AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/389689 389689 Research Article Solving Split Common Fixed-Point Problem of Firmly Quasi-Nonexpansive Mappings without Prior Knowledge of Operators Norms Zhao Jing Zhang Hang Park Sehie College of Science Civil Aviation University of China Tianjin 300300 China cauc.edu.cn 2014 2372014 2014 11 04 2014 28 06 2014 23 7 2014 2014 Copyright © 2014 Jing Zhao and Hang Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Throughout this paper, we always assume that H is a real Hilbert space with the inner product ·,· and the norm ·. Let I denote the identity operator on H. Let T:HH be a mapping. A point xH 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.

Let C 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)xCsuch  thatAxQ, where A:H1H2 is a bounded linear operator. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving  for modeling inverse problems which arise from phase retrievals and in medical image reconstruction .

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  for the details). This implies that we can use fixed point algorithms (see ) to solve SFP. To solve (2), Byrne  proposed his CQ algorithm which generates a sequence {xk} by (3)xk+1=PC(I-γA*(I-PQ)A)xk,kN, where γ(0,2/λ) with λ being the spectral radius of the operator A*A.

Censor and Segal  introduced the following split common fixed-point problem (SCFP): (4)findx*F(U)suchthatAx*F(T), where A:H1H2 is a bounded linear operator and U:H1H1 and T:H2H2 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 ).

To solve (4), Censor and Segal  proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm: (5)xk+1=U(xk+γAt(T-I)Axk),kN, where γ(0,2/λ), with λ being the largest eigenvalue of the matrix AtA (At stands for matrix transposition).

Let H1, H2, and H3 be real Hilbert spaces; let A:H1H3 and B:H2H3 be two bounded linear operators; let U:H1H1 and T:H2H2 be two firmly quasi-nonexpansive operators. In , Moudafi introduced the following split common fixed-point problem (SCFP): (6)findxF(U),yF(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 ). In IMRT, these amounts envisage a weak coupling between the vector of doses absorbed in all voxels and that of the radiation intensity (see ).

If H2=H3 and B=I, then SCFP (6) reduces to SCFP (4). For solving SCFP (6), Moudafi  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  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 , 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  and Zhao and Yang  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,2Axk-Byk2A*(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:xkjx} 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 operator T:HH is said to be

nonexpansive if Tx-Tyx-y for all x,yH,

quasi-nonexpansive if F(T) and if Tx-qx-q for all xH and qF(T),

firmly nonexpansive if Tx-Ty2x-y2-(x-y)-(Tx-Ty)2 for all x,yH,

firmly quasi-nonexpansive if F(T) and if Tx-q2x-q2-x-Tx2 for all xH and qF(T).

Remark 2.

A firmly quasi-nonexpansive operator is also called a separating operator , cutter operator , directed operators [7, 20], or class-T operator which was introduced by Bauschke and Combettes . 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:HR is a firmly quasi-nonexpansive operator. Recall that the subgradient projection T is defined by, assuming that the level set {xH: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 xH).

Particularly, projections are firmly quasi-nonexpansive operators. Recall that, given a closed convex subset C of a Hilbert space H, the projection PC:HC assigns each xH to its closest point from C defined by (12)PCx=argminzCx-z. It is well known that PCx is characterized by the inequality (13)PCxC,x-PCx,z-PCx0,zC.

Lemma 3 (see [<xref ref-type="bibr" rid="B12">19</xref>, <xref ref-type="bibr" rid="B5">21</xref>]).

The fixed point set of a firmly quasi-nonexpansive operator is closed convex.

We also need other classes of operators.

Definition 4.

An operator T:HH 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 operator T may be not nonexpansive; even T-I is demiclosed at origin. See the following example .

Example 5.

Let H=R1 and define a mapping by T:HH by (14)Tx{x2sin1x,x0,0,x=0. Then, F(T)={0} and (15)|Tx-0|2=x24(sin1x)2x2-(x-x2sin1x)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 operator T:HH is called contraction with constant ρ[0,1) if, for any x,yH, (16)Tx-Tyρx-y.

In real Hilbert space, we easily get the following equality: (17)2x,y=x2+y2-x-y2=x+y2-x2-y2,x,yH.

We end this section by the following lemmas, which are important in convergence analysis for our iterative algorithm.

Lemma 7 (see [<xref ref-type="bibr" rid="B15">23</xref>]).

Assume {sk} is a sequence of nonnegative real numbers such that (18)sk+1(1-λk)sk+λkδk,k0,sk+1sk-ηk+μk,k0, 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

Σk=1λk=;

limkμk=0;

limlηkl=0 implies limsuplδkl0 for any subsequence {kl}{k}.

Then, limksk=0.

Lemma 8 (see [<xref ref-type="bibr" rid="B26">24</xref>, 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}j0 of {δn} which satisfies δnj<δnj+1 for all j0. Also consider the sequence of integers {τ(n)}nn0 defined by (19)τ(n)=max{knδk<δk+1}. Then, {τ(n)}nn0 is a nondecreasing sequence verifying limnτ(n)= and, for all nn0, it holds that δτ(n)δτ(n)+1 and we have (20)δnδτ(n)+1.

3. Viscosity Iterative Algorithm without Prior Knowledge of Operator Norms

In this section, we introduce a viscosity iterative algorithm where the stepsizes γk 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.

Let f1:H1H1 and f2:H2H2 be two contractions with constants ρ1, ρ2[0,1), and αk[0,1]. Choose an initial guess x0H1, y0H2 arbitrarily. Assume that the kth iterate xkH1, ykH2 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,2Axk-Byk2A*(Axk-Byk)2+B*(Axk-Byk)2-ϵ),bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbkΩ, otherwise, γk=γ (γ being any nonnegative value), where the set of indexes Ω={k:Axk-Byk0}.

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 set Γ of (6) is nonempty. Then, γk defined by (22) is well defined.

Proof.

Taking (x,y)Γ, that is, xF(U), yF(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-Byk2=A*(Axk-Byk),xk-x+B*(Axk-Byk),y-ykA*(Axk-Byk)·xk-x+B*(Axk-Byk)·y-yk. Consequently, for kΩ, 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.

Let H1, H2, and H3 be real Hilbert spaces. Given two bounded linear operators A:H1H3 and B:H2H3, let U:H1H1 and T:H2H2 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:

ρ1,ρ2[0,1/2);

limkαk=0 and k=0αk=.

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*2-2γkxk-x*,A*(Axk-Byk)+γk2A*(Axk-Byk)2. Using (17), we have (27)-2xk-x*,A*(Axk-Byk)=-2Axk-Ax*,Axk-Byk=-Axk-Ax*2-Axk-Byk2+Byk-Ax*2. By (26) and (27), we obtain (28)uk-x*2xk-x*2-γkAxk-Ax*2-γkAxk-Byk2+γkByk-Ax*2+γk2A*(Axk-Byk)2. Similarly, we have (29)vk-y*2yk-y*2-γkByk-By*2-γkAxk-Byk2+γkAxk-By*2+γk2B*(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*2xk-x*2+yk-y*2-γk[2Axk-Byk211111111-γk(A*(Axk-Byk)21111111111111+B*(Axk-Byk)2)]. With assumption on γk, we obtain (31)uk-x*2+vk-y*2xk-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*2αkf1(xk)-x*2+(1-αk)U(uk)-x*2αk(f1(xk)-f1(x*)+f1(x*)-x*)2+(1-αk)uk-x*2-(1-αk)uk-U(uk)22αk(f1(xk)-f1(x*)2+f1(x*)-x*2)+(1-αk)uk-x*2-(1-αk)uk-U(uk)22αkρ12xk-x*2+2αkf1(x*)-x*2+(1-αk)uk-x*2-(1-αk)uk-U(uk)2,yk+1-y*22αkρ22yk-y*2+2αkf2(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)skmax{s0,21-2ρ2(f1(x*)-x*2+f2(y*)-y*2)}bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbk0, which implies that {xk} and {yk} are bounded. It follows that {uk},{vk},{f1(xk)} and {f2(yk)} are bounded.

Note that U is a firmly quasi-nonexpansive operator; we have (36)xk+1-x*2=αk2f1(xk)-x*2+2αk(1-αk)×f1(xk)-x*,U(uk)-x*+(1-αk)2U(uk)-x*2=αk2f1(xk)-x*2+2αk(1-αk)×f1(xk)-f1(x*),U(uk)-x*+2αk(1-αk)f1(x*)-x*,U(uk)-x*+(1-αk)2U(uk)-x*2αk2f1(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)2U(uk)-x*2αk2f1(xk)-x*2+αk(1-αk)ρ12xk-x*2+αk(1-αk)U(uk)-x*2+2αk(1-αk)f1(x*)-x*,U(uk)-x*+(1-αk)2U(uk)-x*2(1-αk)uk-x*2+αk(1-αk)ρ12xk-x*2+αk[αkf1(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)ρ22yk-y*2+αk[αkf2(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*2f2(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*2αkf1(xk)-x*2+(1-αk)U(uk)-x*2αkf1(xk)-x*2+(1-αk)uk-x*2-(1-αk)U(uk)-uk2,yk+1-y*2αkf2(yk)-y*2+(1-αk)vk-y*2-(1-αk)T(vk)-vk2. Adding up the last two inequalities and using (30), we obtain (41)sk+1uk-x*2+vk-y*2+αk(f1(xk)-x*2+f2(yk)-y*2)-(1-αk)(U(uk)-uk2+T(vk)-vk2)sk-γk[2Axk-Byk2111111111-γk(A*(Axk-Byk)2111111111111111+B*(Axk-Byk)2)]+αk(f1(xk)-x*2+f2(yk)-y*2)-(1-αk)(U(uk)-uk2+T(vk)-vk2). Now, by setting μk=αk(f1(xk)-x*2+f2(yk)-y*2), ηk=γk[2Axk-Byk2-γk(A*(Axk-Byk)2+B*(Axk-Byk)2)], and θk=(1-αk)(U(uk)-uk2+T(vk)-vk2), (41) can be rewritten as the following form: (42)sk+1sk-ηk+μk-θksk-ηk+μk,k0. By the assumption on αk, we get k=0λk= and limkμk=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}kk0 is nonincreasing. In this situation, {sk} is convergent because it is nonnegative so that limk(sk+1-sk)=0; hence, in light of (33) together with αk0 and the boundedness of {sk}, we obtain (43)limkuk-U(uk)=limkvk-T(vk)=0.

To use Lemma 8, it suffices to verify that, for all subsequences {kl}{k}, limlηkl=0 implies (44)limsuplδkl0. It follows from limkηkl=0 that (45)limlγkl[2Axkl-Bykl211111111-γkl(A*(Axkl-Bykl)211111111111111+B*(Axkl-Bykl)2)]=0, which yields limlAxkl-Bykl=0 from the assumption on γk. So, (46)limlukl-xkl=limlγklA*(Axkl-Bykl)=0,limlvkl-ykl=limlγklB*(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~liminflAxkl-Bykl=0; hence, (x~,y~)Γ. So, ωw(xkl,ykl)Γ. Since limkαk(f1(xk)-x*2+f2(yk)-y*2)=0 and limk(1-(1-αk)ρ2)=1-ρ2, to get (44), we only need to verify (48)limsupl(f1(x*)-x*,U(ukl)-x*11111111+f2(y*)-y*,T(vkl)-y*)0. Indeed, from (43) and (46), we have (49)limsupl(f1(x*)-x*,U(ukl)-x*1111111111+f2(y*)-y*,T(vkl)-y*)=limsupl(f1(x*)-x*,ukl-x*1111111111111+f2(y*)-y*,vkl-y*)=limsupl(f1(x*)-x*,xkl-x*1111111111111+f2(y*)-y*,ykl-y*)=-liminfl((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)-liminfl((I-f1)x*,xkl-x*1111111111+(I-f2)y*,ykl-y*(I-f1)x*,xkl-x*)=-limj((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)limsupl(f1(x*)-x*,U(ukl)-x*111111111+f2(y*)-y*,T(vkl)-y*)0. From Lemma 8, it follows (52)limk(xk-x*2+yk-y*2)=0, which implies that xkx* and yky*.

Case 2. Suppose there exists a subsequence {skj}j0 of {sk} such that skj<skj+1 for all j0. 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)0ητ(k)sτ(k)-sτ(k)+1+μτ(k)<μτ(k),k0. So, by limkμk=0, we obtain (54)limkητ(k)=0. Again from (42), we get (55)0θτ(k)μτ(k)-ητ(k); hence, (56)limkθτ(k)=limk(1-ατ(k))×(U(uτ(k))-uτ(k)2+T(vτ(k))-vτ(k)2)=0. In light of αk0, we obtain (57)limkuτ(k)-U(uτ(k))=limkvτ(k)-T(vτ(k))=0. From yτ(k)0, similar to Case 1, we have (58)limkAxτ(k)-Byτ(k)=limkuτ(k)-xτ(k)=limkvτ(k)-yτ(k)=0,ωw(xτ(k),yτ(k))Γ, and (59)limsupk(f1(x*)-x*,U(uτ(k))-x*111111111+f2(y*)-y*,T(vτ(k))-y*)0, which implies (60)limsupkδτ(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 k0. It follows from (60) and (61) that limksτ(k)=0 and hence (62)limkxτ(k)-x*=limkyτ(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 αk0, (57), and (58), implies that (64)limkxτ(k)+1-xτ(k)=0. By (62), we obtain (65)limkxτ(k)+1-x*=0. Similarly, we have limkyτ(k)+1-y*=0; hence, (66)limksτ(k)+1=limk(xτ(k)+1-x*2+yτ(k)+1-y*2)=0. Then, recalling that sksτ(k)+1 (by Lemma 8), we get limksk=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

Taking U=PC and T=PQ, we have that the following viscosity iterative algorithm for split feasibility problem (SFP) under consideration is nothing but (67)findxC,yQ,suchthatAx=By.

Algorithm 13.

Let x0H1, y0H2 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 , 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-Byk2A*(Axk-Byk)2,1111111111Axk-Byk2B*(Axk-Byk)2}-ϵ). It is easy to see that the results of this paper improve and extend the corresponding results of .

4.2. Split Common Null Point Problem

Given a maximal monotone operator M:H12H1, it is well known that its associated resolvent mapping, JμM(x):=(I+μM)-1, is firmly quasi-nonexpansive and 0M(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:H22H2 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)x0H1,y0H2,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),k1, The stepsize γk is chosen as follows: (72)γk((A*(Axk-Byk)2+B*(Axk-Byk)2)-1ϵ,(2Axk-Byk2)×(A*(Axk-Byk)2+B*(Axk-Byk)2)-1-ϵ),111111111111111111111111111111111111111kΩ; otherwise, γk=γ (γ being any nonnegative value), where the set of indexes Ω={k:Axk-Byk0}.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research was supported by Fundamental Research Funds for the Central Universities (Program no. 3122013k004).

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