1 Introduction and preliminaries

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 .

The split feasibility problem (SFP) is to find a point with the property

(1.1)

where is a bounded linear operator.

Assuming that SFP (1.1) is consistent (i.e., (1.1) has a solution), itis not hard to see that solves(1.1) if and only if it solves the following fixed point equation:

(1.2)

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 [1] for modeling inverse problems whicharise from phase retrievals and in medical image reconstruction [2]. 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].

Let and be two mappings satisfying and,respectively; let be a bounded linear operator. The split common fixed point problem (SCFP) formappings S and T is to find a point with the property

(1.3)

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) [5].

Split feasibility problems and split common fixed point problems have been studied bysome authors [8–15]. In 2010,Moudafi [10] proposed the following iterationmethod to approximate a split common fixed point of demi-contractive mappings: forarbitrarily chosen ,

and he proved that converges weakly to a split common fixedpoint , whereand are two demi-contractive mappings, is a bounded linear operator.

Using the iterative algorithm above, in 2011, Moudafi [9] 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 [16], asymptotically quasi-nonexpansive mappings[12], κ-asymptoticallystrictly pseudononspreading mappings [17],asymptotically strictly pseudocontraction mappings [18]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?

In this paper, we introduce the following iterative scheme. Let ,,the sequence is defined as follows:

(1.4)

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 andwithout the assumption of semi-compactness on and.

The following lemma and results are useful for our proofs.

Lemma 1.1[19]

LetEbe a real uniformly convex Banach space, Kbea nonempty closed subset ofE, and letbe an asymptoticallynonexpansive mapping. Thenisdemiclosed at zero, that is, ifconvergesweakly to a pointand,then.

Let C be a closed convex subset of a real Hilbert space H.denotes the metric projection of H onto C. It is well known that ischaracterized by the properties: for and,

(1.5)

and

(1.6)

In a real Hilbert space H, it is also well known that

(1.7)

and

(1.8)

2 Main results

Theorem 2.1Letandbe twoHilbert spaces, bea bounded linear operator, bean asymptotically nonexpansive mapping with the sequencesatisfying,andbean asymptotically nonexpansive mapping with the sequencesatisfying,and, respectively.Let,,and let the sequencebe defined as follows:

(2.1)

wheredenotesthe adjoint ofA, andsatisfies, , .If, thenconverges stronglyto.

Proof We will divide the proof into five steps.

Step 1. We first show that is closedand convex for any .

Since ,so isclosed and convex. Assume that is closedand convex. For any ,since

we know that is closed and convex. Therefore is closedand convex for any .

Step 2. We prove for any .

Let , then from (2.1) we have

(2.2)

where

(2.3)

Substituting (2.3) into (2.2), we can obtain that

(2.4)

In addition, it follows from (2.1) that

(2.5)

Therefore, from (2.4) and (2.5), we know that and for any .

Step 3. We will show that is a Cauchy sequence.

Since and ,then

(2.6)

It means that is bounded. For any , by using(1.6), we have

which implies that . Thus is nondecreasing. Therefore, by the boundednessof , exists. For somepositive integers m, n with , fromand (1.6), we have

(2.7)

Since exists, it followsfrom (2.7) that . Therefore is a Cauchy sequence.

Step 4. We will show that .

Since ,we have

(2.8)
(2.9)
(2.10)

Notice that , it follows from (2.4)that

thus, since is bounded and ,from (2.8) we have

(2.11)

On the other hand, since

we have

Since and ,we know that

(2.12)

In addition, since , we know that. So from

we can obtain that

(2.13)

Similarly, we have

(2.14)

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.

Corollary 2.2Letbe aHilbert space, bean asymptotically nonexpansive mapping with a sequencesatisfying.The sequenceis defined as follows:,

(2.15)

whereandsatisfies.If, thenconverges strongly to a fixedpointofT.

In Theorem 2.1, when and aretwo nonexpansive mappings, the following result holds.

Corollary 2.3Letandbe twoHilbert spaces, bea bounded linear operator, andbetwo nonexpansive mappings such thatand, respectively.Let,,and let the sequencebe defined as follows:

(2.16)

wheredenotesthe adjoint ofA, andsatisfies.If, thenconverges stronglyto.

Remark 2.4 When and aretwo quasi-nonexpansive mappings and and are demiclosed at zero, Corollary 2.3 also holds.

Example 2.5 Let C be a unit ball in a real Hilbert space,and let be amapping defined by

It is proved in Goebel and Kirk [20] that

  1. (i)

    , ;

  2. (ii)

    , , .

Taking ,, itis easy to see that .So we can take ,and ,,then

Therefore is anasymptotically nonexpansive mapping from C into itself with.

Let D be an orthogonal subspace of with thenorm and the inner product for and . For each, we define amapping by

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.

3 Applications and examples

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 pointsuch that for all.The set of its solutions is denoted by .

Lemma 3.1[21]

LetCbe a nonempty closed convex subset of a HilbertspaceH, and letbe a bifunctionsatisfying (A1)-(A4). Letand.Then there existssuchthat

Lemma 3.2[21]

Assume thatsatisfies(A1)-(A4). Forand,define a mappingas follows:

Then

  1. (1)

    is single-valued;

  2. (2)

    is firmly nonexpansive, that is, for all,

  3. (3)

    ;

  4. (4)

    is nonempty, closed and convex.

Theorem 3.3Letandbe twoHilbert spaces, bea bounded linear operator, bea nonexpansive mapping, be a bifunctionsatisfying (A1)-(A4). Assume thatand.Taking,for arbitrarily chosen,the sequenceis defined asfollows:

(3.1)

wheredenotesthe adjoint ofA, ,andsatisfies.If, then thesequenceconverges strongly toa point.

Proof It follows from Lemma 3.2 that , is nonempty, closed and convex andis 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 and be two real Hilbert spaces. Let C be a closed convex subset of,K be a closed convex subset of ,be a bounded linear operator. Assume that F is a bi-function frominto R and G is a bi-function from intoR. The split equilibrium problem (SEP) is to

(3.2)

and

(3.3)

Let denote the solution setof the split equilibrium problem SEP.

Example 3.4[22]

Let , and . Letfor all R, then A is a bounded linear operator. Let and bedefined by and , respectively. Clearly, and. So .

Example 3.5[22]

Let with the standard norm andwith 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 andfor 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 .

Corollary 3.6Letandbe twoHilbert spaces, bea bounded linear operator, be a bifunctionsatisfyingandbe a bifunctionsatisfying.Taking,for arbitrarily chosen,the sequenceis defined asfollows:

(3.4)

wheredenotesthe adjoint ofA, ,andsatisfies.If, then thesequenceconverges 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, andbe two nonexpansive mappings from H to H such that and .

The so-called hierarchical variational inequality problem for nonexpansive mappingwith respect to a nonexpansive mapping isto find a point such that

(3.5)

It is easy to see that (3.5) is equivalent to the following fixed point problem:

(3.6)

where is the metric projection from H onto . Letting and (the fixed point set ofthe mapping )and (theidentity mapping on H), then problem (3.6) is equivalent to the followingsplit feasibility problem:

(3.7)

Hence from Theorem 2.1 we have the following theorem.

Theorem 3.8LetH, ,,CandQbe the same as above.Letand,and let the sequencebe defined asfollows:

(3.8)

whereandsatisfies.If, then thesequenceconverges strongly toa solution of the hierarchical variational inequality problem (3.5).