Approximating common fixed points of averaged self-mappings with applications to the split feasibility problem and maximal monotone operators in Hilbert spaces
© Huang and Hong; licensee Springer 2013
Received: 18 April 2013
Accepted: 5 July 2013
Published: 19 July 2013
In this paper, a modified proximal point algorithm for finding common fixed points of averaged self-mappings in Hilbert spaces is introduced and a strong convergence theorem associated with it is proved. As a consequence, we apply it to study the split feasibility problem, the zero point problem of maximal monotone operators, the minimization problem and the equilibrium problem, and to show that the unique minimum norm solution can be obtained through our algorithm for each of the aforementioned problems. Our results generalize and unify many results that occur in the literature.
MSC:47H10, 47J25, 68W25.
Throughout this paper, ℋ denotes a real Hilbert space with the inner product and the norm , I the identity mapping on ℋ, ℕ the set of all natural numbers and ℝ the set of all real numbers. For a self-mapping T on ℋ, denotes the set of all fixed points of T.
The SFP was first introduced by Censor and Elfving  for modeling inverse problems which arise from phase retrievals and medical image reconstruction. Recently, it has been found that the SFP can also be used to model the intensity-modulated radiation therapy. For details, the readers are referred to Xu  and the references therein.
where , is the adjoint of A, and are the metric projections onto C and Q respectively.
and proved that the sequence converges strongly to a minimum norm solution of SFP (1) provided the parameters and verify some suitable conditions. This regularized method was further investigated by Yao, Jiang and Liou , and Yao, Liou and Shahzad .
where A, B are two bounded linear mappings from to .
can converge strongly to a common zero of A and B.
and are convergent sequences in with limit respectively;
- (ii)there are two nonnegative real-valued functions and on ℕ with
, , and are sequences in with and , ;
, , ;
Based on this main result, we shall deduce many corollaries for averaged mappings in Section 3. Section 4 is devoted to applications. We apply our results in Section 3 to study the split feasibility problem, the zero point problem of maximal monotone operators, the minimization problem and the equilibrium problem, and to show that the unique minimum norm solution can be obtained through our algorithm for each of the aforementioned problems.
- (i)nonexpansive if
- (ii)firmly nonexpansive if
- (iii)λ-averaged by K if
for some and some nonexpansive mapping K.
If is nonexpansive, then the fixed point set of T is closed and convex, cf. . If is averaged, then T is nonexpansive with .
For , the resolvent of maximal monotone operator A on ℋ has the following properties.
is single-valued and firmly nonexpansive;
- (c)(The resolvent identity) for , the following identity holds:
We still need some lemmas that will be quoted in the sequel.
- (b)for any ,
- (c)for with ,
Lemma 2.3 (Demiclosedness principle )
Let T be a nonexpansive self-mapping on a nonempty closed convex subset C of ℋ, and suppose that is a sequence in C such that converges weakly to some and . Then .
Lemma 2.4 
Lemma 2.5 
For any , define . Then as and , .
3 Strong convergence theorems
To establish a strong convergence theorem for averaged mappings , , , on ℋ associated with algorithm (9), we at first need a lemma.
Moreover, when every is the identity mapping I, the result still holds without the condition .
We now carry on with the proof by considering the following two cases: (I) is eventually decreasing, and (II) is not eventually decreasing.
by virtue of (20). Consequently, we conclude via (16) and (21). In addition, note that the condition is used to establish and in (12) and (18) respectively. However, both limits hold trivially without this condition provided every is the identity mapping I.
for all , we see that , and hence follows. This completes the proof. □
The following lemma is easily proved and so its proof is omitted.
Lemma 3.3 For any , suppose that and are averaged self-mappings on a nonempty closed convex subset C of ℋ such that condition (3.1) holds. Then and satisfy condition (3.2) if and only if and satisfy condition (3.2).
If the sequence (resp. ) of averaged mappings consists of a single mapping S (resp. T), then and obviously verify conditions (3.1) and (3.2), and hence from Lemma 3.3 we have the following corollary.
Moreover, when S is the identity mapping I, the result still holds without the condition .
Moreover, when every is the identity mapping I, the result still holds without the condition .
Proof Since any firmly nonexpansive mapping is -averaged, condition (3.1) holds, and hence by Lemma 3.3 we see that all the requirements of Theorem 3.2 are verified. Therefore, the desired conclusion follows. □
If and for all in Theorem 3.2, then we have the following corollary.
In this section, we shall apply some of the strong convergence theorems in Section 3 to approximate a solution of the split feasibility problem, a common zero of maximal monotone operators, a minimizer of a proper lower semicontinuous convex function, and to study the related equilibrium problem.
He proved Lemma 4.1 below.
Lemma 4.1 
A point solves SFP (1) if and only if is a fixed point of the operator (3): .
Moreover, in the proof of Theorem 3.6 of , Xu showed the following lemma.
Lemma 4.2 
For any with , the operator (3): is -averaged.
Invoking Lemmas 4.1 and 4.2, we obtain the theorem below from Corollary 3.4 by putting and for all .
When the point u in the above theorem is taken to be 0, we see that the limit point v of the sequence is the unique minimum norm solution of SFP (1), that is, .
converges weakly to a solution of SFP (1) provided the solution set of SFP (1) is nonempty. It is also interesting to compare Theorem 4.3 with Theorem 5.5 of  and Theorem 3.1 of . Our method is different from those in  and  even in the case of , because our algorithm contains an error term and uses the operator directly without any regularization.
Proof It is clear that this theorem follows from Lemmas 4.1 and 4.2 and Corollary 3.4. □
Replacing and in Theorem 3.2 with the resolvents and of two maximal monotone operators B and A respectively, we have Theorem 4.5 below.
Therefore, condition (3.2) is true for and . □
Putting in Corollary 3.6 (resp. Corollary 3.7) and noting that and verifies condition (3.2) due to , we obtain the following two corollaries.
Corollary 4.7 
Hence for any , and then invoking Corollary 4.6, we obtain the following theorem.
f is monotone, that is, , ;
for all , ;
for all , is convex and lower semicontinuous.
Lemma 4.10 
is firmly nonexpansive;
is closed and convex.
We call the resolvent of f for . Using Lemmas 4.9 and 4.10, Takahashi et al.  established the lemma below.
Lemma 4.11 
is a maximal monotone operator with ;
for all .
Proof The set-valued mappings associated with , , defined in Lemma 4.11 are maximal monotone operators with , and it follows from Lemmas 4.10 and 4.11 that and for any . Putting and in Theorem 4.5, the desired conclusion follows. □
Here, it is worth mentioning, just as the SFP, that the unique minimum norm solution can be obtained through our algorithm for each of the minimization problem and the equilibrium problem by taking in Theorems 4.8 and 4.12.
Proof Putting and in Theorem 4.5, the desired conclusion follows. □
The work was supported by the National Science Council of Taiwan with contract No. NSC101-2221-E-020-031.
- Censor Y, Elfving T: A multiprojection algorithm using Bregman projection in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692MathSciNetView ArticleGoogle Scholar
- Xu HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010., 26: Article ID 105018 10.1088/0266-5611/26/10/105018Google 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
- Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2004, 20: 103–120. 10.1088/0266-5611/20/1/006MathSciNetView ArticleGoogle Scholar
- Yao Y, Jigang W, Liou YC: Regularized methods for the split feasibility problem. Abstr. Appl. Anal. 2012. 10.1155/2012/140679Google Scholar
- Yao Y, Liou YC, Shahzad N: A strongly convergent method for the split feasibility problem. Abstr. Appl. Anal. 2012. 10.1155/2012/125046Google Scholar
- Martinet B: Régularisation d’inéquations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 1970, 4: 154–158.MathSciNetGoogle Scholar
- Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056MathSciNetView ArticleGoogle Scholar
- Güler O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 1991, 29: 403–419. 10.1137/0329022MathSciNetView ArticleGoogle Scholar
- Boikanyo OA, Moroşanu G: Inexact Halpern-type proximal point algorithm. J. Glob. Optim. 2011, 51: 11–26. 10.1007/s10898-010-9616-7View ArticleGoogle Scholar
- Boikanyo OA, Moroşanu G: Four parameter proximal point algorithms. Nonlinear Anal. 2011, 74: 544–555. 10.1016/j.na.2010.09.008MathSciNetView ArticleGoogle Scholar
- Boikanyo OA, Moroşanu G: A proximal point algorithm converging strongly for general errors. Optim. Lett. 2010, 4: 635–641. 10.1007/s11590-010-0176-zMathSciNetView ArticleGoogle Scholar
- Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 2000, 106: 226–240. 10.1006/jath.2000.3493MathSciNetView ArticleGoogle Scholar
- Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 2000, 87: 189–202.MathSciNetGoogle Scholar
- Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 2010, 147: 27–41. 10.1007/s10957-010-9713-2MathSciNetView ArticleGoogle Scholar
- Wang F, Cui H: On the contraction-proximal point algorithms with multi-parameters. J. Glob. Optim. 2012, 54: 485–491. 10.1007/s10898-011-9772-4MathSciNetView ArticleGoogle Scholar
- Xu HK: A regularization method for the proximal point algorithm. J. Glob. Optim. 2006, 36: 115–125. 10.1007/s10898-006-9002-7View ArticleGoogle Scholar
- Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66(2):240–256.View ArticleGoogle Scholar
- Yao Y, Noor MA: On convergence criteria of generalized proximal point algorithm. J. Comput. Appl. Math. 2008, 217: 46–55. 10.1016/j.cam.2007.06.013MathSciNetView ArticleGoogle Scholar
- Marino G, Xu HK: Convergence of generalized proximal point algorithm. Commun. Pure Appl. Anal. 2004, 3: 791–808.MathSciNetView ArticleGoogle Scholar
- Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.View 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
- Rockafellar RT: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 1970, 33: 209–216. 10.2140/pjm.1970.33.209MathSciNetView ArticleGoogle Scholar
- Takahashi W: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama; 2009.Google Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
- Aoyama K, Kimura T, Takahashi W: Maximal monotone operators and maximal monotone functions for equilibrium problems. J. Convex Anal. 2008, 15: 395–409.MathSciNetGoogle Scholar
- Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetGoogle Scholar
- Lin LJ, Takahashi W: A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications. Positivity 2012, 16: 429–453. 10.1007/s11117-012-0161-0MathSciNetView 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.