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Strong convergence theorem of two-step iterative algorithm for split feasibility problems
Journal of Inequalities and Applications volume 2014, Article number: 280 (2014)
The main purpose of this paper is to introduce a two-step iterative algorithm for split feasibility problems such that the strong convergence is guaranteed. Our result extends and improves the corresponding results of He et al. and some others.
MSC:90C25, 90C30, 47J25.
The split feasibility problem (SFP) was proposed by Censor and Elfving in . It can be formulated as the problem of finding a point x satisfying the property
where A is a given real matrix, C and Q are nonempty, closed and convex subsets in and , respectively.
Due to its extraordinary utility and broad applicability in many areas of applied mathematics (most notably, fully-discretized models of problems in image reconstruction from projections, in image processing, and in intensity-modulated radiation therapy), algorithms for solving convex feasibility problems continue to receive great attention (see, for instance, [2–5] and also [6–10]).
We assume that SFP (1.1) is consistent, and let Γ be the solution set, i.e.,
It is not hard to see that Γ is closed convex and if and only if it solves the fixed-point equation
where and are the orthogonal projections onto C and Q, respectively, is any positive constant and denotes the adjoint of A.
To solve (1.2), Byrne  proposed his CQ algorithm which generates a sequence by
The CQ algorithm (1.3) can be obtained from optimization. In fact, if we introduce the convex objective function
and analyze the minimization problem
then the CQ algorithm (1.3) comes immediately as a special case of the gradient projection algorithm (GPA). Since the convex objective function is differentiable and has a Lipschitz gradient, which is given by
the GPA for solving the minimization problem (1.4) generates a sequence recursively as
where is chosen in the interval , and L is the Lipschitz constant of ∇f.
Observe that in algorithms (1.3) and (1.7) mentioned above, in order to implement the CQ algorithm, one has to compute the operator norm , which is in general not an easy work in practice. To overcome this difficulty, some authors proposed different adaptive choices of selecting the (see [11–13]). For instance, López et al. introduced a new way of selecting the stepsize  as follows:
The computation of a projection onto a general closed convex subset is generally difficult. To overcome this difficulty, Fukushima  suggested the so-called relaxed projection method to calculate the projection onto a level set of a convex function by computing a sequence of projections onto half-spaces containing the original level set. In the setting of finite-dimensional Hilbert spaces, this idea was followed by Yang , who introduced the relaxed CQ algorithms for solving SFP (1.1) where the closed convex subsets C and Q are level sets of convex functions.
Recently, for the purpose of generality, SFP (1.1) has been studied in a more general setting. For instance, see [8, 12]. However, their algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. Very recently, He and Zhao  introduced a new relaxed CQ algorithm (1.9) such that the strong convergence is guaranteed in infinite-dimensional Hilbert spaces:
Motivated and inspired by the research going on in this section, the purpose of this article is to study a two-step iterative algorithm for split feasibility problems such that the strong convergence is guaranteed in infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results of He and Zhao  and some others.
2 Preliminaries and lemmas
Throughout the rest of this paper, we assume that H, and all are Hilbert spaces, A is a bounded linear operator from to , and I is the identity operator on H, or . If is a differentiable function, then we denote by ∇f the gradient of the function f. We will also use the following notations: → to denote the strong convergence, ⇀ to denote the weak convergence and
to denote the weak ω-limit set of .
Recall that a mapping is said to be nonexpansive if
is said to be firmly nonexpansive if
A mapping is said to be demiclosed at origin if for any sequence with and , then .
It is easy to prove that if is a firmly nonexpansive mapping, then T is demiclosed at origin.
A function is called convex if
A function is said to be weakly lower semi-continuous (w-lsc) at x if implies
Lemma 2.1 Let be a firmly nonexpansive mapping such that is a convex function from to ℝ, let be a bounded linear operator and
∇f is -Lipschitz: , .
Proof (i) From the definition of f, we know that f is convex. For any given and for any , first we prove that the limit
exists in and satisfies
If fact, if , then
Since f is convex and , it follows that
This shows that this difference quotient is increasing, therefore it has a limit in as :
This implies that f is differential. Taking in (2.1), we have
Next we prove that
In fact, since
Substituting (2.3) into (2.2), simplifying it and then letting , we have
It follows from (2.1) that
Now we prove conclusion (ii). Indeed, it follows from (i) that
Lemma 2.1 is proved. □
Lemma 2.2 Let be an operator. The following statements are equivalent.
T is firmly nonexpansive.
is firmly nonexpansive.
Proof (i) ⇒ (ii): Since T is firmly nonexpansive, we have, for all ,
⇒ (iii): From (ii) we know that for all ,
This implies that is firmly nonexpansive.
⇒ (i): From (iii) we immediately know that T is firmly nonexpansive. □
Lemma 2.3 
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in , and is a sequence in ℝ such that
, or .
Lemma 2.4 
Let X be a real Banach space and be the normalized duality mapping, then, for any , the following inequality holds:
Especially, if X is a real Hilbert space, then we have
3 Main results
In this section, we shall prove our main theorem.
Theorem 3.1 Let , be two real Hilbert spaces, be a bounded linear operator. Let be a firmly nonexpansive mapping, be two firmly nonexpansive mappings such that () is a convex function from to ℝ with and . Let and be a sequence generated by
If the solution set Γ of SPF (1.1) is not empty, and the sequences and satisfy the following conditions:
the sequence is bounded and ,
then the sequence converges strongly to .
Proof Since Γ (≠∅) is the solution set of SPF (1.1), Γ is closed and convex. Thus, the metric projection is well defined. Letting , it follows from Lemma 2.4 that
Since , . Observe that is firmly nonexpansive, from Lemma 2.2(ii) we have that
which implies that
Thus, we have
and so we have
Consequently, we have
It turns out that
This implies that the sequence is bounded. Since the mapping S is firmly nonexpansive, it follows from Lemma 2.4 that
Since , . Observe that is firmly nonexpansive, it is deduced from Lemma 2.2(ii) that
which implies that
Thus, we have
Consequently, we have
It turns out that
Since is bounded, so is . Since and , without loss of generality, we may assume that there is such that
Substituting (3.3) into (3.5), we have
Setting , we get the following inequality:
Now, we prove . For the purpose, we consider two cases.
Case 1: is eventually decreasing, i.e., there exists a sufficiently large positive integer such that holds for all . In this case, must be convergent, and from (3.7) it follows that
where M is a constant such that for all . Using condition (i) and (3.8), we have that
To verify that and , it suffices to show that and are bounded. In fact, it follows from Lemma 2.1(ii) that for all ,
These imply that and are bounded. It yields and , namely
From (3.1) we have
Noting that , are bounded and , , we get
For any , and is a subsequence of such that , then it follows from (3.10) that . Thus we have
On the other hand, from (3.9) we have
Since , are demiclosed at origin, from (3.11) and (3.12) we have that , i.e., .
Next, we turn to prove . For convenience, we set . Since S is firmly nonexpansive, it follows from Lemma 2.4 that
In view of the definition of , we have
From (3.4), (3.13) and (3.14), we have
where is a suitable constant. Clearly, from (3.15) we have
Thus, we assert that . In view of and S is demiclosed at origin, we get , i.e., . Consequently, . Furthermore, we have
From (3.7) we have
From condition (ii) and Lemma 2.3, we obtain .
Case 2: is not eventually decreasing, that is, we can find a positive integer such that . Now we define
It easy to see that is nonempty and satisfies . Let
It is clear that as (otherwise, is eventually decreasing). It is also clear that for all . Moreover, we prove that
In fact, if , then inequality (3.18) is trivial; if , from the definition of , there exists some such that , we deduce that
and inequality (3.18) holds again. Since for all , it follows from (3.8) that
Noting that and are both bounded, we get
By the same argument to the proof in case 1, we have and
On the other hand, noting again, we have from (3.14) and (3.16) that
Letting , we get
From (3.19) and (3.21), we have
Furthermore, we can deduce that
Since , it follows from (3.7) that
Combining (3.23), (3.24) and (3.25), we have
and hence , which together with (3.22) implies that
Noting inequality (3.18), this shows that , that is, . This completes the proof of Theorem 3.1. □
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This study was supported by the Scientific Research Fund of Sichuan Provincial Education Department (13ZA0199), the Scientific Research Fund of Sichuan Provincial Department of Science and Technology (2012JYZ011) and the Scientific Research Project of Yibin University (No. 2013YY06) and partially supported by the National Natural Science Foundation of China (Grant No. 11361070).
The authors declare that they have no competing interests.
All the authors contributed equally to the writing of the present article. They also read and approved final manuscript.
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Tang, J., Chang, Ss. Strong convergence theorem of two-step iterative algorithm for split feasibility problems. J Inequal Appl 2014, 280 (2014). https://doi.org/10.1186/1029-242X-2014-280
- split feasibility problem
- two-step iterative algorithm
- strong convergence
- bounded linear operator