Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces
© Chang and Wang; licensee Springer. 2014
Received: 1 May 2014
Accepted: 19 July 2014
Published: 18 August 2014
The purpose of this paper is to introduce and study a general split variational inclusion problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split variational inclusion problem. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces.
where is a bounded linear operator. In 1994, Censor and Elfving  first introduced the in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction . It has been found that the can also be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning [3–5]. The in an infinite-dimensional real Hilbert space can be found in [2, 4, 6–10]. For comprehensive literature, bibliography and a survey on SFP, we refer to .
where and are the metric projection from onto C and from onto Q, respectively, is a positive constant, and is the adjoint of A.
where with λ being the spectral radius of the operator .
In this case, converges weakly to a minimizer of f. Later, many researchers have studied the convergence problems of the proximal point algorithm in Hilbert spaces (see [14–21] and the references therein).
The following examples are special cases of (GSVIP) (1.4).
converges weakly to a solution of problem (1.5), where λ and γ are given positive numbers.
It is well known that this kind of problems was first introduced by Censor and Elfving  for modeling inverse problems arising from phase retrievals and in medical image reconstruction . Also it can be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning.
where is a contraction mapping with a contractive constant , , and are sequence in satisfying some conditions. Under suitable conditions, some strong convergence theorems for the sequence proposed by (1.11) to a solution for (GSVIP) (1.4) in Hilbert spaces are proved. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces. Our results extend and improve the related results of Censor and Elfving , Byrne , Censor et al. [3–5], Rockafellar , Moudafi [14, 17], Eslamian and Latif , Eslamian , and Chuang .
Lemma 2.1 (demi-closed principle)
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a nonexpansive mapping, and let be a sequence in C. If and , then .
Lemma 2.2 
Lemma 2.3 
Lemma 2.5 
Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that , and are satisfied by all (sufficiently large) numbers . In fact, .
Lemma 2.6 
For each , is a single-valued and firmly nonexpansive mapping;
is a firmly nonexpansive mapping for each ;
- (iv)suppose that , then for each , each and each
suppose that . Then for each and each , and each .
if , then is a nonexpansive mapping.
Proof By Lemma 2.6(iii), the mapping is firmly nonexpansive, hence the conclusions (i) and (ii) are obvious.
Now we prove the conclusion (iii).
This completes the proof of Lemma 2.7. □
3 Main results
The following lemma will be used in proving our main results.
This implies that .
This implies that , and so for each . Hence from (3.1), , i.e., . Hence is a solution of (GSVIP)(1.4).
This completes the proof of Lemma 3.1. □
We are now in a position to prove the following main result.
, and ;
for each ;
then where , where is the metric projection from onto Ω.
Proof (I) First we prove that is bounded.
This implies that is bounded, so is .
Simplifying it, (3.6) is proved.
Now we prove that .
In order to prove that (as ), we consider two cases.
Finally, we prove that .
where , , and . It is easy to see that , , and . Hence by Lemma 2.4, the sequence converges strongly to .
This implies that and is a solution of (GSVIP) (1.4).
This completes the proof of Theorem 3.2. □
In Theorem 3.2, if and , for each , where and are two set-valued maximal monotone mappings, then from Theorem 3.2 we have the following.
, and ;
then where .
In this section we shall utilize the results presented in Theorem 3.2 and Theorem 3.3 to study some problems.
4.1 Application to split optimization problem
This implies that the (SOP) (4.1) is equivalent to the split variational inclusion problem (SVIP) (4.2). From Theorem 3.3 we have the following.
, and ;
then where .
The conclusion of Theorem 4.1 can be obtained from Theorem 3.3 immediately. □
4.2 Application to split feasibility problem
Hence, the following result can be obtained from Theorem 4.1 immediately.
, and ;
then where .
The authors would like to express their thanks to the referees and the editors for their kind and helpful comments and advice. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
- Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692View ArticleMathSciNetGoogle Scholar
- Byrne C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 2002, 18: 441–453. 10.1088/0266-5611/18/2/310View ArticleMathSciNetGoogle Scholar
- Censor Y, Bortfeld T, Martin N, 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
- Censor Y, Elfving T, Kopf N, Bortfeld T: The multiple-sets split feasibility problem and its applications. Inverse Probl. 2005, 21: 2071–2084. 10.1088/0266-5611/21/6/017View ArticleMathSciNetGoogle Scholar
- Censor Y, Motova A, Segal A: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 2007, 327: 1244–1256. 10.1016/j.jmaa.2006.05.010View ArticleMathSciNetGoogle Scholar
- Xu HK: A variable Krasnosel’skii-Mann algorithm and the multiple-sets split feasibility problem. Inverse Probl. 2006, 22: 2021–2034. 10.1088/0266-5611/22/6/007View ArticleGoogle Scholar
- Yang Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 2004, 20: 1261–1266. 10.1088/0266-5611/20/4/014View ArticleGoogle Scholar
- Zhao J, Yang Q: Several solution methods for the split feasibility problem. Inverse Probl. 2005, 21: 1791–1799. 10.1088/0266-5611/21/5/017View ArticleGoogle Scholar
- Chang SS, Cho YJ, Kim JK, Zhang WB, Yang L: Multiple-set split feasibility problems for asymptotically strict pseudocontractions. Abstr. Appl. Anal. 2012., 2012: Article ID 491760 10:1155/2012/491760Google Scholar
- Chang SS, Wang L, Tang YK, Yang L: The split common fixed point problem for total asymptotically strictly pseudocontractive mappings. J. Appl. Math. 2012., 2012: Article ID 385638 10.1155/2012.385638Google Scholar
- Ansari QH, Rehan A: Split feasibility and fixed point problems. In Nonlinear Analysis: Approximation Theory, Optimization and Applications. Birkhäuser, New Delhi; 2014:281–322.Google Scholar
- Martinet B: Régularisation d’inéquations variationnelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér. 1970, 4: 154–158.MathSciNetGoogle Scholar
- Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056View ArticleMathSciNetGoogle Scholar
- Moudafi A: A relaxed alternating CQ algorithm for convex feasibility problems. Nonlinear Anal. 2013, 79: 117–121.View ArticleMathSciNetGoogle Scholar
- Eslamian M, Latif A: General split feasibility problems in Hilbert spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 805104Google Scholar
- Chen RD, Wang J, Zhang HW: General split equality problems in Hilbert spaces. Fixed Point Theory Appl. 2014., 2014: Article ID 35Google Scholar
- Moudafi A: Split monotone variational inclusions. J. Optim. Theory Appl. 2011, 150: 275–283. 10.1007/s10957-011-9814-6View ArticleMathSciNetGoogle Scholar
- Güler O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 1991, 29: 403–419. 10.1137/0329022View ArticleMathSciNetGoogle Scholar
- Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611XView ArticleMathSciNetGoogle Scholar
- Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 2000, 87: 189–202.MathSciNetGoogle Scholar
- Eslamian M: Rockafellar’s proximal point algorithm for a finite family of monotone operators. Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys. 2014, 76(1):43–50.MathSciNetGoogle Scholar
- Chuang C-S: Strong convergence theorems for the split variational inclusion problem in Hilbert spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 350Google Scholar
- Chang SS: On Chidume’s open questions and approximate solutions for multi-valued strongly accretive mapping equations in Banach spaces. J. Math. Anal. Appl. 1997, 216: 94–111. 10.1006/jmaa.1997.5661View ArticleMathSciNetGoogle Scholar
- Chang S-S, Kim JK, Wang XR: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. 2010., 2010: Article ID 869684Google Scholar
- Maingé P-E: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008, 16(7–8):899–912. 10.1007/s11228-008-0102-zView ArticleMathSciNetGoogle Scholar
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