- Open Access
Approximation of the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings
© Zegeye and Shahzad; licensee Springer 2013
- Received: 21 September 2012
- Accepted: 28 November 2012
- Published: 2 January 2013
We introduce an iterative process which converges strongly to the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings. As a consequence, convergence result to a common minimum-norm fixed point of a finite family of nonexpansive mappings is proved.
MSC:47H09, 47H10, 47J05, 47J25.
- asymptotically nonexpansive mappings
- minimum-norm fixed point
- nonexpansive mappings
- split feasibility problem
- strong convergence
where A is bounded linear operator from into . A split feasibility problem in finite dimensional Hilbert spaces was first studied by Censor and Elfving  for modeling inverse problems which arise in medical image reconstruction, image restoration and radiation therapy treatment planing (see, e.g., [1–3]).
Recall that a point is said to be a fixed point of T if . We denote the set of fixed points of T by , i.e., . Therefore, finding a solution to the split feasibility problem (1.1) is equivalent to finding the minimum-norm fixed point of the mapping .
The class of asymptotically nonexpansive mappings was introduced as a generalization of the class of nonexpansive mappings by Goebel and Kirk  who proved that if K is a nonempty closed convex bounded subset of a real uniformly convex Banach spaces which includes Hilbert spaces as a special case and T is an asymptotically nonexpansive self-mapping of K, then T has a fixed point.
In , Browder proved that, as , converges strongly to the nearest point projection of u onto .
converges strongly to the element of which is nearest to u under certain conditions on .
where satisfies certain conditions, converges strongly to the element of which is nearest to u.
It is worth mentioning that the methods studied above are used to approximate the fixed point of T which is closest to the point . These methods can be used to find the minimum-norm fixed point of T if . If, however, , any of the methods above fails to provide the minimum-norm fixed point of T.
They proved that under appropriate conditions on and β, the sequence converges strongly to the minimum-norm fixed point of T in real Hilbert spaces.
More recently, Yao and Xu  have also shown that the explicit scheme , , converges strongly to the minimum-norm fixed point of a nonexpansive self-mapping T provided that satisfies certain conditions.
Let K be a closed convex subset of a real Hilbert space H and let , be a finite family of asymptotically nonexpansive mappings.
It is our purpose in this paper to introduce an explicit iteration process which converges strongly to the common minimum-norm fixed point of . Our theorems improve several results in this direction.
In what follows, we shall make use of the following lemmas.
Lemma 2.2 
Lemma 2.3 
Lemma 2.4 
Let H be a real Hilbert space, K be a closed convex subset of H and be an asymptotically nonexpansive mapping, then is demiclosed at zero, i.e., if is a sequence in K such that and , as , then .
Lemma 2.5 
where , and satisfying the following conditions: , , and , as . Then .
Lemma 2.6 
In fact, .
Proposition 2.7 Let H be a real Hilbert space, let K be a closed convex subset of H, and let T be an asymptotically nonexpansive mapping from K into itself. Then is closed and convex.
and hence, since as , we get that , which implies that . Now, by the continuity of T, we obtain that . Hence, and that is convex. □
We now state and proof our main theorem.
where such that , , for each and , for , satisfying for each . Then converges strongly to the common minimum-norm point of F.
for some , where for all .
Now, we consider the following two cases.
for some . But note that satisfies and . Thus, it follows from (3.15) and Lemma 2.5 that , as . Consequently, .
Thus, from (3.16) and the fact that , we obtain that as . This together with (3.17) gives as . But for all , thus we obtain that . Therefore, from the above two cases, we can conclude that converges strongly to a point of F which is the common minimum-norm fixed point of the family and the proof is complete. □
If in Theorem 3.1 we assume that , then we get the following corollary.
where such that , and , for each . Then converges strongly to the minimum-norm fixed point of T.
If in Theorem 3.1 we assume that each is nonexpansive for , then the method of proof of Theorem 3.1 provides the following corollary.
where such that and , , for , satisfying for each . Then converges strongly to the common minimum-norm point of F.
If in Corollary 3.3 we assume that , then we have the following corollary.
where such that and , for each . Then converges strongly to the minimum-norm point of .
In this section, we study the problem of finding a minimizer of a continuously Fréchet-differentiable convex functional which has the minimum norm in Hilbert spaces.
for all .
Now, we have the following corollary deduced from Corollary 3.2.
where such that , and , for each . Then converges strongly to the minimum-norm solution of the minimization problem (4.1).
Remark 4.2 Our results extend and unify most of the results that have been proved for this important class of nonlinear mappings. In particular, Theorem 3.1 improves Theorem 3.2 of Yang et al.  and of Yao and Xu  to a more general class of a finite family of asymptotically nonexpansive mappings.
The second author gratefully acknowledges the sup- port provided by the Deanship of Scientific Research (DSR), King Abdulaziz University during this research.
- Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692MathSciNetView ArticleGoogle Scholar
- Byrne C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 2002, 18: 441–453. 10.1088/0266-5611/18/2/310MathSciNetView ArticleGoogle Scholar
- Censor Y, Bortfeld T, Martin B, Trofimov A: A unified approach for inversion problem in intensity-modulated radiation therapy. Phys. Med. Biol. 2006, 51: 2353–2365. 10.1088/0031-9155/51/10/001View ArticleGoogle Scholar
- Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleGoogle Scholar
- Browder FE: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 1967, 24: 82–90.MathSciNetView ArticleGoogle Scholar
- Halpern B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0View ArticleGoogle Scholar
- Wittmann R: Approximation of fixed point of nonexpansive mappings. Arch. Math. 1992, 58: 486–491. 10.1007/BF01190119MathSciNetView ArticleGoogle Scholar
- Shimizu T, Takahashi W: Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces. Nonlinear Anal. 1998, 34: 87–99. 10.1016/S0362-546X(97)00682-2MathSciNetView ArticleGoogle Scholar
- Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 1993, 65: 169–179.MathSciNetGoogle Scholar
- Lim TC, Xu HK: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. 1994, 22: 1345–1355. 10.1016/0362-546X(94)90116-3MathSciNetView ArticleGoogle Scholar
- Morales CH, Jung JS: Convergence of paths for pseudo-contractive mappings in Banach spaces. Proc. Am. Math. Soc. 2000, 128: 3411–3419. 10.1090/S0002-9939-00-05573-8MathSciNetView ArticleGoogle Scholar
- Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 1980, 75: 287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleGoogle Scholar
- Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 1991, 158: 407–413. 10.1016/0022-247X(91)90245-UMathSciNetView ArticleGoogle Scholar
- Schu J: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884MathSciNetView ArticleGoogle Scholar
- Shioji N, Takahashi W: A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces. Arch. Math. 1999, 72: 354–359. 10.1007/s000130050343MathSciNetView ArticleGoogle Scholar
- Shioji N, Takahashi W: Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces. J. Approx. Theory 1999, 97: 53–64. 10.1006/jath.1996.3251MathSciNetView ArticleGoogle Scholar
- Takahashi W, Ueda Y: On Reich’s strong convergence theorems for resolvents of accretive operators. J. Math. Anal. Appl. 1984, 104: 546–553. 10.1016/0022-247X(84)90019-2MathSciNetView ArticleGoogle Scholar
- Tan KK, Xu HK: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1994, 122: 733–739. 10.1090/S0002-9939-1994-1203993-5MathSciNetView ArticleGoogle Scholar
- Yang X, Liou Y-C, Yao Y: Finding minimum norm fixed point of nonexpansive mappings and applications. Math. Probl. Eng. 2011., 2011: Article ID 106450. doi:10.1155/2011/106450Google Scholar
- Yao Y, Xu H-K: Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications. Optimization 2011, 60: 645–658. 10.1080/02331930903582140MathSciNetView ArticleGoogle Scholar
- Zegeye H: A hybrid iteration method for equilibrium, variational inequality problems and common fixed point problems in Banach spaces. Nonlinear Anal. 2010, 72(3–4):2136–2146. 10.1016/j.na.2009.10.014MathSciNetView ArticleGoogle Scholar
- Takahashi W: Nonlinear Functional Analysis-Fixed Point Theory and Applications. Yokohama Publishers, Yokohama; 2000.Google Scholar
- Chang SS, Cho YJ, Zhou H: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. J. Korean Math. Soc. 2001, 38: 1245–1260.MathSciNetGoogle Scholar
- Ohara JG, Pillay P, Xu HK: Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear Anal. 2006, 64: 2022–2042. 10.1016/j.na.2005.07.036MathSciNetView ArticleGoogle Scholar
- Maingé PE: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008, 16: 899–912. 10.1007/s11228-008-0102-zMathSciNetView ArticleGoogle 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 cited.