1 Introduction

Due to their extraordinary utility and broad applicability in many areas of appliedmathematics (most notably, fully discretized models of problems in imagereconstruction from projections, in image processing, and in intensity-modulatedradiation therapy), algorithms for solving convex feasibility problems continue toreceive great attention; see for instance [1–11]. Recently, Moudafi [12] introduced a new convex feasibility problem (CFP). Let, , be real Hilbert spaces, let , be two nonempty closed convex sets, let, be two bounded linear operators. The convexfeasibility problem in [12] is to find

(1.1)

which allows asymmetric and partial relations between the variables x andy. The interest is to cover many situations, for instance indecomposition methods for PDEs, applications in game theory and inintensity-modulated radiation therapy (IMRT). In decision sciences, this allows oneto consider agents who interplay only via some components of their decisionvariables, for further details, the interested reader is referred to [13]. In IMRT, this amounts to envisage a weak coupling between the vector ofdoses absorbed in all voxels and that of the radiation intensity, for furtherdetails, the interested reader is referred to [13, 14].

For solving the CFP (1.1), Moudafi [12] studied the fixed point formulation of the solutions of the CFP (1.1).Assume that the CFP (1.1) is consistent (i.e., (1.1) has a solution), if solves (1.1), then it solves the following fixedpoint equation system:

(1.2)

where are any positive constants, and then Moudafiintroduced the following alternating CQ algorithm:

(1.3)

where , and are the spectral radii of and , respectively. The weak convergence of the sequence to a solution of (1.1) under some conditions wasproved.

In [15], Moudafi and Al-Shemas considered the following problem:

(1.4)

and proposed the following simultaneous algorithm:

(1.5)

for firmly quasi-nonexpansive operators U and T, where, and are the spectral radiuses of and , respectively.

Observe that in the algorithms (1.3) and (1.5) mentioned above, the determination ofthe stepsize depends on the operator (matrix) norms and (or the largest eigenvalues of and ). To implement the alternating algorithm (1.3) andthe simultaneous algorithm (1.5), one has first to compute (or, at least, estimate)operator norms of A and B, which is in general not easy inpractice.

To overcome this difficulty, Lopez et al.[16] and Zhao et al.[17] presented useful method for choosing the stepsizes which do not needprior knowledge of the operator norms for solving the split feasibility problems andmultiple-set split feasibility problems, respectively.

Motivated by above results, we introduce a new choice of the stepsize sequence for the simultaneous iterative algorithm to solve(1.4) governed by quasi-nonexpansive mapping as follows:

(1.6)

The advantage of our choice (1.6) of the stepsizes lies in the fact that no priorinformation about the operator norms of A and B is required, andstill convergence is guaranteed.

In this article, we propose the following simultaneous iterative algorithm where thestepsizes do not depend on the operator norms and and prove the weak convergence of the algorithm tosolve (1.4). Let and be two quasi-nonexpansive mappings which are definedby (2.5). We denote by Γ be the set of solutions of (1.4), i.e.,

Algorithm 1.1 Let , be arbitrary and be real number sequences in . Assume that the kth iterate, has been constructed and , then we calculate th iterate via the formula

(1.7)

where the stepsize is chosen by (1.6). If , then is a solution of the problem (1.4) and the iterativeprocess stops. Otherwise, we set and go on to (1.7) to evaluate the next iterate.

Remark 1.1 Notice that in (1.6) the choice of the stepsize is independent of the norms and .

2 Preliminaries

Throughout this paper, we denote by H be a real Hilbert space with innerproduct and induced norm , and denote by C be a nonempty closed convexsubset of H. Let be a mapping. A point is said to be a fixed point of T provided. we use to denote the fixed point set. We write to indicate that the sequence converges weakly to x, implies that converges strongly to x. We use to stand for the weak ω-limit set of. For any , there exists a unique nearest point in C,denoted by , such that

Before proceeding, we need to introduce a few concepts.

A mapping belongs to the set of quasi-nonexpansive, if

(2.1)

A mapping belongs to the set of nonexpansive, if

(2.2)

A mapping belongs to the set of firmly nonexpansive, if

(2.3)

A mapping belongs to the set of firmly quasi-nonexpansive, if

(2.4)

A mapping is called nonspreading, if

A mapping is called k-strictly pseudononspreading ifthere exists such that

Remark 2.1 It is easy to see that and . Furthermore, is well known to contain resolvents and projectionoperators, and includes subgradient projection operators [18]. T is a nonspreading mapping if and only if T is a0-strictly pseudononspreading mapping.

The so-called demiclosedness principle plays an important role in our argument.

A mapping is called demiclosed at the origin if for anysequence which weakly converges to x, and if thesequence strongly converges to 0, then .

To establish our results, we need the following technical lemmas.

Lemma 2.1 ([19])

If, then:

  1. (a)

    .

  2. (b)

    For any,

  3. (c)

    Forwith,

The following definition will be useful for our results.

In 2009, Kangtunyakarn and Suantai [20] introduced T-mapping generated by and as follows.

Definition 2.1 Let C be a nonempty convex subset of real Banachspace. Let be a finite family of mappings of C intoitself, and let be real numbers such that for every . We define a mapping as follows:

(2.5)

Such a mapping T is called the T-mapping generated by and .

Using the above definition, we have the following important lemma.

Lemma 2.2LetCbe a nonempty convex subset of real Banach space. Letbe a finite family of-strictly pseudononspreading mappings ofCinto itself with, and letbe real numbers such thatfor every. IfTis theT-mapping generated byand, thenandTis a quasi-nonexpansive mapping.

Proof It is easy to deduce that . Next, we claim that . Let and . Assume that , for , it follows from being a finite family of -strictly pseudononspreading mappings of Cinto itself that

(2.6)

From the definition of T and (2.6), we have

(2.7)

which means , that is, . Furthermore,

it yields . Applying the same argument, we can conclude that and , for .

Next, we claim that . Indeed,

It follows that . Therefore, , that is, . Hence, . From the definition of T and (2.7), we findthat T is a quasi-nonexpansive mapping. □

Proposition 2.1LetCbe a closed convex subset of a real Hilbert spaceH. IfTis a quasi-nonexpansive mapping fromCinto itself, thenis closed and convex.

Proof Obviously, the continuity of T implies that is closed. Now, we show that is convex. For and , put . Now, we claim that . In fact,

which means that . Hence, and is convex. □

3 Main results

Now, we are in a position to prove our convergence results in this section.

Theorem 3.1Let, , be real Hilbert spaces. Given two bounded linear operators, . Letbe a finite family of-strictly pseudononspreading mappings ofCinto itself with, and Letbe a finite family of-strictly pseudononspreading mappings ofQinto itself with. Suppose thatUis defined by (2.5) which is generated byand, withfor every, and suppose thatTis defined by (2.5) which is generated byand, withfor every, respectively. Assume thatandare demiclosed at the origin. If the solution set Γof (1.4) is nonempty and for small enoughand,

then the sequencegenerated by Algorithm 1.1 weakly converges to a solutionof (1.4). Moreover, , , andas.

Proof It follows from the condition on that

(3.1)

and

(3.2)

On the other hand, from

and

we obtain and . Furthermore,

Inequalities (3.1) and (3.2) lead to and is bounded.

For , by Algorithm 1.1, we obtain

(3.3)

Notice that

(3.4)

Substituting (3.4) into (3.3), one has

(3.5)

Similarly, by Algorithm 1.1, we deduce

(3.6)

Furthermore, adding the two last inequalities, following from the fact, we have

(3.7)

Next, we will estimate and . It follows from U and T being twoquasi-nonexpansive mappings that

(3.8)

and

(3.9)

Thus, (3.8) and (3.9) lead to

(3.10)

Furthermore, it follows from (3.7) that

(3.11)

Now, setting , one has

(3.12)

On the other hand, note that

From the assumptions on and , we see that the sequence being decreasing and lower bounded by 0,consequently, converges to some finite limit, that is, , which means the sequences and are bounded. Thus, we have

(3.13)
(3.14)

and

(3.15)

Now, we show that . Indeed, as is shown below, we break up the proof bydistinguishing two cases.

Case 1. Suppose that there exists such that , for all , we obtain . It yields

Furthermore, (3.13) leads to

(3.16)

Since

and

we deduce

Conversely, suppose that there exists such that , for all , following the above process, we obtain theresults.

Case 2. Suppose that there does not exist such that

or

for all . We can divide the sequence into two sequences: one satisfies, which is denoted by and the other sequence satisfies, which is denoted by . Following the process of Case 1, we show that theresults hold for the subsequences with and . Thus, we obtain .

Let us prove that and are asymptotically regular. Indeed, since

one has

(3.17)

Consequently,

which yields is asymptotically regular. Similarly, and is asymptotically regular, too.

Next, we show that and as . Indeed, since

(3.15) and (3.17) mean that . In the same way as above, we can also show that as .

Taking , from and , we obtain and . Combining with the demiclosednesses of and at 0, one has

which yields and . Thus, and . On the other hand, and lower semicontinuity of the norm imply that

hence .

Finally, we will show the uniqueness of the weak cluster points of and . Indeed, let , be other weak cluster points of and , respectively. From the definition of, we have

(3.18)

Without loss of generality, we may assume that , , and then

(3.19)

Reversing the role of and , we obtain

(3.20)

Equations (3.19) and (3.20) yield

which means and . Hence, the sequence weakly converges to a solution of the problem (1.4),which completes the proof. □

The following conclusions can be obtained from Theorem 3.1 immediately.

Theorem 3.2Let, , be real Hilbert spaces. Given two bounded linear operators, . LetUbe aρ-strictly pseudononspreading mapping ofCinto itself and LetTbe aτ-strictly pseudononspreading mapping ofQinto itself. Assume thatandare demiclosed at the origin. If the solution set Γof (1.4) is nonempty and for small enoughand,

then the sequencegenerated by Algorithm 1.1 weakly converges to a solutionof (1.4). Moreover, , , andas.

Theorem 3.3Let, , be real Hilbert spaces. Given two bounded linear operators, . LetUbe a nonspreading mapping ofCinto itself and letTbe a nonspreading mapping ofQinto itself. Assume thatandare demiclosed at the origin. If the solution set Γof (1.4) is nonempty and for small enoughand,

then the sequencegenerated by Algorithm 1.1 weakly converges to a solutionof (1.4). Moreover, , , andas.

4 Applications

We now pay attention to applying our simultaneous iterative algorithms to some convexand nonlinear analysis notions; see, for example, [21].

4.1 Split feasibility problem

Let C and Q be nonempty closed convex subset of real Hilbertspaces and , respectively. The split feasibility problem(SFP) is to find a point

(4.1)

where is a bounded linear operator. The SFP was firstintroduced by Censor and Elfving [22] for modeling inverse problems which arise from phase retrievals andin medical image reconstruction [23].

If , , then Algorithm 1.1 becomes:

Algorithm 4.1

(4.2)

where the stepsize is chosen by (1.6). If , then is a solution of the problem (4.1) and theiterative process stops. Otherwise, we set and go on to (4.1) to evaluate the next iterate.

Furthermore, if and , then we obtain the following simultaneousiterative algorithm for solving SFP (4.1).

Algorithm 4.2

(4.3)

where the stepsize is chosen by (1.6). If , then is a solution of the problem (4.1) and theiterative process stops. Otherwise, we set and go on to (4.3) to evaluate the next iterate.

4.2 Variational problems via resolvent mappings

Given a maximal monotone operator , it is well known that its associated resolventmapping, , is quasi-nonexpansive and, which implies that zeroes of M areexactly fixed points of its resolvent mapping. If and , where is another maximal monotone operator, the problemunder consideration is nothing but

(4.4)

and the algorithm is applied to the following form.

Algorithm 4.3

(4.5)

where the stepsize is chosen by (1.6). If , then is a solution of the problem (4.4) and theiterative process stops. Otherwise, we set and go on to (4.5) to evaluate the next iterate.