- Open Access
Boundary point algorithms for minimum norm fixed points of nonexpansive mappings
© He and Yang; licensee Springer. 2014
- Received: 11 October 2013
- Accepted: 19 February 2014
- Published: 4 March 2014
Let H be a real Hilbert space and C be a closed convex subset of H. Let be a nonexpansive mapping with a nonempty set of fixed points . If , then Halpern’s iteration process cannot be used for finding a minimum norm fixed point of T since may not belong to C. To overcome this weakness, Wang and Xu introduced the iteration process for finding the minimum norm fixed point of T, where the sequence , arbitrarily and is the metric projection from H onto C. However, it is difficult to implement this iteration process in actual computing programs because the specific expression of cannot be obtained, in general. In this paper, three new algorithms (called boundary point algorithms due to using certain boundary points of C at each iterative step) for finding the minimum norm fixed point of T are proposed and strong convergence theorems are proved under some assumptions. Since the algorithms in this paper do not involve , they are easy to implement in actual computing programs.
MSC:47H09, 47H10, 65K10.
- minimum norm fixed point
- nonexpansive mapping
- metric projection
- boundary point algorithm
- Hilbert space
Let H be a real Hilbert space with the inner product and the norm , and let C be a nonempty closed convex subset of H. Recall that a mapping is nonexpansive if for all . We use to denote a set of fixed points of T, i.e., . Throughout this article, is always assumed to be nonempty.
For every nonempty closed convex subset K of H, the metric (or nearest point) projection indicated by from H onto K can be defined, that is, for each , is the only point in K such that . It is well known (e.g., see ) that is nonexpansive and a characteristic inequality holds.
Since is a closed convex subset of H, so the metric projection is valid and thus there exists a unique element, denoted by , in such that , that is, . is called a minimum norm fixed point of T. Because the minimum norm fixed point of a nonexpansive mapping is closely related to convex optimization problems, it is favored by people.
where is a sequence in the interval and u is some given element in C. For Halpern’s iteration process, a classical result is as follows.
They proved that if satisfies the same conditions in Theorem 1.2, then the sequence generated by (1.2) converges strongly to .
However, it is difficult to implement the iteration process (1.2) in actual computing programs because the specific expression of cannot be obtained, in general.
The purpose of this paper is to propose three new algorithms for finding the minimum norm fixed point of T. The strong convergence theorems are proved under some assumptions. The main advantage of the algorithms in this paper is that they have nothing to do with the metric projection and thus they are easy to implement in actual computing programs. Because the key of our algorithms is replacing a fixed element u in (1.1) by a certain sequence in the boundary of C, they are called boundary point algorithms.
⇀ for weak convergence and → for strong convergence.
denotes the weak ω-limit set of .
means that B is the definition of A.
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma 1.3 ()
Let C be a closed convex subset of a real Hilbert space H, and let be a nonexpansive mapping such that . If a sequence in C is such that and , then .
It is easy to see that and hold for each due to the assumption .
Since our iteration processes will involve the function , it is necessary to explain how to calculate for any given in actual computing programs. In order to get the value for a given , we often need to deal with an algebraic equation. But dealing with an algebraic equation is easier than calculating the metric projection , in general. To illustrate this viewpoint, let us consider the following simple example.
then, using Theorem 1.2, as . Thus we can take approximately for a sufficiently large integer in actual computing programs.
Geometric intuition seems to encourage us to guess as under some certain assumptions. As a matter of fact, it is true.
Theorem 2.1 If satisfies , then generated by (2.1) converges strongly to .
Thus this together with the condition leads to the conclusion. □
Remark 2 Is the condition reasonable? In other words, can we find an example which satisfies this condition? The answer is yes. The following result implies that this condition is not harsh.
Corollary 1 If , then generated by (2.1) converges strongly to .
This implies that holds. □
where , () and is taken in C arbitrarily.
to calculate approximately since as , where can be taken in C arbitrarily, for example, .
Theorem 2.2 Assume that and , then generated by (2.3) converges strongly to .
and is bounded, so are . This together with (2.3) implies that (). Thus it follows from Lemma 1.3 that .
Using Lemma 1.5, we conclude from (2.5) and conditions and that . □
By a similar argument as above, we easily get the following result.
Corollary 2 If and , then generated by (2.3) converges strongly to , where is the range of T.
Note that , it follows that . □
where , () and is taken in C arbitrarily.
Then generated by (2.6) converges strongly to .
This means that is bounded, so are .
Using Lemma 1.5, we conclude from conditions (i)-(iv) that . Noticing the boundedness of and and condition (i), we have from (2.6) that . Consequently, . Using Lemma 1.3, we derive that .
As a matter of fact, this is derived by the same argument as in the proof of Theorem 2.3.
It is easily seen that , by conditions (i) and (ii), and by (2.7). By Lemma 1.5, we conclude that . □
This work was supported in part by the Fundamental Research Funds for the Central Universities (3122013k004) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.
- Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.Google Scholar
- Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0View ArticleGoogle Scholar
- Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5View ArticleMathSciNetGoogle Scholar
- Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleGoogle Scholar
- Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 2003, 279: 372–379. 10.1016/S0022-247X(02)00458-4View ArticleMathSciNetGoogle Scholar
- Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 1979, 67: 274–276. 10.1016/0022-247X(79)90024-6View ArticleMathSciNetGoogle Scholar
- Reich S: Strong convergence theorem for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 1980, 75: 287–292. 10.1016/0022-247X(80)90323-6View ArticleMathSciNetGoogle Scholar
- Reich S, Shemen L: Two algorithms for nonexpansive mappings. Fixed Point Theory 2011, 12: 443–448.MathSciNetGoogle Scholar
- Shioji N, Takahashi W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 1997, 125: 3641–3645. 10.1090/S0002-9939-97-04033-1View ArticleMathSciNetGoogle Scholar
- Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178(2):301–308. 10.1006/jmaa.1993.1309View ArticleMathSciNetGoogle Scholar
- Wittmann R: Approximation of fixed points of nonexpansive mappings. Arch. Math. 1992, 58: 486–491. 10.1007/BF01190119View ArticleMathSciNetGoogle Scholar
- Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332View ArticleGoogle Scholar
- Moudafi A: Viscosity approximation methods for fixed points problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615View ArticleMathSciNetGoogle Scholar
- Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059View ArticleMathSciNetGoogle Scholar
- Wang F, Xu HK: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2010., 2010: Article ID 102085 10.1155/2010/102085Google Scholar
- Diestel J Lecture Notes in Mathematics 485. In Geometry of Banach Spaces - Selected Topics. Springer, Berlin; 1975.Google Scholar
- Beauzamy B: Introduction to Banach Spaces and Their Geometry. North-Holland, Amsterdam; 1982.Google Scholar
- Goebel K, Kirk WA Cambridge Studies in Advanced Mathematics 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.View ArticleGoogle Scholar
- Liu LS: Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289View ArticleMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.