- Research Article
- Open access
- Published:
Strong Convergence of Two Iterative Algorithms for Nonexpansive Mappings in Hilbert Spaces
Fixed Point Theory and Applications volume 2009, Article number: 279058 (2009)
Abstract
We introduce two iterative algorithms for nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to a fixed point of a nonexpansive mapping .
1. Introduction
Let be a nonempty closed convex subset of a real Hilbert space
. Recall that a mapping
is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ1_HTML.gif)
for all . We use
to denote the set of fixed points of
.
Construction of fixed points of nonlinear mappings is an important and active research area. In particular, iterative algorithms for finding fixed points of nonexpansive mappings have received vast investigation (cf. [1, 2]) since these algorithms find applications in a variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing see; [3–8]. Iterative methods for nonexpansive mappings have been extensively investigated in the literature; see [1–7, 9–21].
It is our purpose in this paper to introduce two iterative algorithms for nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to a fixed point of nonexpansive mapping .
2. Preliminaries
Let be a nonempty closed convex subset of
. For every point
, there exists a unique nearest point in
, denoted by
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ2_HTML.gif)
The mapping is called the metric projection of
onto
. It is well known that
is a nonexpansive mapping.
In order to prove our main results, we need the following well-known lemmas.
Lemma 2.1 (see [22], Demiclosed principle).
Let be a nonempty closed convex of a real Hilbert space
. Let
be a nonexpansive mapping. Then
is demiclosed at
, that is, if
and
, then
.
Lemma 2.2 (see [20]).
Let ,
be bounded sequences in a Banach space
, and let
be a sequence in
which satisfies the following condition:
. Suppose that
for all
and
, then
.
Lemma 2.3 (see [22]).
Assume, that is a sequence of nonnegative real numbers such that
, where
is a sequence in
and
is a sequence in
such that
(i),
(ii) or
,
then .
3. Main Results
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a nonexpansive mapping. For each
, we consider the following mapping
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ3_HTML.gif)
It is easy to check that which implies that
is a contraction. Using the Banach contraction principle, there exists a unique fixed point
of
in
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ4_HTML.gif)
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a nonexpansive mapping with
. For each
, let the net
be generated by (3.2). Then, as
, the net
converges strongly to a fixed point of
.
Proof.
First, we prove that is bounded. Take
. From (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ5_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ6_HTML.gif)
Hence, is bounded.
Again from (3.2), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ7_HTML.gif)
Next we show that is relatively norm compact as
. Let
be a sequence such that
as
. Put
. From (3.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ8_HTML.gif)
From (3.2), we get, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ9_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ10_HTML.gif)
where is a constant such that
. In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ11_HTML.gif)
Since is bounded, without loss of generality, we may assume that
converges weakly to a point
. Noticing (3.6) we can use Lemma 2.1 to get
. Therefore we can substitute
for
in (3.9) to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ12_HTML.gif)
Hence, the weak convergence of to
actually implies that
strongly. This has proved the relative norm compactness of the net
as
.
To show that the entire net converges to
, assume
, where
. Put
. Similarly we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ13_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ14_HTML.gif)
Interchange and
to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ15_HTML.gif)
Adding up (3.12) and (3.13) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ16_HTML.gif)
which implies that . This completes the proof.
Theorem 3.2.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a nonexpansive mapping such that
. Let
and
be two real sequences in
. For given
arbitrarily, let the sequence
,
, be generated iteratively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ17_HTML.gif)
Suppose that the following conditions are satisfied:
(i) and
,
(ii),
then the sequence generated by (3.15) strongly converges to a fixed point of
.
Proof.
First, we prove that the sequence is bounded. Take
. From (3.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ18_HTML.gif)
Hence, is bounded and so is
.
Set . It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ19_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ20_HTML.gif)
This together with Lemma 2.2 implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ21_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ22_HTML.gif)
We observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ23_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ24_HTML.gif)
Let the net be defined by (3.2). By Theorem 3.1, we have
as
. Next we prove
. Indeed,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ25_HTML.gif)
where such that
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ26_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ27_HTML.gif)
We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ28_HTML.gif)
This together with and (3.25) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ29_HTML.gif)
Finally we show that . From (3.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F279058/MediaObjects/13663_2009_Article_1128_Equ30_HTML.gif)
We can check that all assumptions of Lemma 2.3 are satisfied. Therefore, . This completes the proof.
References
Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics. Volume 83. Marcel Dekker, New York, NY, USA; 1984.
Reich S: Almost convergence and nonlinear ergodic theorems. Journal of Approximation Theory 1978,24(4):269–272. 10.1016/0021-9045(78)90012-6
Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 2004,20(1):103–120. 10.1088/0266-5611/20/1/006
Combettes PL: On the numerical robustness of the parallel projection method in signal synthesis. IEEE Signal Processing Letters 2001,8(2):45–47. 10.1109/97.895371
Combettes PL: The convex feasibility problem in image recovery. In Advances in Imaging and Electron Physics. Volume 95. Edited by: Hawkes P. Academic Press, New York, NY, USA; 1996:155–270.
Engl HW, Leitão A: A Mann iterative regularization method for elliptic Cauchy problems. Numerical Functional Analysis and Optimization 2001,22(7–8):861–884. 10.1081/NFA-100108313
Podilchuk CI, Mammone RJ: Image recovery by convex projections using a least-squares constraint. Journal of the Optical Society of America 1990,7(3):517–512. 10.1364/JOSAA.7.000517
Youla D: Mathematical theory of image restoration by the method of convex projection. In Image Recovery Theory and Applications. Edited by: Stark H. Academic Press, Orlando, Fla, USA; 1987:29–77.
Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1996,202(1):150–159. 10.1006/jmaa.1996.0308
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6
Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0
Jung JS: Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2006,64(11):2536–2552. 10.1016/j.na.2005.08.032
Kim T-H, Xu H-K: Robustness of Mann's algorithm for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2007,327(2):1105–1115. 10.1016/j.jmaa.2006.05.009
Lions P-L: Approximation de points fixes de contractions. Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 1977,284(21):A1357-A1359.
Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615
Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 1979,67(2):274–276. 10.1016/0022-247X(79)90024-6
Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. Journal of Mathematical Analysis and Applications 1980,75(1):287–292. 10.1016/0022-247X(80)90323-6
Shioji N, Takahashi W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proceedings of the American Mathematical Society 1997,125(12):3641–3645. 10.1090/S0002-9939-97-04033-1
Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory and Applications 2005,2005(1):103–123. 10.1155/FPTA.2005.103
Suzuki T: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proceedings of the American Mathematical Society 2007, 135: 99–106.
Wittmann R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992,58(5):486–491. 10.1007/BF01190119
Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059
Acknowledgment
The second author was partially supposed by the Grant NSC 98-2622-E-230-006-CC3 and NSC 98-2923-E-110-003-MY3.
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, Y.C. & Marino, G. Strong Convergence of Two Iterative Algorithms for Nonexpansive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2009, 279058 (2009). https://doi.org/10.1155/2009/279058
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/279058