`Abstract and Applied AnalysisVolume 2012, Article ID 469270, 12 pageshttp://dx.doi.org/10.1155/2012/469270`
Research Article

## Iterative Schemes for Fixed Point Computation of Nonexpansive Mappings

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 18 December 2011; Accepted 3 February 2012

Copyright © 2012 Rudong Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Fixed point (especially, the minimum norm fixed point) computation is an interesting topic due to its practical applications in natural science. The purpose of the paper is devoted to finding the common fixed points of an infinite family of nonexpansive mappings. We introduce an iterative algorithm and prove that suggested scheme converges strongly to the common fixed points of an infinite family of nonexpansive mappings under some mild conditions. As a special case, we can find the minimum norm common fixed point of an infinite family of nonexpansive mappings.

#### 1. Introduction

In many problems, it is needed to find a solution with minimum norm. In an abstract way, we may formulate such problems as finding a point with the property where is a nonempty closed convex subset of a real Hilbert space . In other words, is the (nearest point or metric) projection of the origin onto , where is the metric (or nearest point) projection from onto .

A typical example is the least-squares solution to the constrained linear inverse problem [1] where is a bounded linear operator from to another real Hilbert space and is a given point in . The least-squares solution to (1.3) is the least-norm minimizer of the minimization problem Recently, some authors consider the minimum norm solution problem by using the iterative algorithm. For some related works, please refer to [24]. Yao and Xu [3] introduced the following algorithm: They proved that the sequence converges in norm to the unique solution of VI , . Particularly, the sequence defined by converges to the minimum norm fixed point of . We note that the authors added an additional assumption, that is, . Iterative algorithm for finding the fixed points of nonexpansive mappings has been considered by many authors, see [520].

The purpose of this paper is to extend Yao and Xu’s result to an infinite family of nonexpansive mappings . We suggest a new algorithm. Particularly, we drop the above additional assumption and prove the suggested algorithm converges strongly to the common fixed points of . As a special case, we can find the minimum norm fixed point of .

#### 2. Preliminaries

Let be a real Hilbert space with inner product and norm , respectively, and let be a nonempty closed convex subset of . We call a -contraction if there exists a constant such that for all . A bounded linear operator is said to be strongly positive on if there exists a constant such that Recall that the (nearest point or metric) projection from onto , denoted by , is defined in such a way that, for each , is the unique point in with the property It is known that satisfies Moreover, is characterized by the following properties: for all and .

We also need other sorts of nonlinear operators which are introduced below. Let be a nonlinear operator.(a) is nonexpansive if for all .(b) is firmly nonexpansive if is nonexpansive. Equivalently, , where is nonexpansive. Alternatively, is firmly nonexpansive if and only if (c) is averaged if , where and is nonexpansive. In this case, we also say that is -averaged. A firmly nonexpansive mapping is -averaged.

It is well known that both and are firmly nonexpansive. We will need to use the following notation:(i) stands for the set of fixed points of ;(ii) stands for the weak convergence of to ;(iii) stands for the strong convergence of to .

Let be infinite mappings of into itself, and let be real numbers such that for every . For any , define a mapping of into itself as follows: Such is called the -mapping generated by and . For the iterative algorithm for a finite family of nonexpansive mappings, we refer the reader to [21].

We have the following crucial lemmas concerning which can be found in [22].

Lemma 2.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for any . Then, for every and , the limit exists.

Lemma 2.2. Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for any . Then, .

The following remark [23] is important to prove our main results.

Remark 2.3. Using Lemma 2.1, one can define a mapping of into itself as , for every . If is a bounded sequence in , then one has

Throughout this paper, we will assume that for every .

Lemma 2.4 (see [24]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with . Then is demiclosed on , that is, if weakly and , then .

Lemma 2.5 (see [25]). Let and be bounded sequences in a Banach space , and let be a sequence in with . Suppose for all integers and . Then, .

Lemma 2.6 (see [26]). Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that (1);(2) or .Then .

#### 3. Main Result

In this section, we introduce our algorithm and prove its strong convergence.

Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings from to such that the common fixed point set . Let be a -contraction and be a self-adjoint, strongly positive bounded linear operator with coefficient . Let be a constant such that . For an arbitrary initial point belonging to , one defines a sequence iteratively where is a real sequence in . Assume the sequence satisfies the following conditions:(C1);(C2).Then the sequence generated by (3.1) converges in norm to the unique solution which solves the following variational inequality:

Proof. Let . From (3.1), we have It follows by induction that This indicates that is bounded. It is easy to deduce that , , and are also bounded.
Set . It is known that is nonexpansive. Note that . Then, we can rewrite (3.1) as Note that From (3.5), we have where Set and for all . Then It follows that Thus, From the nonexpansivity of , we get Since and are nonexpansive, we have where is a constant such that for all . So, Hence, Since , we have , , and . Therefore, By Lemma 2.5, we get Hence, from (3.7), we deduce Observe that From (3.18) and (3.19), we deduce Next we prove where is the unique solution of VI (3.2).
Indeed, we can choose a subsequence of such that Since is bounded, there exists a subsequence of which converges weakly to a point . Without loss of generality, we may assume that converges weakly to . Therefore, from (3.20) and Lemma 2.4, we have . Therefore, Finally, we show that . We observe that Since , we get It follows that Hence, all conditions of Lemma 2.6 are satisfied. Therefore, we immediately deduce that . This completes the proof.

From (3.1) and Theorem 3.1, we can deduce easily the following result.

Corollary 3.2. Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings from to such that the common fixed point set . For an arbitrary initial point , one defines a sequence iteratively where is a real sequence in . Assume the sequence satisfies the following conditions:(C1);(C2). Then the sequence generated by (3.27) converges to the minimum norm common fixed point of .

#### Acknowledgment

The paper was supported by NSFC 11071279.

#### References

1. A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative least-squares and regularization,” IEEE Transactions on Signal Processing, vol. 46, no. 9, pp. 2345–2352, 1998.
2. Y.-L. Cui and X. Liu, “Notes on Browder's and Halpern's methods for nonexpansive mappings,” Fixed Point Theory, vol. 10, no. 1, pp. 89–98, 2009.
3. Y. Yao and H.-K. Xu, “Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications,” Optimization, vol. 60, no. 6, pp. 645–658, 2011.
4. Y. Yao, R. Chen, and H.-K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 7-8, pp. 3447–3456, 2010.
5. H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 202, no. 1, pp. 150–159, 1996.
6. F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967.
7. F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.
8. H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
9. S. A. Hirstoaga, “Iterative selection methods for common fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1020–1035, 2006.
10. P.-E. Mainge, “Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 469–479, 2007.
11. Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 6, pp. 1687–1693, 2008.
12. M. Kikkawa and W. Takahashi, “Approximating fixed points of infinite nonexpansive mappings by the hybrid method,” Journal of Optimization Theory and Applications, vol. 117, no. 1, pp. 93–101, 2003.
13. H. Zegeye and N. Shahzad, “Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 155–163, 2007.
14. Y. Yao, Y. C. Liou, and G. Marino, “Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, Article ID 279058, 7 pages, 2009.
15. Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242–250, 2011.
16. S.-S. Chang, “Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1402–1416, 2006.
17. M. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004.
18. M. A. Noor, K. I. Noor, and E. Al-Said, “Iterative methods for solving nonconvex equilibrium variational inequalities,” Applied Mathematics & Information Sciences, vol. 6, no. 1, pp. 65–69, 2012.
19. M. A. Noor, “Extended general variational inequalities,” Applied Mathematics Letters, vol. 22, no. 2, pp. 182–186, 2009.
20. M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 303569, 16 pages, 2012.
21. P. K. F. Kuhfittig, “Common fixed points of nonexpansive mappings by iteration,” Pacific Journal of Mathematics, vol. 97, no. 1, pp. 137–139, 1981.
22. K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001.
23. Y. Yao, Y.-C. Liou, and J.-C. Yao, “Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings,” Fixed Point Theory and Applications, Article ID 64363, 12 pages, 2007.
24. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
25. T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, no. 1, pp. 103–123, 2005.
26. H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.