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Simultaneous iterative algorithms for the split common fixed-point problem of generalized asymptotically quasi-nonexpansive mappings without prior knowledge of operator norms
Fixed Point Theory and Applications volume 2014, Article number: 73 (2014)
Let , , be real Hilbert spaces, let , be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms (Trans. Math. Program. Appl. 1:1-11, 2013) with weak convergence for the following split common fixed-point problem:
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). In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of generalized asymptotically quasi-nonexpansive mappings. To estimate the norm of an operator is a very difficult, if it is not an impossible task. The purpose of this paper is to propose a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information as regards the operator norms.
MSC:47H09, 47H10, 47J05, 54H25.
1 Introduction and preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with the inner product and the norm . Let be a mapping. A point is said to be a fixed point of T provided . In this paper, we use to denote the fixed point set and use → and ⇀ to denote the strong convergence and weak convergence, respectively. We use stand for the weak ω-limit set of .
Let C and Q be nonempty closed convex subset of real Hilbert spaces and , respectively. The split feasibility problem (SFP) is to find a point
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 . Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph and radiation therapy treatment planning [3–5]. Various algorithms have been invented to solve it, etc. (see [6–9] and references therein).
Note that if the split feasibility problem (1.1) is consistent (i.e., (1.1) has a solution), then (1.1) can be formulated as a fixed point equation by using the fact
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 [10–13]) 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.
In , Censor and Segal consider the following split common fixed-point problem (SCFP):
where is a bounded linear operator, and are two nonexpansive operators with nonempty fixed-point sets. To solve (1.3), Censor and Segal  proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm:
where , with λ being the largest eigenvalue of the matrix ( stands for matrix transposition). SCFP (1.3) is in itself at the core of the modeling of many inverse problems in various areas of mathematics and physical sciences and has been used to model significant real-world inverse problems in sensor networks, in radiation therapy treatment planning, in resolution enhancement, in wavelet-based denoising, in antenna design, in computerized tomography, in materials science, in watermarking, in data compression, in magnetic resonance imaging, in holography, in color imaging, in optics and neural networks, in graph matching, etc. (see [15, 16]).
Let , , be real Hilbert spaces, let , be two bounded linear operators, let and be two firmly quasi-nonexpansive operators. In , Moudafi introduced the following split common fixed-point problem:
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 SCFP (1.4) reduces to SCFP (1.3).
For solving the SCFP (1.4), Moudafi  introduced the following alternating algorithm:
for firmly quasi-nonexpansive operators U and T, where non-decreasing sequence , , stand for the spectral radiuses of and , respectively.
Very recently, Moudafi  introduced the following simultaneous iterative method to solve SCFP (1.4):
for firmly quasi-nonexpansive operators U and T, where , , stand for the spectral radiuses of and , respectively.
Note that in the algorithms (1.5) and (1.6) mentioned above, the determination of the stepsize depends on the operator (matrix) norms and (or the largest eigenvalues of and ). In order to implement the alternating algorithm (1.5) and the simultaneous algorithm (1.6) for solving SCFP (1.4), one has first to compute (or, at least, estimate) operator norms of A and B, which is in general not an easy work in practice. To overcome this difficulty, López et al.  and Zhao and Yang  presented a helpful method for estimating the stepsizes which do not need prior knowledge of the operator norms for solving the split feasibility problems and multiple-set split feasibility problems, respectively. Inspired by them, in this paper, we introduce a new choice of the stepsize sequence for the simultaneous iterative algorithm to solve SCFP (1.4) governed by generalized asymptotically quasi-nonexpansive operators as follows:
The advantage of our choice (1.7) of the stepsizes lies in the fact that no prior information as regards the operator norms of A and B is required, and still convergence is guaranteed.
Now let us recall some definitions, notations and conclusions which will be needed in proving our main results.
Definition 1.1 A mapping is called demiclosed at the origin if, for any sequence which weakly converges to x, if the sequence strongly converges to 0, then .
Definition 1.2 (1) A mapping is said to be nonexpansive if
(2) A mapping is said to be firmly nonexpansive if
(3) A mapping is said to be quasi-nonexpansive if such that
(4) A mapping is said to be firmly quasi-nonexpansive if such that
(5) A mapping is said to be asymptotically nonexpansive if there exists a nonnegative real sequence with such that for each ,
(6) A mapping is said to be asymptotically quasi-nonexpansive if and there exists a nonnegative real sequence with such that for each ,
(7) A mapping is said to be generalized asymptotically quasi-nonexpansive mapping with if , and there exist nonnegative real sequences , with and such that for each ,
It is clear from this definition that every firmly nonexpansive mapping is nonexpansive, nonexpansive mapping with a fixed point is quasi-nonexpansive, each firmly quasi-nonexpansive mapping is quasi-nonexpansive, and each quasi-nonexpansive mapping is asymptotically quasi-nonexpansive. We remark that an asymptotically nonexpansive mapping with a nonempty fixed point set is an asymptotically quasi-nonexpansive mapping, but the converse may be not true. The class of generalized asymptotically quasi-nonexpansive mappings is more general than the class of asymptotically quasi-nonexpansive mappings and asymptotically nonexpansive mappings. The following example shows that the inclusion is proper. Let and define (see ) if and if . Then uniformly but T is not Lipschitzian. Notice that . For each fixed k, define for . Set . Then and
This shows that T is a generalized asymptotically quasi-nonexpansive but it is not asymptotically quasi-nonexpansive and asymptotically nonexpansive because it is not Lipschitzian.
Definition 1.3 A mapping is said to be uniformly L-Lipschitzian if there exists a constant such that for each ,
In real Hilbert space, we easily get the following equality:
In what follows, we give some preliminary results needed for the convergence analysis of our algorithms.
Lemma 1.4 ()
Let H be a real Hilbert space. Then for all and ,
Lemma 1.5 ()
Let , and be sequences of nonnegative real numbers satisfying
If and , then the limit exists.
2 Simultaneous iterative algorithm without prior knowledge of operator norms
In this section we introduce a simultaneous iterative algorithm where the stepsizes don‘t depend on the operator norms and and prove weak convergence of the algorithm to solve SCFP (1.4) governed by generalized asymptotically quasi-nonexpansive operators.
We always assume that , , are real Hilbert spaces and , are two bounded linear operators. Let and be two generalized asymptotically quasi-nonexpansive mappings with . In the sequel, we use Γ to denote the set of solutions of SCFP (1.4), i.e.,
Algorithm 2.1 Let , be arbitrary and . Assume that the kth iterate , has been constructed; then we calculate the th iterate via the formula:
The stepsize is chosen in such a way that
otherwise, (γ being any nonnegative value), where the set of indices .
Remark 2.2 Note that in (2.2) the choice of the stepsize is independent of the norms and . The value of γ does not influence the considered algorithm, but it was introduced just for the sake of clarity. Furthermore, we will see from Lemma 2.3 that is well defined.
Lemma 2.3 If Γ is nonempty, then defined by (2.2) is well defined.
Proof Taking , i.e., , and . We have
By adding the two above equalities and by taking into account the fact that , we obtain
Consequently, for , that is, , we have or . This leads to the conclusion that is well defined. □
Theorem 2.4 Assume that , are demiclosed at origin, and U, T are uniformly L-Lipschitzian. Let the sequence be generated by Algorithm 2.1. Assume Γ is nonempty and for small enough ,
where . Then weakly converges to a solution of (1.4) provided that and for small enough . Moreover, and are asymptotically regular and .
Proof From the condition on , we have
On the other hand, from and we obtain is lower bounded by and so
It follows from (2.3) that and is bounded.
Taking , i.e., ; and . We have
Using equality (1.8), we have
By (2.4) and (2.5), we obtain
Similarly, by (2.1) we have
By adding the two last inequalities, and by taking into account fact that , we obtain
By the fact that U and T are generalized asymptotically quasi-nonexpansive mappings with , it follows from Lemma 1.4 that
So, by (2.8), we have
Now, by setting , we obtain the following inequality:
where and . By the condition , we have and . It follows from the condition on that
By Lemma 1.5, the following limit exists:
So the sequences and are bounded. Now, we rewrite (2.10) as follows:
It follows from the assumption
(Note that if .) So, we obtain
Similarly, by the conditions on , we obtain
Using the assumption on , (2.11), (2.12), and the convergence of we have
and is bounded, we have . It follows from that . So,
as , from which one infers that is asymptotically regular, namely . Noting
from (2.14) and (2.15) we have
Similarly, , and are asymptotically regular, too.
Next, we prove that and as . In fact, since U is uniformly L-Lipschitzian continuous, it follows from (2.13) and (2.16) that
Since T is uniformly L-Lipschitzian continuous, in the same way as above, we can also prove that as .
Taking , from and , we have and . Combined with the demiclosednesses of and at 0,
yields and . So, and . On the other hand, and weakly lower semicontinuity of the norm imply that
Finally, we will show the uniqueness of the weak cluster points of and . Indeed, let , be other weak cluster points of and , respectively, then . From the definition of , we have
Without loss of generality, we may assume that and . By passing to the limit in relation (2.17), we obtain
Reversing the role of and , we also have
By adding the two last equalities, we obtain and , which implies that the whole sequence weakly converges to a solution of problem (1.4). This completes the proof. □
The following conclusions can be obtained from Theorem 2.4 immediately.
Theorem 2.5 Let and be two asymptotically quasi-nonexpansive mappings with . Assume that , are demiclosed at origin, and U, T are uniformly L-Lipschitzian. Let the sequence be generated by Algorithm 2.1. Assume Γ is nonempty and for small enough ,
where . Then weakly converges to a solution of (1.4) provided that and for small enough . Moreover, and are asymptotically regular and .
Theorem 2.6 Let and be two quasi-nonexpansive mappings. Assume that , are demiclosed at origin, and U, T are uniformly L-Lipschitzian. Let the sequence be generated by Algorithm 2.1. Assume Γ is nonempty and for small enough ,
where . Then weakly converges to a solution of (1.4) provided that for small enough . Moreover, and are asymptotically regular and .
Remark 2.7 When , Algorithm 2.1 becomes
where the stepsize is chosen in such a way that
otherwise (γ being any nonnegative value), where the set of indices . This solves SCFP (1.3) for generalized asymptotically quasi-nonexpansive operators, asymptotically quasi-nonexpansive operators, quasi-nonexpansive operators, and firmly quasi-nonexpansive operators without prior knowledge of operator norm .
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The research was supported by the Fundamental Research Funds for the Central Universities (Program No. 3122013k004), it was also supported by the Fundamental Research Funds for the Central Universities (Program No. 3122013C002).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Zhao, J., He, S. Simultaneous iterative algorithms for the split common fixed-point problem of generalized asymptotically quasi-nonexpansive mappings without prior knowledge of operator norms. Fixed Point Theory Appl 2014, 73 (2014). https://doi.org/10.1186/1687-1812-2014-73
- split common fixed-point problem
- generalized asymptotically quasi-nonexpansive mappings
- weak convergence
- simultaneous iterative algorithm
- Hilbert space