# Solving Split Common Fixed-Point Problem of Firmly Quasi-Nonexpansive Mappings without Prior Knowledge of Operators Norms

Authors: Jing Zhao; Hang Zhang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Very recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms for the split common fixed-point problem concerned two bounded linear operators. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. It is the purpose of this paper to introduce a viscosity iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. We prove the strong convergence of the proposed algorithms for split common fixed-point problem governed by the firmly quasi-nonexpansive operators. As a consequence, we obtain strong convergence theorems for split feasibility problem and split common null point problems of maximal monotone operators. Our results improve and extend the corresponding results announced by many others.


---

## Body

## 1. Introduction

Throughout this paper, we always assume thatH is a real Hilbert space with the inner product 〈·,·〉 and the norm ∥·∥. Let I denote the identity operator on H. Let T:H→H be a mapping. A point x∈H is said to be a fixed point of T provided Tx=x. In this paper, we use F(T) to denote the fixed point set of T.LetC and Q be nonempty closed convex subsets of real Hilbert spaces H1 and H2, respectively. The split feasibility problem (SFP) is to find a point as follows:
(1)x∈CsuchthatAx∈Q,
where A:H1→H2 is a bounded linear operator. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2].Note that if the split feasibility problem (1) is consistent (i.e., (1) has a solution), then (1) can be formulated as a fixed point equation by using the following fact:
(2)PC(I-γA*(I-PQ)A)x*=x*,
where PC and PQ are the (orthogonal) projections onto C and Q, respectively, γ>0 is any positive constant, and A* denotes the adjoint of A. That is, x* solves SFP (1) if and only if x* solves fixed point equation (2) (see [3] for the details). This implies that we can use fixed point algorithms (see [3–6]) to solve SFP. To solve (2), Byrne [2] proposed his CQ algorithm which generates a sequence {xk} by
(3)xk+1=PC(I-γA*(I-PQ)A)xk,k∈N,
where γ∈(0,2/λ) with λ being the spectral radius of the operator A*A.Censor and Segal [7] introduced the following split common fixed-point problem (SCFP):
(4)findx*∈F(U)suchthatAx*∈F(T),
where A:H1→H2 is a bounded linear operator and U:H1→H1 and T:H2→H2 are two nonexpansive operators with nonempty fixed-point sets F(U)=C and F(T)=Q. SCFP is in itself at the core of the modeling of many inverse problems in various areas of mathematics and physical sciences and has been used to model significant real-world inverse problems in many areas (see [8]).To solve (4), Censor and Segal [7] proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm:
(5)xk+1=U(xk+γAt(T-I)Axk),k∈N,
where γ∈(0,2/λ), with λ being the largest eigenvalue of the matrix AtA (At stands for matrix transposition).LetH1, H2, and H3 be real Hilbert spaces; let A:H1→H3 and B:H2→H3 be two bounded linear operators; let U:H1→H1 and T:H2→H2 be two firmly quasi-nonexpansive operators. In [9], Moudafi introduced the following split common fixed-point problem (SCFP):
(6)findx∈F(U),y∈F(T),suchthatAx=By,
which allows asymmetric and partial relations between the variables x and y. The interest is to cover many situations, for instance in decomposition methods for PDE’s, in a applications in game theory, and in intensity-modulated radiation therapy (IMRT). In decision sciences, this allows to consider agents who interplay only via some components of their decision variables (see [10]). In IMRT, these amounts envisage a weak coupling between the vector of doses absorbed in all voxels and that of the radiation intensity (see [11]).IfH2=H3 and B=I, then SCFP (6) reduces to SCFP (4). For solving SCFP (6), Moudafi [9] introduced the following alternating algorithm:
(7)xk+1=U(xk-γkA*(Axk-Byk)),yk+1=T(yk+γkB*(Axk+1-Byk)),
for firmly quasi-nonexpansive operators U and T, where nondecreasing sequence γk∈(ɛ,min⁡(1/λA,1/λB)-ɛ) and λA and λB stand for the spectral radius of A*A and B*B, respectively.Very recently, Moudafi and Al-Shemas [12] introduced the following simultaneous iterative method to solve SCFP (6):
(8)xk+1=U(xk-γkA*(Axk-Byk)),yk+1=T(yk+γkB*(Axk-Byk)),
for firmly quasi-nonexpansive operators U and T, where γk∈(ɛ,(2/(λA+λB))-ɛ) and λA and λB stand for the spectral radius of A*A and B*B, respectively.In [13], Zhao and He introduced the following alternating mann iterative algorithms for SCFP (6) governed by quasi-nonexpansive mappings and obtained weak convergence results:
(9)uk=xk-γkA*(Axk-Byk),xk+1=αkuk+(1-αk)U(uk),vk+1=yk+γkB*(Axk+1-Byk),yk+1=βkvk+1+(1-βk)T(vk+1).Note that, in (7), (8), and (9) mentioned above, the determination of the stepsize {γk} depends on the operator (matrix) norms ∥A∥ and ∥B∥ (or the largest eigenvalues of A*A and B*B). In order to implement the above algorithms for solving SCFP (6), one has first to compute (or, at least, estimate) operator norms of A and B, which is in general not an easy work in practice. To overcome this difficulty, López et al [14] and Zhao and Yang [15] presented a helpful method for estimating the stepsizes which do not need prior knowledge of the operator norms for solving the split feasibility problems and multiple-set split feasibility problems, respectively. Inspired by them, in this paper, we introduce a new choice of the stepsize sequence {γk} for the viscosity iterative algorithm to solve SCFP (6) governed by firmly quasi-nonexpansive operators as follows:
(10)γk∈(ϵ,2∥Axk-Byk∥2∥A*(Axk-Byk)∥2+∥B*(Axk-Byk)∥2-ϵ).
The advantage of our choice (9) of the stepsizes lies in the fact that no prior information about the operator norms of A and B is required, and still convergence is guaranteed.Some algorithms have been invented to solve SCFP (6) (see [16, 17] and references therein). In this paper, inspired and motivated by the works mentioned above, to get the strong convergence of the algorithm, we introduce the viscosity iterative algorithm without prior knowledge of operator norms for solving SCFP (6) governed by firmly quasi-nonexpansine operators. The organization of this paper is as follows. Some useful definitions and results are listed for the convergence analysis of the iterative algorithm in Section 2. In Section 3, the strong convergence theorem of the proposed viscosity iterative algorithm is obtained. At last, we provide some applications.


## 2. Preliminaries

In this paper, we use→ and ⇀ to denote the strong convergence and weak convergence, respectively. We use ωw(xk)={x:∃xkj⇀x} to stand for the weak ω-limit set of {xk} and use Γ to stand for the solution set of SCFP (6).Definition 1.
An operatorT:H→H is said to be(i)
nonexpansive if∥Tx-Ty∥≤∥x-y∥ for all x,y∈H,(ii)
quasi-nonexpansive ifF(T)≠∅ and if ∥Tx-q∥≤∥x-q∥ for all x∈H and q∈F(T),(iii)
firmly nonexpansive if∥Tx-Ty∥2≤∥x-y∥2-∥(x-y)-(Tx-Ty)∥2 for all x,y∈H,(iv)
firmly quasi-nonexpansive ifF(T)≠∅ and if ∥Tx-q∥2≤∥x-q∥2-∥x-Tx∥2 for all x∈H and q∈F(T).Remark 2.
A firmly quasi-nonexpansive operator is also called a separating operator [18], cutter operator [19], directed operators [7, 20], or class-T operator which was introduced by Bauschke and Combettes [21]. Firmly quasi-nonexpansive operators are important because they include many types of nonlinear operators arising in applied mathematics such as approximation and convex optimization. For instance, the subgradient projection T of a continuous convex function f:H→R is a firmly quasi-nonexpansive operator. Recall that the subgradient projection T is defined by, assuming that the level set {x∈H:f(x)≤0}≠∅,
(11)Tx≔{x-f(x)∥g(x)∥2g(x),f(x)>0,x,f(x)≤0,
where g is a selection of the subdifferential ∂f (i.e., g(x)∈∂f(x) for all x∈H).Particularly, projections are firmly quasi-nonexpansive operators. Recall that, given a closed convex subsetC of a Hilbert space H, the projection PC:H→C assigns each x∈H to its closest point from C defined by
(12)PCx=argmin⁡z∈C⁡∥x-z∥.
It is well known that PCx is characterized by the inequality
(13)PCx∈C,〈x-PCx,z-PCx〉≤0,z∈C.Lemma 3 (see [19, 21]).
The fixed point set of a firmly quasi-nonexpansive operator is closed convex.We also need other classes of operators.Definition 4.
An operatorT:H→H called demiclosed at the origin if whenever the sequence {xn} converges weakly to x and the sequence {Txn} converges strongly to 0, then Tx=0.We remark here that a firmly quasi-nonexpansive operatorT may be not nonexpansive; even T-I is demiclosed at origin. See the following example [22].Example 5.
LetH=R1 and define a mapping by T:H→H by
(14)Tx≔{x2sin⁡1x,x≠0,0,x=0.
Then, F(T)={0} and
(15)|Tx-0|2=x24(sin⁡1x)2≤x2-(x-x2sin⁡1x)2=|x-0|2-|x-Tx|2.
So, T is firmly quasi-nonexpansive but not nonexpansive. It is easy to see that T-I is demiclosed at origin.Definition 6.
An operatorT:H→H is called contraction with constant ρ∈[0,1) if, for any x,y∈H,
(16)∥Tx-Ty∥≤ρ∥x-y∥.In real Hilbert space, we easily get the following equality:(17)2〈x,y〉=∥x∥2+∥y∥2-∥x-y∥2=∥x+y∥2-∥x∥2-∥y∥2,∀x,y∈H.We end this section by the following lemmas, which are important in convergence analysis for our iterative algorithm.Lemma 7 (see [23]).
Assume{sk} is a sequence of nonnegative real numbers such that
(18)sk+1≤(1-λk)sk+λkδk,k≥0,sk+1≤sk-ηk+μk,k≥0,
where {λk} is a sequence in (0,1), {ηk} is a sequence of nonnegative real numbers, and {δk} and {μk} are two sequences in R such that(i)
Σk=1∞λk=∞;(ii)
lim⁡k→∞μk=0;(iii)
lim⁡l→∞ηkl=0 implies limsup⁡l→∞δkl≤0 for any subsequence {kl}⊂{k}.Then, lim⁡k→∞sk=0.Lemma 8 (see [24, Lemma 1.3]).
Let{δn} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {δnj}j≥0 of {δn} which satisfies δnj<δnj+1 for all j≥0. Also consider the sequence of integers {τ(n)}n≥n0 defined by
(19)τ(n)=max⁡⁡{k≤n∣δk<δk+1}.
Then, {τ(n)}n≥n0 is a nondecreasing sequence verifying lim⁡n→∞τ(n)=∞ and, for all n≥n0, it holds that δτ(n)≤δτ(n)+1 and we have
(20)δn≤δτ(n)+1.


## 3. Viscosity Iterative Algorithm without Prior Knowledge of Operator Norms

In this section, we introduce a viscosity iterative algorithm where the stepsizesγk do not depend on the operator norms ∥A∥ and ∥B∥ and prove the strong convergence of algorithm without prior knowledge of operator norms.Algorithm 9.
Letf1:H1→H1 and f2:H2→H2 be two contractions with constants ρ1, ρ2∈[0,1), and αk∈[0,1]. Choose an initial guess x0∈H1, y0∈H2 arbitrarily. Assume that the kth iterate xk∈H1, yk∈H2 has been constructed; then, we calculate the (k+1)th iterate (xk+1,yk+1) via the formula
(21)uk=xk-γkA*(Axk-Byk),xk+1=αkf1(xk)+(1-αk)U(uk),vk=yk+γkB*(Axk-Byk),yk+1=αkf2(xk)+(1-αk)T(vk).
The stepsize γk is chosen in such a way that
(22)γk∈(ϵ,2∥Axk-Byk∥2∥A*(Axk-Byk)∥2+∥B*(Axk-Byk)∥2-ϵ),bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbk∈Ω,
otherwise, γk=γ (γ being any nonnegative value), where the set of indexes Ω={k:Axk-Byk≠0}.Remark 10.
Note that, in (22), the choice of the stepsize γk is independent of the norms ∥A∥ and ∥B∥. The value of γ does not influence the considered algorithm, but it was introduced just for the sake of clarity. Furthermore, we will see from Lemma 3 that γk is well defined.Lemma 11.
Assume the solution setΓ of (6) is nonempty. Then, γk defined by (22) is well defined.Proof.
Taking(x,y)∈Γ, that is, x∈F(U), y∈F(T), and Ax=By, we have
(23)〈A*(Axk-Byk),xk-x〉=〈Axk-Byk,Axk-Ax〉,〈B*(Axk-Byk),y-yk〉=〈Axk-Byk,By-Byk〉.
By adding the two above equalities and by taking into account the fact that Ax=By, we obtain
(24)∥Axk-Byk∥2=〈A*(Axk-Byk),xk-x〉+〈B*(Axk-Byk),y-yk〉≤∥A*(Axk-Byk)∥·∥xk-x∥+∥B*(Axk-Byk)∥·∥y-yk∥.
Consequently, for k∈Ω, that is, ∥Axk-Byk∥>0, we have ∥A*(Axk-Byk)∥≠0 or ∥B*(Axk-Byk)∥≠0. This leads that γk is well defined.Theorem 12.
LetH1, H2, and H3 be real Hilbert spaces. Given two bounded linear operators A:H1→H3 and B:H2→H3, let U:H1→H1 and T:H2→H2 be firmly quasi-nonexpansive operators with the solution set Γ of (6) being nonempty. Let the sequence {(xk,yk)} be generated by Algorithm 9. Assume that the following conditions are satisfied:(1)
ρ1,ρ2∈[0,1/2);(2)
lim⁡k→∞αk=0 and ∑k=0∞‍αk=∞.(3)
U-I and T-I are demiclosed at origin;
Then, sequence{(xk,yk)} strongly converges to a solution (x*,y*) of (6) which solves the variational inequality problem
(25)〈(I-f1)x*,x-x*〉≥0,〈(I-f2)y*,y-y*〉≥0,Proof.
Let(x*,y*)∈Γ be the solution of the variational inequality problem (25). Then, x*∈F(U), y*∈F(T), and Ax*=By*. We have
(26)∥uk-x*∥2=∥xk-γkA*(Axk-Byk)-x*∥2=∥xk-x*∥2-2γk〈xk-x*,A*(Axk-Byk)〉+γk2∥A*(Axk-Byk)∥2.
Using (17), we have
(27)-2〈xk-x*,A*(Axk-Byk)〉=-2〈Axk-Ax*,Axk-Byk〉=-∥Axk-Ax*∥2-∥Axk-Byk∥2+∥Byk-Ax*∥2.
By (26) and (27), we obtain
(28)∥uk-x*∥2≤∥xk-x*∥2-γk∥Axk-Ax*∥2-γk∥Axk-Byk∥2+γk∥Byk-Ax*∥2+γk2∥A*(Axk-Byk)∥2.
Similarly, we have
(29)∥vk-y*∥2≤∥yk-y*∥2-γk∥Byk-By*∥2-γk∥Axk-Byk∥2+γk∥Axk-By*∥2+γk2∥B*(Axk-Byk)∥2.
By adding the two last inequalities and by taking into account the fact that Ax*=By*, we obtain
(30)∥uk-x*∥2+∥vk-y*∥2≤∥xk-x*∥2+∥yk-y*∥2-γk[2∥Axk-Byk∥211111111-γk(∥A*(Axk-Byk)∥21111111111111+∥B*(Axk-Byk)∥2)].
With assumption on γk, we obtain
(31)∥uk-x*∥2+∥vk-y*∥2≤∥xk-x*∥2+∥yk-y*∥2.
Setting ρ=max⁡{ρ1,ρ1}, we have ρ∈[0,1/2). By U and T being firmly quasi-nonexpansive operators, it follows that
(32)∥xk+1-x*∥2≤αk∥f1(xk)-x*∥2+(1-αk)∥U(uk)-x*∥2≤αk(∥f1(xk)-f1(x*)∥+∥f1(x*)-x*∥)2+(1-αk)∥uk-x*∥2-(1-αk)∥uk-U(uk)∥2≤2αk(∥f1(xk)-f1(x*)∥2+∥f1(x*)-x*∥2)+(1-αk)∥uk-x*∥2-(1-αk)∥uk-U(uk)∥2≤2αkρ12∥xk-x*∥2+2αk∥f1(x*)-x*∥2+(1-αk)∥uk-x*∥2-(1-αk)∥uk-U(uk)∥2,∥yk+1-y*∥2≤2αkρ22∥yk-y*∥2+2αk∥f2(y*)-y*∥2+(1-αk)∥vk-y*∥2-(1-αk)∥vk-T(vk)∥2.
Adding up the last two inequalities and using (31), setting sk=∥xk-x*∥2+∥yk-y*∥2, we get
(33)sk+1≤(1-αk(1-2ρ2))sk+2αk(∥f1(x*)-x*∥2+∥f2(y*)-y*∥2)-(1-αk)(∥uk-U(uk)∥2+∥vk-T(vk)∥2),
which implies
(34)sk+1≤(1-αk(1-2ρ2))sk+αk(1-2ρ2)21-2ρ2×(∥f1(x*)-x*∥2+∥f2(y*)-y*∥2).
It follows from induction that
(35)sk≤max⁡{s0,21-2ρ2(∥f1(x*)-x*∥2+∥f2(y*)-y*∥2)}bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbk≥0,
which implies that {xk} and {yk} are bounded. It follows that {uk},{vk},{f1(xk)} and {f2(yk)} are bounded.
Note thatU is a firmly quasi-nonexpansive operator; we have
(36)∥xk+1-x*∥2=αk2∥f1(xk)-x*∥2+2αk(1-αk)×〈f1(xk)-x*,U(uk)-x*〉+(1-αk)2∥U(uk)-x*∥2=αk2∥f1(xk)-x*∥2+2αk(1-αk)×〈f1(xk)-f1(x*),U(uk)-x*〉+2αk(1-αk)〈f1(x*)-x*,U(uk)-x*〉+(1-αk)2∥U(uk)-x*∥2≤αk2∥f1(xk)-x*∥2+αk(1-αk)×(∥f1(xk)-f1(x*)∥2+∥U(uk)-x*∥2)+2αk(1-αk)〈f1(x*)-x*,U(uk)-x*〉+(1-αk)2∥U(uk)-x*∥2≤αk2∥f1(xk)-x*∥2+αk(1-αk)ρ12∥xk-x*∥2+αk(1-αk)∥U(uk)-x*∥2+2αk(1-αk)〈f1(x*)-x*,U(uk)-x*〉+(1-αk)2∥U(uk)-x*∥2≤(1-αk)∥uk-x*∥2+αk(1-αk)ρ12∥xk-x*∥2+αk[αk∥f1(xk)-x*∥2+2(1-αk)111111×〈f1(x*)-x*,U(uk)-x*〉∥f1(xk)-x*∥2].
Similarly, we have
(37)∥yk+1-y*∥2≤(1-αk)∥vk-y*∥2+αk(1-αk)ρ22∥yk-y*∥2+αk[αk∥f2(yk)-y*∥2+2(1-αk)1111111111×〈f2(y*)-y*,T(vk)-y*〉∥f2(yk)-y*∥2].
So, by (31), (36), and (37), we obtain
(38)sk+1≤(1-αk)sk+αk(1-αk)ρ2sk+αk[αk(∥f1(xk)-x*∥2+∥f2(yk)-y*∥2)11111111+2(1-αk)(〈f1(x*)-x*,U(uk)-x*〉1111111111111111111+〈f2(y*)-y*,T(vk)-y*〉)∥f1(xk)-x*∥2∥f2(yk)-y*∥2]=(1-λk)sk+λkδk,
where
(39)λk=αk(1-(1-αk)ρ2),δk=(2(1-αk)(〈f1(x*)-x*,U(uk)-x*〉11111111111111+〈f2(y*)-y*,T(vk)-y*〉))×(1-(1-αk)ρ2)-11111+αk(∥f1(xk)-x*∥2+∥f2(yk)-y*∥2)1-(1-αk)ρ2.
On the other hand, from (21), we have
(40)∥xk+1-x*∥2≤αk∥f1(xk)-x*∥2+(1-αk)∥U(uk)-x*∥2≤αk∥f1(xk)-x*∥2+(1-αk)∥uk-x*∥2-(1-αk)∥U(uk)-uk∥2,∥yk+1-y*∥2≤αk∥f2(yk)-y*∥2+(1-αk)∥vk-y*∥2-(1-αk)∥T(vk)-vk∥2.
Adding up the last two inequalities and using (30), we obtain
(41)sk+1≤∥uk-x*∥2+∥vk-y*∥2+αk(∥f1(xk)-x*∥2+∥f2(yk)-y*∥2)-(1-αk)(∥U(uk)-uk∥2+∥T(vk)-vk∥2)≤sk-γk[2∥Axk-Byk∥2111111111-γk(∥A*(Axk-Byk)∥2111111111111111+∥B*(Axk-Byk)∥2)]+αk(∥f1(xk)-x*∥2+∥f2(yk)-y*∥2)-(1-αk)(∥U(uk)-uk∥2+∥T(vk)-vk∥2).
Now, by setting μk=αk(∥f1(xk)-x*∥2+∥f2(yk)-y*∥2), ηk=γk[2∥Axk-Byk∥2-γk(∥A*(Axk-Byk)∥2+∥B*(Axk-Byk)∥2)], and θk=(1-αk)(∥U(uk)-uk∥2+∥T(vk)-vk∥2), (41) can be rewritten as the following form:
(42)sk+1≤sk-ηk+μk-θk≤sk-ηk+μk,k≥0.
By the assumption on αk, we get ∑k=0∞‍λk=∞ and lim⁡k→∞μk=0 which thanks to the boundedness of {f1(xk)} and {f2(yk)}.
The rest of the proof will be divided into two parts.
Case
1. Suppose that there exists k0 such that {sk}k≥k0 is nonincreasing. In this situation, {sk} is convergent because it is nonnegative so that lim⁡k→∞(sk+1-sk)=0; hence, in light of (33) together with αk→0 and the boundedness of {sk}, we obtain
(43)lim⁡k→∞∥uk-U(uk)∥=lim⁡k→∞∥vk-T(vk)∥=0.
To use Lemma8, it suffices to verify that, for all subsequences {kl}⊂{k}, lim⁡l→∞ηkl=0 implies
(44)limsup⁡l→∞δkl≤0.
It follows from lim⁡k→∞ηkl=0 that
(45)lim⁡l→∞γkl[2∥Axkl-Bykl∥211111111-γkl(∥A*(Axkl-Bykl)∥211111111111111+∥B*(Axkl-Bykl)∥2)]=0,
which yields lim⁡l→∞∥Axkl-Bykl∥=0 from the assumption on γk. So,
(46)lim⁡l→∞∥ukl-xkl∥=lim⁡l→∞γkl∥A*(Axkl-Bykl)∥=0,lim⁡l→∞∥vkl-ykl∥=lim⁡l→∞γkl∥B*(Axkl-Bykl)∥=0.
Taking (x~,y~)∈ωw(xkl,ykl), from (46), we have (x~,y~)∈ωw(ukl,vkl). Combined with the demiclosednesses of U-I and T-I at 0, (43) yields Ux~=x~ and Ty~=y~. So, x~∈F(U) and y~∈F(T). On the other hand, Ax~-By~∈ωw(Axkl-Bykl) and weakly lower semicontinuity of the norm imply
(47)∥Ax~-By~∥≤liminf⁡l→∞∥Axkl-Bykl∥=0;
hence, (x~,y~)∈Γ. So, ωw(xkl,ykl)⊂Γ. Since lim⁡k→∞αk(∥f1(xk)-x*∥2+∥f2(yk)-y*∥2)=0 and lim⁡k→∞(1-(1-αk)ρ2)=1-ρ2, to get (44), we only need to verify
(48)limsup⁡l→∞(〈f1(x*)-x*,U(ukl)-x*〉11111111+〈f2(y*)-y*,T(vkl)-y*〉)≤0.
Indeed, from (43) and (46), we have
(49)lim⁡sup⁡l→∞(〈f1(x*)-x*,U(ukl)-x*〉1111111111+〈f2(y*)-y*,T(vkl)-y*〉)=lim⁡sup⁡l→∞(〈f1(x*)-x*,ukl-x*〉1111111111111+〈f2(y*)-y*,vkl-y*〉)=lim⁡sup⁡l→∞(〈f1(x*)-x*,xkl-x*〉1111111111111+〈f2(y*)-y*,ykl-y*〉)=-lim⁡inf⁡l→∞(〈(I-f1)x*,xkl-x*〉11111111111111+〈(I-f2)y*,ykl-y*〉).
We can take subsequence {(xklj,yklj)} of {(xkl,ykl)} such that (xklj,yklj)⇀(x~,y~) as j→∞ and
(50)-lim⁡inf⁡l→∞(〈(I-f1)x*,xkl-x*〉1111111111+〈(I-f2)y*,ykl-y*〉〈(I-f1)x*,xkl-x*〉)=-lim⁡j→∞(〈(I-f1)x*,xklj-x*〉11111111+〈(I-f2)y*,yklj-y*〉)=-(〈(I-f1)x*,x~-x*〉+〈(I-f2)y*,y~-y*〉).
Since ωw(xkl,ykl)⊂Γ and (x*,y*) is the solution of the variational inequality problem (25), from (49) and (50), we obtain
(51)limsup⁡l→∞(〈f1(x*)-x*,U(ukl)-x*〉111111111+〈f2(y*)-y*,T(vkl)-y*〉)≤0.
From Lemma 8, it follows
(52)lim⁡k→∞(∥xk-x*∥2+∥yk-y*∥2)=0,
which implies that xk→x* and yk→y*.
Case
2. Suppose there exists a subsequence {skj}j≥0 of {sk} such that skj<skj+1 for all j≥0. In this situation, we consider the sequence of indices {τ(k)} as defined in Lemma 8. It follows that sτ(k)+1-sτ(k)>0. From (42), we have
(53)0≤ητ(k)≤sτ(k)-sτ(k)+1+μτ(k)<μτ(k),k≥0.
So, by lim⁡k→∞μk=0, we obtain
(54)lim⁡k→∞ητ(k)=0.
Again from (42), we get
(55)0≤θτ(k)≤μτ(k)-ητ(k);
hence,
(56)lim⁡k→∞θτ(k)=lim⁡k→∞(1-ατ(k))×(∥U(uτ(k))-uτ(k)∥2+∥T(vτ(k))-vτ(k)∥2)=0.
In light of αk→0, we obtain
(57)lim⁡k→∞∥uτ(k)-U(uτ(k))∥=lim⁡k→∞∥vτ(k)-T(vτ(k))∥=0.
From yτ(k)→0, similar to Case 1, we have
(58)lim⁡k→∞∥Axτ(k)-Byτ(k)∥=lim⁡k→∞∥uτ(k)-xτ(k)∥=lim⁡k→∞∥vτ(k)-yτ(k)∥=0,ωw(xτ(k),yτ(k))⊂Γ, and
(59)lim⁡sup⁡k→∞(〈f1(x*)-x*,U(uτ(k))-x*〉111111111+〈f2(y*)-y*,T(vτ(k))-y*〉)≤0,
which implies
(60)lim⁡sup⁡k→∞δτ(k)≤0.
From sτ(k)+1-sτ(k)>0 and (38), it follows that
(61)λτ(k)sτ(k)≤λτ(k)δτ(k).
Since sτ(k)+1-sτ(k)>0, again from (38), we may assume λτ(k)>0 for all k≥0. It follows from (60) and (61) that lim⁡k→∞sτ(k)=0 and hence
(62)lim⁡k→∞∥xτ(k)-x*∥=lim⁡k→∞∥yτ(k)-y*∥=0.
On the other hand, it follows that
(63)∥xτ(k)+1-xτ(k)∥=∥+(1-ατ(k))(U(uτ(k))-xτ(k))+(1-ατ(k))(U(uτ(k))-xτ(k))ατ(k)(f(xτ(k))-xτ(k))1111+(1-ατ(k))(U(uτ(k))-xτ(k))∥ατ(k)(f(xτ(k))-xτ(k))1111+(1-ατ(k))(U(uτ(k))-xτ(k))∥≤ατ(k)∥f(xτ(k))-xτ(k)∥+(1-ατ(k))×[∥U(uτ(k))-uτ(k)∥+∥uτ(k)-xτ(k)∥],
which, by αk→0, (57), and (58), implies that
(64)lim⁡k→∞∥xτ(k)+1-xτ(k)∥=0.
By (62), we obtain
(65)lim⁡k→∞∥xτ(k)+1-x*∥=0.
Similarly, we have lim⁡k→∞∥yτ(k)+1-y*∥=0; hence,
(66)lim⁡k→∞sτ(k)+1=lim⁡k→∞(∥xτ(k)+1-x*∥2+∥yτ(k)+1-y*∥2)=0.
Then, recalling that sk≤sτ(k)+1 (by Lemma 8), we get lim⁡k→∞sk=0.
So, sequence{(xk,yk)} strongly converges to the solution (x*,y*) of (6) which solves the variational inequality problem (25).


## 4. Another Split Problem Deduced from SCFP

We now turn our attention to providing some algorithms for solving another split problem without prior knowledge of operator norms.


### 4.1. Split Feasibility Problem

TakingU=PC and T=PQ, we have that the following viscosity iterative algorithm for split feasibility problem (SFP) under consideration is nothing but
(67)findx∈C,y∈Q,suchthatAx=By.Algorithm 13.
Letx0∈H1, y0∈H2 be arbitrary. Consider
(68)uk=xk-γkA*(Axk-Byk),xk+1=αkf1(xk)+(1-αk)PC(uk),vk=yk+γkB*(Axk-Byk),yk+1=αkf2(xk)+(1-αk)PQ(vk),
where the stepsize γk is chosen by (22) in Algorithm 9.In [16], Dong et al. introduced Algorithm 13 for SFP (67) without prior knowledge of operator norms. The stepsize γk is chosen in such a way that
(69)γk∈(ϵ,min⁡{∥Axk-Byk∥2∥A*(Axk-Byk)∥2,1111111111∥Axk-Byk∥2∥B*(Axk-Byk)∥2}-ϵ).
It is easy to see that the results of this paper improve and extend the corresponding results of [16].


### 4.2. Split Common Null Point Problem

Given a maximal monotone operatorM:H1→2H1, it is well known that its associated resolvent mapping, JμM(x):=(I+μM)-1, is firmly quasi-nonexpansive and 0∈M(x)⇔x=JμM(x). In other words, zeroes of M are exactly fixed-points of its resolvent mapping. By taking U=JμM, T=JνN, where N:H2→2H2 is another maximal monotone operator, the problem under consideration is nothing but
(70)findx*∈M-1(0),y*∈N-1(0)suchthatAx*=By*,
and the algorithms take the following equivalent form:
(71)∀x0∈H1,y0∈H2,uk=xk-γkA*(Axk-Byk),xk+1=αkf1(xk)+(1-αk)JμM(uk),vk=yk+γkB*(Axk-Byk),yk+1=βkf2(xk)+(1-βk)JνN(vk),k≥1,
The stepsize γk is chosen as follows:
(72)γk∈((∥A*(Axk-Byk)∥2+∥B*(Axk-Byk)∥2)-1ϵ,(2∥Axk-Byk∥2)×(∥A*(Axk-Byk)∥2+∥B*(Axk-Byk)∥2)-1-ϵ),111111111111111111111111111111111111111k∈Ω;
otherwise, γk=γ (γ being any nonnegative value), where the set of indexes Ω={k:Axk-Byk≠0}.


## 4.1. Split Feasibility Problem

TakingU=PC and T=PQ, we have that the following viscosity iterative algorithm for split feasibility problem (SFP) under consideration is nothing but
(67)findx∈C,y∈Q,suchthatAx=By.Algorithm 13.
Letx0∈H1, y0∈H2 be arbitrary. Consider
(68)uk=xk-γkA*(Axk-Byk),xk+1=αkf1(xk)+(1-αk)PC(uk),vk=yk+γkB*(Axk-Byk),yk+1=αkf2(xk)+(1-αk)PQ(vk),
where the stepsize γk is chosen by (22) in Algorithm 9.In [16], Dong et al. introduced Algorithm 13 for SFP (67) without prior knowledge of operator norms. The stepsize γk is chosen in such a way that
(69)γk∈(ϵ,min⁡{∥Axk-Byk∥2∥A*(Axk-Byk)∥2,1111111111∥Axk-Byk∥2∥B*(Axk-Byk)∥2}-ϵ).
It is easy to see that the results of this paper improve and extend the corresponding results of [16].


## 4.2. Split Common Null Point Problem

Given a maximal monotone operatorM:H1→2H1, it is well known that its associated resolvent mapping, JμM(x):=(I+μM)-1, is firmly quasi-nonexpansive and 0∈M(x)⇔x=JμM(x). In other words, zeroes of M are exactly fixed-points of its resolvent mapping. By taking U=JμM, T=JνN, where N:H2→2H2 is another maximal monotone operator, the problem under consideration is nothing but
(70)findx*∈M-1(0),y*∈N-1(0)suchthatAx*=By*,
and the algorithms take the following equivalent form:
(71)∀x0∈H1,y0∈H2,uk=xk-γkA*(Axk-Byk),xk+1=αkf1(xk)+(1-αk)JμM(uk),vk=yk+γkB*(Axk-Byk),yk+1=βkf2(xk)+(1-βk)JνN(vk),k≥1,
The stepsize γk is chosen as follows:
(72)γk∈((∥A*(Axk-Byk)∥2+∥B*(Axk-Byk)∥2)-1ϵ,(2∥Axk-Byk∥2)×(∥A*(Axk-Byk)∥2+∥B*(Axk-Byk)∥2)-1-ϵ),111111111111111111111111111111111111111k∈Ω;
otherwise, γk=γ (γ being any nonnegative value), where the set of indexes Ω={k:Axk-Byk≠0}.



---

*Source: 389689-2014-07-23.xml*