- Open Access
The strong convergence theorems for split common fixed point problem ofasymptotically nonexpansive mappings in Hilbert spaces
© Zhang et al.; licensee Springer. 2015
- Received: 12 April 2014
- Accepted: 11 December 2014
- Published: 7 January 2015
In this paper, an iterative algorithm is introduced to solve the split common fixedpoint problem for asymptotically nonexpansive mappings in Hilbert spaces. Theiterative algorithm presented in this paper is shown to possess strong convergencefor the split common fixed point problem of asymptotically nonexpansive mappingsalthough the mappings do not have semi-compactness. Our results improve and developprevious methods for solving the split common fixed point problem.
MSC: 47H09, 47J25.
- split common fixed point problem
- asymptotically nonexpansive mapping
- strong convergence
- Hilbert space
Throughout this paper, let and bereal Hilbert spaces whose inner product and norm are denoted by and , respectively; let C andQ be nonempty closed convex subsets of and ,respectively. A mapping is said to benonexpansive if for any .A mapping is said to bequasi-nonexpansive if for any and , where is the set of fixed pointsof T. A mapping is calledasymptotically nonexpansive if there exists a sequence satisfying such that for any .A mapping is semi-compact if, forany bounded sequence with , there exists asubsequence such that converges strongly to somepoint .
where is a bounded linear operator.
where and arethe (orthogonal) projections onto C and Q, respectively, isany positive constant, and denotes the adjoint of A.
The SFP in finite-dimensional Hilbert spaces was first introduced by Censor andElfving  for modeling inverse problems whicharise from phase retrievals and in medical image reconstruction . Recently, it has been found that the SFP can also beused in various disciplines such as image restoration, computer tomograph, and radiationtherapy treatment planning [2–7].
We use Γ to denote the set of solutions of SCFP (1.3), that is, .
Since each closed and convex subset may be considered as a fixed point set of aprojection on the subset, hence the split common fixed point problem (SCFP) isa generalization of the split feasibility problem (SFP) and the convexfeasibility problem (CFP) .
and he proved that converges weakly to a split common fixedpoint , where and are two demi-contractive mappings, is a bounded linear operator.
Using the iterative algorithm above, in 2011, Moudafi  also obtained a weak convergence theorem for the split commonfixed point problem of quasi-nonexpansive mappings in Hilbert spaces. After that, someauthors also proposed some iterative algorithms to approximate a split common fixedpoint of other nonlinear mappings, such as nonspreading type mappings , asymptotically quasi-nonexpansive mappings, κ-asymptoticallystrictly pseudononspreading mappings ,asymptotically strictly pseudocontraction mappings etc., but they just obtained weak convergence theoremswhen those mappings do not have semi-compactness. This naturally brings us to thefollowing question.
Can we construct an iterative scheme which can guarantee the strong convergence forsplit common fixed point problems without assumption ofsemi-compactness?
where and are two asymptotically nonexpansive mappings, is a bounded linear operator, denotes the adjoint of A. Under some suitable conditions on parameters, theiterative scheme is shown to converge strongly to a splitcommon fixed point of asymptotically nonexpansive mappings and without the assumption of semi-compactness on and .
The following lemma and results are useful for our proofs.
LetEbe a real uniformly convex Banach space, Kbea nonempty closed subset ofE, and let be an asymptoticallynonexpansive mapping. Then isdemiclosed at zero, that is, if convergesweakly to a point and ,then .
where denotesthe adjoint ofA, and satisfies , , .If , then converges stronglyto .
Proof We will divide the proof into five steps.
Step 1. We first show that is closedand convex for any .
we know that is closed and convex. Therefore is closedand convex for any .
Step 2. We prove for any .
Therefore, from (2.4) and (2.5), we know that and for any .
Step 3. We will show that is a Cauchy sequence.
Since exists, it followsfrom (2.7) that . Therefore is a Cauchy sequence.
Step 4. We will show that .
Step 5. We will show that converges strongly to an element ofΓ.
Since is a Cauchy sequence, we may assume that ,from (2.8) we have ,which implies that .So it follows from (2.13) and Lemma 1.1 that .
In addition, since A is a bounded linear operator, we have that . Hence, itfollows from (2.14) and Lemma 1.1 that . This means that and converges strongly to . The proof iscompleted. □
In Theorem 2.1, as and ,we have the following result.
where and satisfies .If , then converges strongly to a fixedpoint ofT.
In Theorem 2.1, when and aretwo nonexpansive mappings, the following result holds.
where denotesthe adjoint ofA, and satisfies .If , then converges stronglyto .
Remark 2.4 When and aretwo quasi-nonexpansive mappings and and are demiclosed at zero, Corollary 2.3 also holds.
, , .
Therefore is anasymptotically nonexpansive mapping from C into itself with .
It is easy to show that or for any .Therefore is anasymptotically nonexpansive mapping from D into itself with since for anysequence with .
Obviously, C and D are closed convex subsets of and ,respectively. Let be defined by for . ThenA is a bounded linear operator with adjoint operator for . Clearly, , .
Taking , , , , and , . Itfollows from Theorem 2.1 that converges strongly to .
Application to the equilibrium problem
Let H be a real Hilbert space, C be a nonempty closed and convexsubset of H, and let the bifunction satisfy the following conditions:
(A1) , ;
(A2) , ;
(A3) For all , ;
(A4) For each , thefunction is convex and lower semi-continuous.
The so-called equilibrium problem for F is to find a point such that for all .The set of its solutions is denoted by .
is nonempty, closed and convex.
where denotesthe adjoint ofA, , and satisfies .If , then thesequence converges strongly toa point .
Proof It follows from Lemma 3.2 that , is nonempty, closed and convex and is a firmly nonexpansive mapping. Hence all conditions in Corollary 2.3 aresatisfied. The conclusion of Theorem 3.3 can be directly obtained fromCorollary 2.3. □
Let denote the solution setof the split equilibrium problem SEP.
Let , and . Let for all R, then A is a bounded linear operator. Let and bedefined by and , respectively. Clearly, and . So .
Let with the standard norm and with the norm for some . and . Define a bi-function ,where , , thenF is a bi-function from intoR with . For each ,let ,then A is a bounded linear operator from into .In fact, it is also easy to verify that and for some and .Now define another bi-function G as follows: for all .Then G is a bi-function from intoR with .
Clearly, when , we have . So .
where denotesthe adjoint ofA, , and satisfies .If , then thesequence converges strongly toa point .
Remark 3.7 Since Example 3.4 and Example 3.5 satisfy the conditionsof Corollary 2.3, the split equilibrium problems in Example 3.4 andExample 3.5 can be solved by algorithm (3.4).
Application to the hierarchial variational inequality problem
Let H be a real Hilbert space, and be two nonexpansive mappings from H to H such that and .
Hence from Theorem 2.1 we have the following theorem.
where and satisfies .If , then thesequence converges strongly toa solution of the hierarchical variational inequality problem (3.5).
The authors would like to express their thanks to the reviewers and editors for theirhelpful suggestions and advice. This work was supported by the National NaturalScience Foundation of China (Grant No. 11361070) and the Scientific ResearchFoundation of Postgraduate of Yunnan University of Finance and Economics.
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