Research Article | Open Access
Solving Split Common Fixed-Point Problem of Firmly Quasi-Nonexpansive Mappings without Prior Knowledge of Operators Norms
Very recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms for the split common fixed-point problem concerned two bounded linear operators. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. It is the purpose of this paper to introduce a viscosity iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. We prove the strong convergence of the proposed algorithms for split common fixed-point problem governed by the firmly quasi-nonexpansive operators. As a consequence, we obtain strong convergence theorems for split feasibility problem and split common null point problems of maximal monotone operators. Our results improve and extend the corresponding results announced by many others.
Throughout this paper, we always assume that is a real Hilbert space with the inner product and the norm . Let denote the identity operator on . Let be a mapping. A point is said to be a fixed point of provided . In this paper, we use to denote the fixed point set of .
Let and be nonempty closed convex subsets of real Hilbert spaces and , respectively. The split feasibility problem (SFP) is to find a point as follows: 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 .
Note that if the split feasibility problem (1) is consistent (i.e., (1) has a solution), then (1) can be formulated as a fixed point equation by using the following fact: where and are the (orthogonal) projections onto and , respectively, is any positive constant, and denotes the adjoint of . That is, solves SFP (1) if and only if solves fixed point equation (2) (see  for the details). This implies that we can use fixed point algorithms (see [3–6]) to solve SFP. To solve (2), Byrne  proposed his CQ algorithm which generates a sequence by where with being the spectral radius of the operator .
Censor and Segal  introduced the following split common fixed-point problem (SCFP): where is a bounded linear operator and and are two nonexpansive operators with nonempty fixed-point sets and . SCFP 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 many areas (see ).
To solve (4), Censor and Segal  proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm: where , with being the largest eigenvalue of the matrix ( stands for matrix transposition).
Let , , and be real Hilbert spaces; let and be two bounded linear operators; let and be two firmly quasi-nonexpansive operators. In , Moudafi introduced the following split common fixed-point problem (SCFP): which allows asymmetric and partial relations between the variables and . The interest is to cover many situations, for instance in decomposition methods for PDE’s, in a 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, these amounts envisage a weak coupling between the vector of doses absorbed in all voxels and that of the radiation intensity (see ).
If and , then SCFP (6) reduces to SCFP (4). For solving SCFP (6), Moudafi  introduced the following alternating algorithm: for firmly quasi-nonexpansive operators and , where nondecreasing sequence and and stand for the spectral radius of and , respectively.
Very recently, Moudafi and Al-Shemas  introduced the following simultaneous iterative method to solve SCFP (6): for firmly quasi-nonexpansive operators and , where and and stand for the spectral radius of and , respectively.
Note that, in (7), (8), and (9) 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 above algorithms for solving SCFP (6), one has first to compute (or, at least, estimate) operator norms of and , 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 viscosity iterative algorithm to solve SCFP (6) governed by firmly quasi-nonexpansive operators as follows: The advantage of our choice (9) of the stepsizes lies in the fact that no prior information about the operator norms of and is required, and still convergence is guaranteed.
Some algorithms have been invented to solve SCFP (6) (see [16, 17] and references therein). In this paper, inspired and motivated by the works mentioned above, to get the strong convergence of the algorithm, we introduce the viscosity iterative algorithm without prior knowledge of operator norms for solving SCFP (6) governed by firmly quasi-nonexpansine 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, the strong convergence theorem of the proposed viscosity iterative algorithm is obtained. At last, we provide some applications.
In this paper, we use and to denote the strong convergence and weak convergence, respectively. We use to stand for the weak -limit set of and use to stand for the solution set of SCFP (6).
Definition 1. An operator is said to be(i)nonexpansive if for all ,(ii)quasi-nonexpansive if and if for all and ,(iii)firmly nonexpansive if for all ,(iv)firmly quasi-nonexpansive if and if for all and .
Remark 2. A firmly quasi-nonexpansive operator is also called a separating operator , cutter operator , directed operators [7, 20], or class- operator which was introduced by Bauschke and Combettes . Firmly quasi-nonexpansive operators are important because they include many types of nonlinear operators arising in applied mathematics such as approximation and convex optimization. For instance, the subgradient projection of a continuous convex function is a firmly quasi-nonexpansive operator. Recall that the subgradient projection is defined by, assuming that the level set , where is a selection of the subdifferential (i.e., for all ).
Particularly, projections are firmly quasi-nonexpansive operators. Recall that, given a closed convex subset of a Hilbert space , the projection assigns each to its closest point from defined by It is well known that is characterized by the inequality
We also need other classes of operators.
Definition 4. An operator called demiclosed at the origin if whenever the sequence converges weakly to and the sequence converges strongly to 0, then .
We remark here that a firmly quasi-nonexpansive operator may be not nonexpansive; even is demiclosed at origin. See the following example .
Example 5. Let and define a mapping by by Then, and So, is firmly quasi-nonexpansive but not nonexpansive. It is easy to see that is demiclosed at origin.
Definition 6. An operator is called contraction with constant if, for any ,
In real Hilbert space, we easily get the following equality:
We end this section by the following lemmas, which are important in convergence analysis for our iterative algorithm.
Lemma 7 (see ). Assume is a sequence of nonnegative real numbers such that where is a sequence in , is a sequence of nonnegative real numbers, and and are two sequences in such that(i);(ii);(iii) implies for any subsequence .Then, .
Lemma 8 (see [24, Lemma 1.3]). Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of which satisfies for all . Also consider the sequence of integers defined by Then, is a nondecreasing sequence verifying and, for all , it holds that and we have
3. Viscosity Iterative Algorithm without Prior Knowledge of Operator Norms
In this section, we introduce a viscosity iterative algorithm where the stepsizes do not depend on the operator norms and and prove the strong convergence of algorithm without prior knowledge of operator norms.
Algorithm 9. Let and be two contractions with constants , , and . Choose an initial guess , arbitrarily. Assume that the th iterate , has been constructed; then, we calculate the ()th iterate via the formula The stepsize is chosen in such a way that otherwise, ( being any nonnegative value), where the set of indexes .
Remark 10. Note that, in (22), the choice of the stepsize is independent of the norms and . The value of does not influence the considered algorithm, but it was introduced just for the sake of clarity. Furthermore, we will see from Lemma 3 that is well defined.
Proof. Taking , that is, , , 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 that is well defined.
Theorem 12. Let , , and be real Hilbert spaces. Given two bounded linear operators and , let and be firmly quasi-nonexpansive operators with the solution set of (6) being nonempty. Let the sequence be generated by Algorithm 9. Assume that the following conditions are satisfied:(1);(2) and .(3) and are demiclosed at origin;
Then, sequence strongly converges to a solution of (6) which solves the variational inequality problem
Proof. Let be the solution of the variational inequality problem (25). Then, , , and . We have
Using (17), we have
By (26) and (27), we obtain
Similarly, we have
By adding the two last inequalities and by taking into account the fact that , we obtain
With assumption on , we obtain
Setting , we have . By and being firmly quasi-nonexpansive operators, it follows that
Adding up the last two inequalities and using (31), setting , we get
It follows from induction that
which implies that and are bounded. It follows that and are bounded.
Note that is a firmly quasi-nonexpansive operator; we have Similarly, we have So, by (31), (36), and (37), we obtain where
On the other hand, from (21), we have Adding up the last two inequalities and using (30), we obtain Now, by setting , , and , (41) can be rewritten as the following form: By the assumption on , we get and which thanks to the boundedness of and .
The rest of the proof will be divided into two parts.
Case 1. Suppose that there exists such that is nonincreasing. In this situation, is convergent because it is nonnegative so that ; hence, in light of (33) together with and the boundedness of , we obtain
To use Lemma 8, it suffices to verify that, for all subsequences , implies It follows from that which yields from the assumption on . So, Taking , from (46), we have . Combined with the demiclosednesses of and at , (43) yields and . So, and . On the other hand, and weakly lower semicontinuity of the norm imply hence, . So, . Since and , to get (44), we only need to verify Indeed, from (43) and (46), we have We can take subsequence of such that as and Since and is the solution of the variational inequality problem (25), from (49) and (50), we obtain From Lemma 8, it follows which implies that and .
Case 2. Suppose there exists a subsequence of such that for all . In this situation, we consider the sequence of indices as defined in Lemma 8. It follows that . From (42), we have So, by , we obtain Again from (42), we get hence, In light of , we obtain From , similar to Case 1, we have , and which implies From and (38), it follows that Since , again from (38), we may assume for all . It follows from (60) and (61) that and hence On the other hand, it follows that which, by , (57), and (58), implies that By (62), we obtain Similarly, we have ; hence, Then, recalling that (by Lemma 8), we get .
So, sequence strongly converges to the solution of (6) which solves the variational inequality problem (25).
4. Another Split Problem Deduced from SCFP
We now turn our attention to providing some algorithms for solving another split problem without prior knowledge of operator norms.
4.1. Split Feasibility Problem
Taking and , we have that the following viscosity iterative algorithm for split feasibility problem (SFP) under consideration is nothing but
In , Dong et al. introduced Algorithm 13 for SFP (67) without prior knowledge of operator norms. The stepsize is chosen in such a way that It is easy to see that the results of this paper improve and extend the corresponding results of .
4.2. Split Common Null Point Problem
Given a maximal monotone operator , it is well known that its associated resolvent mapping, , is firmly quasi-nonexpansive and . In other words, zeroes of are exactly fixed-points of its resolvent mapping. By taking , , where is another maximal monotone operator, the problem under consideration is nothing but and the algorithms take the following equivalent form: The stepsize is chosen as follows: otherwise, ( being any nonnegative value), where the set of indexes .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research was supported by Fundamental Research Funds for the Central Universities (Program no. 3122013k004).
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Copyright © 2014 Jing Zhao and Hang Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.