudocontractive self-mappings of
such that
. Let
, and
be three real sequences in
satisfying the following conditions:
(i);
(ii) and
;
(iii).
For and given
arbitrarily, let the sequence
be defined by (1.9). Then
converges strongly to a common fixed point
of the mappings
, where
is the unique solution of the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ13_HTML.gif)
Proof.
First, we observe that is bounded. Indeed, if we take a fixed point
of
, noting that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ14_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ15_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ16_HTML.gif)
Now, an induction yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ17_HTML.gif)
Hence is bounded, so are
and
for all
.
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ18_HTML.gif)
Set for all
, it is well known that
are all nonexpansive mappings and
as a consequence of [10, Theorem 6]. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ19_HTML.gif)
It also follows from (3.6) that .
Next, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ20_HTML.gif)
where with
being the fixed point of
(see Lemma 2.1).
Indeed, solves the fixed point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ21_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ22_HTML.gif)
Thus we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ23_HTML.gif)
Noting that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ24_HTML.gif)
Thus (3.11) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ25_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ26_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ27_HTML.gif)
Letting in (3.15) and noting (3.14) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ28_HTML.gif)
where is a constant.
For (3.9), since strongly converges to
, then
is bounded. Hence we obtain immediately that the set
is bounded. At the same time, we note that the duality map
is norm-to-norm uniformly continuous on bounded sets of
. By letting
in (3.16), it is not hard to find that the two limits can be interchanged and (3.8) is thus proven.
Finally, we show that strongly.
Indeed, using Lemma 2.2 and noting that (3.4), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ29_HTML.gif)
where and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ30_HTML.gif)
We observe that , then
and
is bounded. At the same time, from
, we have that
. This implies that
.
Now, we apply Lemma 2.3 and use (3.8) to see that . This completes the proof.
Remark 3.2.
Theorem 3.1 proves the strong convergence in the framework of real uniformly smooth Banach spaces. Our theorem extends Theorem O1 to the more general real Banach spaces. Our result improves Theorem O2 without condition and at the same time extends the mappings from nonexpansive mappings to pseudocontractive mappings.
Corollary 3.3.
Let be a uniformly smooth Banach space and let
be a nonempty closed convex subset of
. Let
be
pseudocontractive self-mappings of
such that
. Let
, and
be three real sequences in
satisfying the following conditions:
(i);
(ii) and
;
(iii).
For fixed and given
arbitrarily, let the sequence
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ31_HTML.gif)
Then converges strongly to a common fixed point
of the mappings
, where
is the unique solution of the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ32_HTML.gif)
where is a sunny nonexpansive retraction from
onto
.
Corollary 3.4.
Let be a uniformly smooth Banach space and let
be a nonempty closed convex subset of
. Let
be
nonexpansive self-mappings of
such that
. Let
, and
be three real sequences in
satisfying the following conditions:
(i);
(ii) and
;
(iii).
For and given
arbitrarily, let the sequence
be defined by (1.9). Then
converges strongly to a common fixed point
of the mappings
, where
is the unique solution of the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ33_HTML.gif)
Corollary 3.5.
Let be a uniformly smooth Banach space and let
be a nonempty closed convex subset of
. Let
be
nonexpansive self-mappings of
such that
. Let
, and
be three real sequences in
satisfying the following conditions:
(i);
(ii) and
;
(iii).
For fixed and given
arbitrarily, let the sequence
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ34_HTML.gif)
Then converges strongly to a common fixed point
of the mappings
, where
is the unique solution of the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F280908/MediaObjects/13660_2007_Article_1791_Equ35_HTML.gif)
where is a sunny nonexpansive retraction from
onto
.
Remark 3.6.
Corollary 3.5 improves Theorem CS without compactness assumption of mappings.
References
Xu HK, Ori RG: An implicit iteration process for nonexpansive mappings. Numerical Functional Analysis and Optimization 2001,22(5–6):767–773. 10.1081/NFA-100105317
Osilike MO: Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. Journal of Mathematical Analysis and Applications 2004,294(1):73–81. 10.1016/j.jmaa.2004.01.038
Chidume CE, Shahzad N: Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings. Nonlinear Analysis 2005,62(6):1149–1156. 10.1016/j.na.2005.05.002
Liou YC, Yao Y, Chen R: Iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive Mappings. Fixed Point Theory and Applications 2007, 2007:-10.
Liou YC, Yao Y, Kimura K: Strong convergence to common fixed points of a finite family of nonexpansive mappings. Journal of Inequalities and Applications 2007, 2007:-10.
Ceng LC, Wong NC, Yao JC: Implicit predictor-corrector iteration process for finitely many asymptotically (quasi-)nonexpansive mappings. Journal of Inequalities and Applications 2006, 2006:-11.
Xu HK: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059
Reich S: An iterative procedure for constructing zeros of accretive sets in Banach spaces. Nonlinear Analysis 1978,2(1):85–92. 10.1016/0362-546X(78)90044-5
Xu HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003,116(3):659–678. 10.1023/A:1023073621589
Martin RH Jr.: Differential equations on closed subsets of a Banach space. Transactions of the American Mathematical Society 1973, 179: 399–414.
Acknowledgments
The authors are extremely grateful to the referee for his/her careful reading. The first author was partially supposed by National Natural Science Foundation of China, Grant no. 10771050. The second author was partially supposed by the Grant no. NSC 96-2221-E-230-003.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Yao, Y., Liou, YC. Strong Convergence of an Implicit Iteration Algorithm for a Finite Family of Pseudocontractive Mappings. J Inequal Appl 2008, 280908 (2007). https://doi.org/10.1155/2008/280908
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2008/280908