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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)
Abstract
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 pseudocontractive mappings and the quasinonexpansive 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.
1 Introduction
This article we devote to the split common fixed point problem and study it for the pseudocontractive and quasinonexpansive mappings. The split common fixed point problem is a generalization of the convex feasibility problem which is to find a point {x}^{\ast} satisfying the following:
where m\ge 1 is an integer, and each {C}_{i} 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 {x}^{\ast} such that
where C and Q are two closed convex subsets of two Hilbert spaces {H}_{1} and {H}_{2}, respectively, and A:{H}_{1}\to {H}_{2} is a bounded linear operator. Such problems arise in the field of intensitymodulated radiation therapy when one attempts to describe physical dose constraints and equivalent uniform dose constraints within a single model; see [4]. The problem with only a single pair of sets C\in {\mathbb{R}}^{N} and Q\in {\mathbb{R}}^{M} was first introduced by Censor and Elfving [5]. They used their simultaneous multiprojections algorithm to solve the split feasibility problem. Their algorithms, as well as others, see, e.g., Byrne [6], involve matrix inversion at each iterative step. Calculating inverses of matrices is very timeconsuming, particularly if the dimensions are large. Therefore, a new algorithm for solving the split feasibility problem was devised by Byrne [7], called the CQalgorithm:
where \tau \in (0,\frac{2}{L}) with L being the largest eigenvalue of the matrix {A}^{\ast}A, I is the unit matrix or operator and {P}_{C} and {P}_{Q} 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 subgradient at the current point. This was done by Yang, in his recent paper [8], where he proposed a relaxed version of the CQalgorithm in which orthogonal projections are replaced by subgradient projections, which are easily executed when the sets C and Q are given as lower level sets of convex functions; see also [9]. 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 pseudocontractive and quasinonexpansive mappings. Weak convergence theorem is given. Our results improve and develop previously discussed feasibility problems and related algorithms.
2 Preliminaries
Let H be a real Hilbert space with inner product \u3008\cdot ,\cdot \u3009 and norm \parallel \cdot \parallel, respectively. Let C be a nonempty closed convex subset of H.
Definition 2.1 A mapping S:C\to C is called pseudocontractive if
for all x,y\in C.
We will use Fix(S) to denote the set of fixed points of S, that is, Fix(S)=\{x\in C:x=Sx\}.
Remark 2.2 It is clear that S is pseudocontractive if and only if
for all x,y\in C.
Interest in pseudocontractive mappings stems mainly from their firm connection with the class of nonlinear accretive operators. It is a classical result, see Deimling [26], that if S is an accretive operator, then the solutions of the equations Sx=0 correspond to the equilibrium points of some evolution systems.
Definition 2.3 A mapping T:C\to C is called LLipschitzian if there exists L>0 such that
for all x,y\in C.
Remark 2.4 We call T nonexpansive if L=1 and T is contractive if L<1.
Definition 2.5 A mapping T:C\to C is called quasinonexpansive if
Remark 2.6 It is obvious that if T is nonexpansive with Fix(T)\ne \mathrm{\varnothing}, then T is quasinonexpansive.
Usually, the convergence of fixed point algorithms requires some additional smoothness properties of the mapping T such as demiclosedness.
Definition 2.7 A mapping T is said to be demiclosed if, for any sequence \{{x}_{k}\} which weakly converges to \tilde{x}, and if the sequence \{T({x}_{k})\} strongly converges to z, then T(\tilde{x})=z.
It is well known that in a real Hilbert space H, the following equality holds:
for all x,y\in H and t\in [0,1].
Lemma 2.8 ([27])
Let H be a real Hilbert space, C a closed convex subset of H. Let U:C\to C be a continuous pseudocontractive mapping. Then

(i)
Fix(U) is a closed convex subset of C,

(ii)
(IU) is demiclosed at zero.
Lemma 2.9 ([28])
Let H be a Hilbert space and let \{{u}_{n}\} be a sequence in H such that there exists a nonempty set \mathrm{\Omega}\subset H satisfying the following:

(i)
for every u\in \mathrm{\Omega}, {lim}_{n}\parallel {u}_{n}u\parallel exists,

(ii)
any weakcluster point of the sequence \{{u}_{n}\} belongs in Ω.
Then there exists {x}^{\u2020}\in \mathrm{\Omega} such that \{{u}_{n}\} weakly converges to {x}^{\u2020}.
In the sequel we shall use the following notation:

1.
{\omega}_{w}({u}_{n})=\{x:\mathrm{\exists}{u}_{{n}_{j}}\to x\text{weakly}\} denote the weak ωlimit set of \{{u}_{n}\};

2.
{u}_{n}\rightharpoonup x stands for the weak convergence of \{{u}_{n}\} to x;

3.
{u}_{n}\to x stands for the strong convergence of \{{u}_{n}\} to x.
3 Main results
In this section, we will focus our attention on the following general twooperator split common fixed point problem:
where A:{H}_{1}\to {H}_{2} is a bounded linear operator, U:{H}_{1}\to {H}_{1} is a pseudocontractive mapping and T:{H}_{2}\to {H}_{2} is a quasinonexpansive mapping with nonempty fixed point sets Fix(U)=C and Fix(T)=Q, and we denote the solution set of the twooperator split common fixed point problem by
To solve (3.1), Censor and Segal [12] proposed and proved, in finitedimensional spaces, the convergence of the following algorithm:
where \gamma \in (0,\frac{2}{\lambda}), with λ being the largest eigenvalue of the matrix {A}^{\ast}A.
Moudafi [16] extended (3.2) to the following relaxed algorithm:
where \beta \in (0,1), {\alpha}_{k}\in (0,1) are relaxation parameters.
Inspired by their works, we introduce the following algorithm.
Algorithm 3.1 Let {H}_{1} and {H}_{2} be two real Hilbert spaces. Let A:{H}_{1}\to {H}_{2} be a bounded linear operator. Let U:{H}_{1}\to {H}_{1} be a pseudocontractive mapping with Lipschitzian constant L and T:{H}_{2}\to {H}_{2} be a quasinonexpansive mapping with nonempty Fix(U)=C and Fix(T)=Q. Let {x}_{0}\in {H}_{1}. Define a sequence \{{u}_{n}\} as follows:
for all n\in \mathbb{N}, where γ and ν are two constants, \{{\alpha}_{n}\}, \{{\delta}_{n}\}, and \{{\xi}_{n}\} are three sequences in [0,1].
In the sequel, we assume the parameters satisfy the following restrictions.
Parameters restrictions:
(R_{1}): 0<\nu <1 and 0<\gamma <\frac{1}{\lambda \nu}, where λ is the largest eigenvalue of the matrix {A}^{\ast}A;
(R_{2}): 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;
(R_{3}): 0<k\le 1{\delta}_{n}\le {\xi}_{n}<\frac{1}{\sqrt{1+{L}^{2}}+1} for all n\in \mathbb{N}, where L is the Lipschitz constant of U.
Remark 3.2 Without loss of generality, we may assume that the Lipschitz constant L>1. It is obvious that \frac{1}{\sqrt{1+{L}^{2}}+1}<\frac{1}{L} for all n\ge 1. Since {\xi}_{n}<\frac{1}{\sqrt{1+{L}^{2}}+1}, we have 12{\xi}_{n}{\xi}_{n}^{2}{L}^{2}>0 for all n\in \mathbb{N}.
Theorem 3.3 Let {H}_{1} and {H}_{2} be two real Hilbert spaces. Let A:{H}_{1}\to {H}_{2} be a bounded linear operator. Let U:{H}_{1}\to {H}_{1} be a pseudocontractive mapping with Lipschitzian constant L and T:{H}_{2}\to {H}_{2} be a quasinonexpansive mapping with nonempty Fix(U)=C and Fix(T)=Q. Assume that TI is demiclosed at 0 and \mathrm{\Gamma}\ne \mathrm{\varnothing}. Then the sequence \{{u}_{n}\} generated by algorithm (3.3) weakly converges to a split common fixed point \mu \in \mathrm{\Gamma}.
Proof Let {x}^{\ast}\in \mathrm{\Gamma}. Then we get {x}^{\ast}\in Fix(U) and A{x}^{\ast}\in Fix(T). From (2.2) and (3.3), we have
Since {x}^{\ast}\in Fix(U), we have from (2.1)
for all x\in C.
By (3.4) and (3.5), we obtain
Note that U is LLipschitzian and {x}_{n}{y}_{n}={\xi}_{n}(U{x}_{n}{x}_{n}). Then we have
Substituting (3.6) into (3.4), we have
Since λ is the spectral radius of the operator A{A}^{\ast}, we deduce
This together with (3.1) implies that
Since T is quasinonexpansive and A{x}^{\ast}\in Fix(T), we have
At the same time, we have the following equality in Hilbert spaces:
In (3.9), picking up x=(TI)A{u}_{n} and y=TA{u}_{n}A{x}^{\ast} to deduce
It follows that
Thus,
From (3.7), (3.8), and (3.10), we get
We deduce immediately that
Hence, {lim}_{n\to \mathrm{\infty}}\parallel {u}_{n}{x}^{\ast}\parallel exists. This implies that \{{u}_{n}\} is bounded. Consequently, we have
Therefore,
Since \{{u}_{n}\} is bounded, {\omega}_{w}({u}_{n})\ne \mathrm{\varnothing}. We can take \mu \in {\omega}_{w}({u}_{n}), that is, there exists \{{u}_{{n}_{j}}\} such that \omega {lim}_{j\to \mathrm{\infty}}{u}_{{n}_{j}}=\mu. Noting that TI is demiclosed at 0, from (3.12), we obtain
Thus, A\mu \in Fix(T).
From (3.11), we deduce
Since 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1, 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})<1. Then we have
Note that lim\hspace{0.17em}sup{\delta}_{n}<1; we get immediately
Since U is LLipschitzian, we have
It follows that
So
From (3.3), (3.12), and (3.13), we have {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{u}_{n}\parallel =0. Thus, \omega {lim}_{j\to \mathrm{\infty}}{x}_{{n}_{j}}=\mu. By the demiclosedness of UI at 0 (Lemma 2.8), we get
Hence, \mu \in Fix(U). Therefore, \mu \in \mathrm{\Gamma}. Since there is no more than one weakcluster point, the weak convergence of the whole sequence \{{u}_{n}\} follows by applying Lemma 2.9 with \mathrm{\Omega}=\mathrm{\Gamma}. This completes the proof. □
Example 3.4 Let H=\mathbb{R} with the inner product defined by \u3008x,y\u3009=xy for all x,y\in \mathbb{R} and the standard norm \cdot . Let C=[0,+\mathrm{\infty}) and Tx=\frac{{x}^{2}+2}{1+x} for all x\in C. Obviously, Fix(T)=2. It is easy to see that
for all x\in C and
Hence, T is a continuous quasinonexpansive mapping but not nonexpansive.
Example 3.5 Let H=\mathbb{R} with the inner product defined by \u3008x,y\u3009=xy for all x,y\in \mathbb{R} and the standard norm \cdot . Let C=[0,+\mathrm{\infty}) and let Ux=x1+\frac{4}{x+1} for all x\in C. Observe that Fix(U)=3. It is easy to see that
and
for all x,y\in C.
But
Hence, U is a Lipschitzian pseudocontractive mapping but it is not nonexpansive.
References
Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Rev. 1996, 38: 367–426. 10.1137/S0036144593251710
Combettes PL: The convex feasibility problem in image recovery. 95. In Advances in Imaging and Electron Physics. Edited by: Hawkes P. Academic Press, New York; 1996:155–270.
Stark H (Ed): Image Recovery Theory and Applications. Academic Press, Orlando; 1987.
Censor Y, Bortfeld T, Martin B, Trofimov A: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 2006, 51: 2353–2365. 10.1088/00319155/51/10/001
Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692
Byrne C: BregmanLegendre multidistance projection algorithms for convex feasibility and optimization. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Edited by: Butnariu D, Censor Y, Reich S. Elsevier, Amsterdam; 2001:87–100.
Byrne C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 2002, 18: 441–453. 10.1088/02665611/18/2/310
Yang Q: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 2004, 20: 1261–1266. 10.1088/02665611/20/4/014
Zhao J, Yang Q: Several solution methods for the split feasibility problem. Inverse Probl. 2005, 21: 1791–1799. 10.1088/02665611/21/5/017
Ceng LC, Ansari QH, Yao JC: An extragradient method for split feasibility and fixed point problems. Comput. Math. Appl. 2012, 64: 633–642. 10.1016/j.camwa.2011.12.074
Censor Y, Motova A, Segal A: Perturbed projections and subgradient projections for the multiplesets split feasibility problem. J. Math. Anal. Appl. 2007, 327: 1244–1256. 10.1016/j.jmaa.2006.05.010
Censor Y, Segal A: The split common fixed point problem for directed operators. J. Convex Anal. 2009, 16: 587–600.
Dang Y, Gao Y: The strong convergence of a KMCQlike algorithm for a split feasibility problem. Inverse Probl. 2011., 27: Article ID 015007 10.1088/02665611/27/1/015007
Deepho J, Kumam P: A modified Halpern’s iterative scheme for solving split feasibility problems. Abstr. Appl. Anal. 2012., 2012: Article ID 876069 10.1155/2012/876069
López G, MartínMárquez V, Xu HK: Iterative algorithms for the multiplesets split feasibility problem. In Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems. Edited by: Censor Y, Jiang M, Wang G. Medical Physics Publishing, Madison; 2009:243–279.
Moudafi A: A note on the split common fixedpoint problem for quasinonexpansive operators. Nonlinear Anal. 2011, 74: 4083–4087. 10.1016/j.na.2011.03.041
Wang F, Xu HK: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2010., 2010: Article ID 102085 10.1155/2010/102085
Wang F, Xu HK: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 2011, 74: 4105–4111. 10.1016/j.na.2011.03.044
Xu HK: A variable Krasnosel’skiiMann algorithm and the multipleset split feasibility problem. Inverse Probl. 2006, 22: 2021–2034. 10.1088/02665611/22/6/007
Xu HK: Iterative methods for the split feasibility problem in infinitedimensional Hilbert spaces. Inverse Probl. 2010., 26: Article ID 105018 10.1088/02665611/26/10/105018
Yao Y, Kim TH, Chebbi S, Xu HK: A modified extragradient method for the split feasibility and fixed point problems. J. Nonlinear Convex Anal. 2012, 13: 383–396.
Yao Y, Wu J, Liou YC: Regularized methods for the split feasibility problem. Abstr. Appl. Anal. 2012., 2012: Article ID 140679 10.1155/2012/140679
Yu X, Shahzad N, Yao Y: Implicit and explicit algorithms for solving the split feasibility problem. Optim. Lett. 2012, 6: 1447–1462. 10.1007/s1159001103400
Zhao J, Yang Q: Selfadaptive projection methods for the multiplesets split feasibility problem. Inverse Probl. 2011., 27: Article ID 035009 10.1088/02665611/27/3/035009
Zhang W, Han D, Li Z: A selfadaptive projection method for solving the multiplesets split feasibility problem. Inverse Probl. 2009., 25: Article ID 115001 10.1088/02665611/25/11/115001
Deimling K: Zeros of accretive operators. Manuscr. Math. 1974, 13: 365–374. 10.1007/BF01171148
Zhou H: Strong convergence of an explicit iterative algorithm for continuous pseudocontractions in Banach spaces. Nonlinear Anal. 2009, 70: 4039–4046. 10.1016/j.na.2008.08.012
Bauschke HH, Combettes PL: A weaktostrong convergence principle for Fejérmonotone methods in Hilbert spaces. Math. Oper. Res. 2001, 26: 248–264. 10.1287/moor.26.2.248.10558
Acknowledgements
The authors are grateful to the five reviewers for their valuable comments and suggestions. LiJun Zhu was supported in part by NNSF of China (61362033) and NZ13087. YeongCheng Liou was supported in part by NSC 1012628E230001MY3 and NSC 1012622E230005CC3.
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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/10.1186/168718122014172
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DOI: https://doi.org/10.1186/168718122014172
Keywords
 split common fixed point problem
 pseudocontractive mappings
 quasinonexpansive mapping
 weak convergence