Alternating mann iterative algorithms for the split common fixed-point problem of quasi-nonexpansive mappings
© Zhao and He; licensee Springer. 2013
Received: 29 May 2013
Accepted: 25 September 2013
Published: 8 November 2013
Very recently, Moudafi (Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal. ) introduced an alternating CQ-algorithm with weak convergence for the following split common fixed-point problem. Let , , be real Hilbert spaces, let , be two bounded linear operators.
where and are two firmly quasi-nonexpansive operators with nonempty fixed-point sets and . Note that by taking and , we recover the split common fixed-point problem originally introduced in Censor and Segal (J. Convex Anal. 16:587-600, 2009) and used to model many significant real-world inverse problems in sensor net-works and radiation therapy treatment planning. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce two alternating Mann iterative algorithms and prove the weak convergence of algorithms. At last, we provide some applications. Our results improve and extend the corresponding results announced by many others.
MSC:47H09, 47H10, 47J05, 54H25.
Keywordssplit common fixed-point problem quasi-nonexpansive mapping weak convergence Mann iterative algorithm Hilbert space
where is a bounded linear operator. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving  for modeling inverse problems which arise from phase retrievals and in medical image reconstruction . The SFP attracts many authors’ attention due to its application in signal processing. Various algorithms have been invented to solve it (see [3–12] and references therein).
That is, solves the SFP (1.1) if and only if solves the fixed point equation (1.2) (see  for the details). This implies that we can use fixed point algorithms (see [6, 13–15]) to solve SFP. A popular algorithm that solves the SFP (1.1) is due to Byrne’s CQ algorithm , which is found to be a gradient-projection method (GPM) in convex minimization. Subsequently, Byrne  applied KM iteration to the CQ algorithm and Zhao  applied KM iteration to the perturbed CQ algorithm to solve the SFP.
which allows asymmetric and partial relations between the variables x and y. The interest is to cover many situations, for instance, in decomposition methods for PDEs, applications in game theory and in intensity-modulated radiation therapy (IMRT). In decision sciences, this allows to consider agents who interplay only via some components of their decision variables (see ). In (IMRT), this amounts to envisaging a weak coupling between the vector of doses absorbed in all voxels and that of the radiation intensity (see ). If and , then the convex feasibility problem (1.3) reduces to the split feasibility problem (1.1).
where , and are the spectral radiuses of and , respectively.
where , , and , with λ being the spectral radius of the operator . Moudafi proved weak convergence result of the algorithm in Hilbert spaces.
for firmly quasi-nonexpansive operators U and T. Moudafi  obtained the following result.
Theorem 1.1 Let , , be real Hilbert spaces, let , be two firmly quasi-nonexpansive operators such that , are demiclosed at 0. Let , be two bounded linear operators. Assume that the solution set Γ is nonempty, is a positive non-decreasing sequence such that , where , stand for the spectral radiuses of and , respectively. Then the sequence generated by (1.9) weakly converges to a solution of (1.8). Moreover, , , and as .
In this paper, inspired and motivated by the works mentioned above, firstly, we introduce the following alternating Mann iterative algorithm for solving the SCFP (1.8) for the general class of quasi-nonexpansive operators.
By taking , we recover (1.8) clearly the classical SCFP (1.6). In addition, if and in Algorithm 1.1, we have and . Thus, Algorithm 1.1 reduces to and , which is algorithm (1.7) proposed by Moudafi .
The CQ algorithm is a special case of the K-M algorithm. Due to the fixed point formulation (1.4) of the CFP (1.3), we can apply the K-M algorithm to obtain the following other alterative Mann iterative sequence for solving the SCFP (1.8) for quasi-nonexpansive operators.
The organization of this paper is as follows. Some useful definitions and results are listed for the convergence analysis of the iterative algorithm in Section 2. In Section 3, we prove the weak convergence of the alternating Mann iterative Algorithms 1.1 and 1.2. At last, we provide some applications of Algorithms 1.1 and 1.2.
A mapping belongs to the general class of (possibly discontinuous) quasi-nonexpansive mappings if
A mapping belongs to the set of nonexpansive mappings if
A mapping belongs to the set of firmly nonexpansive mappings if
A mapping belongs to the set of firmly quasi-nonexpansive mappings if
It is easily observed that and that . Furthermore, is well known to include resolvents and projection operators, while contains subgradient projection operators (see, for instance,  and the reference therein).
A mapping is called demiclosed at the origin if, for any sequence which weakly converges to x, and if the sequence strongly converges to 0, then .
In what follows, we give some key properties of the relaxed operator which will be needed in the convergence analysis of our algorithms.
Lemma 2.1 ()
Remark 2.2 Let , where is a quasi-nonexpansive mapping and . We have and . It follows from (ii) of Lemma 2.1 that , which implies that is firmly quasi-nonexpansive when . On the other hand, if is a firmly quasi-nonexpansive mapping, we can obtain , where T is quasi-nonexpansive. This is proved by the following inequalities.
where is firmly quasi-nonexpansive.
Lemma 2.3 ()
3 Convergence of the alternating Mann iterative Algorithms 1.1 and 1.2
Theorem 3.1 Let , , be real Hilbert spaces. Given two bounded linear operators , , let and be quasi-nonexpansive mappings with nonempty fixed point set and . Assume that , are demiclosed at origin, and the solution set Γ of (1.8) is nonempty. Let be a positive non-decreasing sequence such that , where , stand for the spectral radiuses of and , respectively, and ε is small enough. Then, the sequence generated by Algorithm 1.1 weakly converges to a solution of (1.8), provided that and for small enough . Moreover, , and as .
which ensures that both sequences and are bounded thanks to the fact that converges to a finite limit.
Hence and , this implies that the whole sequence weakly converges to a solution of problem (1.8), which completes the proof. □
Remark 3.2 Taking in Algorithm 1.1, it follows from Remark 2.2 that Theorem 3.1 becomes Theorem 1.1, which is proved by Moudafi .
Theorem 3.3 Let , , be real Hilbert spaces. Given two bounded linear operators , , let and be quasi-nonexpansive mappings with nonempty fixed point set and . Assume that , are demiclosed at origin, and the solution set Γ of (1.8) is nonempty. Let be a positive non-decreasing sequence such that , where , stand for the spectral radiuses of and , respectively, and ε is small enough. Then the sequence generated by Algorithm 1.2 weakly converges to a solution of (1.8), provided that is an non-increasing sequence such that for small enough . Moreover, , and as .
Proof Taking , i.e., ; and . By repeating the proof of Theorem 3.1, we have that (3.6) is true.
that is asymptotically regular. Similarly, and is asymptotically regular, too.
The rest of the proof is analogous to that of Theorem 3.1. □
We now turn our attention to providing some applications relying on some convex and nonlinear analysis notions, see, for example, .
4.1 Convex feasibility problem (1.3)
Taking and , we have the following alterative Mann iterative algorithms for CFP (1.3).
4.2 Variational problems via resolvent mappings
and the algorithms take the following equivalent form.
The research was supported by the Fundamental Research Funds for the Central Universities (Program No. 3122013k004), it was also supported by the science research foundation program in the Civil Aviation University of China (2012KYM04). The authors would also like to thank the referees for careful reading of the manuscript.
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