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Damped projection method for split common fixed point problems
Journal of Inequalities and Applicationsvolume 2013, Article number: 123 (2013)
The paper deals with the split common fixed-point problem (SCFP) introduced by Censor and Segal. Motivated by Eicke’s damped projection method, we propose a cyclic iterative scheme and prove its strong convergence to a solution of SCFP under some mild assumptions. An application of the proposed method to multiple-set split feasibility problems is also included.
The split feasibility problem (SFP)  consists of finding an element satisfying
where C and Q are closed convex subsets in Hilbert spaces ℋ and , respectively. Moreover, if C and Q are the intersections of finitely many closed convex subsets, then the problem is known as the multiple-set split feasibility problem (MSFP) . Note that SFP and MSFP model image retrieval  and intensity-modulated radiation therapy , and they have recently been investigated by many researchers (see, e.g., [5–11]). One method for solving SFP is Byrne’s CQ algorithm : For any initial guess , define recursively by
where stands for the metric projection onto C, I is the identity operator on and λ is the step-size satisfying . By using Hundal’s counterexample, Xu  showed the CQ algorithm does not converge strongly in infinite-dimensional spaces. Motivated by Byrne’s CQ algorithm, Wang and Xu  proposed the following iterative method: For any initial guess , define recursively by
where satisfies ; ; either or . It is worth noting that this algorithm is in fact a generalization of Eicke’s damped projection method  for solving convexly constrained linear inverse problems (see ). Motivated by Krasnosel’skii-Mann’s iteration, Dang and Gao  proposed the following algorithm: For any initial guess , define recursively by
where satisfies (i) , ; (ii) ; (iii) . It is clear that such an algorithm is an extension of (3). However, algorithm (4) fails to include the original one (3) because of condition (iii).
In the case where C and Q in (1) are the intersections of finitely many fixed-point sets of nonlinear operators, problem (1) is called by Censor and Segal  the split common fixed-point problem (SCFP). More precisely, SCFP requires to seek an element satisfying
where , and denote the fixed point sets of two classes of nonlinear operators , and , . In this situation, Byrne’s CQ algorithm does not work because the metric projection onto fixed point sets is generally not easy to calculate. To solve the two-set SCFP, that is, in (5), Censor and Segal  proposed the following iterative method: For any initial guess , define recursively by
where is known as the step-size. They proved that if U and T in (6) are directed operators, then λ should be chosen in . Some further generations of this algorithm were studied by Moudafi  for demicontractive operators and by Wang-Xu  for finitely many directed operators.
We note that the existing algorithms for SCFP have only weak convergence in the framework of infinite-dimensional spaces (see [18, 19]). However, as pointed by Bauschke and Combettes , norm convergence of the algorithm is much more desirable than weak convergence in some applied sciences. It is therefore of interest to seek modifications of these algorithms so that strong convergence is guaranteed. Following the damped projection method, we propose in this paper a new iterative scheme and prove its strong convergence to a solution of SCFP. An application of our method to multiple-set split feasibility problems is also included. This enables us to cover some recent results on split feasibility problems.
2 Preliminary and notation
Throughout this paper, I denotes the identity operator on ℋ, the set of fixed points of an operator T, ‘→’ strong convergence, and ‘⇀’ weak convergence. Given a positive integer p, denote by the mod function taking values in .
Definition 1 An operator is called nonexpansive if , ; firmly nonexpansive if , .
Definition 2 Assume that is a nonlinear operator. Then is said to be demiclosed at zero, if, for any in ℋ, the following implication holds:
Clearly, firm nonexpansiveness implies nonexpansiveness. It is well known that nonexpansive operators are demiclosed at zero (cf. ).
Definition 3 Let be an operator with . Then T is called directed if , , ; ν-demicontractive with if , , .
Lemma 1 (Bauschke-Combettes )
An operator is directed if and only if one of following inequalities holds for all and :
It is clear that demicontractive operators include directed operators, while the latter include firmly nonexpansive operators with nonempty fixed-point sets. The concept of directed operators was introduced by Bauschke and Combettes . Such a class of operators is important because they include many types of nonlinear operators arising in applied mathematics. For instance, the metric projections onto a closed convex subset. Recall that the metric projection, denoted by , is defined by
It is well known that is characterized by the variational inequality
Lemma 2 (Wang-Xu )
Assume that is a bounded linear operator and is a directed operator. Let with . Then
whenever is nonempty.
Lemma 3 Assume that is a bounded linear operator and is a directed operator. Let with . If is nonempty, then
for all and .
Proof Since , it follows from (8) that
Hence the proof is complete. □
We end this section by a useful lemma.
Lemma 4 (Xu )
Let be a nonnegative real sequence satisfying
where and are real sequences. Then provided that
3 Algorithm and its convergence analysis
In this section, we consider the following problem.
Problem 1 Find an element satisfying
where p is a positive integer and , are two classes of directed operators such that and are demiclosed at zero for every .
We remark here that problem (11) is a special case of (5). However, this is not restrictive. Indeed, following an idea in , one can easily extend the results to the general case. We now present our algorithm for SCFP: Take and define a sequence by the iterative procedure:
where , and , , are properly chosen real sequences.
Theorem 1 Assume that the following conditions hold:
If the solution set of problem (11) denoted by Ω is nonempty, then the sequence generated by (12) converges strongly to .
Proof We first show the boundedness of . To see this, let and set , . Hence
Since is directed, it follows that
Adding up these inequalities, we have
By induction, the sequence is bounded, and so is .
Next we show the following key inequality:
Indeed, in view of Lemma 3, we arrive at
On the other hand, we deduce that
where we use the subdifferential inequality, and also that
Adding up (14)-(16), we thus get inequality (13).
Finally, we prove . To see this, let be a subsequence such that it includes all elements in with the property: each of them is less than or equal to the term after it. Following an idea developed by Maingé , we consider two possible cases on such a sequence.
Case 1. Assume that is finite. Then there exists such that for all , and therefore must be convergent. It follows from (13) that
where is a sufficiently large real number. Consequently, both and converge to zero. We have
Take a subsequence of so that
Without loss of generality, we assume that weakly converges to an element . Let an index be fixed. Noticing that the pool of indexes is finite, we can find a subsequence of such that and for all k. Since , we thus use the demiclosedness of at zero to conclude that . On the other hand, we deduce from (8) that
As , the weak continuity of A yields that , which together with the demiclosedness of at zero enables us to deduce . Since the index i is arbitrary, we therefore conclude . Consequently,
where the inequality uses (9). It then follows from (13) that
We therefore apply Lemma 4 to conclude .
Case 2. Assume now that is infinite. Let be fixed. Then there exists such that . By the choice of , we see that is the largest one among ; in particular,
Then we deduce from (13) that so that
In a similar way to case 1, we deduce and
Since by (17) , it follows from (13) that
Hence so that . Moreover,
which immediately implies . Consequently, follows from (17) and the proof is complete. □
We next use our algorithm to approximate a solution to the two-set SCFP: Find an element such that
where and are directed operators so that and are demiclosed at zero.
Corollary 2 Suppose that the following conditions hold:
Then the sequence , generated by
converges strongly to , whenever such point exists.
4 Some applications
In this section, we extend our result to SCFP for demicontractive operators recently considered by Moudafi .
Problem 2 Find an element satisfying
where p is a positive integer and , are respectively -demicontractive and -demicontractive operator so that and are demiclosed at zero for every .
The following lemma states a relation between directed and demicontractive operators.
Lemma 5 Let and . If T is ν-demicontractive, then is directed.
Proof For , we deduce that
Then the result follows from Lemma 1. □
We now propose an algorithm to solve problem (22). Take and define a sequence by the iterative procedure
where , and . By using the previous lemma, we can easily extend our result to demicontractive operators.
Theorem 3 Let and for every . Assume that the following conditions hold:
If the solution set of problem (22) denoted by Ω is nonempty, then the sequence generated by (23) converges strongly to .
Remark 1 Theorem 3 also holds true if we relax hypothesis (ii) above as .
We next consider the multiple-set split feasibility problem (MSFP): Find an element satisfying
where and are closed convex subsets in ℋ and , respectively. Take and define a sequence by the iterative procedure
where , , and , , are properly chosen real sequences.
Theorem 4 Assume that the following conditions hold:
If the solution set of MSFP denoted by Ω is nonempty, then the sequence generated by (25) converges strongly to .
Proof We note that the metric projection is firmly nonexpansive, which implies is directed and is demiclosed at zero. Hence, by using Theorem 1, one can immediately get the desired result. □
Remark 2 Theorem 4 covers [, Theorem 3.1], and we relax the condition on as . Moreover, the choice of variable is more flexible than the fixed one. Also, we cover the result of  and remove one condition posed on : either or .
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We would like to express our sincere thanks to the referees for their valuable suggestions. This work is supported by the National Natural Science Foundation of China, Tianyuan Foundation (11226227), the Basic Science and Technological Frontier Project of Henan (122300410268) and the Foundation of Henan Educational Committee (12A110016).
The authors declare that they have no competing interests.
All authors contributed equally and significantly to writing this manuscript. All authors read and approved the manuscript.