Open Access

# Strong Convergence of Two Iterative Algorithms for Nonexpansive Mappings in Hilbert Spaces

Fixed Point Theory and Applications20092009:279058

DOI: 10.1155/2009/279058

Accepted: 12 September 2009

Published: 12 October 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

(1.1)

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; [38]. Iterative methods for nonexpansive mappings have been extensively investigated in the literature; see [17, 921].

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

(2.1)

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

(3.1)

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,

(3.2)

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
(3.3)
that is,
(3.4)

Hence, is bounded.

Again from (3.2), we obtain
(3.5)
Next we show that is relatively norm compact as . Let be a sequence such that as . Put . From (3.5), we have
(3.6)
From (3.2), we get, for ,
(3.7)
Hence,
(3.8)
where is a constant such that . In particular,
(3.9)
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
(3.10)

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
(3.11)
Therefore,
(3.12)
Interchange and to obtain
(3.13)
Adding up (3.12) and (3.13) yields
(3.14)

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
(3.15)

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
(3.16)

Hence, is bounded and so is .

Set . It follows that
(3.17)
Hence,
(3.18)
This together with Lemma 2.2 implies that
(3.19)
Therefore,
(3.20)
We observe that
(3.21)
that is,
(3.22)
Let the net be defined by (3.2). By Theorem 3.1, we have as . Next we prove . Indeed,
(3.23)
where such that . It follows that
(3.24)
Therefore,
(3.25)
We note that
(3.26)
This together with and (3.25) implies that
(3.27)
Finally we show that . From (3.15), we have
(3.28)

We can check that all assumptions of Lemma 2.3 are satisfied. Therefore, . This completes the proof.

## Declarations

### 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.

## Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University
(2)
Department of Information Management, Cheng Shiu University
(3)
Dipartimento di Matematica, Universitá della Calabria

## References

1. 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.
2. Reich S: Almost convergence and nonlinear ergodic theorems. Journal of Approximation Theory 1978,24(4):269–272. 10.1016/0021-9045(78)90012-6
3. 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
4. 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
5. 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.Google Scholar
6. 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
7. 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
8. 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.Google Scholar
9. 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
10. 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
11. Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0
12. 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
13. 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
14. Lions P-L: Approximation de points fixes de contractions. Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 1977,284(21):A1357-A1359.
15. 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
16. 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
17. 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
18. 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
19. 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
20. Suzuki T: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proceedings of the American Mathematical Society 2007, 135: 99–106.
21. Wittmann R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992,58(5):486–491. 10.1007/BF01190119
22. 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