Strong convergence theorem of two-step iterative algorithm for split feasibility problems
© Tang and Chang; licensee Springer. 2014
Received: 20 February 2014
Accepted: 11 July 2014
Published: 15 August 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.
Keywordssplit feasibility problem two-step iterative algorithm strong convergence bounded linear operator
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]).
where and are the orthogonal projections onto C and Q, respectively, is any positive constant and denotes the adjoint of A.
where is chosen in the interval , and L is the Lipschitz constant of ∇f.
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.
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
to denote the weak ω-limit set of .
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.
∇f is -Lipschitz: , .
Lemma 2.1 is proved. □
T is firmly nonexpansive.
is firmly nonexpansive.
- (ii)⇒ (iii): From (ii) we know that for all ,
⇒ (i): From (iii) we immediately know that T is firmly nonexpansive. □
Lemma 2.3 
, or .
Lemma 2.4 
3 Main results
In this section, we shall prove our main theorem.
the sequence is bounded and ,
then the sequence converges strongly to .
Now, we prove . For the purpose, we consider two cases.
Since , are demiclosed at origin, from (3.11) and (3.12) we have that , i.e., .
From condition (ii) and Lemma 2.3, we obtain .
Noting inequality (3.18), this shows that , that is, . This completes the proof of Theorem 3.1. □
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).
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