- Research Article
- Open access
- Published:
A Strong Convergence Theorem for a Family of Quasi-
-Nonexpansive Mappings in a Banach Space
Fixed Point Theory and Applications volume 2009, Article number: 351265 (2009)
Abstract
The purpose of this paper is to propose a modified hybrid projection algorithm and prove a strong convergence theorem for a family of quasi--nonexpansive mappings. The strong convergence theorem is proven in the more general reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the results of S. Matsushita and W. Takahashi (2005), X. L. Qin and Y. F. Su (2007), and others.
1. Introduction
It is well known that, in an infinite-dimensional Hilbert space, the normal Mann's iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Consequently, in order to obtain strong convergence, one has to modify the normal Mann's iteration algorithm, the so-called hybrid projection iteration method is such a modification.
The hybrid projection iteration algorithm (HPIA) was introduced initially by Haugazeau [1] in 1968. For 40 years, HPIA has received rapid developments. For details, the readers are referred to papers [2–7] and the references therein.
In 2005, Matsushita and Takahashi [5] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping in a Banach space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ1_HTML.gif)
They proved the following convergence theorem.
Theorem 1 MT.
Let be a uniformly convex and uniformly smooth Banach space, let
be a nonempty closed convex subset of
, let
be a relatively nonexpansive mapping from
into itself, and let
be a sequence of real numbers such that
and
. Suppose that
is given by (1.1), where
is the normalized duality mapping on
. If
is nonempty, then
converges strongly to
, where
is the generalized projection from
onto
.
In 2007, Qin and Su [2] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping in a Banach space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ2_HTML.gif)
They proved the following convergence theorem.
Theorem 1 QS.
Let be a uniformly convex and uniformly smooth Banach space, let
be a nonempty closed convex subset of
, let
be a relatively nonexpansive mapping from
into itself such that
. Assume that
and
are sequences in
such that
and
. Suppose that
is given by (1.2). If
is uniformly continuous, then
converges strongly to
.
Question 1.
Can both Theorems MT and QS be extended to more general reflexive, strictly convex, and smooth Banach spaces with the property (K)?
Question 2.
Can both Theorems MT and QS be extended to more general class of quasi--nonexpansive mappings?
The purpose of this paper is to give some affirmative answers to the questions mentioned previously, by introducing a modified hybrid projection iteration algorithm and by proving a strong convergence theorem for a family of closed and quasi--nonexpansive mappings by using new analysis techniques in the setting of reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the results of Matsushita and Takahashi [5], Qin and Su [2], and others.
2. Preliminaries
In this paper, we denote by and
a Banach space and the dual space of
, respectively. Let
be a nonempty closed convex subset of
. We denote by
the normalized duality mapping from
to
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ3_HTML.gif)
where denotes the generalized duality pairing between
and
. It is well known that if
is reflexive, strictly convex, and smooth, then
is single-valued, demi-continuous and strictly monotone (see, e.g., [8, 9]).
It is also very well known that if is a nonempty closed convex subset of a Hilbert space
and
is the metric projection of
onto
, then
is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [10] recently introduced a generalized projection operator
in a Banach space
which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that is a real reflexive, strictly convex, and smooth Banach space. Let us consider the functional defined as in [4, 5] by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ4_HTML.gif)
Observe that, in a Hilbert space , (2.2) reduces to
The generalized projection is a map that assigns to an arbitrary point
the unique minimum point of the functional
; that is,
where
is the unique solution to the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ5_HTML.gif)
Remark 2.1.
The existence and uniqueness of the element follow from the reflexivity of
, the properties of the functional
and strict monotonicity of the mapping
(see, e.g., [8–12]). In Hilbert spaces,
It is obvious from the definition of function
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ6_HTML.gif)
Remark 2.2.
If is a reflexive, strictly convex, and smooth Banach space, then for
,
if and only if
. It is sufficient to show that if
then
. From (2.4), we have
. This in turn implies that
From the smoothness of
, we know that
is single valued, and hence we have
. Since
is strictly convex,
is strictly monotone, in particular,
is one to one, which implies that
one may consult [8, 9] for the details.
Let be a closed convex subset of
, and
a mapping from
into itself. A point
in
is said to be asymptotic fixed point of
[13] if
contains a sequence
which converges weakly to
such that
. The set of asymptotic fixed point of
will be denoted by
. A mapping
from
into itself is said to be relatively nonexpansive [5, 14–16] if
and
for all
and
. The asymptotic behavior of a relatively nonexpansive mapping was studied in [14–16].
is said to be quasi-
-nonexpansive if
and
for all
and
.
Remark 2.3.
The class of quasi--nonexpansive mappings is more general than the class of relatively nonexpansive mappings [5, 14–16] which requires the strong restriction:
.
We present two examples which are closed and quasi--nonexpansive.
Example 2.4.
Let be the generalized projection from a smooth, strictly convex, and reflexive Banach space
onto a nonempty closed convex subset
of
. Then,
is a closed and quasi-
-nonexpansive mapping from
onto
with
.
Example 2.5.
Let be a reflexive, strictly convex, and smooth Banach space, and
is a maximal monotone mapping such that its zero set
is nonempty. Then,
is a closed and quasi-
-nonexpansive mapping from
onto
and
.
Recall that a Banach space has the property (K) if for any sequence
and
, if
weakly and
, then
. For more information concerning property (K) the reader is referred to [17] and references cited therein.
In order to prove our main result of this paper, we need to the following facts.
Lemma 2.6 (see, e.g., [10–12]).
Let be a convex subset of a real smooth Banach space
,
and
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ7_HTML.gif)
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ8_HTML.gif)
Lemma 2.7 (see, e.g., [10–12]).
Let be a convex subset of a real reflexive, strictly convex, and smooth Banach space
. Then the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ9_HTML.gif)
for all and
.
Now we are in a proposition to prove the main results of this paper.
3. Main Results
Theorem 3.1.
Let be a reflexive, strictly convex, smooth Banach space such that
and
have the property (K). Assume that
is a nonempty closed convex subset of
. Let
be an infinitely countable family of closed and quasi-
-nonexpansive mappings such that
. Assume that
are real sequences in
such that
. Define a sequence
in
by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ10_HTML.gif)
Then converges strongly to
, where
is the generalized projection from
onto
.
Proof.
We split the proof into six steps.
Step 1.
Show that is well defined for every
.
To this end, we prove first that is closed and convex for any
. Let
be a sequence in
with
as
, we prove that
. From the definition of quasi-
-nonexpansive mappings, one has
, which implies that
as
. Noticing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ11_HTML.gif)
By taking limit in (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ12_HTML.gif)
Hence . It implies that
for all
. We next show that
is convex. To this end, for arbitrary
, putting
, we prove that
. Indeed, by using the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ13_HTML.gif)
This implies that . Hence
is closed and convex for all
and consequently
is closed and convex. By our assumption that
, we have
is well defined for every
.
Step 2.
Show that is closed and convex for each
.
It suffices to show that for any ,
is closed and convex for every
. This can be proved by induction on
. In fact, for
,
is closed and convex. Assume that
is closed and convex for some
. For
, one obtains that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ14_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ15_HTML.gif)
It is easy to see that is closed and convex. Then, for all
,
is closed and convex. Consequently,
is closed and convex for all
.
Step 3.
Show that .
It suffices to show that for any ,
for every
. For any
, from the definition of quasi-
-nonexpansive mappings, we have
, for all
and
. Noting that for any
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ16_HTML.gif)
which implies that and consequently
. So
. Hence
is well defined for each
. Therefore, the iterative algorithm (3.1) is well defined.
Step 4.
Show that , where
.
From Steps 2 and 3, we obtain that is a nonempty, closed, and convex subset of
. Hence
is well defined for every
. From the construction of
, we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ17_HTML.gif)
Let , where
is the unique element that satisfies
. Since
, by Lemma 2.7, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ18_HTML.gif)
By the reflexivity of , we can assume that
weakly. Since
, for
, we have
for
. Since
is closed and convex, by the Marzur theorem,
for any
. Hence
. Moreover, by using the weakly lower semicontinuity of the norm on
and (3.9), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ19_HTML.gif)
which implies that . By using Lemma 2.6, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ20_HTML.gif)
and hence , since
is strictly monotone.
Further, by the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ21_HTML.gif)
which shows that . By the property (K) of
, we have
, where
.
Step 5.
Show that , for any
.
Since for all
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ22_HTML.gif)
Since and consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ23_HTML.gif)
Note that . Hence
and consequently
. This implies that
is bounded. Since
is reflexive,
is also reflexive. So we can assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ24_HTML.gif)
weakly. On the other hand, in view of the reflexivity of , one has
, which means that for
, there exists
, such that
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ25_HTML.gif)
where we used the weakly lower semicontinuity of the norm on . From (3.14), we have
and consequently
, which implies that
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ26_HTML.gif)
weakly. Since and
has the property (K), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ27_HTML.gif)
Since , noting that
is demi-continuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ28_HTML.gif)
weakly. Noticing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ29_HTML.gif)
which implies that . By using the property (K) of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ30_HTML.gif)
From (3.1), (3.18), (3.21), and , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ31_HTML.gif)
Since is demi-continuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ32_HTML.gif)
weakly in . Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ33_HTML.gif)
which implies that . By the property (K) of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ34_HTML.gif)
From and the closeness property of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ35_HTML.gif)
which implies that .
Step 6.
Show that .
It follows from Steps 3, 4, and 5 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ36_HTML.gif)
which implies that . Hence,
. Then
converges strongly to
. This completes the proof.
From Theorem 3.1, we can obtain the following corollary.
Corollary 3.2.
Let be a reflexive, strictly convex and smooth Banach space such that both
and
have the property (K). Assume that
is a nonempty closed convex subset of
. Let
be a closed and quasi-
-nonexpansive mapping. Assume that
is a sequence in
such that
. Define a sequence
in
by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ37_HTML.gif)
Then converges strongly to
, where
is the generalized projection from
onto
.
Remark 3.3.
Theorem 3.1 and its corollary improve and extend Theorems MT and QS at several aspects.
(i)From uniformly convex and uniformly smooth Banach spaces extend to reflexive, strictly convex and smooth Banach spaces with the property (K). In Theorem 3.1 and its corollary the hypotheses on are weaker than the usual assumptions of uniform convexity and uniform smoothness. For example, any strictly convex, reflexive and smooth Musielak-Orlicz space satisfies our assumptions [17] while, in general, these spaces need not to be uniformly convex or uniformly smooth.
(ii)From relatively nonexpansive mappings extend to closed and quasi--non-expansive mappings.
(iii)The continuity assumption on mapping in Theorem QS is removed.
(iv)Relax the restriction on from
to
.
Remark 3.4.
Corollary 3.2 presents some affirmative answers to Questions 1 and 2.
4. Applications
In this section, we present some applications of the main results in Section 3.
Theorem 4.1.
Let be a reflexive, strict, and smooth Banach space that both
and
have the property (K), and let
be a nonempty closed convex subset of
. Let
be a family of proper, lower semicontinuous, and convex functionals. Assume that the common fixed point set
is nonempty, where
for
and
. Let
be a sequence generated by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ38_HTML.gif)
where satisfies the restriction:
and
. Then
defined by (4.1) converges strongly to a minimizer
of the family
.
Proof.
By a result of Rockafellar [18], we see that is a maximal monotone mapping for every
. It follows from Example 2.5 that
is a closed and quasi-
-nonexpansive mapping for every
. Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ39_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ40_HTML.gif)
and the last inclusion relation is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F351265/MediaObjects/13663_2009_Article_1134_Equ41_HTML.gif)
Now the desired conclusion follows from Theorem 3.1. This completes the proof.
References
Haugazeau Y: Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes, Thése. Université de Paris, Paris, France;
Qin XL, Su YF: Strong convergence theorems for relatively nonexpansive mappings in a Banach space. Nonlinear Analysis 2007,67(6):1958–1965. 10.1016/j.na.2006.08.021
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4
Carlos MY, Xu HK: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis 2006,64(11):2400–2411. 10.1016/j.na.2005.08.018
Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. Journal of Approximation Theory 2005,134(2):257–266. 10.1016/j.jat.2005.02.007
Su YF, Qin XL: Strong convergence of modified Ishikawa iterations for nonlinear mappings. Proceedings of the Indian Academy of Sciences, Mathematical Sciences 2007,117(1):97–107. 10.1007/s12044-007-0008-y
Su YF, Wang DX, Shang MJ: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, Article ID 284613, 2008:-8.
Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Volume 62. Kluwer Academic Publishers Group, Dordrecht, The Netherlands; 1990.
Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000.
Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Volume 178. Edited by: Kartsatos AG. Marcel Dekker, New York, NY, USA; 1996:15–50.
Alber YaI, Reich S: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panamerican Mathematical Journal 1994,4(2):39–54.
Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal on Optimization 2002,13(3):938–945. 10.1137/S105262340139611X
Reich S: A weak convergence theorem for the alternating method with Bregman distances. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathmatics. Volume 178. Edited by: Kartsatos AG. Marcel Dekker, New York, NY, USA; 1996:313–318.
Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. Journal of Applied Analysis 2001,7(2):151–174. 10.1515/JAA.2001.151
Butnariu D, Reich S, Zaslavski AJ: Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numerical Functional Analysis and Optimization 2003,24(5–6):489–508. 10.1081/NFA-120023869
Censor Y, Reich S: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 1996,37(4):323–339. 10.1080/02331939608844225
Hudzik H, Kowalewski W, Lewicki G: Approximate compactness and full rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces. Zeitschrift fuer Analysis und ihre Anwendungen 2006,25(2):163–192.
Rockafellar RT: On the maximal monotonicity of subdifferential mappings. Pacific Journal of Mathematics 1970, 33: 209–216.
Acknowledgment
This work was supported by the National Natural Science Foundation of China under Grant (10771050).
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
Zhou, H., Gao, X. A Strong Convergence Theorem for a Family of Quasi--Nonexpansive Mappings in a Banach Space.
Fixed Point Theory Appl 2009, 351265 (2009). https://doi.org/10.1155/2009/351265
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/351265