Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 386930, 5 pages http://dx.doi.org/10.1155/2013/386930 Research Article Inertial Iteration for Split Common Fixed-Point Problem for Quasi-Nonexpansive Operators Yazheng Dang 1,2 and Yan Gao 1 1 School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China 2 College of Computer Science and Technology, Henan Polytechnic University, Jiaozuo 454000, China Correspondence should be addressed to Yazheng Dang; [email protected] Received 14 March 2013; Accepted 6 May 2013 Academic Editor: Ru Dong Chen Copyright © 2013 Y. Dang and Y. Gao. is 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. Inspired by the note on split common fixed-point problem for quasi-nonexpansive operators presented by Moudafi (2011), based on the very recent work by Dang et al. (2012), in this paper, we propose an inertial iterative algorithm for solving the split common fixed-point problem for quasi-nonexpansive operators in the Hilbert space. We also prove the asymptotical convergence of the algorithm under some suitable conditions. e results improve and develop previously discussed feasibility problems and related algorithms. 1. Introduction e convex feasibility problem (CFP), as an important opti- mization problem [1], is to find a common point in the intersection of finitely many convex sets. It has been applied to many areas, for instance, approximation theory [2], image reconstruction from projections [3, 4], control [5], and so on. When there are only two sets and constraints are imposed on the solutions in the domain of a linear operator as well as in this operator’s ranges, the problem is said to be a split feasibility problem (SFP) which has the following formula: finding a point satisfying ∈ , ∈ , (1) where is a closed convex subset of a Hilbert space 1 , is a closed convex subset of a Hilbert space 2 , and : 1 → 2 is a bounded linear operator. e SFP was originally introduced in [6], and it has also broad applications in many fields, such as image reconstruction problem, signal processing, and radiation therapy. Many projection methods have also been developed for solving the SFP; see [7–9]. Denote by the orthogonal projection onto ; that is, () = arg min ∈ ‖ − ‖, over all ∈. Assuming that the SFP is consistent (i.e., (1) has a solution), it is not hard to see that ∈ solves (1) if and only if it solves the fixed-point equation: = [( − ∗ ( − ) ) ()] , (2) where 0< is any positive constant and ∗ denotes the adjoint of . To solve (2), in [10], Byrne introduced the so-called CQ algorithm, which generates a sequence { } by +1 = [( − ∗ ( − ) ) ( )] , (3) where 0 < < 2/( ) and ( ) is the spectral radius of ∗ . e split common fixed-point problem (SCFP) is a gener- alization of the split feasibility problem (SFP) and the convex feasibility problem (CFP); see [11]. Our main purpose here is to give an extension of the results developed in [12] to the split common fixed-point problem for quasi-nonexpansive oper- ators, and we will introduce weak symposium convergence
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 386930 5 pageshttpdxdoiorg1011552013386930
Research ArticleInertial Iteration for Split Common Fixed-Point Problem forQuasi-Nonexpansive Operators
Yazheng Dang12 and Yan Gao1
1 School of Management University of Shanghai for Science and Technology Shanghai 200093 China2 College of Computer Science and Technology Henan Polytechnic University Jiaozuo 454000 China
Correspondence should be addressed to Yazheng Dang jgdyz163com
Received 14 March 2013 Accepted 6 May 2013
Academic Editor Ru Dong Chen
Copyright copy 2013 Y Dang and Y GaoThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Inspired by the note on split common fixed-point problem for quasi-nonexpansive operators presented by Moudafi (2011) basedon the very recent work by Dang et al (2012) in this paper we propose an inertial iterative algorithm for solving the split commonfixed-point problem for quasi-nonexpansive operators in the Hilbert space We also prove the asymptotical convergence of thealgorithm under some suitable conditions The results improve and develop previously discussed feasibility problems and relatedalgorithms
1 Introduction
The convex feasibility problem (CFP) as an important opti-mization problem [1] is to find a common point in theintersection of finitely many convex sets It has been appliedto many areas for instance approximation theory [2] imagereconstruction from projections [3 4] control [5] and so onWhen there are only two sets and constraints are imposedon the solutions in the domain of a linear operator as wellas in this operatorrsquos ranges the problem is said to be a splitfeasibility problem (SFP) which has the following formulafinding a point 119909 satisfying
119909 isin 119862 119860119909 isin 119876 (1)
where 119862 is a closed convex subset of a Hilbert space 1198671 119876
is a closed convex subset of a Hilbert space 1198672 and 119860
1198671
rarr 1198672is a bounded linear operator The SFP was
originally introduced in [6] and it has also broad applicationsin many fields such as image reconstruction problem signalprocessing and radiation therapy Many projection methodshave also been developed for solving the SFP see [7ndash9]Denote by 119875
119862the orthogonal projection onto 119862 that is
119875119862(119909) = argmin
119910isin119862119909 minus 119910 over all 119909 isin 119862 Assuming that
the SFP is consistent (ie (1) has a solution) it is not hard tosee that 119909 isin 119862 solves (1) if and only if it solves the fixed-pointequation
119909 = 119875119862[(119868 minus 120574119860
lowast(119868 minus 119875
119876) 119860) (119909)] (2)
where 0 lt 120574 is any positive constant and 119860lowast denotes the
adjoint of 119860To solve (2) in [10] Byrne introduced the so-called CQ
algorithm which generates a sequence 119909119896 by
119909119896+1
= 119875119862[(119868 minus 120574119860
lowast(119868 minus 119875
119876) 119860) (119909
119896)] (3)
where 0 lt 120574 lt 2120588(119860119879119860) and 120588(119860
119879119860) is the spectral radius
of 119860lowast119860The split common fixed-point problem (SCFP) is a gener-
alization of the split feasibility problem (SFP) and the convexfeasibility problem (CFP) see [11] Our main purpose here isto give an extension of the results developed in [12] to the splitcommon fixed-point problem for quasi-nonexpansive oper-ators and we will introduce weak symposium convergence
2 Abstract and Applied Analysis
result of the algorithm under some suitable conditions Thiswill be done in the context of general Hilbert spaces
The paper is organized as follows In Section 2 we recallsome preliminaries In Section 3 we present an inertial CQalgorithm and show its convergence
2 Preliminaries
Throughout the rest of the paper 119868 denotes the identityoperator and Fix(119879) denotes the set of the fixed points of anoperator 119879 that is Fix(119879) = 119909 | 119909 = 119879(119909)
Recall that a mapping 119879 is said to be quasi-nonexpansive(120576Q) if
1003817100381710038171003817119879119909 minus 119902
1003817100381710038171003817le1003817100381710038171003817119909 minus 119902
1003817100381710038171003817 forall (119909 119902) isin 119867 times Fix (119879) (4)
A mapping 119879 is called nonexpansive (120576N) if
1003817100381710038171003817119879 (119909) minus 119879 (119910)
1003817100381710038171003817le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 forall (119909 119910) isin 119867 times 119867 (5)
A mapping 119879 is called firmly nonexpansive (120576FN) if
1003817100381710038171003817119879(119909) minus 119879(119910)
1003817100381710038171003817
2
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
minus1003817100381710038171003817(119909 minus 119910) minus (119879 (119909) minus 119879 (119910))
1003817100381710038171003817
2
forall (119909 119910) isin 119867 times 119867
(6)
A mapping 119879 is called firmly quasi-nonexpansive (120576FQ) if
1003817100381710038171003817119879(119909) minus 119902
1003817100381710038171003817
2
le1003817100381710038171003817119909 minus 119902
1003817100381710038171003817
2
minus 119909 minus 119879 (119909)2
forall (119909 119902) isin 119867 times Fix (119867)
(7)
It is easily observed that 120576FN sub 120576N sub 120576Q and that 120576FN sub 120576FQ sub
120576Q Furthermore 120576FN is well known to include resolvents andprojection operators while 120576FQ contains subgradient projec-tion operators (see eg [13] and the references therein)
Recently Bauschke and Combettes [14] have considereda class of mappings satisfying the condition
⟨119902 minus 119879119909 119909 minus 119879119909⟩ le 0 forall (119909 119902) isin 119867 times Fix (119879) (8)
It can easily be seen that the class of mappings satisfyingthe latter condition coincides with that of firmly quasi-nonexpansive mappings
Usually the convergence of fixed-point algorithmsrequires some additional smoothness properties of themapping 119879 such as demiclosedness
Definition 1 A mapping 119879 is said to be demiclosed if forany sequence 119909
119896 which weakly converges to 119910 and if the
sequence 119879(119909119896) strongly converges to 119911 then 119879(119910) = 119911In what follows only the particular case of demiclosed-
ness at zero will be used which is the particular case when119911 = 0
The following lemmas will be needed in the proof of theconvergence of the algorithm
Lemma 2 Let 119879 be a quasi-nonexpansive mapping Set 119879120572=
(1 minus 120572)119868 + 120572119879 Then it is immediate that for all (119909 119902) isin 119867 times
Fix(119879)
(1) ⟨119909minus119879(119909) 119909minus119902⟩ ge (12)119909 minus 119879(119909)2 and ⟨119909minus119879(119909) 119902minus
119879(119909)⟩ le (12)119909 minus 119879(119909)2
(2) 119879120572(119909) minus 119902
2le 119909 minus 119902
2minus 120572(1 minus 120572)119909 minus 119879(119909)
(3) ⟨119909 minus 119879120572(119909) 119909 minus 119902⟩ ge (1205722)119909 minus 119879(119909)
2
Lemma 3 (see [8]) Assume 120593119896isin [0infin) and 120575
119896isin [0infin)
satisfy
(1) 120593119896+1
minus 120593119896le 120579119896(120593119896minus 120593119896minus1
) + 120575119896
(2) sum+infin119896=1
120575119896lt infin
(3) 120579119896 sub [0 120579] where 120579 isin [0 1)
Then the sequence 120593119896 is convergent withsum+infin
119896=1[120593119896+1
minus120593119896]+lt
infin where [119905]+= max119905 0 (for any 119905 isin 119877)
3 The Inertial Algorithm andIts Asymptotic Convergence
In what follows we will focus our attention on the followinggeneral two-operator split common fixed-point problem
find 119909lowastisin 119862 such that 119860119909lowast isin 119876 (9)
where 119860 1198671rarr 119867
2is a bounded linear operator and 119880
1198671rarr 119867
1and 119879 119867
2rarr 119867
2are two quasi-nonexpansive
operators with nonempty fixed-point sets Fix(119880) = 119862 andFix(119879) = 119876 and denote the solution set of the two-operatorSCFP by
Γ = 119910 isin 119862 119860119910 isin 119876 (10)
31 The Inertial Algorithm To solve (9) Moudafi [15] pro-posed and proved in finite-dimensional spaces the conver-gence of the following algorithm
119909119896+1
= 119880120572119896(119909119896+ 120574 (119860
lowast119879120573minus 119868)119860 (119909
119896)) 119896 isin 119873 (11)
where 120573 isin (0 1) 120572119896isin (0 1) are relaxation parameters and
120574 gt 0 Inspired by the inertial technique we introduce thefollowing inertial method and then present its convergenceanalysis
Algorithm 4
Initialization Let 1199090 isin 1198671 be arbitrary
Iterative step For 119896 isin 119873 set 119906 = 119868 + 120574120578119860lowast(119879 minus 119868)119860
and let
119910119896= 119909119896+ 120579119896(119909119896minus 119909119896minus1
)
119909119896+1
= (1 minus 120572119896) 119906 (119910
119896) + 120572119896119880(119906 (119910
119896)) 119896 isin 119873
(12)
Abstract and Applied Analysis 3
where 120578 isin (0 1) 120572119896isin (0 1) and 120574 isin (0 1(120582120578)) with 120582 being
the spectral radius of the operator 119860lowast119860 120579119896isin [0 1)
32 Asymptotic Convergence of the Inertial Algorithm Inthis subsection we establish the asymptotic convergence ofAlgorithm 4
Lemma 5 (Opial [16]) Let 119867 be a Hilbert space and let 119909119896be a sequence in119867 such that there exists a nonempty set 119878 sub 119867
satisfying
(1) for every 119909lowast lim119896119909119896minus 119909lowast exists
(2) any weak cluster point of the sequence 119909119896 belongs to 119878Then there exists 119911 isin 119878 such that 119909119896weakly convergesto 119911
Theorem 6 Given a bounded linear operator 119860 1198671rarr 1198672
let 119880 1198671
rarr 1198671be a quasi-nonexpansive operator with
nonempty Fix(119880) = 119862 and let 119879 1198672
rarr 1198672be a quasi-
nonexpansive operator with nonempty Fix(119879) = 119876 Assumethat 119880 minus 119868 and 119879 minus 119868 are demiclosed at 0 If Γ = 0 then anysequence 119909119896 generated by Algorithm 4 weakly converges to asplit common fixed point provided that we choose 120579
119896satisfying
120579119896isin [0 120579
119896] with 120579
119896= min120579 1(119896119909119896 minus 119909
119896minus1)
2
120579 isin [0 1)120574 isin (0 1(120582120578)) and 120572
119896isin (120575 1 minus 120575) for a small enough 120575 gt 0
Proof Taking 119911 isin Γ and using (2) in Lemma 2 we obtain
10038171003817100381710038171003817119909119896+1
minus 119911
10038171003817100381710038171003817
2
=
10038171003817100381710038171003817(1 minus 120572
119896) 119906 (119910
119896) + 120572119896119880(119906 (119910
119896)) minus 119911
10038171003817100381710038171003817
2
le
10038171003817100381710038171003817119906 (119910119896) minus 119911
10038171003817100381710038171003817
2
minus 120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(13)
On the other hand we have
10038171003817100381710038171003817119906 (119910119896) minus 119911
10038171003817100381710038171003817
2
=
10038171003817100381710038171003817119910119896+ 120574120578119860
lowast(119879 minus 119868) (119860119910
119896) minus 119911
10038171003817100381710038171003817
2
=
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
+ 1205742120578210038171003817100381710038171003817119860lowast(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
+ 2120574120578 ⟨119910119896minus 119911 119860
lowast(119879 minus 119868) (119860119910
119896)⟩
le
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
+ 1205821205742120578210038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
+ 2120574120578 ⟨119860119910119896minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
(14)
that is
10038171003817100381710038171003817119906(119910119896) minus 119911
10038171003817100381710038171003817
2
le
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
+ 1205821205742120578210038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
+ 2120574120578 ⟨119860119910119896minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
(15)
Now by setting 120592 = 2120574120578⟨119860119910119896minus 119860119911 (119879 minus 119868)(119860119910
119896)⟩ and using
(1) of Lemma 2 we obtain
120592 = 2120574120578 ⟨119860119910119896minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
= 2120574120578 ⟨119860119910119896minus 119860119911 + (119879 minus 119868) (119860119910
119896)
minus (119879 minus 119868) (119860119910119896) (119879 minus 119868) (119860119910
119896)⟩
= 2120574120578 (⟨119879 (119860119910119896) minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
minus
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
)
le 2120574120578 (
1
2
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
)
le minus 120574120578
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
(16)
Combining the key inequality above with (15) yields10038171003817100381710038171003817119909119896+1
minus 119911
10038171003817100381710038171003817
2
le
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
minus 120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus 120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(17)
Define the auxiliary real sequence 120593119896
= (12)119909119896minus 119911
2
Therefore from (17) we have
120593119896+1
le
1
2
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
minus
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(18)
By deducing we have1
2
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
=
1
2
10038171003817100381710038171003817119909119896+ 120579119896(119909119896minus 119909119896minus1
) minus 119911
10038171003817100381710038171003817
2
=
1
2
10038171003817100381710038171003817119909119896minus 119911
10038171003817100381710038171003817
2
+ 120579119896⟨119909119896minus 119911 119909
119896minus 119909119896minus1
⟩
+
1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
= 120593119896+ 120579119896⟨119909119896minus 119911 119909
119896minus 119909119896minus1
⟩
+
1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(19)
It is easy to check that 120593119896= 120593119896minus1
+ ⟨119909119896minus 119911 119909
119896minus 119909119896minus1
⟩ minus
(12)119909119896minus 119909119896minus1
2
Hence
1
2
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
= 120593119896+ 120579119896(120593119896minus 120593119896minus1
)
+
120579119896+ 1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(20)
4 Abstract and Applied Analysis
Putting (20) into (18) we get
120593119896+1
le 120593119896+ 120579119896(120593119896minus 120593119896minus1
)
+
120579119896+ 1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
minus
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(21)
Since 120574 isin (0 1(120582120578)) according to 1205792
119896le 120579119896 120572119896isin (0 1) and
(21) we derive
120593119896+1
le 120593119896+ 120579119896(120593119896minus 120593119896minus1
) + 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(22)
Evidently+infin
sum
119896=1
120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
lt infin (23)
due to 120579119896119909119896minus 119909119896minus1
2
le 11198962 Let 120575
119896= 120579119896119909119896minus 119909119896minus1
2
in Lemma 3 We deduce that the sequence 119909119896minus 119911 is
convergent (hence 119909119896 is bounded) By (23) and Lemma 3we obtain sum
+infin
119896=1[119909119896minus 119911
2
minus 119909119896minus1
minus 119911
2
]+lt infin By reason of
(21) we have1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
le 120593119896minus 120593119896+1
+ 120579119896(120593119896minus 120593119896minus1
)
+ 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
le 120593119896minus 120593119896+1
+ 120579119896(120593119896minus 120593119896minus1
)
+ 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(24)
Hence+infin
sum
119896=1
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
lt infin
+infin
sum
119896=1
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
lt infin
(25)
By 120574 isin (0 1(120582120578)) and the assumption on 120572119896 we get
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
997888rarr 0 (26)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
997888rarr 0 (27)
Denoting by 119909lowast a weak-cluster point 119909
119896 let 119909
119896120590 be a
subsequence of 119909119896 Obviously
119908 minus lim120590
119910119896120590
= 119908 minus lim120590
119909119896120590
= 119909lowast (28)
Then from (26) and the demiclosedness of 119879 minus 119868 at 0 weobtain
119879 (119860119909lowast) = 119860119909
lowast (29)
it follows that 119860119909lowast isin 119876
Now by setting 119906119896= 119910119896+ 120574120578119860
lowast(119879 minus 119868)(119860119910
119896) it follows
that 119908 minus lim120590119906119896120590
= 119909lowast By the demiclosedness of 119880 minus 119868 at 0
from (27) we have
119880(119909lowast) = 119909lowast (30)
Hence 119909lowast isin 119862 and therefore 119909lowast isin ΓSince there is no more than one weak-cluster point the
weak convergence of the whole sequence 119909119896 follows by
applying Lemma 5 with 119878 = Γ
Remark 7 Since the current value of 119909119896 minus 119909119896minus1
is knownbefore choosing the parameter 120579
119896 then 120579
119896is well-defined in
Theorem 6 In fact from the process of proof for Theorem 6we can get the following assert the convergence result ofTheorem 6 always holds provided that we take 120579
119896isin [0 120579]
120579 isin [0 1) for all 119896 ge 0 with+infin
sum
119896=1
120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
lt infin (31)
To conclude we have proposed an algorithm for solvingthe SCFP in the wide class of quasi-nonexpansive operatorsand proved its convergence in general Hilbert spaces Nextwe will improve the algorithm to assure the strong conver-gence in infinite Hilbert spaces
Acknowledgments
This work was supported by the National Science Foun-dation of China (under Grant no 11171221) ShanghaiMunicipal Committee of Science and Technology (underGrant no 10550500800) Shanghai Leading Academic Dis-cipline (under Grant no XTKX 2012) Basic and FrontierResearch Program of the Science and Technology Depart-ment of Henan Province (under Grant nos 112300410277and 082300440150) and China Coal Industry AssociationScientific andTechnicalGuidance to Project (underGrant noMTKJ-2011-403)
References
[1] J W Chinneck ldquoThe constraint consensus method for find-ing approximately feasible points in nonlinear programsrdquoINFORMS Journal on Computing vol 16 no 3 pp 255ndash2652004
[2] F Deutsch ldquoThemethod of alternating orthogonal projectionsrdquoinApproximationTheory Spline Functions andApplications vol356 of NATO Advanced Science Institutes Series C pp 105ndash121 Kluwer Academic Publishers Dordrecht The Netherlands1992
[3] Y Censor ldquoParallel application of block-iterative methods inmedical imaging and radiation therapyrdquo Mathematical Pro-gramming vol 42 no 2 pp 307ndash325 1988
Abstract and Applied Analysis 5
[4] G T Herman Image Reconstruction from Projections TheFundamentals of Computerized Tomography Academic PressNew York NY USA 1980
[5] Y Gao ldquoDetermining the viability for a affine nonlinear controlsystemrdquo Journal of Control Theory amp Applications vol 26 no 6pp 654ndash656 2009 (Chinese)
[6] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquoNumerical Algorithmsvol 8 no 2ndash4 pp 221ndash239 1994
[7] Y Dang and Y Gao ldquoThe strong convergence of a KM-CQ-likealgorithm for a split feasibility problemrdquo Inverse Problems vol27 no 1 Article ID 015007 2011
[8] P-E Mainge ldquoConvergence theorems for inertial KM-typealgorithmsrdquo Journal of Computational and Applied Mathemat-ics vol 219 no 1 pp 223ndash236 2008
[9] B Qu and N Xiu ldquoA note on the 119862119876 algorithm for the splitfeasibility problemrdquo Inverse Problems vol 21 no 5 pp 1655ndash1665 2005
[10] C Byrne ldquoIterative oblique projection onto convex sets and thesplit feasibility problemrdquo Inverse Problems vol 18 no 2 pp 441ndash453 2002
[11] Y Censor and A Segal ldquoThe split common fixed point problemfor directed operatorsrdquo Journal of Convex Analysis vol 16 no2 pp 587ndash600 2009
[12] Y Dang Y Gao and Y Han ldquoA perturbed projection algorithmwith inertial technique for split feasibility problemrdquo Journal ofApplied Mathematics vol 2012 Article ID 207323 10 pages2012
[13] S Maruster and C Popirlan ldquoOn the Mann-type iteration andthe convex feasibility problemrdquo Journal of Computational andApplied Mathematics vol 212 no 2 pp 390ndash396 2008
[14] H H Bauschke and P L Combettes ldquoA weak-to-strong conver-gence principle for Fejer-monotonemethods in Hilbert spacesrdquoMathematics of Operations Research vol 26 no 2 pp 248ndash2642001
[15] A Moudafi ldquoA note on the split common fixed-point problemfor quasi-nonexpansive operatorsrdquo Nonlinear Analysis vol 74no 12 pp 4083ndash4087 2011
[16] Z Opial ldquoWeak convergence of the sequence of successiveapproximations for nonexpansive mappingsrdquo Bulletin of theAmerican Mathematical Society vol 73 pp 591ndash597 1967
Submit your manuscripts athttpwwwhindawicom
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
result of the algorithm under some suitable conditions Thiswill be done in the context of general Hilbert spaces
The paper is organized as follows In Section 2 we recallsome preliminaries In Section 3 we present an inertial CQalgorithm and show its convergence
2 Preliminaries
Throughout the rest of the paper 119868 denotes the identityoperator and Fix(119879) denotes the set of the fixed points of anoperator 119879 that is Fix(119879) = 119909 | 119909 = 119879(119909)
Recall that a mapping 119879 is said to be quasi-nonexpansive(120576Q) if
1003817100381710038171003817119879119909 minus 119902
1003817100381710038171003817le1003817100381710038171003817119909 minus 119902
1003817100381710038171003817 forall (119909 119902) isin 119867 times Fix (119879) (4)
A mapping 119879 is called nonexpansive (120576N) if
1003817100381710038171003817119879 (119909) minus 119879 (119910)
1003817100381710038171003817le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 forall (119909 119910) isin 119867 times 119867 (5)
A mapping 119879 is called firmly nonexpansive (120576FN) if
1003817100381710038171003817119879(119909) minus 119879(119910)
1003817100381710038171003817
2
le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
minus1003817100381710038171003817(119909 minus 119910) minus (119879 (119909) minus 119879 (119910))
1003817100381710038171003817
2
forall (119909 119910) isin 119867 times 119867
(6)
A mapping 119879 is called firmly quasi-nonexpansive (120576FQ) if
1003817100381710038171003817119879(119909) minus 119902
1003817100381710038171003817
2
le1003817100381710038171003817119909 minus 119902
1003817100381710038171003817
2
minus 119909 minus 119879 (119909)2
forall (119909 119902) isin 119867 times Fix (119867)
(7)
It is easily observed that 120576FN sub 120576N sub 120576Q and that 120576FN sub 120576FQ sub
120576Q Furthermore 120576FN is well known to include resolvents andprojection operators while 120576FQ contains subgradient projec-tion operators (see eg [13] and the references therein)
Recently Bauschke and Combettes [14] have considereda class of mappings satisfying the condition
⟨119902 minus 119879119909 119909 minus 119879119909⟩ le 0 forall (119909 119902) isin 119867 times Fix (119879) (8)
It can easily be seen that the class of mappings satisfyingthe latter condition coincides with that of firmly quasi-nonexpansive mappings
Usually the convergence of fixed-point algorithmsrequires some additional smoothness properties of themapping 119879 such as demiclosedness
Definition 1 A mapping 119879 is said to be demiclosed if forany sequence 119909
119896 which weakly converges to 119910 and if the
sequence 119879(119909119896) strongly converges to 119911 then 119879(119910) = 119911In what follows only the particular case of demiclosed-
ness at zero will be used which is the particular case when119911 = 0
The following lemmas will be needed in the proof of theconvergence of the algorithm
Lemma 2 Let 119879 be a quasi-nonexpansive mapping Set 119879120572=
(1 minus 120572)119868 + 120572119879 Then it is immediate that for all (119909 119902) isin 119867 times
Fix(119879)
(1) ⟨119909minus119879(119909) 119909minus119902⟩ ge (12)119909 minus 119879(119909)2 and ⟨119909minus119879(119909) 119902minus
119879(119909)⟩ le (12)119909 minus 119879(119909)2
(2) 119879120572(119909) minus 119902
2le 119909 minus 119902
2minus 120572(1 minus 120572)119909 minus 119879(119909)
(3) ⟨119909 minus 119879120572(119909) 119909 minus 119902⟩ ge (1205722)119909 minus 119879(119909)
2
Lemma 3 (see [8]) Assume 120593119896isin [0infin) and 120575
119896isin [0infin)
satisfy
(1) 120593119896+1
minus 120593119896le 120579119896(120593119896minus 120593119896minus1
) + 120575119896
(2) sum+infin119896=1
120575119896lt infin
(3) 120579119896 sub [0 120579] where 120579 isin [0 1)
Then the sequence 120593119896 is convergent withsum+infin
119896=1[120593119896+1
minus120593119896]+lt
infin where [119905]+= max119905 0 (for any 119905 isin 119877)
3 The Inertial Algorithm andIts Asymptotic Convergence
In what follows we will focus our attention on the followinggeneral two-operator split common fixed-point problem
find 119909lowastisin 119862 such that 119860119909lowast isin 119876 (9)
where 119860 1198671rarr 119867
2is a bounded linear operator and 119880
1198671rarr 119867
1and 119879 119867
2rarr 119867
2are two quasi-nonexpansive
operators with nonempty fixed-point sets Fix(119880) = 119862 andFix(119879) = 119876 and denote the solution set of the two-operatorSCFP by
Γ = 119910 isin 119862 119860119910 isin 119876 (10)
31 The Inertial Algorithm To solve (9) Moudafi [15] pro-posed and proved in finite-dimensional spaces the conver-gence of the following algorithm
119909119896+1
= 119880120572119896(119909119896+ 120574 (119860
lowast119879120573minus 119868)119860 (119909
119896)) 119896 isin 119873 (11)
where 120573 isin (0 1) 120572119896isin (0 1) are relaxation parameters and
120574 gt 0 Inspired by the inertial technique we introduce thefollowing inertial method and then present its convergenceanalysis
Algorithm 4
Initialization Let 1199090 isin 1198671 be arbitrary
Iterative step For 119896 isin 119873 set 119906 = 119868 + 120574120578119860lowast(119879 minus 119868)119860
and let
119910119896= 119909119896+ 120579119896(119909119896minus 119909119896minus1
)
119909119896+1
= (1 minus 120572119896) 119906 (119910
119896) + 120572119896119880(119906 (119910
119896)) 119896 isin 119873
(12)
Abstract and Applied Analysis 3
where 120578 isin (0 1) 120572119896isin (0 1) and 120574 isin (0 1(120582120578)) with 120582 being
the spectral radius of the operator 119860lowast119860 120579119896isin [0 1)
32 Asymptotic Convergence of the Inertial Algorithm Inthis subsection we establish the asymptotic convergence ofAlgorithm 4
Lemma 5 (Opial [16]) Let 119867 be a Hilbert space and let 119909119896be a sequence in119867 such that there exists a nonempty set 119878 sub 119867
satisfying
(1) for every 119909lowast lim119896119909119896minus 119909lowast exists
(2) any weak cluster point of the sequence 119909119896 belongs to 119878Then there exists 119911 isin 119878 such that 119909119896weakly convergesto 119911
Theorem 6 Given a bounded linear operator 119860 1198671rarr 1198672
let 119880 1198671
rarr 1198671be a quasi-nonexpansive operator with
nonempty Fix(119880) = 119862 and let 119879 1198672
rarr 1198672be a quasi-
nonexpansive operator with nonempty Fix(119879) = 119876 Assumethat 119880 minus 119868 and 119879 minus 119868 are demiclosed at 0 If Γ = 0 then anysequence 119909119896 generated by Algorithm 4 weakly converges to asplit common fixed point provided that we choose 120579
119896satisfying
120579119896isin [0 120579
119896] with 120579
119896= min120579 1(119896119909119896 minus 119909
119896minus1)
2
120579 isin [0 1)120574 isin (0 1(120582120578)) and 120572
119896isin (120575 1 minus 120575) for a small enough 120575 gt 0
Proof Taking 119911 isin Γ and using (2) in Lemma 2 we obtain
10038171003817100381710038171003817119909119896+1
minus 119911
10038171003817100381710038171003817
2
=
10038171003817100381710038171003817(1 minus 120572
119896) 119906 (119910
119896) + 120572119896119880(119906 (119910
119896)) minus 119911
10038171003817100381710038171003817
2
le
10038171003817100381710038171003817119906 (119910119896) minus 119911
10038171003817100381710038171003817
2
minus 120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(13)
On the other hand we have
10038171003817100381710038171003817119906 (119910119896) minus 119911
10038171003817100381710038171003817
2
=
10038171003817100381710038171003817119910119896+ 120574120578119860
lowast(119879 minus 119868) (119860119910
119896) minus 119911
10038171003817100381710038171003817
2
=
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
+ 1205742120578210038171003817100381710038171003817119860lowast(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
+ 2120574120578 ⟨119910119896minus 119911 119860
lowast(119879 minus 119868) (119860119910
119896)⟩
le
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
+ 1205821205742120578210038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
+ 2120574120578 ⟨119860119910119896minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
(14)
that is
10038171003817100381710038171003817119906(119910119896) minus 119911
10038171003817100381710038171003817
2
le
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
+ 1205821205742120578210038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
+ 2120574120578 ⟨119860119910119896minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
(15)
Now by setting 120592 = 2120574120578⟨119860119910119896minus 119860119911 (119879 minus 119868)(119860119910
119896)⟩ and using
(1) of Lemma 2 we obtain
120592 = 2120574120578 ⟨119860119910119896minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
= 2120574120578 ⟨119860119910119896minus 119860119911 + (119879 minus 119868) (119860119910
119896)
minus (119879 minus 119868) (119860119910119896) (119879 minus 119868) (119860119910
119896)⟩
= 2120574120578 (⟨119879 (119860119910119896) minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
minus
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
)
le 2120574120578 (
1
2
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
)
le minus 120574120578
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
(16)
Combining the key inequality above with (15) yields10038171003817100381710038171003817119909119896+1
minus 119911
10038171003817100381710038171003817
2
le
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
minus 120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus 120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(17)
Define the auxiliary real sequence 120593119896
= (12)119909119896minus 119911
2
Therefore from (17) we have
120593119896+1
le
1
2
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
minus
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(18)
By deducing we have1
2
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
=
1
2
10038171003817100381710038171003817119909119896+ 120579119896(119909119896minus 119909119896minus1
) minus 119911
10038171003817100381710038171003817
2
=
1
2
10038171003817100381710038171003817119909119896minus 119911
10038171003817100381710038171003817
2
+ 120579119896⟨119909119896minus 119911 119909
119896minus 119909119896minus1
⟩
+
1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
= 120593119896+ 120579119896⟨119909119896minus 119911 119909
119896minus 119909119896minus1
⟩
+
1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(19)
It is easy to check that 120593119896= 120593119896minus1
+ ⟨119909119896minus 119911 119909
119896minus 119909119896minus1
⟩ minus
(12)119909119896minus 119909119896minus1
2
Hence
1
2
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
= 120593119896+ 120579119896(120593119896minus 120593119896minus1
)
+
120579119896+ 1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(20)
4 Abstract and Applied Analysis
Putting (20) into (18) we get
120593119896+1
le 120593119896+ 120579119896(120593119896minus 120593119896minus1
)
+
120579119896+ 1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
minus
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(21)
Since 120574 isin (0 1(120582120578)) according to 1205792
119896le 120579119896 120572119896isin (0 1) and
(21) we derive
120593119896+1
le 120593119896+ 120579119896(120593119896minus 120593119896minus1
) + 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(22)
Evidently+infin
sum
119896=1
120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
lt infin (23)
due to 120579119896119909119896minus 119909119896minus1
2
le 11198962 Let 120575
119896= 120579119896119909119896minus 119909119896minus1
2
in Lemma 3 We deduce that the sequence 119909119896minus 119911 is
convergent (hence 119909119896 is bounded) By (23) and Lemma 3we obtain sum
+infin
119896=1[119909119896minus 119911
2
minus 119909119896minus1
minus 119911
2
]+lt infin By reason of
(21) we have1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
le 120593119896minus 120593119896+1
+ 120579119896(120593119896minus 120593119896minus1
)
+ 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
le 120593119896minus 120593119896+1
+ 120579119896(120593119896minus 120593119896minus1
)
+ 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(24)
Hence+infin
sum
119896=1
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
lt infin
+infin
sum
119896=1
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
lt infin
(25)
By 120574 isin (0 1(120582120578)) and the assumption on 120572119896 we get
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
997888rarr 0 (26)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
997888rarr 0 (27)
Denoting by 119909lowast a weak-cluster point 119909
119896 let 119909
119896120590 be a
subsequence of 119909119896 Obviously
119908 minus lim120590
119910119896120590
= 119908 minus lim120590
119909119896120590
= 119909lowast (28)
Then from (26) and the demiclosedness of 119879 minus 119868 at 0 weobtain
119879 (119860119909lowast) = 119860119909
lowast (29)
it follows that 119860119909lowast isin 119876
Now by setting 119906119896= 119910119896+ 120574120578119860
lowast(119879 minus 119868)(119860119910
119896) it follows
that 119908 minus lim120590119906119896120590
= 119909lowast By the demiclosedness of 119880 minus 119868 at 0
from (27) we have
119880(119909lowast) = 119909lowast (30)
Hence 119909lowast isin 119862 and therefore 119909lowast isin ΓSince there is no more than one weak-cluster point the
weak convergence of the whole sequence 119909119896 follows by
applying Lemma 5 with 119878 = Γ
Remark 7 Since the current value of 119909119896 minus 119909119896minus1
is knownbefore choosing the parameter 120579
119896 then 120579
119896is well-defined in
Theorem 6 In fact from the process of proof for Theorem 6we can get the following assert the convergence result ofTheorem 6 always holds provided that we take 120579
119896isin [0 120579]
120579 isin [0 1) for all 119896 ge 0 with+infin
sum
119896=1
120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
lt infin (31)
To conclude we have proposed an algorithm for solvingthe SCFP in the wide class of quasi-nonexpansive operatorsand proved its convergence in general Hilbert spaces Nextwe will improve the algorithm to assure the strong conver-gence in infinite Hilbert spaces
Acknowledgments
This work was supported by the National Science Foun-dation of China (under Grant no 11171221) ShanghaiMunicipal Committee of Science and Technology (underGrant no 10550500800) Shanghai Leading Academic Dis-cipline (under Grant no XTKX 2012) Basic and FrontierResearch Program of the Science and Technology Depart-ment of Henan Province (under Grant nos 112300410277and 082300440150) and China Coal Industry AssociationScientific andTechnicalGuidance to Project (underGrant noMTKJ-2011-403)
References
[1] J W Chinneck ldquoThe constraint consensus method for find-ing approximately feasible points in nonlinear programsrdquoINFORMS Journal on Computing vol 16 no 3 pp 255ndash2652004
[2] F Deutsch ldquoThemethod of alternating orthogonal projectionsrdquoinApproximationTheory Spline Functions andApplications vol356 of NATO Advanced Science Institutes Series C pp 105ndash121 Kluwer Academic Publishers Dordrecht The Netherlands1992
[3] Y Censor ldquoParallel application of block-iterative methods inmedical imaging and radiation therapyrdquo Mathematical Pro-gramming vol 42 no 2 pp 307ndash325 1988
Abstract and Applied Analysis 5
[4] G T Herman Image Reconstruction from Projections TheFundamentals of Computerized Tomography Academic PressNew York NY USA 1980
[5] Y Gao ldquoDetermining the viability for a affine nonlinear controlsystemrdquo Journal of Control Theory amp Applications vol 26 no 6pp 654ndash656 2009 (Chinese)
[6] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquoNumerical Algorithmsvol 8 no 2ndash4 pp 221ndash239 1994
[7] Y Dang and Y Gao ldquoThe strong convergence of a KM-CQ-likealgorithm for a split feasibility problemrdquo Inverse Problems vol27 no 1 Article ID 015007 2011
[8] P-E Mainge ldquoConvergence theorems for inertial KM-typealgorithmsrdquo Journal of Computational and Applied Mathemat-ics vol 219 no 1 pp 223ndash236 2008
[9] B Qu and N Xiu ldquoA note on the 119862119876 algorithm for the splitfeasibility problemrdquo Inverse Problems vol 21 no 5 pp 1655ndash1665 2005
[10] C Byrne ldquoIterative oblique projection onto convex sets and thesplit feasibility problemrdquo Inverse Problems vol 18 no 2 pp 441ndash453 2002
[11] Y Censor and A Segal ldquoThe split common fixed point problemfor directed operatorsrdquo Journal of Convex Analysis vol 16 no2 pp 587ndash600 2009
[12] Y Dang Y Gao and Y Han ldquoA perturbed projection algorithmwith inertial technique for split feasibility problemrdquo Journal ofApplied Mathematics vol 2012 Article ID 207323 10 pages2012
[13] S Maruster and C Popirlan ldquoOn the Mann-type iteration andthe convex feasibility problemrdquo Journal of Computational andApplied Mathematics vol 212 no 2 pp 390ndash396 2008
[14] H H Bauschke and P L Combettes ldquoA weak-to-strong conver-gence principle for Fejer-monotonemethods in Hilbert spacesrdquoMathematics of Operations Research vol 26 no 2 pp 248ndash2642001
[15] A Moudafi ldquoA note on the split common fixed-point problemfor quasi-nonexpansive operatorsrdquo Nonlinear Analysis vol 74no 12 pp 4083ndash4087 2011
[16] Z Opial ldquoWeak convergence of the sequence of successiveapproximations for nonexpansive mappingsrdquo Bulletin of theAmerican Mathematical Society vol 73 pp 591ndash597 1967
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
where 120578 isin (0 1) 120572119896isin (0 1) and 120574 isin (0 1(120582120578)) with 120582 being
the spectral radius of the operator 119860lowast119860 120579119896isin [0 1)
32 Asymptotic Convergence of the Inertial Algorithm Inthis subsection we establish the asymptotic convergence ofAlgorithm 4
Lemma 5 (Opial [16]) Let 119867 be a Hilbert space and let 119909119896be a sequence in119867 such that there exists a nonempty set 119878 sub 119867
satisfying
(1) for every 119909lowast lim119896119909119896minus 119909lowast exists
(2) any weak cluster point of the sequence 119909119896 belongs to 119878Then there exists 119911 isin 119878 such that 119909119896weakly convergesto 119911
Theorem 6 Given a bounded linear operator 119860 1198671rarr 1198672
let 119880 1198671
rarr 1198671be a quasi-nonexpansive operator with
nonempty Fix(119880) = 119862 and let 119879 1198672
rarr 1198672be a quasi-
nonexpansive operator with nonempty Fix(119879) = 119876 Assumethat 119880 minus 119868 and 119879 minus 119868 are demiclosed at 0 If Γ = 0 then anysequence 119909119896 generated by Algorithm 4 weakly converges to asplit common fixed point provided that we choose 120579
119896satisfying
120579119896isin [0 120579
119896] with 120579
119896= min120579 1(119896119909119896 minus 119909
119896minus1)
2
120579 isin [0 1)120574 isin (0 1(120582120578)) and 120572
119896isin (120575 1 minus 120575) for a small enough 120575 gt 0
Proof Taking 119911 isin Γ and using (2) in Lemma 2 we obtain
10038171003817100381710038171003817119909119896+1
minus 119911
10038171003817100381710038171003817
2
=
10038171003817100381710038171003817(1 minus 120572
119896) 119906 (119910
119896) + 120572119896119880(119906 (119910
119896)) minus 119911
10038171003817100381710038171003817
2
le
10038171003817100381710038171003817119906 (119910119896) minus 119911
10038171003817100381710038171003817
2
minus 120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(13)
On the other hand we have
10038171003817100381710038171003817119906 (119910119896) minus 119911
10038171003817100381710038171003817
2
=
10038171003817100381710038171003817119910119896+ 120574120578119860
lowast(119879 minus 119868) (119860119910
119896) minus 119911
10038171003817100381710038171003817
2
=
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
+ 1205742120578210038171003817100381710038171003817119860lowast(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
+ 2120574120578 ⟨119910119896minus 119911 119860
lowast(119879 minus 119868) (119860119910
119896)⟩
le
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
+ 1205821205742120578210038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
+ 2120574120578 ⟨119860119910119896minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
(14)
that is
10038171003817100381710038171003817119906(119910119896) minus 119911
10038171003817100381710038171003817
2
le
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
+ 1205821205742120578210038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
+ 2120574120578 ⟨119860119910119896minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
(15)
Now by setting 120592 = 2120574120578⟨119860119910119896minus 119860119911 (119879 minus 119868)(119860119910
119896)⟩ and using
(1) of Lemma 2 we obtain
120592 = 2120574120578 ⟨119860119910119896minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
= 2120574120578 ⟨119860119910119896minus 119860119911 + (119879 minus 119868) (119860119910
119896)
minus (119879 minus 119868) (119860119910119896) (119879 minus 119868) (119860119910
119896)⟩
= 2120574120578 (⟨119879 (119860119910119896) minus 119860119911 (119879 minus 119868) (119860119910
119896)⟩
minus
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
)
le 2120574120578 (
1
2
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
)
le minus 120574120578
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
(16)
Combining the key inequality above with (15) yields10038171003817100381710038171003817119909119896+1
minus 119911
10038171003817100381710038171003817
2
le
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
minus 120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus 120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(17)
Define the auxiliary real sequence 120593119896
= (12)119909119896minus 119911
2
Therefore from (17) we have
120593119896+1
le
1
2
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
minus
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(18)
By deducing we have1
2
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
=
1
2
10038171003817100381710038171003817119909119896+ 120579119896(119909119896minus 119909119896minus1
) minus 119911
10038171003817100381710038171003817
2
=
1
2
10038171003817100381710038171003817119909119896minus 119911
10038171003817100381710038171003817
2
+ 120579119896⟨119909119896minus 119911 119909
119896minus 119909119896minus1
⟩
+
1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
= 120593119896+ 120579119896⟨119909119896minus 119911 119909
119896minus 119909119896minus1
⟩
+
1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(19)
It is easy to check that 120593119896= 120593119896minus1
+ ⟨119909119896minus 119911 119909
119896minus 119909119896minus1
⟩ minus
(12)119909119896minus 119909119896minus1
2
Hence
1
2
10038171003817100381710038171003817119910119896minus 119911
10038171003817100381710038171003817
2
= 120593119896+ 120579119896(120593119896minus 120593119896minus1
)
+
120579119896+ 1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(20)
4 Abstract and Applied Analysis
Putting (20) into (18) we get
120593119896+1
le 120593119896+ 120579119896(120593119896minus 120593119896minus1
)
+
120579119896+ 1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
minus
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(21)
Since 120574 isin (0 1(120582120578)) according to 1205792
119896le 120579119896 120572119896isin (0 1) and
(21) we derive
120593119896+1
le 120593119896+ 120579119896(120593119896minus 120593119896minus1
) + 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(22)
Evidently+infin
sum
119896=1
120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
lt infin (23)
due to 120579119896119909119896minus 119909119896minus1
2
le 11198962 Let 120575
119896= 120579119896119909119896minus 119909119896minus1
2
in Lemma 3 We deduce that the sequence 119909119896minus 119911 is
convergent (hence 119909119896 is bounded) By (23) and Lemma 3we obtain sum
+infin
119896=1[119909119896minus 119911
2
minus 119909119896minus1
minus 119911
2
]+lt infin By reason of
(21) we have1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
le 120593119896minus 120593119896+1
+ 120579119896(120593119896minus 120593119896minus1
)
+ 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
le 120593119896minus 120593119896+1
+ 120579119896(120593119896minus 120593119896minus1
)
+ 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(24)
Hence+infin
sum
119896=1
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
lt infin
+infin
sum
119896=1
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
lt infin
(25)
By 120574 isin (0 1(120582120578)) and the assumption on 120572119896 we get
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
997888rarr 0 (26)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
997888rarr 0 (27)
Denoting by 119909lowast a weak-cluster point 119909
119896 let 119909
119896120590 be a
subsequence of 119909119896 Obviously
119908 minus lim120590
119910119896120590
= 119908 minus lim120590
119909119896120590
= 119909lowast (28)
Then from (26) and the demiclosedness of 119879 minus 119868 at 0 weobtain
119879 (119860119909lowast) = 119860119909
lowast (29)
it follows that 119860119909lowast isin 119876
Now by setting 119906119896= 119910119896+ 120574120578119860
lowast(119879 minus 119868)(119860119910
119896) it follows
that 119908 minus lim120590119906119896120590
= 119909lowast By the demiclosedness of 119880 minus 119868 at 0
from (27) we have
119880(119909lowast) = 119909lowast (30)
Hence 119909lowast isin 119862 and therefore 119909lowast isin ΓSince there is no more than one weak-cluster point the
weak convergence of the whole sequence 119909119896 follows by
applying Lemma 5 with 119878 = Γ
Remark 7 Since the current value of 119909119896 minus 119909119896minus1
is knownbefore choosing the parameter 120579
119896 then 120579
119896is well-defined in
Theorem 6 In fact from the process of proof for Theorem 6we can get the following assert the convergence result ofTheorem 6 always holds provided that we take 120579
119896isin [0 120579]
120579 isin [0 1) for all 119896 ge 0 with+infin
sum
119896=1
120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
lt infin (31)
To conclude we have proposed an algorithm for solvingthe SCFP in the wide class of quasi-nonexpansive operatorsand proved its convergence in general Hilbert spaces Nextwe will improve the algorithm to assure the strong conver-gence in infinite Hilbert spaces
Acknowledgments
This work was supported by the National Science Foun-dation of China (under Grant no 11171221) ShanghaiMunicipal Committee of Science and Technology (underGrant no 10550500800) Shanghai Leading Academic Dis-cipline (under Grant no XTKX 2012) Basic and FrontierResearch Program of the Science and Technology Depart-ment of Henan Province (under Grant nos 112300410277and 082300440150) and China Coal Industry AssociationScientific andTechnicalGuidance to Project (underGrant noMTKJ-2011-403)
References
[1] J W Chinneck ldquoThe constraint consensus method for find-ing approximately feasible points in nonlinear programsrdquoINFORMS Journal on Computing vol 16 no 3 pp 255ndash2652004
[2] F Deutsch ldquoThemethod of alternating orthogonal projectionsrdquoinApproximationTheory Spline Functions andApplications vol356 of NATO Advanced Science Institutes Series C pp 105ndash121 Kluwer Academic Publishers Dordrecht The Netherlands1992
[3] Y Censor ldquoParallel application of block-iterative methods inmedical imaging and radiation therapyrdquo Mathematical Pro-gramming vol 42 no 2 pp 307ndash325 1988
Abstract and Applied Analysis 5
[4] G T Herman Image Reconstruction from Projections TheFundamentals of Computerized Tomography Academic PressNew York NY USA 1980
[5] Y Gao ldquoDetermining the viability for a affine nonlinear controlsystemrdquo Journal of Control Theory amp Applications vol 26 no 6pp 654ndash656 2009 (Chinese)
[6] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquoNumerical Algorithmsvol 8 no 2ndash4 pp 221ndash239 1994
[7] Y Dang and Y Gao ldquoThe strong convergence of a KM-CQ-likealgorithm for a split feasibility problemrdquo Inverse Problems vol27 no 1 Article ID 015007 2011
[8] P-E Mainge ldquoConvergence theorems for inertial KM-typealgorithmsrdquo Journal of Computational and Applied Mathemat-ics vol 219 no 1 pp 223ndash236 2008
[9] B Qu and N Xiu ldquoA note on the 119862119876 algorithm for the splitfeasibility problemrdquo Inverse Problems vol 21 no 5 pp 1655ndash1665 2005
[10] C Byrne ldquoIterative oblique projection onto convex sets and thesplit feasibility problemrdquo Inverse Problems vol 18 no 2 pp 441ndash453 2002
[11] Y Censor and A Segal ldquoThe split common fixed point problemfor directed operatorsrdquo Journal of Convex Analysis vol 16 no2 pp 587ndash600 2009
[12] Y Dang Y Gao and Y Han ldquoA perturbed projection algorithmwith inertial technique for split feasibility problemrdquo Journal ofApplied Mathematics vol 2012 Article ID 207323 10 pages2012
[13] S Maruster and C Popirlan ldquoOn the Mann-type iteration andthe convex feasibility problemrdquo Journal of Computational andApplied Mathematics vol 212 no 2 pp 390ndash396 2008
[14] H H Bauschke and P L Combettes ldquoA weak-to-strong conver-gence principle for Fejer-monotonemethods in Hilbert spacesrdquoMathematics of Operations Research vol 26 no 2 pp 248ndash2642001
[15] A Moudafi ldquoA note on the split common fixed-point problemfor quasi-nonexpansive operatorsrdquo Nonlinear Analysis vol 74no 12 pp 4083ndash4087 2011
[16] Z Opial ldquoWeak convergence of the sequence of successiveapproximations for nonexpansive mappingsrdquo Bulletin of theAmerican Mathematical Society vol 73 pp 591ndash597 1967
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Putting (20) into (18) we get
120593119896+1
le 120593119896+ 120579119896(120593119896minus 120593119896minus1
)
+
120579119896+ 1205792
119896
2
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
minus
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
minus
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
(21)
Since 120574 isin (0 1(120582120578)) according to 1205792
119896le 120579119896 120572119896isin (0 1) and
(21) we derive
120593119896+1
le 120593119896+ 120579119896(120593119896minus 120593119896minus1
) + 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(22)
Evidently+infin
sum
119896=1
120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
lt infin (23)
due to 120579119896119909119896minus 119909119896minus1
2
le 11198962 Let 120575
119896= 120579119896119909119896minus 119909119896minus1
2
in Lemma 3 We deduce that the sequence 119909119896minus 119911 is
convergent (hence 119909119896 is bounded) By (23) and Lemma 3we obtain sum
+infin
119896=1[119909119896minus 119911
2
minus 119909119896minus1
minus 119911
2
]+lt infin By reason of
(21) we have1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
le 120593119896minus 120593119896+1
+ 120579119896(120593119896minus 120593119896minus1
)
+ 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
le 120593119896minus 120593119896+1
+ 120579119896(120593119896minus 120593119896minus1
)
+ 120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
(24)
Hence+infin
sum
119896=1
1
2
120574120578 (1 minus 120582120574120578)
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
lt infin
+infin
sum
119896=1
1
2
120572119896(1 minus 120572
119896)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
lt infin
(25)
By 120574 isin (0 1(120582120578)) and the assumption on 120572119896 we get
10038171003817100381710038171003817(119879 minus 119868) (119860119910
119896)
10038171003817100381710038171003817
2
997888rarr 0 (26)
10038171003817100381710038171003817119880 (119906 (119910
119896)) minus 119906 (119910
119896)
10038171003817100381710038171003817
2
997888rarr 0 (27)
Denoting by 119909lowast a weak-cluster point 119909
119896 let 119909
119896120590 be a
subsequence of 119909119896 Obviously
119908 minus lim120590
119910119896120590
= 119908 minus lim120590
119909119896120590
= 119909lowast (28)
Then from (26) and the demiclosedness of 119879 minus 119868 at 0 weobtain
119879 (119860119909lowast) = 119860119909
lowast (29)
it follows that 119860119909lowast isin 119876
Now by setting 119906119896= 119910119896+ 120574120578119860
lowast(119879 minus 119868)(119860119910
119896) it follows
that 119908 minus lim120590119906119896120590
= 119909lowast By the demiclosedness of 119880 minus 119868 at 0
from (27) we have
119880(119909lowast) = 119909lowast (30)
Hence 119909lowast isin 119862 and therefore 119909lowast isin ΓSince there is no more than one weak-cluster point the
weak convergence of the whole sequence 119909119896 follows by
applying Lemma 5 with 119878 = Γ
Remark 7 Since the current value of 119909119896 minus 119909119896minus1
is knownbefore choosing the parameter 120579
119896 then 120579
119896is well-defined in
Theorem 6 In fact from the process of proof for Theorem 6we can get the following assert the convergence result ofTheorem 6 always holds provided that we take 120579
119896isin [0 120579]
120579 isin [0 1) for all 119896 ge 0 with+infin
sum
119896=1
120579119896
10038171003817100381710038171003817119909119896minus 119909119896minus110038171003817
100381710038171003817
2
lt infin (31)
To conclude we have proposed an algorithm for solvingthe SCFP in the wide class of quasi-nonexpansive operatorsand proved its convergence in general Hilbert spaces Nextwe will improve the algorithm to assure the strong conver-gence in infinite Hilbert spaces
Acknowledgments
This work was supported by the National Science Foun-dation of China (under Grant no 11171221) ShanghaiMunicipal Committee of Science and Technology (underGrant no 10550500800) Shanghai Leading Academic Dis-cipline (under Grant no XTKX 2012) Basic and FrontierResearch Program of the Science and Technology Depart-ment of Henan Province (under Grant nos 112300410277and 082300440150) and China Coal Industry AssociationScientific andTechnicalGuidance to Project (underGrant noMTKJ-2011-403)
References
[1] J W Chinneck ldquoThe constraint consensus method for find-ing approximately feasible points in nonlinear programsrdquoINFORMS Journal on Computing vol 16 no 3 pp 255ndash2652004
[2] F Deutsch ldquoThemethod of alternating orthogonal projectionsrdquoinApproximationTheory Spline Functions andApplications vol356 of NATO Advanced Science Institutes Series C pp 105ndash121 Kluwer Academic Publishers Dordrecht The Netherlands1992
[3] Y Censor ldquoParallel application of block-iterative methods inmedical imaging and radiation therapyrdquo Mathematical Pro-gramming vol 42 no 2 pp 307ndash325 1988
Abstract and Applied Analysis 5
[4] G T Herman Image Reconstruction from Projections TheFundamentals of Computerized Tomography Academic PressNew York NY USA 1980
[5] Y Gao ldquoDetermining the viability for a affine nonlinear controlsystemrdquo Journal of Control Theory amp Applications vol 26 no 6pp 654ndash656 2009 (Chinese)
[6] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquoNumerical Algorithmsvol 8 no 2ndash4 pp 221ndash239 1994
[7] Y Dang and Y Gao ldquoThe strong convergence of a KM-CQ-likealgorithm for a split feasibility problemrdquo Inverse Problems vol27 no 1 Article ID 015007 2011
[8] P-E Mainge ldquoConvergence theorems for inertial KM-typealgorithmsrdquo Journal of Computational and Applied Mathemat-ics vol 219 no 1 pp 223ndash236 2008
[9] B Qu and N Xiu ldquoA note on the 119862119876 algorithm for the splitfeasibility problemrdquo Inverse Problems vol 21 no 5 pp 1655ndash1665 2005
[10] C Byrne ldquoIterative oblique projection onto convex sets and thesplit feasibility problemrdquo Inverse Problems vol 18 no 2 pp 441ndash453 2002
[11] Y Censor and A Segal ldquoThe split common fixed point problemfor directed operatorsrdquo Journal of Convex Analysis vol 16 no2 pp 587ndash600 2009
[12] Y Dang Y Gao and Y Han ldquoA perturbed projection algorithmwith inertial technique for split feasibility problemrdquo Journal ofApplied Mathematics vol 2012 Article ID 207323 10 pages2012
[13] S Maruster and C Popirlan ldquoOn the Mann-type iteration andthe convex feasibility problemrdquo Journal of Computational andApplied Mathematics vol 212 no 2 pp 390ndash396 2008
[14] H H Bauschke and P L Combettes ldquoA weak-to-strong conver-gence principle for Fejer-monotonemethods in Hilbert spacesrdquoMathematics of Operations Research vol 26 no 2 pp 248ndash2642001
[15] A Moudafi ldquoA note on the split common fixed-point problemfor quasi-nonexpansive operatorsrdquo Nonlinear Analysis vol 74no 12 pp 4083ndash4087 2011
[16] Z Opial ldquoWeak convergence of the sequence of successiveapproximations for nonexpansive mappingsrdquo Bulletin of theAmerican Mathematical Society vol 73 pp 591ndash597 1967
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
[4] G T Herman Image Reconstruction from Projections TheFundamentals of Computerized Tomography Academic PressNew York NY USA 1980
[5] Y Gao ldquoDetermining the viability for a affine nonlinear controlsystemrdquo Journal of Control Theory amp Applications vol 26 no 6pp 654ndash656 2009 (Chinese)
[6] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquoNumerical Algorithmsvol 8 no 2ndash4 pp 221ndash239 1994
[7] Y Dang and Y Gao ldquoThe strong convergence of a KM-CQ-likealgorithm for a split feasibility problemrdquo Inverse Problems vol27 no 1 Article ID 015007 2011
[8] P-E Mainge ldquoConvergence theorems for inertial KM-typealgorithmsrdquo Journal of Computational and Applied Mathemat-ics vol 219 no 1 pp 223ndash236 2008
[9] B Qu and N Xiu ldquoA note on the 119862119876 algorithm for the splitfeasibility problemrdquo Inverse Problems vol 21 no 5 pp 1655ndash1665 2005
[10] C Byrne ldquoIterative oblique projection onto convex sets and thesplit feasibility problemrdquo Inverse Problems vol 18 no 2 pp 441ndash453 2002
[11] Y Censor and A Segal ldquoThe split common fixed point problemfor directed operatorsrdquo Journal of Convex Analysis vol 16 no2 pp 587ndash600 2009
[12] Y Dang Y Gao and Y Han ldquoA perturbed projection algorithmwith inertial technique for split feasibility problemrdquo Journal ofApplied Mathematics vol 2012 Article ID 207323 10 pages2012
[13] S Maruster and C Popirlan ldquoOn the Mann-type iteration andthe convex feasibility problemrdquo Journal of Computational andApplied Mathematics vol 212 no 2 pp 390ndash396 2008
[14] H H Bauschke and P L Combettes ldquoA weak-to-strong conver-gence principle for Fejer-monotonemethods in Hilbert spacesrdquoMathematics of Operations Research vol 26 no 2 pp 248ndash2642001
[15] A Moudafi ldquoA note on the split common fixed-point problemfor quasi-nonexpansive operatorsrdquo Nonlinear Analysis vol 74no 12 pp 4083ndash4087 2011
[16] Z Opial ldquoWeak convergence of the sequence of successiveapproximations for nonexpansive mappingsrdquo Bulletin of theAmerican Mathematical Society vol 73 pp 591ndash597 1967
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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