# An Iterative Method for Solving Split Monotone Variational Inclusion Problems and Finite Family of Variational Inequality Problems in Hilbert Spaces

Authors: Wanna Sriprad; Somnuk Srisawat
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

The purpose of this paper is to study the convergence analysis of an intermixed algorithm for finding the common element of the set of solutions of split monotone variational inclusion problem (SMIV) and the set of a finite family of variational inequality problems. Under the suitable assumption, a strong convergence theorem has been proved in the framework of a real Hilbert space. In addition, by using our result, we obtain some additional results involving split convex minimization problems (SCMPs) and split feasibility problems (SFPs). Also, we give some numerical examples for supporting our main theorem.


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## Body

## 1. Introduction

LetH1 and H2 be real Hilbert spaces whose inner product and norm are denoted by ·,· and ·, respectively, and let C, Q be nonempty closed convex subsets of H1 and H2, respectively. For a mapping S:C⟶C, we denoted by FS the set of fixed points of S (i.e., FS=x∈C:Sx=x). Let A:C⟶H be a nonlinear mapping. The variational inequality problem (VIP) is to find x∗∈C such that(1)Ax∗,y−x∗≥0,∀y∈C,and the solution set of problem (1) is denoted by VIC,A. It is known that the variational inequality, as a strong and great tool, has already been investigated for an extensive class of optimization problems in economics and equilibrium problems arising in physics and many other branches of pure and applied sciences. Recall that a mapping A:C⟶C is said to be α-inverse strongly monotone if there exists α>0 such that(2)Ax−Ay,x−y≥αAx−Ay2,∀x,y∈C.A multivalued mappingM:H1⟶2H1 is called monotone if for all x,y∈H1, x−y,u−v≥0, for any u∈Mx and v∈My. A monotone mapping M:H1⟶2H1 is maximal if the graph GM for M is not properly contained in the graph of any other monotone mapping. It is generally known that M is maximal if and only if for x,u∈H1×H1,x−y,u−v≥0 for all y,v∈GM implies u∈Mx. Let M:H1⟶2H1 be a multivalued maximal monotone mapping. The resolvent mapping JλM:H1⟶H1 associated with M is defined by(3)JλMx≔I+λM−1x,∀x∈H1,λ>0,where I stands for the identity operator on H1. We note that for all λ>0, the resolvent JλM is single-valued, nonexpansive, and firmly nonexpansive.In 2011, Moudafi [1] introduced the following split monotone variational inclusion problem (SMVI):(4)findx∗∈H1such thatθ∈A1x∗+M1x∗(5)and such thaty∗=Tx∗∈H2solvesθ∈A2y∗+M2y∗,where θ is the zero vector in H1 and H2, M1:H1⟶2H1 and M2:H2⟶2H2 are multivalued mappings on H1 and H2, A1:H1⟶H1 and A2:H2⟶H2 are two given single-valued mappings, and T:H1⟶H2 is a bounded linear operator with adjoint T∗ of T. We note that if (4) and (5) are considered separately, we have that (4) is a variational inclusion problem with its solution set VIH1,A1,M1 and (5) is a variational inclusion problem with its solution set VIH1,A2,M2. We denoted the set of all solutions of (SMVI) by Ω=x∗∈VIH1,A1,M1:Tx∗∈VIH2,A2,M2.It is worth noticing that by takingM1=NCandM2=NQ normal cones to closed convex sets C and Q, then (SMVI) (4) and (5) reduce to the split variational inequality problem (SVIP) that was introduced by Censor et at. [2]. In [1], they mentioned that (SMVI) (4) and (5) contain many special cases, such as split minimization problem (SMP), split minimax problem (SMMP), and split equilibrium problem (SEP). Some related works can be found in [1, 3–10].For solving (SMVI) (4) and (5), Modafi  [1] proposed the following algorithm.Algorithm 1.
Letλ>0, x0∈H1, and the sequence xn be generated by(6)xn+1=JλM11−λfxn+γT∗JλM2I−λg−ITxn,n∈ℕ,where γ∈1,1/L with L being the spectral radius of the operator T∗T.He obtained the following weak convergence theorem for algorithm (6).Theorem 1.
(see [1]). Let H1, H2 be real Hilbert spaces. Let T:H1⟶H2 be a bounded linear operator with adjoint T∗. For i=1,2, let Ai:Hi⟶Hi be αi-inverse strongly monotone with α=minα1,α2 and let Mi:Hi⟶2Mi be two maximal monotone operators. Then, the sequence generated by (6) converges weakly to an element x∗∈Ω provided that Ω≠∅, λ∈0,2α, and γ∈1,1/L with L being the spectral radius of the operator T∗T.Since then, because of a lot of applications of (SMVI), it receives much attention from many authors. They presented many approximation methods for solving (SMVI) (4) and (5). Also the iterative methods for solving (SMVIP) (4) and (5) and fixed-point problems of some nonlinear mappings have been investigated (see [11–19]).On the other hand, Yao et al. [20] presented an intermixed Algorithm 1.3 for two strict pseudo-contractions in real Hilbert spaces. They also showed that the suggested algorithms converge strongly to the fixed points of two strict pseudo-contractions, independently. As a special case, they can find the common fixed points of two strict pseudo-contractions in Hilbert spaces (i.e., a mapping S:C⟶C is said to be κ-strictly pseudo-contractive if there exists a constant κ∈0,1 such that Sx−Sy2≤x−y2+kI−Sx−I−S,0.3cm∀x,y∈C).Algorithm 2.
For arbitrarily givenx0,y0∈C, let the sequences xn and yn be generated iteratively by(7)xn+1=1−βnxn+βnPCαnfyn+1−k−αnxn+kTxn,n≥0,yn+1=1−βnyn+βnPCαnfxn+1−k−αnyn+kSyn,n≥0,where αn and βn are two sequences of the real number in 0,1, T,S:C⟶C are λ-strictly pseudo-contractions, f:C⟶H is a ρ1-contraction, g:C⟶H is a ρ2-contraction, and k∈0,1−λ is a constant.Under some control conditions, they proved that the sequencexn converges strongly to PFTfy∗ and yn converges strongly to PFSfx∗, respectively, where x∗∈FT, y∗∈FS, and PFTandPFS are the metric projection of H onto FT and FS, respectively. After that, many authors have developed and used this algorithm to solve the fixed-point problems of many nonlinear operators in real Hilbert spaces (see for example [21–27]). Question: can we prove the strong convergence theorem of two sequences of split monotone variational inclusion problems and fixed-point problems of nonlinear mappings in real Hilbert spaces?The purpose of this paper is to modify an intermixed algorithm to answer the question above and prove a strong convergence theorem of two sequences for finding a common element of the set of solutions of (SMVI) (4) and (5) and the set of solutions of a finite family of variational inequality problems in real Hilbert spaces. Furthermore, by applying our main result, we obtain some additional results involving split convex minimization problems (SCMPs) and split feasibility problems (SFPs). Finally, we give some numerical examples for supporting our main theorem.


## 2. Preliminaries

LetH be a real Hilbert space and C be a nonempty closed convex subset of H. We denote the strong convergence of xn to x and the weak convergence of xn to x by notations “xn⟶x as n⟶∞” and “xn⇀x as n⟶∞,” respectively. For each x,y,z∈H and α,β,γ∈0,1 with α+β+γ=1, we have(8)x+y2≤x2+2y,x+y,αx+βy+γz2=αx2+βy2+γx2−αβx−y2−αγx−z2−βγy−z2.Definition 1.
LetH be a real Hilbert space and C be a closed convex subset of H. Let S:C⟶C be a mapping. Then, S is said to be(1)
Monotone, ifSx−Sy,x−y≥0,∀x,y∈H(2)
Firmly nonexpansive, ifSx−Sy,x−y≥Sx−Sy2,∀x,y∈H(3)
Lipschitz continuous, if there exists a constantL>0 such that Sx−Sy≤Lx−y,∀x,y∈H(4)
Nonexpansive, if =Sx−Sy≤x−y,∀x,y∈HIt is well known that ifS is α-inverse strongly monotone, then it is 1/α-Lipschitz continuous and every nonexpansive mapping S is 1-Lipschitz continuous. We note that if S:H⟶H is a nonexpansive mapping, then it satisfies the following inequality (see Theorem 3 in [28] and Theorem 1 in [29]):(9)Sy−Sx,I−Sx−I−Sy≤12I−Sx−I−Sy2,∀x,y∈H.Particularly, for every x∈H and y∈FS, we have(10)y−Sx,I−Sx≤12I−Sx2.For everyx∈H, there is a unique nearest point PCx in C such that(11)x−PCx≤x−y,∀y∈C.Such an operatorPC is called the metric projection of H onto C.Lemma 1.
(see [30]). For a given z∈H and u∈C,(12)u=PCz⇔u−z,v−u≥0,∀v∈C.
Furthermore,PC is a firmly nonexpansive mapping of H onto C and satisfies(13)PCx−PCy2≤PCx−PCy,x−y,∀x,y∈H.Moreover, we also have the following lemma.Lemma 2.
(see [31]). Let H be a real Hilbert space, let C be a nonempty closed convex subset of H, and let A be a mapping of C into H. Let u∈C. Then, for λ>0,(14)u∈VIC,A⇔u=PCI−λAu,where PC is the metric projection of H onto C.Lemma 3.
LetC be a nonempty closed and convex subset of a real Hilbert space H. For every i=1,2,…,N, let Ai:C⟶H be the αi-inverse strongly monotone with α¯=mini=1,2,…,Nαi. If ∩i=1NVIC,Ai≠∅, then(15)VIC,∑i=1NaiAi=∩i=1NVIC,Ai,where 0<ai<1 for all i=1,2,…,N and ∑i=1Nai=1. Moreover, I−λ∑i=1NaiAi is a nonexpansive mapping for all λ∈0,2α¯.Proof.
By Lemma 4.3 of [32], we have that VIC,∑i=1NaiAi=∩i=1NVIC,Ai. Let λ∈0,2α¯ and let x,y∈C. As the same argument as in the proof of Lemma 8 in [16], we have I−λ∑i=1NaiAi as nonexpansive.Lemma 4.
(see [33]). Let H be a real Hilbert space, A:H⟶H be a single-valued nonlinear mapping, and M:H⟶2H be a set-valued mapping. Then, a point u∈H is a solution of variational inclusion problem if and only if u=JλMI−λAu,∀λ>0, i.e.,(16)VIH,A,M=FJλMI−λA,∀λ>0.
Furthermore, ifA is α-inverse strongly monotone and λ∈0,2α, then VIH,A,M is a closed convex subset of H.Lemma 5.
(see [33]). The resolvent operator JλM associated with M is single-valued, nonexpansive, and 1-inverse strongly monotone for all λ>0.The following two lemmas are the particular case of Lemmas7 and 8 in [16].Lemma 6.
(see [16]). For every i=1,2, let Hi be real Hilbert spaces, let Mi:Hi⟶2Hi be a multivalued maximal monotone mapping, and let Ai:Hi⟶Hi be an αi-inverse strongly monotone mapping. Let T:H1⟶H2 be a bounded linear operator with adjoint T∗ of T, and let G˜:H1⟶H1 be a mapping defined by G˜x=Jλ1M1I−λ1A1x−γT∗I−Jλ2M2I−λ2A2Tx, for all x∈H1. Then, G˜x−G˜y2≤x−y2−γ1−γLI−Jλ2M2I−λ2A2Tx−I−Jλ2M2I−λ2A2Ty2, for all x,y∈H1, where L is the spectral radius of the operator T∗T, λ1∈0,2α1,λ2∈0,2α2, and γ>0. Furthermore, if 0<γ<1/L, then G˜ is a nonexpansive mapping.Lemma 7.
(see [16]). Let H1 and H2 be Hilbert spaces. For i=1,2, let Mi:Hi⟶2Hi be a multivalued maximal monotone mapping and let Ai:Hi⟶Hi be an αi-inverse strongly monotone mapping. Let T:H1⟶H2 be a bounded linear operator with adjoint T∗. Assume that Ω≠ϕ. Then, x∗∈Ω if and only if x∗=G˜x∗, where G˜:H1⟶H1 is a mapping defined by(17)G˜x=Jλ1M1I−λ1A1x−γT∗I−Jλ2M2I−λ2A2Tx,for all x∈H1, λ1∈0,2α1,λ2∈0,2α2, and 0<γ<1/L,whereL is the spectral radius of the operator T∗T.Next, we give an example to support Lemma7.Example 1.
Letℝ be a set of real number and H1=H2=ℝ2, and let ·,·:ℝ2×ℝ2⟶ℝ be inner product defined by x,y=x·y=x1y1+x2y2, for all x=x1,x2∈ℝ2 and y=y1,y2∈ℝ2 and the usual norm ·:ℝ2⟶ℝ given by x=x12+x22, for all x=x1,x2∈ℝ2. Let T:H1⟶H2 be defined by Tx=2x1,2x2 for all x=x1,x2∈ℝ2 and T∗:ℝ2⟶ℝ2 be defined by T∗z=2z1,2z2 for all z=z1,z2∈ℝ2. Let M1,M2:ℝ2⟶2ℝ2 be defined by M1x=3x1−5,3x2−5andM2x=x1/3−2,x2/3−2, respectively, for all x=x1,x2∈ℝ2. Let the mapping A1,A2:ℝ2⟶ℝ2 be defined by A1x=x1−4/2,x2−4/2andA2x=x1−2/3,x2−2/3, respectively, for all x=x1,x2∈ℝ2. Then, 2,2 is a fixed point of G˜. That is, 2,2∈FG˜.Proof.
It is obvious to see thatΩ=2,2, A1 is 2-inverse strongly monotone, and A2 is 3-inverse strongly monotone. Choose λ1=1/3. Since M1x=3x1−5,3x2−5 and the resolvent of M1,Jλ1M1x=I+λ1M1−1x for all x=x1,x2∈ℝ2, we obtain that(18)Jλ1M1x=x2+56,for allx=x1,x2∈ℝ2.
Chooseλ2=1. Since M2x=x1/3−2,x2/3−2 and the resolvent of M2,Jλ2M2x=I+λ2M2−1x for all x=x1,x2∈ℝ2, we obtain that(19)Jλ2M2x=3x4+32,for allx=x1,x2∈ℝ2.
Since the spectral radius of the operatorT∗T is 4, we choose γ=0.1. Then, from (18) and (19), we get that(20)G˜x=J1/3M1I−13A1x−0.1T∗I−J1M2I−A2Tx,=x3+43,for all x=x1,x2∈ℝ2. Then, by Lemma 7, we have that 2,2∈FG˜.Lemma 8.
(see [34]). Let sn be a sequence of nonnegative real numbers satisfying sn+1≤1−αnsn+δn,∀n≥0 where αn is a sequence in 0,1 and δn is a sequence in ℝ such that(1)
∑n=0∞αn=∞.(2)
limsupn⟶∞αn/δn≤0 or ∑n=0∞δn<∞. Then limn⟶∞sn=0.


## 3. Main Results

In this section, we introduce an iterative algorithm of two sequences which depend on each other by using the intermixed method. Then, we prove a strong convergence theorem for solving two split monotone variational inclusion problems and a finite family of variational inequality problems.Theorem 2.
LetH1 and H2 be Hilbert spaces, and let C be a nonempty closed convex subset of H1. Let T:H1⟶H2 be a bounded linear operator, and let f,g:H1⟶H1 be ρf,ρg-contraction mappings with ρ=maxρf,ρg. For i=1,2, let Mix,Miy:Hi⟶2Hi be multivalued maximal monotone mappings and let Aix,Aiy:Hi⟶Hi be αix,αiy−inverse strongly monotone mappings, respectively. For i=1,2,…,N, let Bix,Biy:H1⟶H1 be βix,βiy−inverse strongly monotone mappings, respectively, β¯x=mini=1,2,…,Nβix, and β¯y=mini=1,2,…,Nβiy. Let G˜x,G˜y:H1⟶H1 be defined by G˜xx=Jλ1xM1xI−λ1xA1xx−γxT∗I−Jλ2xM2xI−λ2xA2xTx,∀x∈H1, and G˜yy=Jλ1yM1yI−λ1yA1yy−γyT∗I−Jλ2yM2yI−λ2yA2yTy,∀y∈H1, respectively, where λix∈0,2αix,λiy∈0,2αiy, and 0<γx,γy<1/L with L being a spectral radius of T∗T. Assume that ℱx=Ωx∩∩i=1NVIC,Bix≠∅ and ℱy=Ωy∩∩i=1NVIC,Biy≠∅. Let xn and yn be sequences generated by x1,y1∈H1 and(21)xn+1=δnxn+σnPCI−μnx∑i=1NaixBixxn+ηnαnfyn+1−αnG˜xxn,yn+1=δnyn+σnPCI−μny∑i=1NaiyBiyyn+ηnαngxn+1−αnG˜yyn,for all n≥1 where δn,σn,ηn,αn⊆0,1 with δn+σn+ηn=1, a1x,a2x,…,aNx,a1y,a2y,…,aNy⊂0,1, and μnx,μny⊂0,∞. Assume the following condition holds:(1)
∑n=1∞μnx<∞,∑n=1∞μny<∞, and 0<a<μnx≤2βx¯,0<b<μny≤2βy¯, for some a,b∈ℝ.(2)
∑n=1∞αn=∞,limn⟶∞αn=0.(3)
∑n=1Naix=∑n=1Naiy=1.(4)
0<a¯≤δn,σn,ηn≤b¯<1, for all n∈ℕ, for some a¯,b¯>0.(5)
∑n=1∞δn+1−δn,∑n=1∞σn+1−σn,and∑n=1∞αn+1−αn<∞. Then, xn converges strongly to x˜=Pℱxfy˜ and yn converges strongly to y˜=Pℱygx˜.Proof.
We divided the proof into five steps.Step 1.
We will show thatxn and yn are bounded. Let x∗∈ℱx and y∗∈ℱy. Then, from Lemma 7 and Lemma 6, we get(22)G˜xxn−x∗=Jλ1xM1xI−λ1xA1xxn−γxT∗I−Jλ2xM2xI−λ2xA2xTxn−x∗,≤xn−x∗.
From (21), Lemma 3, and (22), we have(23)xn+1−x∗=δnxn+σnPCI−μnx∑i=1NaixBixxn+ηnαnfyn+1−αnG˜xxn−x∗,≤δnxn−x∗+σnPCI−μnx∑i=1NaixBixxn−x∗+ηnαnfyn+1−αnG˜xxn−x∗≤δnxn−x∗+σnxn−x∗+ηnαnfyn−x∗+1−αnG˜xxn−x∗≤1−ηnxn−x∗+ηnαnfyn−x∗+1−αnG˜xxn−x∗≤1−ηnxn−x∗+ηnαnfyn−fy∗+αnfy∗−x∗+1−αnxn−x∗≤1−ηnxn−x∗+ηnαnρfyn−y∗+αnfy∗−x∗+1−αnxn−x∗≤1−ηnxn−x∗+ηnαnρyn−y∗+ηnαnfy∗−x∗+ηn1−αnxn−x∗=1−ηnαnxn−x∗+ηnαnρyn−y∗+ηnαnfy∗−x∗.
Similarly, from definition ofyn, we have(24)yn+1−y∗≤1−ηnαnyn−y∗+ηnαnρxn−x∗+ηnαngx∗−y∗.
Hence, from (23) and (24), we obtain(25)xn+1−x∗+yn+1−y∗≤1−ηnαnxn−x∗+yn−y∗+ηnαnρxn−x∗+yn−y∗+ηnαnfy∗−x∗+gx∗−y∗=1−1−ρηnαnxn−x∗+yn−y∗+ηnαnfy∗−x∗+gx∗−y∗.
By induction, we have(26)xn−x∗+yn−y∗≤maxx1−x∗+y1−y∗,fy∗−x∗+gx∗−y∗1−ρ,for every n∈ℕ. Thus, xn and yn are bounded.Step 2.
We will show thatlimn⟶∞xn+1−xn=limn⟶∞yn+1−yn=0. Put un=PCI−μnx∑i=1NaixBixxn, vn=PCI−μny∑i=1NaiyBiyyn, zn=αnfyn+1−αnG˜xxn, and wn=αnfxn+1−αnG˜yyn, for all n≥1. From Lemma 6, we have(27)zn−zn−1=αnfyn+1−αnG˜xxn−αn−1fyn−1+1−αn−1G˜xxn−1,≤αnfyn−fyn−1+1−αnG˜xxn−G˜xxn−1+αn−αn−1fyn−1+αn−αn−1G˜xxn−1≤αnρfyn−yn−1+1−αnxn−xn−1+αn−αn−1fyn−1+αn−αn−1G˜xxn−1≤αnρyn−yn−1+1−αnxn−xn−1+αn−αn−1fyn−1+G˜xxn−1.
By applying Lemma3, we get that(28)un−un−1=PCI−μnx∑i=1NaixBixxn−PCI−μn−1x∑i=1NaixBixxn−1,≤xn−xn−1+PCI−μnx∑i=1NaixBixxn−1−PCI−μn−1x∑i=1NaixBixxn−1≤xn−xn−1+μnx−μn−1x∑i=1NaixBixxn−1.
From the definition ofxn, (27), and (28), we have(29)xn+1−xn=δnxn+σnun+ηnzn−δn−1xn−1+σn−1un−1+ηn−1zn−1,≤δnxn−xn−1+δn−δn−1xn−1+σnun−un−1+σn−σn−1un−1+ηnzn−zn−1+ηn−ηn−1zn−1≤δnxn−xn−1+δn−δn−1xn−1+σnxn−xn−1+μnx−μn−1x∑i=1NaixBixxn−1+σn−σn−1un−1+ηnαnρyn−yn−1+1−αnxn−xn−1+αn−αn−1fyn−1+G˜xxn−1+ηn−ηn−1zn−1=1−ηnαnxn−xn−1+ηnαnρyn−yn−1+σnμnx−μn−1x∑i=1NaixBixxn−1+δn−δn−1xn−1+σn−σn−1un−1+ηnαn−αn−1fyn−1+G˜xxn−1+ηn−ηn−1zn−1.
By the same argument as in (27) and (29), we also have(30)wn−wn−1≤αnρxn−xn−1+1−αnyn−yn−1+αn−αn−1gxn−1+G˜yyn−1.(31)yn+1−yn≤1−ηnαnyn−yn−1+ηnαnρxn−xn−1+σnμny−μn−1y∑i=1NaiyBiyyn−1+δn−δn−1yn−1+σn−σn−1vn−1+ηnαn−αn−1gxn−1+G˜yyn−1+ηn−ηn−1wn−1.
From (29) and (31), we obtain that(32)xn+1−xn+yn+1−yn≤1−1−ρηnαnxn−xn−1+yn−yn−1+σnμnx−μn−1x∑i=1NaixBixxn−1+∑i=1NaiyBiyyn−1+|δn−δn−1‖xn−1+yn−1+σn−σn−1un−1+vn−1+ηnαn−αn−1fyn−1+gxn−1+G˜xxn−1+G˜yyn−1+ηn−ηn−1zn−1+wn−1≤1−1−ρa¯αnxn−xn−1+yn−yn−1+b¯μnx−μn−1x∑i=1NaixBixxn−1+∑i=1NaiyBiyyn−1+|δn−δn−1‖xn−1+yn−1+σn−σn−1un−1+vn−1+b¯αn−αn−1fyn−1+gxn−1+G˜xxn−1+G˜yyn−1+ηn−ηn−1zn−1+wn−1.
From (32), conditions 1, 2, and 5, and Lemma 8, we obtain that(33)limn⟶∞xn+1−xn=0,(34)limn⟶∞yn+1−yn=0.Step 3.
We show thatlimn⟶∞zn−PCI−μnx∑i=1NaixBixzn=limn⟶∞zn−G˜xzn=0. From (21), we have that(35)xn+1−x∗2=δnxn−x∗+σnPCI−μnx∑i=1NaixBixxn−x∗+ηnzn−x∗2,≤δnxn−x∗2+σnPCI−μnx∑i=1NaixBixxn−x∗2−δnσnxn−PCI−μnx∑i=1NaixBixxn2+ηnzn−x∗2≤1−ηnxn−x∗2−δnσnxn−PCI−μnx∑i=1NaixBixxn2+ηnαnfyn−G˜xxn+G˜xxn−x∗2≤1−ηnxn−x∗2−δnσnxn−PCI−μnx∑i=1NaixBixxn2+ηnG˜xxn−x∗2+2αnfyn−G˜xxn,zn−x∗≤xn−x∗2−δnσnxn−PCI−μnx∑i=1NaixBixxn2+2ηnαnfyn−G˜xxnzn−x∗,(36)δnσnxn−PCI−μnx∑i=1NaixBixxn2≤xn−x∗2−xn+1−x∗2+2ηnαnfyn−G˜xxnzn−x∗,≤xn−xn+1xn−x∗+xn+1−x∗+2ηnαnfyn−G˜xxnzn−x∗.
Then, we have(37)xn−PCI−μnx∑i=1NaixBixxn⟶0asn⟶∞.
Observe that(38)xn+1−xn=σnPCI−μnx∑i=1NaixBixxn−xn+ηnzn−xn.
From (33) and (37), we have(39)zn−xn⟶0asn⟶∞.
By the same argument as above, we also have that(40)wn−xn⟶0asn⟶∞.
Note that(41)zn−PCI−μnx∑i=1NaixBixzn≤zn−xn+xn−PCI−μnx∑i=1NaixBixxn+PCI−μnx∑i=1NaixBixxn−PCI−μnx∑i=1NaixBixzn,≤zn−xn+xn−PCI−μnx∑i=1NaixBixxn+xn−zn=2zn−xn+xn−PCI−μnx∑i=1NaixBixxn,
By (37) and (39), we get that(42)zn−PCI−μnx∑i=1NaixBixzn⟶0asn⟶∞.
By the same argument as (41), we also obtain(43)wn−PCI−μny∑i=1NaiyBiywn⟶0asn⟶∞.
Consider(44)xn+1−zn≤xn+1−xn+xn−zn.
By (33) and (39), we get that(45)xn+1−zn⟶0asn⟶∞.
However,(46)xn−G˜xxn≤xn−xn+1+xn+1−zn+zn−G˜xxn,=xn−xn+1+xn+1−zn+αnfyn+1−αnG˜xxn−G˜xxn=xn−xn+1+xn+1−zn+αnfyn−G˜xxn,
It follows from (33) and (45) that(47)xn−G˜xxn⟶0asn⟶∞.
Consider(48)zn−G˜xzn≤zn−xn+xn−G˜xxn+G˜xxn−G˜xzn,≤2zn−xn+xn−G˜xxn.
From (39) and (47), we obtain(49)zn−G˜xzn⟶0asn⟶∞.
Applying the same method as (48), we also have(50)wn−G˜ywn⟶0asn⟶∞.Step 4.
We will show thatlimsupn⟶∞fy˜−x˜,zn−x˜≤0 and limsupn⟶∞gx˜−y˜,zn−y˜≤0, where x˜=Pℱxfy˜ and y˜=Pℱyfx˜. First, we take a subsequence znk of zn such that(51)limsupn⟶∞fy˜−x˜,zn−x˜=limk⟶∞fy˜−x˜,znk−x˜.
Sincexn is bounded, there exists a subsequence xnk of xn such that xnk⇀q1 as k⟶∞. From (39), we get that znk⇀q1. Next, we need to show that q1∈ℱx=Ωx∩∩i=1NVIC,Bix. Assume that q1∉Ωx. By Lemma 7, we get that q1≠G˜xq1. Applying Opial’s condition and (49), we get that(52)liminfk⟶∞znk−q1<liminfk⟶∞znk−G˜xq1,≤liminfn⟶∞znk−G˜xznk+liminfk⟶∞G˜xznk−G˜xq1≤liminfk⟶∞znk−q1.
This is a contradiction. Thus,q1∈Ωx.
Assume thatq1∉∩i=1NVIC,Bix. Then, from Lemma 3 and Lemma 2, we have q1∉FPCI−μnx∑i=1NaiBix. From Opial’s condition and (42), we obtain(53)liminfk⟶∞znk−q1<liminfk⟶∞znk−PCI−μnkx∑i=1NaixBixq1,≤liminfk⟶∞znk−PCI−μnkx∑i=1NaixBixznk+liminfk⟶∞PCI−μnkx∑i=1NaixBixznk−PCI−μnk∑i=1NaixBixq1≤liminfk⟶∞znk−q1.
This is a contradiction. Thus,q1∈∩i=1NVIC,Bix, and so,(54)q1∈ℱx=Ωx∩∩i=1NVIC,Bix.
However,znk⇀q1. From (54) and Lemma 1, we can derive that(55)limsupn⟶∞fy˜−x˜,zn−x˜=limk⟶∞fy˜−x˜,znk−x˜,=fy˜−x˜,q1−x˜≤0.
By the same method as (55), we also obtain that(56)limsupn⟶∞gx˜−y˜,zn−y˜≤0.Step 5.
Finally, we show that the sequencesxn and yn converges strongly to x˜=Pℱxfy˜ and y˜=Pℱyfx˜, respectively. From the definition of zn, we have(57)zn−x˜2=αnfyn−x˜+1−αnG˜xxn−x˜,zn−x˜,=αnfyn−x˜,zn−x˜+1−αnG˜xxn−x˜,zn−x˜≤αnfyn−fy˜,zn−x˜+αnfy˜−x˜,zn−x˜+1−αnG˜xxn−x˜zn−x˜≤αnρyn−y˜zn−x˜+αnfy˜−x˜,zn−x˜+1−αnxn−x˜zn−x˜≤αnρ2yn−y˜2+zn−x˜2+αnfy˜−x˜,zn−x˜+1−αn2xn−x˜2+zn−x˜2,which implies that(58)zn−x˜2≤αnρ1+αn1−ρyn−y˜2+1−αn1+αn1−ρxn−x˜2+2αn1+αn1−ρfy˜−x˜,zn−x˜.
From the definition ofxn and (58), we get(59)xn+1−x˜2≤δnxn−x˜2+σnPCI−μnx∑i=1NaixBixxn−x˜2+ηnzn−x˜2,≤1−ηnxn−x˜2+αnηnρ1+αn1−ρyn−y˜2+1−αnηn1+αn1−ρxn−x˜2+2αnηn1+αn1−ρfy˜−x˜,zn−x˜=1−αnηn2−ρ1+αn1−ρxn−x˜2+αnηnρ1+αn1−ρyn−y˜2+2αnηn1+αn1−ρfy˜−x˜,zn−x˜.
Applying the same argument as in (58) and (59), we get(60)yn+1−y˜2≤1−αnηn2−ρ1+αn1−ρyn−y˜2+αnηnρ1+αn1−ρxn−x˜2+2αnηn1+αn1−ρgx˜−y˜,zn−y˜.
From (58) and (59), we have(61)xn+1−x˜2+yn+1−y˜2≤1−αnηn2−ρ1+αn1−ρxn−x˜2+yn−y˜2+αnηnρ1+αn1−ρxn−x˜2+yn−y˜2+2αnηn1+αn1−ρfy˜−x˜,zn−x˜+gx˜−y˜,zn−y˜,≤1−2αnηn1−ρ1+αn1−ρxn−x˜2+yn−y˜2+2αnηn1+αn1−ρfy˜−x˜,zn−x˜+gx˜−y˜,zn−y˜.
According to condition2 and 4, (61), and Lemma 8, we can conclude that xn and yn converge strongly to x˜=Pℱxfy˜ and y˜=Pℱygx˜, respectively. Furthermore, from (39) and (40), we get that zn and wn converge strongly to x˜=Pℱxfy˜ and y˜=Pℱygx˜, respectively. This completes the proof. □One of the great special cases of the SMVIP is the split variational inclusion problem that has a wide variety of application backgrounds, such as split minimization problems and split feasibility problems.If we setAix=0 and Aiy=0 in Theorem 2, for all i=1,2, then we get the strong convergence theorem for the split variational inclusion problem and the finite families of the variational inequality problems as follows:Corollary 1.
LetH1 and H2 be Hilbert spaces, and let C be a nonempty closed convex subset of H1. Let T:H1⟶H2 be a bounded linear operator, and let f,g:H1⟶H1 be ρf,ρg-contraction mappings with ρ=maxρf,ρg. For every i=1,2, let Mix,Miy:Hi⟶2Hi be multivalued maximal monotone mappings. For i=1,2,…,N, let Bix,Biy:H1⟶H1 be βix,βiy−inverse strongly monotone with β¯x=mini=1,2,…,Nβix and β¯y=mini=1,2,…,Nβiy. Let Sx=x∗∈H1:0∈M1xx∗,x˜=Tx∗∈H2:0∈M2xx˜ and Sy=y∗∈H1:0∈M1yy∗,y˜=Ty∗∈H2:0∈M2yy˜. Assume that ℱx=Sx∩∩i=1NVIC,Bix≠∅ and ℱy=Sy∩∩i=1NVIC,Biy≠∅. Let xn and yn be sequences generated by x1,y1∈H1 and(62)xn+1=δnxn+σnPCI−μnx∑i=1NaixBixxn+ηnαnfyn+1−αnJλ1xM1xx−γxT∗I−Jλ2xM2xTxn,yn+1=δnyn+σnPCI−μnx∑i=1NaiyBiyyn+ηnαngxn+1−αnJλ1yM1yy−γyT∗I−Jλ2yM2yTyn,for all n≥1, where δn,σn,ηn,αn⊆0,1 with δn+σn+ηn=1, a1x,a2x,…aNx,a1y,a2y,…,aNy⊂0,1, λix,λiy∈0,∞ for all i=1,2, and 0<γx,γy<1/L with L being a spectral radius of T∗T. Assume the following conditions hold:(1)
∑n=1∞μnx<∞,∑n=1∞μny<∞, and 0<a<μnx≤2βx¯,0<b<μny≤2βy¯, for some a,b∈ℝ.(2)
∑n=1∞αn=∞,limn⟶∞αn=0.(3)
∑n=1Naix=∑n=1Naiy=1.(4)
0<a¯≤δn,σn,ηn≤b¯<1, for all n∈ℕ, for some a¯,b¯>0.(5)
∑n=1∞δn+1−δn,∑n=1∞σn+1−σn,and∑n=1∞αn+1−αn<∞. Then, xn converges strongly to x˜=Pℱxfy˜ and yn converges strongly to y˜=Pℱygx˜.


## 4. Applications

In this section, by applying our main result in Theorem2, we can prove strong convergence theorems for approximating the solution of the split convex minimization problems and split feasibility problems.


### 4.1. Split Convex Minimization Problems

Letφ:H⟶ℝ be a convex and differentiable function and ψ:H⟶−∞,∞ be a proper convex and lower semicontinuous function. It is well known that if ∇φ is 1/α-Lipschitz continuous, then it is α-inverse strongly monotone, where ∇φ is the gradient of φ (see [10]). It is also known that the subdifferential ∂ψ of ψ is maximal monotone (see [35]). Moreover,(63)φx∗+ψx∗=minx∈Hφx+ψx⇔0∈∇φx∗+∂φx∗.Next, we consider the following the split convex minimization problem (SCMP): find(64)x∗∈H1,such thatφ1x∗+ψ1x∗=minx∈H1φ1x+ψ1xand such that y∗=Tx∗∈H2 solves(65)φ2y∗+ψ2y∗=miny∈H2φ2y+ψ2y,where T:H1⟶H2 is a bounded linear operator with adjoint T∗, φi,andψi defined as above, for i=1,2. We denoted the set of all solutions of (64) and (65) by Θ. That is, Θ=x∗which solves64:Tx∗solves65.If we setAix=∇φix,Aiy=∇φiy, and Mix=∂ψix,Miy=∂ψiy, for i=1,2, in Theorem 2, then we get the strong convergence theorem for finding the common solution of the split convex minimization problems and the finite families of the variational inequality problems as follows.Theorem 3.
LetH1 and H2 be Hilbert spaces, and let C be a nonempty closed convex subset of H1. Let T:H1⟶H2 be a bounded linear operator, and let f,g:H1⟶H1 be ρf,ρg-contraction mappings with ρ=maxρf,ρg. For i=1,2, let ψix,ψiy:Hi⟶−∞,∞ be proper convex and lower semicontinuous functions, and let φix,φiy:Hi⟶ℝ be convex and differentiable function such that ∇φix and ∇φiy be 1/αix-Lipschitz continuous and 1/αiy-Lipschitz continuous, respectively. For i=1,2,…,N, let Bix,Biy:H1⟶H1 be βix,βiy−inverse strongly monotone with β¯x=mini=1,2,…,Nβix and β¯y=mini=1,2,…,Nβiy. Let G˜x,G˜y:H1⟶H1 be defined by G˜xx=Jλ1x∂ψ1xI−λ1x∇φ1xx−γxT∗I−Jλ2x∂ψ2xI−λ2x∇φ2xTx,∀x∈H1, and G˜yy=Jλ1y∂ψ1yI−λ1y∇φ1yy−γyT∗I−Jλ2y∂ψ2yI−λ2y∇φ2yTy,∀y∈H1, respectively, where λix∈0,2αix,λiy∈0,2αiy and 0<γx,γy<1/L with L is a spectral radius of T∗T. Assume that ℱx=Θx∩∩i=1NVIC,Bix≠∅ and ℱy=Θy∩∩i=1NVIC,Biy≠∅. Let xn and yn be sequences generated by x1,y1∈H1 and(66)xn+1=δnxn+σnPCI−μnx∑i=1NaixBixxn+ηnαnfyn+1−αnG˜xxn,yn+1=δnyn+σnPCI−μny∑i=1NaiyBiyyn+ηnαngxn+1−αnG˜yyn,for all n≥1 where δn,σn,ηn,αn⊆0,1 with δn+σn+ηn=1, a1x,a2x,…aNx,a1y,a2y,…,aNy⊂0,1, and μnx,μny⊂0,∞. Assume the following condition holds:(1)
∑n=1∞μnx<∞,∑n=1∞μny<∞, and 0<a<μnx≤2βx¯,0<b<μny≤2βy¯, for some a,b∈ℝ.(2)
∑n=1∞αn=∞,limn⟶∞αn=0.(3)
∑n=1Naix=∑n=1Naiy=1.(4)
0<a¯≤δn,σn,ηn≤b¯<1, for all n∈ℕ, for some a¯,b¯>0 .(5)
∑n=1∞δn+1−δn,∑n=1∞σn+1−σn,and∑n=1∞αn+1−αn<∞. Then, xn converges strongly to x˜=Pℱxfy˜ and yn converges strongly to y˜=Pℱygx˜.


### 4.2. The Split Feasibility Problem

LetH1 and H2 be two real Hilbert spaces. Let C and Q be the nonempty closed convex subset of H1 and H2, respectively. The split feasibility problem (SFP) is to find(67)a pointx∈C,such thatAx∈Q.The set of all solutions (SFP) is denoted byΨ=x∈C:Ax∈Q. This problem was introduced by Censor and Elfving [8] in 1994. The split feasibility problem was investigated extensively as a widely important tool in many fields such as signal processing, intensity-modulated radiation therapy problems, and computer tomography (see [36–38] and the references therein).LetH be a real Hilbert space, and let h be a proper lower semicontinuous convex function of H into −∞,+∞. The subdifferential ∂h of h is defined by ∂hx=z∈H:hx+z,u−x≤hu,∀u∈H for all x∈H. Then, ∂h is a maximal monotone operator [39]. Let C be a nonempty closed convex subset of H, and let iC be the indicator function of C, i.e., iCx=0 if x∈C and iCx=∞ if x∉C. Then, iC is a proper, lower semicontinuous and convex function on H, and so the subdifferential ∂iC of iC is a maximal monotone operator. Then, we can define the resolvent operator Jλ∂iC of ∂iC for λ>0, by Jλ∂iCx=I+λ∂iC−1x,for allx∈H.Recall that the normal coneNCu of C at a point u in H is defined by NCu=z∈H:z,u−v≤0,∀v∈C if u∈C and NCu=∅ if u∉C. We note that ∂iC=NC, and for λ>0, we have that u=Jλ∂iCx if and only if u=PCx (see [31]).SettingM1=∂iC, M2=∂iQ, and in (SMVI) (4) and (5), then (SMVI) (4) and (5) are reduced to the split feasibility problem (SFP) (67)Now, by applying Theorem2, we get the following strong convergence theorem to approximate a common solution of SFP (67) and a finite family of variational inequality problems.Theorem 4.
LetH1 and H2 be Hilbert spaces, and let C and Q be the nonempty closed convex subset of H1 and H2, respectively. Let T:H1⟶H2 be a bounded linear operator with adjoint T∗, and let f,g:H1⟶H1 be ρf,ρg-contraction mappings with ρ=maxρf,ρg. For i=1,2,…,N, let Bix,Biy:H1⟶H1 be βix,βiy−inverse strongly monotone with β¯x=mini=1,2,…,Nβix and β¯y=mini=1,2,…,Nβiy. Assume that ℱx=Ψx∩∩i=1NVIC,Bix≠∅ and ℱy=Ψy∩∩i=1NVIC,Biy≠∅. Let xn and yn be sequences generated by x1,y1∈H1 and(68)xn+1=δnxn+σnPCI−μnx∑i=1NaixBixxn+ηnαnfyn+1−αnPCx−γxT∗I−PQTxn,yn+1=δnyn+σnPCI−μnx∑i=1NaiyBiyyn+ηnαngxn+1−αnPCy−γyT∗I−PQTyn,for all n≥1, where δn,σn,ηn,αn⊆0,1 with δn+σn+ηn=1, a1x,a2x,…,aNx,a1y,a2y,…,aNy⊂0,1, μnx,μny⊂0,∞, λix,λiy∈0,∞ for all i=1,2, and 0<γx,γy<1/L with L being a spectral radius of T∗T. Assume the following condition holds:(1)
∑n=1∞μnx<∞,∑n=1∞μny<∞, and 0<a<μnx≤2βx¯,0<b<μny≤2βy¯, for some a,b∈ℝ.(2)
∑n=1∞αn=∞,limn⟶∞αn=0.(3)
∑n=1Naix=∑n=1Naiy=1.(4)
0<a¯≤δn,σn,ηn≤b¯<1, for all n∈ℕ, for some a¯,b¯>0.(5)
∑n=1∞δn+1−δn,∑n=1∞σn+1−σn,and∑n=1∞αn+1−αn<∞. Then, xn converges strongly to x˜=Pℱxfy˜ and yn converges strongly to y˜=Pℱygx˜.Proof.
SetM1x=∂iC,M1y=∂iC, M2x=∂iQ,M2y=∂iQ, and Aix=0andAiy=0 in Theorem 2. Then, we get the result.The split feasibility problem is a significant part of the split monotone variational inclusion problem. It is extensively used to solve practical problems in numerous situations. Many excellent results have been obtained. In what follows, an example of a signal recovery problem is introduced.Example 2.
In signal recovery, compressed sensing can be modeled as the following under-determined linear equation system:(69)y=Ax+δ,where x∈ℝN is a vector with m non-zero components to be recovered, y∈ℝM is the observed or measured data with noisy δ, and A:ℝN⟶ℝMM<N is a bounded linear observation operator. An essential point of this problem is that the signal x is sparse; that is, the number of nonzero elements in the signal x is much smaller than the dimension of the signal x. To solve this situation, a classical model, convex constraint minimization problem, is used to describe the above problem. It is known that problem (69) can be seen as solving the following LASSO problem [40]:(70)minx∈ℝN12y−Ax22,subject tox1≤t,where t>0 is a given constant and ·1 is ℓ1 norm. In particular, LASSO problem (70) is equivalent to the split feasibility problem (SFP) (67) when C=x∈ℝN:x1≤t and Q=y.


## 4.1. Split Convex Minimization Problems

Letφ:H⟶ℝ be a convex and differentiable function and ψ:H⟶−∞,∞ be a proper convex and lower semicontinuous function. It is well known that if ∇φ is 1/α-Lipschitz continuous, then it is α-inverse strongly monotone, where ∇φ is the gradient of φ (see [10]). It is also known that the subdifferential ∂ψ of ψ is maximal monotone (see [35]). Moreover,(63)φx∗+ψx∗=minx∈Hφx+ψx⇔0∈∇φx∗+∂φx∗.Next, we consider the following the split convex minimization problem (SCMP): find(64)x∗∈H1,such thatφ1x∗+ψ1x∗=minx∈H1φ1x+ψ1xand such that y∗=Tx∗∈H2 solves(65)φ2y∗+ψ2y∗=miny∈H2φ2y+ψ2y,where T:H1⟶H2 is a bounded linear operator with adjoint T∗, φi,andψi defined as above, for i=1,2. We denoted the set of all solutions of (64) and (65) by Θ. That is, Θ=x∗which solves64:Tx∗solves65.If we setAix=∇φix,Aiy=∇φiy, and Mix=∂ψix,Miy=∂ψiy, for i=1,2, in Theorem 2, then we get the strong convergence theorem for finding the common solution of the split convex minimization problems and the finite families of the variational inequality problems as follows.Theorem 3.
LetH1 and H2 be Hilbert spaces, and let C be a nonempty closed convex subset of H1. Let T:H1⟶H2 be a bounded linear operator, and let f,g:H1⟶H1 be ρf,ρg-contraction mappings with ρ=maxρf,ρg. For i=1,2, let ψix,ψiy:Hi⟶−∞,∞ be proper convex and lower semicontinuous functions, and let φix,φiy:Hi⟶ℝ be convex and differentiable function such that ∇φix and ∇φiy be 1/αix-Lipschitz continuous and 1/αiy-Lipschitz continuous, respectively. For i=1,2,…,N, let Bix,Biy:H1⟶H1 be βix,βiy−inverse strongly monotone with β¯x=mini=1,2,…,Nβix and β¯y=mini=1,2,…,Nβiy. Let G˜x,G˜y:H1⟶H1 be defined by G˜xx=Jλ1x∂ψ1xI−λ1x∇φ1xx−γxT∗I−Jλ2x∂ψ2xI−λ2x∇φ2xTx,∀x∈H1, and G˜yy=Jλ1y∂ψ1yI−λ1y∇φ1yy−γyT∗I−Jλ2y∂ψ2yI−λ2y∇φ2yTy,∀y∈H1, respectively, where λix∈0,2αix,λiy∈0,2αiy and 0<γx,γy<1/L with L is a spectral radius of T∗T. Assume that ℱx=Θx∩∩i=1NVIC,Bix≠∅ and ℱy=Θy∩∩i=1NVIC,Biy≠∅. Let xn and yn be sequences generated by x1,y1∈H1 and(66)xn+1=δnxn+σnPCI−μnx∑i=1NaixBixxn+ηnαnfyn+1−αnG˜xxn,yn+1=δnyn+σnPCI−μny∑i=1NaiyBiyyn+ηnαngxn+1−αnG˜yyn,for all n≥1 where δn,σn,ηn,αn⊆0,1 with δn+σn+ηn=1, a1x,a2x,…aNx,a1y,a2y,…,aNy⊂0,1, and μnx,μny⊂0,∞. Assume the following condition holds:(1)
∑n=1∞μnx<∞,∑n=1∞μny<∞, and 0<a<μnx≤2βx¯,0<b<μny≤2βy¯, for some a,b∈ℝ.(2)
∑n=1∞αn=∞,limn⟶∞αn=0.(3)
∑n=1Naix=∑n=1Naiy=1.(4)
0<a¯≤δn,σn,ηn≤b¯<1, for all n∈ℕ, for some a¯,b¯>0 .(5)
∑n=1∞δn+1−δn,∑n=1∞σn+1−σn,and∑n=1∞αn+1−αn<∞. Then, xn converges strongly to x˜=Pℱxfy˜ and yn converges strongly to y˜=Pℱygx˜.


## 4.2. The Split Feasibility Problem

LetH1 and H2 be two real Hilbert spaces. Let C and Q be the nonempty closed convex subset of H1 and H2, respectively. The split feasibility problem (SFP) is to find(67)a pointx∈C,such thatAx∈Q.The set of all solutions (SFP) is denoted byΨ=x∈C:Ax∈Q. This problem was introduced by Censor and Elfving [8] in 1994. The split feasibility problem was investigated extensively as a widely important tool in many fields such as signal processing, intensity-modulated radiation therapy problems, and computer tomography (see [36–38] and the references therein).LetH be a real Hilbert space, and let h be a proper lower semicontinuous convex function of H into −∞,+∞. The subdifferential ∂h of h is defined by ∂hx=z∈H:hx+z,u−x≤hu,∀u∈H for all x∈H. Then, ∂h is a maximal monotone operator [39]. Let C be a nonempty closed convex subset of H, and let iC be the indicator function of C, i.e., iCx=0 if x∈C and iCx=∞ if x∉C. Then, iC is a proper, lower semicontinuous and convex function on H, and so the subdifferential ∂iC of iC is a maximal monotone operator. Then, we can define the resolvent operator Jλ∂iC of ∂iC for λ>0, by Jλ∂iCx=I+λ∂iC−1x,for allx∈H.Recall that the normal coneNCu of C at a point u in H is defined by NCu=z∈H:z,u−v≤0,∀v∈C if u∈C and NCu=∅ if u∉C. We note that ∂iC=NC, and for λ>0, we have that u=Jλ∂iCx if and only if u=PCx (see [31]).SettingM1=∂iC, M2=∂iQ, and in (SMVI) (4) and (5), then (SMVI) (4) and (5) are reduced to the split feasibility problem (SFP) (67)Now, by applying Theorem2, we get the following strong convergence theorem to approximate a common solution of SFP (67) and a finite family of variational inequality problems.Theorem 4.
LetH1 and H2 be Hilbert spaces, and let C and Q be the nonempty closed convex subset of H1 and H2, respectively. Let T:H1⟶H2 be a bounded linear operator with adjoint T∗, and let f,g:H1⟶H1 be ρf,ρg-contraction mappings with ρ=maxρf,ρg. For i=1,2,…,N, let Bix,Biy:H1⟶H1 be βix,βiy−inverse strongly monotone with β¯x=mini=1,2,…,Nβix and β¯y=mini=1,2,…,Nβiy. Assume that ℱx=Ψx∩∩i=1NVIC,Bix≠∅ and ℱy=Ψy∩∩i=1NVIC,Biy≠∅. Let xn and yn be sequences generated by x1,y1∈H1 and(68)xn+1=δnxn+σnPCI−μnx∑i=1NaixBixxn+ηnαnfyn+1−αnPCx−γxT∗I−PQTxn,yn+1=δnyn+σnPCI−μnx∑i=1NaiyBiyyn+ηnαngxn+1−αnPCy−γyT∗I−PQTyn,for all n≥1, where δn,σn,ηn,αn⊆0,1 with δn+σn+ηn=1, a1x,a2x,…,aNx,a1y,a2y,…,aNy⊂0,1, μnx,μny⊂0,∞, λix,λiy∈0,∞ for all i=1,2, and 0<γx,γy<1/L with L being a spectral radius of T∗T. Assume the following condition holds:(1)
∑n=1∞μnx<∞,∑n=1∞μny<∞, and 0<a<μnx≤2βx¯,0<b<μny≤2βy¯, for some a,b∈ℝ.(2)
∑n=1∞αn=∞,limn⟶∞αn=0.(3)
∑n=1Naix=∑n=1Naiy=1.(4)
0<a¯≤δn,σn,ηn≤b¯<1, for all n∈ℕ, for some a¯,b¯>0.(5)
∑n=1∞δn+1−δn,∑n=1∞σn+1−σn,and∑n=1∞αn+1−αn<∞. Then, xn converges strongly to x˜=Pℱxfy˜ and yn converges strongly to y˜=Pℱygx˜.Proof.
SetM1x=∂iC,M1y=∂iC, M2x=∂iQ,M2y=∂iQ, and Aix=0andAiy=0 in Theorem 2. Then, we get the result.The split feasibility problem is a significant part of the split monotone variational inclusion problem. It is extensively used to solve practical problems in numerous situations. Many excellent results have been obtained. In what follows, an example of a signal recovery problem is introduced.Example 2.
In signal recovery, compressed sensing can be modeled as the following under-determined linear equation system:(69)y=Ax+δ,where x∈ℝN is a vector with m non-zero components to be recovered, y∈ℝM is the observed or measured data with noisy δ, and A:ℝN⟶ℝMM<N is a bounded linear observation operator. An essential point of this problem is that the signal x is sparse; that is, the number of nonzero elements in the signal x is much smaller than the dimension of the signal x. To solve this situation, a classical model, convex constraint minimization problem, is used to describe the above problem. It is known that problem (69) can be seen as solving the following LASSO problem [40]:(70)minx∈ℝN12y−Ax22,subject tox1≤t,where t>0 is a given constant and ·1 is ℓ1 norm. In particular, LASSO problem (70) is equivalent to the split feasibility problem (SFP) (67) when C=x∈ℝN:x1≤t and Q=y.


## 5. Numerical Examples

In this section, we give some examples for supporting Theorem2. In example 3, we give the computer programming to support our main result.Example 3.
Letℝ be a set of real number and H1=H2=ℝ2. Let C=−20,20×−20,20, and let ·,·:ℝ2×ℝ2⟶ℝ be inner product defined by x,y=x·y=x1y1+x2y2, for all x=x1,x2∈ℝ2 and y=y1,y2∈ℝ2 and the usual norm ·:ℝ2⟶ℝ given by x=x12+x22, for all x=x1,x2∈ℝ2. Let T:ℝ2⟶ℝ2 be defined by Tx=2x1,2x2 for all x=x1,x2∈ℝ2 and T∗:ℝ2⟶ℝ2 be defined by T∗z=2z1,2z2 for all z=z1,z2∈ℝ2. Let M1x,M1y,M2x,M2y:ℝ2⟶2ℝ2 be defined by M1xx=3x1−2,3x2−2,M1yx=2x1,2x2,M2x=x1/3+2,x2/3+2,andM2yx=x1/3+3,x2/3+3, respectively, for all x=x1,x2∈ℝ2. Let the mapping A1x,A1y,A2x,A2y:ℝ2⟶ℝ2 be defined by A1xx=x1−3/2,x2−3/2,A1yx=x1+6,x2+6,A2xx=x1−2/3,x2−2/3,andA2y=x1−1/3,x2−1/3, respectively, for all x=x1,x2∈ℝ2. For every i=1,2,…,N, let the mappings Bix,Biy:ℝ2⟶ℝ2 be defined by Bixx=x1−1/3i,x2−1/3iandBiyx=x1+2/5i,x2+2/5i, respectively, for all x=x1,x2∈ℝ2, and let aix=2/3i+1/N3Nandaiy=4/5i+1/N5N. Let the mappings f,g:ℝ2⟶ℝ2 be defined by fx=x1/7,x2/7andgx=x1/9,x2/9, respectively, for all x=x1,x2∈ℝ2.
Chooseγxandγy=0.1,  λ1x=2,λ1y=1.2,λ2x=0.1,andλ2y=1.9. Setting δn=n/9n+3, σn=4n+2/3/9n+3, ηn=4n+7/3/9n+3, αn=1/20n, μnx=1/7n2, and μny=1/5n2. Let x1=x11,x12 and y1=y11,y12∈ℝ2, and let the sequences xn and yn be generated by (21) as follows:(71)xn+1=n9n+3xn+4n+2/39n+3PCI−17n2∑i=1N23i+1N3NBixxn+4n+7/39n+3120nfyn+20n−120nG˜xxn,yn+1=n9n+3yn+4n+2/39n+3PCI−15n2∑i=1N45i+1N5NBiyyn+4n+7/39n+3120ngxn+20n−120nG˜yyn,G˜xxn=J2M1xI−2A1xx−0.1T∗I−J0.1M2xI−0.1A2xTxn,G˜yyn=J1.2M1yI−1.2A1yy−0.1T∗I−J1.9M2yI−1.9A2yTyn,for all n≥1, where xn=xn1,xn2 and yn=yn1,yn2. By the definition of Mix,Miy,Aix,andAiy, for all i=1,2,  BixandBiy, for all i=1,2,…,N, and f and g, we have that 1,1∈Ωx∩∩i=1NVIC,Bix and −2,−2∈Ωy∩∩i=1NVIC,Biy. Also, it is easy to see that all parameters satisfy all conditions in Theorem 2. Then, by Theorem 2, we can conclude that the sequence xn converges strongly to 1,1 and yn converges strongly to −2,−2.Table1 and Figure 1 show the numerical results of xn and yn where x1=−10,10, y1=−10,10, and n=N=50.Table 1
Values ofxn and yn with initial values x1=−10,10,y1=−10,10, and n=N=50.
nxn=xn1,xn2yn=yn1,yn21−10.000000,10.000000−10.000000,10.0000002−4.093342,5.126154−5.481385,3.3381823−1.544910,3.037093−3.639877,0.5751014−0.307347,2.029037−2.789106,−0.719370⋮⋮⋮300.998745,0.996821−1.996611,−1.998243⋮⋮⋮470.998973,0.998235−1.998394,−1.999022480.998986,0.998280−1.998454,−1.999055490.999000,0.998324−1.998511,−1.999087500.999013,0.998365−1.998566,−1.999118Figure 1
Convergence ofxn and yn with initial values x1=−10,10,y1=−10,10, and n=N=50.Next, in Example4, we only show an example in infinite-dimensional Hilbert space for supporting Theorem 2. We omit the computer programming.Example 4.
LetH1=H2=C=ℓ2 be the linear space whose elements consist of all 2-summable sequence x1,x2,…,xj,… of scalars, i.e.,(72)ℓ2=x:x=x1,x2,…,xj,…and∑j=1∞xj2<∞,with an inner product ·,·:ℓ2×ℓ2⟶ℝ defined by x,y=∑j=1∞xjyj, where x=xjj=1∞andy=yjj=1∞∈ℓ2, and a norm ·:ℓ2⟶ℝ defined by x2=∑j=1∞xj21/2 where x=xjj=1∞∈ℓ2. Let T:ℓ2⟶ℓ2 be defined by Tx=x1/2,x2/2,…,xj/2,… for all x=xjj=1∞∈ℓ2. And, T∗:ℓ2⟶ℓ2 be defined by Tx=z1/2,z2/2,…,zj/2,… for all x=zjj=1∞∈ℓ2. Let M1x,M1y,M2x,M2y:ℓ2⟶ℓ2 be defined by M1xx=2x1,2x2,…,2xj,…,M1yx=x1−1,x2−1,…,xj−1,…,M2x=3x1,3x2,…,3xj,…, and M2yx=2x1−1,2x2−1,…,2xj−1,…, respectively, for all x=xjj=1∞∈ℓ2. Let the mapping A1x,A1y,A2x,A2y:ℓ2⟶ℓ2 be defined by A1xx=x1/3,x2/3,…,xj/3,…,A1yx=x1−1/2,x2−1/2,…,xj−1/2,…,A2xx=x1/4,x2/4,…,xj/4,…,andA2y=2x1−1/3,2x2−1/3,…,2xj−1/3 respectively, for all x=xjj=1∞∈ℓ2. For every i=1,2,…,N, let the mappings Bix,Biy:ℓ2⟶ℓ2 be defined by Bixx=2x1/3i,2x2/3i,…,2xj/3i,… and Biyx=2x1−1/4i,2x2−1/4i,…,2xj−1/4i, respectively, for all x=xjj=1∞∈ℓ2, and let aix=5/6i+1/N6Nandaiy=7/8i+1/N8N. Let the mappings f,g:ℓ2⟶ℓ2 be defined by fx=x1/5,x2/5,…,xj/5,…,gx=x1/4,x2/4,…,xj/4,…, respectively, for all x=xjj=1∞∈ℓ2.
Letλ1x=1,λ1y=1,λ2x=0.5,andλ2y=2. Since L=1/4, we choose γxandγy=0.5. Let x1=x11,x12,…,x1j,… and y1=y11,y12,…,y1j,…∈ℓ2, and let xn and yn be the sequences generated by (2) as follows:(73)xn+1=n5n+2xn+2n+2/35n+2PCI−13n2∑i=1N56i+1N6NBixxn+2n+4/35n+3110nfyn+10n−120nG˜xxn,yn+1=n5n+2yn+2n+2/35n+2PCI−14n2∑i=1N78i+1N8NBiyyn+2n+4/35n+2110ngxn+10n−120nG˜yyn,G˜xxn=J1M1xI−A1xx−0.5T∗I−J0.5M2xI−0.5A2xTxn,G˜yyn=J1M1yI−A1yy−0.5T∗I−J2M2yI−2A2yTyn,for all n≥1, where xn=xn1,xn2,…,xnj,… and yn=yn1,yn2,…,ynj,…. It is easy to see that Mix,Miy,Aix,andAiy,∀i=1,2, BixandBiy,∀i=1,2,…,N, T, f, g, and all parameters satisfy Theorem 2. Furthermore, we have that 0∈Ωx∩∩i=1NVIC,Bix and 1∈Ωy∩∩i=1NVIC,Biy. Then, by Theorem 2, we can conclude that the sequence xn converges strongly to 0 and yn converges strongly to 1.


## 6. Conclusion

(1).
Table1 and Figure 1 in Example 3 show that the sequence xn converges to 1,1∈Ωx∩∩i=1NVIC,Bix and yn converges to −2,−2∈Ωy∩∩i=1NVIC,Biy(2)
Example4 is an example in infinite-dimensional Hilbert space for supporting Theorem 2(3)
Theorem2 guarantees the convergence of xn and yn in Example 3 and Example 4



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