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Algorithmic and analytical approach to the split common fixed points problem
Fixed Point Theory and Applications volume 2014, Article number: 172 (2014)
The split problem, especially the split common fixed point problem, has been studied by many authors. In this paper, we study the split common fixed point problem for the pseudo-contractive mappings and the quasi-nonexpansive mappings. We suggest and analyze an iterative algorithm for solving this split common fixed point problem. A weak convergence theorem is given.
MSC:49J53, 49M37, 65K10, 90C25.
This article we devote to the split common fixed point problem and study it for the pseudo-contractive and quasi-nonexpansive mappings. The split common fixed point problem is a generalization of the convex feasibility problem which is to find a point satisfying the following:
where is an integer, and each is a nonempty closed convex subset of a Hilbert space H. Note that the convex feasibility problem has received a lot of attention due to its extensive applications in many applied disciplines as diverse as approximation theory, image recovery and signal processing, control theory, biomedical engineering, communications, and geophysics (see [1–3] and the references therein). A special case of the convex feasibility problem is the split feasibility problem, which is to find a point such that
where C and Q are two closed convex subsets of two Hilbert spaces and , respectively, and is a bounded linear operator. Such problems arise in the field of intensity-modulated radiation therapy when one attempts to describe physical dose constraints and equivalent uniform dose constraints within a single model; see . The problem with only a single pair of sets and was first introduced by Censor and Elfving . They used their simultaneous multi-projections algorithm to solve the split feasibility problem. Their algorithms, as well as others, see, e.g., Byrne , involve matrix inversion at each iterative step. Calculating inverses of matrices is very time-consuming, particularly if the dimensions are large. Therefore, a new algorithm for solving the split feasibility problem was devised by Byrne , called the CQ-algorithm:
where with L being the largest eigenvalue of the matrix , I is the unit matrix or operator and and denote the orthogonal projections onto C and Q, respectively. In the case of nonlinear constraint sets, orthogonal projections may demand a great amount of work of solving a nonlinear optimization problem to minimize the distance between the point and the constraint set. However, it can easily be estimated by linear approximation using the current constraint violation and the sub-gradient at the current point. This was done by Yang, in his recent paper , where he proposed a relaxed version of the CQ-algorithm in which orthogonal projections are replaced by sub-gradient projections, which are easily executed when the sets C and Q are given as lower level sets of convex functions; see also . There are a large number of references on the CQ method for the split feasibility problem in the literature; see, for instance, [10–25].
It is our main purpose in this paper to develop algorithms for the split common fixed point for the pseudo-contractive and quasi-nonexpansive mappings. Weak convergence theorem is given. Our results improve and develop previously discussed feasibility problems and related algorithms.
Let H be a real Hilbert space with inner product and norm , respectively. Let C be a nonempty closed convex subset of H.
Definition 2.1 A mapping is called pseudo-contractive if
for all .
We will use to denote the set of fixed points of S, that is, .
Remark 2.2 It is clear that S is pseudo-contractive if and only if
for all .
Interest in pseudo-contractive mappings stems mainly from their firm connection with the class of nonlinear accretive operators. It is a classical result, see Deimling , that if S is an accretive operator, then the solutions of the equations correspond to the equilibrium points of some evolution systems.
Definition 2.3 A mapping is called L-Lipschitzian if there exists such that
for all .
Remark 2.4 We call T nonexpansive if and T is contractive if .
Definition 2.5 A mapping is called quasi-nonexpansive if
Remark 2.6 It is obvious that if T is nonexpansive with , then T is quasi-nonexpansive.
Usually, the convergence of fixed point algorithms requires some additional smoothness properties of the mapping T such as demi-closedness.
Definition 2.7 A mapping T is said to be demi-closed if, for any sequence which weakly converges to , and if the sequence strongly converges to z, then .
It is well known that in a real Hilbert space H, the following equality holds:
for all and .
Lemma 2.8 ()
Let H be a real Hilbert space, C a closed convex subset of H. Let be a continuous pseudo-contractive mapping. Then
is a closed convex subset of C,
is demi-closed at zero.
Lemma 2.9 ()
Let H be a Hilbert space and let be a sequence in H such that there exists a nonempty set satisfying the following:
for every , exists,
any weak-cluster point of the sequence belongs in Ω.
Then there exists such that weakly converges to .
In the sequel we shall use the following notation:
denote the weak ω-limit set of ;
stands for the weak convergence of to x;
stands for the strong convergence of to x.
3 Main results
In this section, we will focus our attention on the following general two-operator split common fixed point problem:
where is a bounded linear operator, is a pseudo-contractive mapping and is a quasi-nonexpansive mapping with nonempty fixed point sets and , and we denote the solution set of the two-operator split common fixed point problem by
To solve (3.1), 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 .
Moudafi  extended (3.2) to the following relaxed algorithm:
where , are relaxation parameters.
Inspired by their works, we introduce the following algorithm.
Algorithm 3.1 Let and be two real Hilbert spaces. Let be a bounded linear operator. Let be a pseudo-contractive mapping with Lipschitzian constant L and be a quasi-nonexpansive mapping with nonempty and . Let . Define a sequence as follows:
for all , where γ and ν are two constants, , , and are three sequences in .
In the sequel, we assume the parameters satisfy the following restrictions.
(R1): and , where λ is the largest eigenvalue of the matrix ;
(R3): for all , where L is the Lipschitz constant of U.
Remark 3.2 Without loss of generality, we may assume that the Lipschitz constant . It is obvious that for all . Since , we have for all .
Theorem 3.3 Let and be two real Hilbert spaces. Let be a bounded linear operator. Let be a pseudo-contractive mapping with Lipschitzian constant L and be a quasi-nonexpansive mapping with nonempty and . Assume that is demi-closed at 0 and . Then the sequence generated by algorithm (3.3) weakly converges to a split common fixed point .
Proof Let . Then we get and . From (2.2) and (3.3), we have
Since , we have from (2.1)
for all .
By (3.4) and (3.5), we obtain
Note that U is L-Lipschitzian and . Then we have
Substituting (3.6) into (3.4), we have
Since λ is the spectral radius of the operator , we deduce
This together with (3.1) implies that
Since T is quasi-nonexpansive and , we have
At the same time, we have the following equality in Hilbert spaces:
In (3.9), picking up and to deduce
It follows that
From (3.7), (3.8), and (3.10), we get
We deduce immediately that
Hence, exists. This implies that is bounded. Consequently, we have
Since is bounded, . We can take , that is, there exists such that . Noting that is demi-closed at 0, from (3.12), we obtain
From (3.11), we deduce
Since , . Then we have
Note that ; we get immediately
Since U is L-Lipschitzian, we have
It follows that
From (3.3), (3.12), and (3.13), we have . Thus, . By the demi-closedness of at 0 (Lemma 2.8), we get
Hence, . Therefore, . Since there is no more than one weak-cluster point, the weak convergence of the whole sequence follows by applying Lemma 2.9 with . This completes the proof. □
Example 3.4 Let with the inner product defined by for all and the standard norm . Let and for all . Obviously, . It is easy to see that
for all and
Hence, T is a continuous quasi-nonexpansive mapping but not nonexpansive.
Example 3.5 Let with the inner product defined by for all and the standard norm . Let and let for all . Observe that . It is easy to see that
for all .
Hence, U is a Lipschitzian pseudo-contractive mapping but it is not nonexpansive.
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The authors are grateful to the five reviewers for their valuable comments and suggestions. Li-Jun Zhu was supported in part by NNSF of China (61362033) and NZ13087. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Zhu, LJ., Liou, YC., Kang, S.M. et al. Algorithmic and analytical approach to the split common fixed points problem. Fixed Point Theory Appl 2014, 172 (2014). https://doi.org/10.1186/1687-1812-2014-172
- split common fixed point problem
- pseudo-contractive mappings
- quasi-nonexpansive mapping
- weak convergence