# The Conjugate Gradient Viscosity Approximation Algorithm for Split Generalized Equilibrium and Variational Inequality Problems

Authors: Meixia Li; Haitao Che; Jingjing Tan
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this paper, we study a kind of conjugate gradient viscosity approximation algorithm for finding a common solution of split generalized equilibrium problem and variational inequality problem. Under mild conditions, we prove that the sequence generated by the proposed iterative algorithm converges strongly to the common solution. The conclusion presented in this paper is the generalization, extension, and supplement of the previously known results in the corresponding references. Some numerical results are illustrated to show the feasibility and efficiency of the proposed algorithm.


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## Body

## 1. Introduction

LetH1 and H2 be real Hilbert spaces with inner product ·,· and norm ·. Let C and Q be nonempty closed convex subsets of H1 and H2, respectively. Let xn be a sequence in H1; then xn→x (respectively, xn⇀x) will denote strong (respectively, weak) convergence of the sequence {xn}. Assume wω(xk)=x:∃xkj⇀x to stand for the weak ω-limit set of xk.The split feasibility problem (SFP) originally introduced by Censor and Elfving [1] is to find(1)x∈Csuch  thatAx∈Q,where A:H1→H2 is a bounded linear operator. It serves as a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in these operator’s ranges. The applications of the split feasibility problem are very comprehensive such as CT in medicine, intelligence antennas, and the electronic warning systems in military, the development of fast image processing technology and HDTV, etc. Many authors generalize SFP to a lot of important problems, such as multiple-sets split feasibility problem, split equality fixed point problem, split variational inequality problem, split variational inclusion problem, and split equilibrium problem, and the theories and algorithms are studied and details can be seen in [2–15] and references therein.The fixed point problem(FPP) for the mapping T is to find x∈C such that(2)Tx=x.We denote Fix(T)≔x∈C:Tx=x the set of solution of FPP.LetB:C→H1 be a nonlinear mapping. The variational inequality problem (VIP) is to find x∈C such that (3)Bx,y-x≥0,∀y∈C.The solution set of VIP is denoted by VI(C,B). It is well known that if B is strongly monotone and Lipschitz continuous mapping on C, then VIP has a unique solution.For finding a common problem ofFix(T)∩VI(C,B), Takahashi and Toyoda [16] introduced the following iterative scheme:(4)x0chosen  arbitrary,xn+1=αnxn+1-αnTPCxn-λnBxn,∀n≥0,where B is ρ-inverse-strongly monotone, {αn} is a sequence in (0,1), and {λn} is a sequence in (0,2ρ). They showed that if Fix(T)∩VI(C,B)≠∅, then the sequence {xn} generated by (4) converges weakly to z0∈Fix(T)∩VI(C,B).On the other hand, there are several numerical methods for solving variational inequalities and related optimization problems; see [5, 17–24] and the references therein.In 2013, Kazmi and Rizvi [25] introduced the split generalized equilibrium problem (SGEP). Let F1,h1:C×C→R and F2,h2:Q×Q→R be nonlinear bifunctions and A:H1→H2 be a bounded linear operator; then the split generalized equilibrium problem (SGEP) is to find x∗∈C such that(5)F1x∗,x+h1x∗,x≥0,∀x∈Cand such that(6)y∗=Ax∗∈QsolvesF2y∗,y+h2y∗,y≥0,∀y∈Q.Denote the solution sets of generalized equilibrium problem(GEP) (5) and GEP (6) by GEP(F1,h1) and GEP(F2,h2), respectively. The solution set of SGEP is denoted by Γ=x∗∈GEPF1,h1:Ax∗∈GEPF2,h2. They proposed the following iterative method for finding a common solution of split generalized equilibrium and fixed point problem.Let{xn} and {un} be the sequences generated by x0∈C and(7)un=TrnF1,h1xn+δA∗TrnF2,h2-IAxn,xn+1=αnγfxn+βnxn+1-βnI-αnD1sn∫0snTsunds,where S=Ts:0≤s<∞ is a nonexpansive semigroup on C and Fix(S)∩Γ≠∅, sn is a positive real sequence which diverges to +∞, {αn}, {βn}⊂(0,1), {rn}⊂(0,∞), and δ∈(0,1/L), L is the spectral radius of the operator A∗A, A∗ is the adjoint of A, and(8)TrF1,h1x=z∈C:F1z,y+h1z,y+1ry-z,z-x≥0,∀y∈C,(9)TsF2,h2w=d∈Q:F2d,e+h2d,e+1se-d,d-w≥0,∀e∈Q.Under suitable conditions, they proved a strong convergence theorem for the sequence generated by the proposed iterative scheme. But the calculation of integral is generally not easy. Therefore, it is necessary to reconsider the algorithm for solving this kind of problem.Motivated by Kazmi and Rivi [25] as well as Che and Li [2], we introduce and study a kind of conjugate gradient viscosity approximation algorithm for finding a common solution of split generalized equilibrium problem and variational inequality problem. Under mild conditions, we prove that the sequence generated by the proposed iterative algorithm converges strongly to the common solution of VI(C,B) and SGEP. The results presented in this paper are the generalization, extension, and supplement of the previously known results in the corresponding references. Numerical results show the feasibility and efficiency of the proposed algorithm.


## 2. Preliminaries

In this section, we introduce some concepts and results which are needed in sequel.A mappingT:H1→H1 is called(1) contraction, if there exists a constantα∈(0,1) such that(10)Tx-Ty≤αx-y,∀x,y∈H1.If α=1, then T is called nonexpansive.(2) monotone, if(11)Tx-Ty,x-y≥0,∀x,y∈H1.(3)η-strongly monotone, if there exists a positive constant η such that(12)Tx-Ty,x-y≥ηx-y2,∀x,y∈H1.(4)α-inverse strongly monotone (α-ism), if there exists a constant α>0 such that(13)Tx-Ty,x-y≥αTx-Ty2,∀x,y∈H1.(5) firmly nonexpansive, if(14)Tx-Ty,x-y≥Tx-Ty2,∀x,y∈H1.A mappingPC is said to be metric projection of H1 onto C if, for every point x∈H1, there exists a unique nearest point in C denoted by PCx such that(15)x-PCx≤x-y,∀y∈C.It is well known thatPC is a nonexpansive mapping and is characterized by the following properties:(16)PCx-PCy2≤x-y,PCx-PCy,∀x,y∈H1,(17)x-PCx,y-PCx≤0,∀x∈H1,y∈C,(18)x-y2≥x-PCx2+y-PCx2,∀x∈H1,y∈Cand(19)x-y-PCx-PCy2≥x-y2-PCx-PCy2,∀x,y∈H1.A linear bounded operatorB is strongly positive if there exists a constant γ¯>0 with the property(20)Bx,x≥γ¯x2,∀x∈H1.A mappingT:H1→H1 is said to be averaged if and only if it can be written as the average of the identity mapping and a nonexpansive mapping; i.e., (21)T=1-αI+αS,where α∈(0,1), S:H1→H1 is nonexpansive, and I is the identity operator on H1. More precisely, we say that T is α-averaged. We note that averaged mapping is nonexpansive. Furthermore, firmly nonexpansive mapping (in particular, projection on nonempty closed and convex subset) is averaged.LetB be a monotone mapping of C into H1. In the context of the variational inequality problem, the characterization of projection (17) implies the following relation:(22)u∈VIC,B⇔u=PCu-λBu,λ>0.In the proof of our results, we need the following assumptions and lemmas.Lemma 1 (see [26]).
Ifx,y,z∈H1, then
(i)x+y2≤x2+2〈y,x+y〉.
(ii) For anyλ∈[0,1], (23)λx+1-λy2=λx2+1-λy2-λ1-λx-y2.
(iii) Fora,b,c∈[0,1] with a+b+c=1, (24)ax+by+cz2=ax2+by2+cz2-abx-y2-acx-z2-bcy-z2.Assumption 2.
LetF:C×C→R be a bifunction satisfying the following assumption:
(i)F(x,x)≥0,∀x∈C.
(ii)F is monotone; i.e., F(x,y)+F(y,x)≤0,∀x∈C.
(iii)F is upper hemicontinuous; i.e., for each x,y,z∈C,(25)limsupt→0Ftz+1-tx,y≤Fx,y.
(iv) For eachx∈C fixed, the function y↦F(x,y) is convex and lower semicontinuous.
Leth:C×C→R such that
(i)h(x,x)≥0,∀x∈C
(ii) for eachy∈C fixed, the function x→h(x,y) is upper semicontinuous
(iii) for eachx∈C fixed, the function y→h(x,y) is convex and lower semicontinuous
And assume that, for fixedr>0 and z∈C, there exists a nonempty compact convex subset K of H1 and x∈C∩K such that (26)Fy,x+hy,x+1ry-x,x-z<0,∀y∈C∖K.Lemma 3 (see [25]).
Assume thatF1,h1:C×C→R satisfy Assumption 2. Let r>0 and x∈H1. Then, there exists z∈C such that(27)F1z,y+h1z,y+1ry-z,z-x≥0,∀y∈C.Lemma 4 (see [1]).
Assume that the bifunctionsF1,h1:C×C→R satisfy Assumption 2 and h1 is monotone. For r>0 and for all x∈H1, define a mapping Tr(F1,h1):H1→C as (8). Then, the following conclusions hold.
(1)Tr(F1,h1) is single-valued.
(2)Tr(F1,h1) is firmly nonexpansive.
(3)Fix(Tr(F1,h1))=GEP(F1,h1).
(4)GEP(F1,h1) is compact and convex.Furthermore, assume thatF2,h2:Q×Q→R satisfy Assumption 2. For s>0 and, for all w∈H2, Ts(F2,h2):H2→Q is defined as (9). By Lemma 4, we easily observe that Ts(F2,h2) is single-valued and firmly nonexpansive, GEP(F2,h2,Q) is compact and convex, and Fix(Ts(F2,h2))=GEP(F2,h2,Q), where GEP(F2,h2,Q) is the solution set of the following generalized equilibrium problem, which is to find y∗∈Q such that F2(y∗,y)+h2(y∗,y)≥0,∀y∈Q.We observe thatGEP(F2,h2)⊂GEP(F2,h2,Q). Further, it is easy to prove that Γ is a closed and convex set.Lemma 5 (see [27, 28]).
Assume thatT:H1→H1 is nonexpansive operator. For all (x,y)∈H1×H1, the following inequality is true(28)x-Tx-y-Ty,Ty-Tx≤12Tx-x-Ty-y2.And for all (x,y)∈H1×Fix(T), one has(29)x-Tx,y-Tx≤12Tx-x2.Lemma 6 (see [29]).
Assume A is a strongly positive linear bounded operator on Hilbert spaceH1 with coefficient γ¯>0 and 0<ρ≤A-1. Then I-ρA≤1-ργ¯.Lemma 7 (see [30, 31]).
Assume that{an} is a sequence of nonnegative real numbers such that(30)an+1≤1-γnan+γnδn+βn,n≥0,where {γn} and {βn} are sequences in (0,1) and {δn} is a sequence in R, such that
(i)∑n=0∞γn=∞.
(ii)limsupn→∞δn≤0 or ∑n=0∞γnδn<∞.
(iii)∑n=0∞βn<∞.
Thenlimn→∞an=0.According to [28], it is easy to prove the following Lemma.Lemma 8.
LetF1,h1:C×C→R be bifunctions satisfying Assumption 2 and, for r>0, the mapping Tr(F1,h1) is defined as (8). Let x,y∈H1, and r1,r2>0. Then(31)Tr2F1,h1y-Tr1F1,h1x≤y-x+r2-r1r2Tr2F1,h1y-y.Lemma 9 (see [32, 33]).
LetT:H→H be given. We have the following:
(i)T is nonexpansive, iff the complement I-T is 1/2-ism
(ii) ifT is v-ism, then, for γ>0, γT is v/γ-ism
(iii)T is averaged, iff the complement I-T is v-ism for some v>1/2; indeed, for α∈(0,1), T is α-averaged, iff I-T is 1/2α-ismLemma 10 (see [4, 32]).
Let the operatorsS,T,V:H→H be given.
(i) IfT=(1-α)S+αV, where S is averaged, V is nonexpansive, and α∈(0,1), then T is averaged.
(ii)T is firmly nonexpansive, iff the complement I-T is firmly nonexpansive.
(iii) IfT=(1-α)S+αV, where S is firmly nonexpansive, V is nonexpansive, and α∈(0,1), then T is averaged.
(iv) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings{Ti}i=1N is averaged, then so is the composite T1,…,TN. In particular, if T1 is α1-averaged and T2 is α2-averaged, where α1,α2∈(0,1), then the composite T1T2 is α-averaged, where α=α1+α2-α1α2.
(v) If the mappings{Ti}i=1N are averaged and have a common fixed point, then (32)⋂i=1NFixTi=FixT1,…,TN.In the following, we give the relation between the projection operator and average mapping.Lemma 11.
Assume that the variational inequality problem (3) is solvable. If B is β-ism from C into H1, then PC(I-λB) is 2β+λ/4β-averaged.Proof.
Note thatB is β-ism, which implies that λB is β/λ-ism; i.e., I-(I-λB) is β/λ-ism. By Lemma 9(iii), we can see that I-λB is λ/2β-averaged. Since the projection PC is 1/2-averaged, it is easy to see from Lemma 10 that the composite PC(I-λB) is 2β+λ/4β-averaged for 0<λ<2β according to (33)12+λ2β-12·λ2β=2β+λ4β,which completes the proof.As a result we have that, for eachn, PC(I-λnB) is 2β+λn/4β-averaged. Therefore, we can write(34)PCI-λnB=2β-λn4βI+2β+λn4βTn=1-bnI+bnTn,where Tn is nonexpansive and bn=2β+λn/4β∈[1/2,1].Lemma 12 (see [28]).
Let{xn} and {zn} be bounded sequence in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose xn+1=(1-βn)zn+βnxn for all integers n≥0 and limsupn→∞(zn+1-zn-xn+1-xn)≤0. Then limn→∞zn-xn=0.


## 3. Main Results

In this section, we give the main results of this paper. First, we describe the algorithm for finding a common solution of split generalized equilibrium and variational inequality problems.Throughout the rest of this paper, letf be a contraction of H1 into itself with coefficient η∈(0,1), A be a bounded linear operator, B be a β-inverse strongly monotone mapping from C into H1, and D be a strongly positive linear bounded self-adjoint operator on H1 with coefficient γ¯>0 and 0<γ<γ¯/η.Now, we give the description of the algorithm.Algorithm 13.
Letx0∈H1 be arbitrary and α>0. Assume that {αn}, {βn}, {γn}⊂(0,1), {λn}⊂(0,2β), {rn}⊂(0,∞), and ξ∈(0,1/L), where L is the spectral radius of the operator AA∗ and A∗ is the adjoint of A. Calculate sequences {un}, {yn}, and {xn} by the following iteration formula.(35)un=TrnF1,h1xn+ξA∗TrnF2,h2-IAxn,yn=un+αdn+1,xn+1=αnγfxn+βnxn+1-βnI-αnDyn,where dn+1=1/α(Tnun-un)+γndn, d0=1/α(T0u0-u0) and Tn is defined by (34).As follows, we propose the convergence analysis of Algorithm13.Theorem 14.
LetH1 and H2 be two real Hilbert spaces and C⊂H1, Q⊂H2 be nonempty closed convex subsets. Let F1,h1:C×C→R and F2,h2:Q×Q→R satisfy Assumption 2; h1, h2 are monotone and F2 is upper semicontinuous in the first argument. Assume that Ω≔VI(C,B)∩Γ≠∅ and {xn}, {un}, and {yn} are generated by (35). Suppose that {αn},{βn}⊂(0,1), {γn}⊂(0,1/2), {λn}⊂(0,2β), {rn}⊂(0,∞) satisfy the following conditions:
(C1)limn→∞αn=0, ∑n=0∞αn=∞.
(C2)0<liminfn→∞βn≤limsupn→∞βn<1.
(C3)γn=o(αn).
(C4)liminfn→∞λn>0, limn→∞|λn+1-λn|=0.
(C5)limn→∞|rn+1-rn|/rn+1=0.
(C6){Tnun-un} is bounded.
Then the sequence{xn} converges strongly to q∈Ω, where q=PΩ(I-D+γf)q, which is the unique solution of the variational inequality problem (36)D-γfq,x-q≥0,∀x∈Ω,or equivalently, q is the unique solution to the minimization problem (37)minx∈Ω⁡12Dx,x-hx,where h is a potential function for γf such that h′(x)=γf(x) for x∈H1.Proof.
Some equalities and inequalities in the following can be obtained according to the proof of Theorem 1 in [25]. However, we give the detailed proof process in order to read handily.
From (C1) and (C2), without loss of generality we assume thatαn≤1-βnD-1 for all n∈N. By Lemma 6, we have I-ρD≤1-ργ¯ if 0<ρ≤D-1. Now suppose that I-D≤1-γ¯. Since D is a strongly positive linear bounded self-adjoint operator on H1, we obtain (38)D=supDx,x:x∈H1,x=1.Notice that (39)1-βnI-αnDx,x=1-βn-αnDx,x≥1-βn-αnD≥0,which shows that (1-βn)I-αnD is positive definite. Furthermore, we have (40)1-βnI-αnD=sup1-βnI-αnDx,x:x∈H1,x=1=sup1-βn-αnDx,x:x∈H1,x=1≤1-βn-αnγ¯.Since f is a contraction mapping with constant η∈(0,1), for all x,y∈H1, we have (41)PΩI-D+γfx-PΩI-D+γfy≤I-D+γfx-I-D+γfy≤I-Dx-y+γfx-fy≤1-γ¯x-y+γηx-y≤1-γ¯-γηx-y,which implies that PΩ(I-D+γf) is a contraction mapping from H1 into itself. It follows from the Banach contraction principle that there exists an element q∈Ω such that q=PΩ(I-D+γf)q.
Step 1 (we show that { x n } is bounded). Let x∗∈Ω; i.e., x∗∈Γ; we have x∗=Trn(F1,h1)x∗ and Ax∗=Trn(F2,h2)Ax∗.
In the following, we compute(42)un-x∗2=TrnF1,h1xn+ξA∗TrnF2,h2-IAxn-x∗2=TrnF1,h1xn+ξA∗TrnF2,h2-IAxn-TrnF1,h1x∗2≤xn+ξA∗TrnF2,h2-IAxn-x∗2=xn-x∗2+ξ2A∗TrnF2,h2-IAxn2+2ξxn-x∗,A∗TrnF2,h2-IAxn.Thus, we have(43)un-x∗2≤xn-x∗2+ξ2TrnF2,h2-IAxn,AA∗TrnF2,h2-IAxn+2ξxn-x∗,A∗TrnF2,h2-IAxn.
On the other hand, we have(44)ξ2TrnF2,h2-IAxn,AA∗TrnF2,h2-IAxn≤Lξ2TrnF2,h2-IAxn2.and(45)2ξxn-x∗,A∗TrnF2,h2-IAxn=2ξAxn-x∗,TrnF2,h2-IAxn=2ξAxn-x∗+TrnF2,h2-IAxn-TrnF2,h2-IAxn,TrnF2,h2-IAxn=2ξTrnF2,h2Axn-Ax∗,TrnF2,h2-IAxn-TrnF2,h2-IAxn2≤2ξ12TrnF2,h2-IAxn2-TrnF2,h2-IAxn2≤-ξTrnF2,h2-IAxn2,where the first inequality is derived from (29).
From (43)-(45), we have(46)un-x∗2≤xn-x∗2+ξLξ-1TrnF2,h2-IAxn2.
Noticing thatξ∈(0,1/L), we obtain(47)un-x∗2≤xn-x∗2.
As follows, we prove that{dn} is bounded. The proof is by induction. It is true trivially for n=0. Let M1=maxd0,2/αsupn∈NTnun-un. From (C6), it is shown that M1<∞. Assume that dn≤M1 for some n; we prove that it holds for n+1. According to the triangle inequality, we obtain (48)dn+1=1αTnun-un+γndn≤1αTnun-un+γndn≤1α·α2M1+M12=M1,which implies that dn≤M1 for all n∈N; i.e., {dn} is bounded.
It is easy to see thatx∗∈VI(C,B) according to x∗∈Ω. By (22), we have PC(I-λB)x∗=x∗, which, together with (34), implies that (49)bnTnx∗=PCI-λBx∗-1-bnx∗=bnx∗;that is,(50)Tnx∗=x∗.
By the definition of{yn}, (47), and {Tn} being nonexpansive, we have(51)yn-x∗=un+αdn+1-x∗=Tnun+αγndn-Tnx∗≤un-x∗+αγnM1≤xn-x∗+αγnM1.
As a result, it follows from (51), Lemma 6,and the fact that αn→0 and γn=o(αn) that when n is large enough, (52)xn+1-x∗=αnγfxn-Dx∗+βnxn-x∗+1-βnI-αnDyn-x∗≤αnγfxn-Dx∗+βnxn-x∗+1-βn-αnγ¯xn-x∗+αγnM1≤αnγfxn-fx∗+αnγfx∗-Dx∗+βnxn-x∗+1-βn-αnγ¯xn-x∗+αγnM1≤αnγηxn-x∗+αnγfx∗-Dx∗+1-αnγ¯xn-x∗+1-βn-αnγ¯αγnM1≤1-αnγ¯-γηxn-x∗+αnγ¯-γηαM1+γfx∗-Dx∗γ¯-γη≤max⁡xn-x∗,αM1+γfx∗-Dx∗γ¯-γη,where the third inequality is true because βn∈(0,1), αn→0, and γn=o(αn). As a result, (53)1-βn-αnγ¯αγnM1≤ααnM1,when n is large enough.
Hence,{xn} is bounded and so are {un} and {yn}.
Step 2 (we show that l i m n → ∞ x n + 1 - x n = 0). Since Trn+1(F1,h1) and Trn+1(F2,h2) both are firmly nonexpansive, for ξ∈(0,1/L), the mapping Trn+1(F1,h1)(I+ξA∗(Trn+1(F2,h2)-I)A) is nonexpansive; see [34, 35]. Noticing that un=Trn(F1,h1)(xn+ξA∗(Trn(F2,h2)-I)Axn) and un+1=Trn+1(F1,h1)(xn+1+ξA∗(Trn+1(F2,h2)-I)Axn+1), we have from Lemma 8 that(54)un+1-un≤Trn+1F1,h1xn+1+ξA∗Trn+1F2,h2-IAxn+1-Trn+1F1,h1xn+ξA∗Trn+1F2,h2-IAxn+Trn+1F1,h1xn+ξA∗Trn+1F2,h2-IAxn-TrnF1,h1xn+ξA∗TrnF2,h2-IAxn≤xn+1-xn+xn+ξA∗Trn+1F2,h2-IAxn-xn+ξA∗TrnF2,h2-IAxn+1-rnrn+1Trn+1F1,h1xn+ξA∗Trn+1F2,h2-IAxn-xn+ξA∗Trn+1F2,h2-IAxn≤xn+1-xn+ξATrn+1F2,h2Axn-TrnF2,h2Axn+ζn≤xn+1-xn+ξA1-rnrn+1Trn+1F2,h2Axn-Axn+ζn=xn+1-xn+ξAσn+ζn,where σn≔|1-rn/rn+1|Trn+1(F2,h2)Axn-Axn and ζn≔|1-rn/rn+1|Trn+1(F1,h1)(xn+ξA∗(Trn(F2,h2)-I)Axn)-(xn+ξA∗(Trn+1(F2,h2)-I)Axn).
Furthermore, one has(55)yn+1-yn=un+1+αdn+2-un+αdn+1≤Tn+1un+1-Tnun+αγn+1dn+1-γndn≤Tn+1un+1-Tn+1un+Tn+1un-Tnun+αγn+1dn+1-γndn≤un+1-un+Tn+1un-Tnun+αM1γn+1+γnIt follows from (34) that(56)Tn+1un-Tnun=PCI-λn+1B-1-bn+1Ibn+1un-PCI-λnB-1-bnIbnun=4βPCI-λn+1B-2β-λn+1I2β+λn+1un-4βPCI-λnB-2β-λnI2β+λnun=4βPCI-λn+1B2β+λn+1un-4βPCI-λnB2β+λnun+2β-λn2β+λnun-2β-λn+12β+λn+1un≤4β2β+λnPCI-λn+1B-2β+λn+1PCI-λnB2β+λn+12β+λnun+4βλn+1-λn2β+λn+12β+λnun≤4βλn-λn+1PCI-λn+1B2β+λn+12β+λnun+4β2β+λn+1PCI-λn+1B-PCI-λnB2β+λn+12β+λnun+4βλn+1-λn2β+λn+12β+λnun≤1βλn-λn+1PCI-λn+1Bun+4βBun+un≤M2λn+1-λn,where M2=supn∈N1/βPCI-λn+1Bun+4βBun+un.
Hence from (54)-(56), we get (57)yn+1-yn≤xn+1-xn+ξAσn+ζn+M2λn+1-λn+αM1γn+1+γn.
Setxn+1=(1-βn)zn+βnxn; it follows that(58)zn=xn+1-βnxn1-βn=αnγfxn+1-βnI-αnDyn1-βn.
As a result,(59)zn+1-zn=αn+1γfxn+1+1-βn+1I-αn+1Dyn+11-βn+1-αnγfxn+1-βnI-αnDyn1-βn=αn+1γfxn+1-Dyn+11-βn+1-αnγfxn-Dyn1-βn+yn+1-yn≤αn+11-βn+1γfxn+1-Dyn+1+αn1-βnγfxn-Dyn+yn+1-yn≤αn+11-βn+1γfxn+1-Dyn+1+αn1-βnγfxn-Dyn+xn+1-xn+ξAσn+ζn+M2λn+1-λn+αM1γn+1+γn.Letting n→∞, from (C1)-(C5), we have (60)limsupn→∞zn+1-zn-xn+1-xn≤0.
By Lemma12, we obtain(61)limn→∞zn-xn=0.Further,(62)limn→∞xn+1-xn=limn→∞1-βnzn-xn=0.
Step 3 (we show that l i m n → ∞ u n - x n = 0). Since x∗∈Ω, x∗=Trn(F1,h1)x∗, and Trn(F1,h1) is firmly nonexpansive, we obtain (63)un-x∗2=TrnF1,h1xn+ξA∗TrnF2,h2-IAxn-x∗2=TrnF1,h1xn+ξA∗TrnF2,h2-IAxn-TrnF1,h1x∗2≤un-x∗,xn+ξA∗TrnF2,h2-IAxn-x∗=12un-x∗2+xn+ξA∗TrnF2,h2-IAxn-x∗2-un-x∗-xn+ξA∗TrnF2,h2-IAxn-x∗2=12un-x∗2+xn-x∗2-un-xn2+2ξun-x∗,A∗TrnF2,h2-IAxn.Hence, we obtain(64)un-x∗2≤xn-x∗2-un-xn2+2ξAun-x∗TrnF2,h2-IAxn.Furthermore,(65)yn-x∗2=un+αdn+1-x∗2≤un-x∗2+2αγnyn-x∗,dn≤un-x∗2+M3γn,where M3=supn∈N2αyn-x∗,dn.
By Lemma1(iii), (46), and (65), we have (66)xn+1-x∗2=αnγfxn+βnxn+1-βnI-αnDyn-x∗2=αnγfxn-Dx∗+βnxn-x∗+1-βnI-αnDyn-x∗2≤αnγfxn-Dx∗+βnxn-x∗+1-βn-αnγ¯yn-x∗2=αnγ¯1γ¯γfxn-Dx∗+βnxn-x∗+1-βn-αnγ¯yn-x∗2≤αnγ¯1γ¯2γfxn-Dx∗2+βnxn-x∗2+1-βn-αnγ¯yn-x∗2≤αnγ¯γfxn-Dx∗2+βnxn-x∗2+1-βn-αnγ¯yn-x∗2≤αnγ¯γfxn-Dx∗2+βnxn-x∗2+1-βn-αnγ¯xn-x∗2+ξLξ-1TrnF2,h2-IAxn2+M3γn=αnγ¯γfxn-Dx∗2+βnxn-x∗2+1-βn-αnγ¯xn-x∗2-1-βn-αnγ¯ξ1-LξTrnF2,h2-IAxn2+1-βn-αnγ¯M3γn.As a result, (67)1-βn-αnγ¯ξ1-LξTrnF2,h2-IAxn2≤xn-x∗2-xn+1-x∗2+αnγ¯γfxn-Dx∗2+1-βn-αnγ¯M3γn≤xn-x∗+xn+1-x∗xn-xn+1+αnγ¯γfxn-Dx∗2+1-βn-αnγ¯M3γn.
According toαn→0, γn=o(αn), (1-βn-αnγ¯)ξ(1-Lξ)>0, and limn→∞xn+1-xn=0, we obtain(68)limn→∞TrnF2,h2-IAxn=0.
From (65), (64), and Lemma 1(iii), we obtain (69)xn+1-x∗2=αnγfxn+βnxn+1-βnI-αnDyn-x∗2=αnγfxn-Dx∗+βnxn-x∗+1-βnI-αnDyn-x∗2≤αnγfxn-Dx∗+βnxn-x∗+1-βn-αnγ¯yn-x∗2=αnγ¯1γ¯γfxn-Dx∗+βnxn-x∗+1-βn-αnγ¯yn-x∗2≤αnγ¯1γ¯2γfxn-Dx∗2+βnxn-x∗2+1-βn-αnγ¯yn-x∗2≤αnγ¯γfxn-Dx∗2+βnxn-x∗2+1-βn-αnγ¯yn-x∗2≤αnγ¯γfxn-Dx∗2+βnxn-x∗2+1-βn-αnγ¯un-x∗2+M3γn≤αnγ¯γfxn-Dx∗2+βnxn-x∗2+1-βn-αnγ¯xn-x∗2-un-xn2+2ξAun-x∗TrnF2,h2-IAxn+M3γn=αnγ¯γfxn-Dx∗2+βnxn-x∗2+1-βn-αnγ¯xn-x∗2-1-βn-αnγ¯un-xn2+2ξ1-βn-αnγ¯Aun-x∗TrnF2,h2-IAxn+1-βn-αnγ¯M3γn.Therefore, one has (70)1-βn-αnγ¯un-xn2≤αnγ¯γfxn-Dx∗2+1-αnγ¯xn-x∗2-xn+1-x∗2+2ξ1-βn-αnγ¯Aun-x∗TrnF2,h2-IAxn+1-βn-αnγ¯M3γn≤αnγ¯γfxn-Dx∗2+xn-x∗2-xn+1-x∗2+2ξ1-βn-αnγ¯Aun-x∗TrnF2,h2-IAxn+1-βn-αnγ¯M3γn≤αnγ¯γfxn-Dx∗2+xn-xn+1xn-x∗+xn+1-x∗+2ξ1-βn-αnγ¯Aun-x∗TrnF2,h2-IAxn+1-βn-αnγ¯M3γn.According to αn→0, γn=o(αn), (62), and (68), we obtain(71)limn→∞un-xn=0.
Step 4 (we show that l i m n → ∞ T n u n - u n = 0). From (58), we have (72)zn-yn=αn1-βnγfxn-αn1-βnDyn.Hence, (73)zn-yn≤αnγ1-βnfxn+αn1-βnDyn.
From (C1) and (C2), one has(74)limn→∞zn-yn=0.By (74) and (61), we have(75)limn→∞xn-yn=0.
Combining (71) and (75), one has(76)limn→∞un-yn=0.Noting that (77)yn-un=Tnun-un+αγndn,one has (78)Tnun-un=yn-un-αγndn.
It follows from (76) and αn→0, γn=o(αn) that(79)limn→∞Tnun-un=0.
We claim that(80)limsupn→∞D-γfq,q-xn≤0,where q is the unique solution of the variational inequality 〈(D-γf)q,x-q〉≥0,∀x∈Ω.
To show this inequality, we choose a subsequence{xni} of {xn} such that(81)limsupn→∞D-γfq,q-xn=limi→∞D-γfq,q-xni.
Since{xni} is bounded, there exists a subsequence of {xni} which converges weakly to z∈C. Without loss of generality, we can assume that xni⇀z.
Step 5 (we show that z ∈ Ω). First, we show that z∈VI(C,B).
LetM:H1→2H1 be a set-valued mapping defined by (82)Mv=Bv+NCv,v∈C,∅,v∉C,where NCv≔z∈H1:v-u,z≥0,∀u∈C is the normal cone to C at v∈C. Then M is maximal monotone and 0∈Mv if and only if v∈VI(C,B) (see [36]). Let (v,u)∈G(M). Therefore, we have (83)u∈Mv=Bv+NCv,and so (84)u-Bv∈NCv.According to un∈C, we obtain(85)v-un,u-Bv≥0.On the other hand, according to (86)PCI-λnBun=un+bnTnun-un,where bn=2β+λn/4β, for ∀n∈N, and v∈H1, we have (87)v-un-bnTnun-un,un+bnTnun-un-un-λnBun≥0.
Therefore,(88)v-un,bnTnun-un+λnBun-bnTnun-un2-bnTnun-un,λnBun≥0.As a result,(89)v-un,bnλnTnun-un+Bun-bnTnun-un,Bun≥0.
Furthermore, according to (85) and (89), for ∀n∈N, one has (90)v-un,u≥v-un,Bv-v-un,bnλnTnun-un+Bun+bnTnun-un,Bun=v-un,Bv-Bun-v-un,bnλnTnun-un+bnTnun-un,Bun.
Replacingn by ni, one has(91)v-uni,u≥v-uni,Bv-Buni-v-uni,bniλniTniuni-uni+bniTniuni-uni,Buni.Since un-xn→0 and xni⇀z, we have uni⇀z. Noting that B is β-ism, from (91) and (79), we have (92)v-z,u≥0.
SinceM is maximal monotone, one has z∈M-10. Hence z∈VI(C,B).
Next, we provez∈Γ.
According to Algorithm13, we have (93)uni=TrniF1,h1xni+ξA∗TrniF2,h2-IAxni.
By (8), for any w∈C, one has(94)0≤F1uni,w+h1uni,w+1rniw-uni,uni-xni+ξA∗TrniF2,h2-IAxni=F1uni,w+h1uni,w+1rniw-uni,uni-xni-ξrniAw-Auni,TrniF2,h2-IAxni≤F1uni,w+h1uni,w+1rniw-uniuni-xni+ξrniAw-AuniTrniF2,h2-IAxni
According to the monotonicity ofF1, we have (95)F1w,uni≤h1uni,w+1rniw-uniuni-xni+ξrniAw-AuniTrniF2,h2-IAxni
From Assumption2(iv) on F and (ii) on h, (68), and (71), one has (96)F1w,z≤h1z,w.It follows from the monotonicity of h1 that (97)F1w,z+h1w,z≤0,∀w∈C.For any t∈(0,1] and w∈C, let wt=tw+(1-t)z. Since z∈C and C is convex, we obtain that wt∈C. Hence (98)F1wt,z+h1wt,z≤0.From Assumption 2(i), (iv) on F and (i), (iii) on h, we have (99)0≤F1wt,wt+h1wt,wt≤tF1wt,w+h1wt,w+1-tF1wt,z+h1wt,z≤tF1wt,w+h1wt,w,which implies that (100)F1wt,w+h1wt,w≥0,∀w∈C.Letting t→0 and by Assumption 2(iii) on F and (ii) on h, we obtain (101)F1z,w+h1z,w≥0,∀w∈C;that is, z∈GEP(F1,h1).
As follows, we proveAz∈GEP(F2,h2).
SinceA is a bounded linear operator, one has Axni⇀Az.
Now, setζni=Axni-Trni(F2,h2)Axni. It follows from (68) that limi→∞ζni=0. Since Axni-ζni=Trni(F2,h2)Axni, by (9) we have (102)F2Axni-ζni,z~+h2Axni-ζni,z~+1rniz~-Axni-ζni,Axni-ζni-Axni≥0,∀z~∈Q.
Furthermore, one has(103)F2Axni-ζni,z~+h2Axni-ζni,z~+1rniz~-Axni+ζni,-ζni≥0,∀z~∈Q.
From the upper semicontinuity ofF2(x,y) and h2(x,y) on x, we have (104)F2Az,z~+h2Az,z~≥0,∀z~∈Q,which means that Az∈GEP(F2,h2). As a result, z∈Γ.
Therefore,z∈Ω.
Sinceq=PΩ(I-D+γf)q is the unique solution of the variational inequality problem 〈(D-γf)q,x-q〉≥0,∀x∈Ω, by (81) and z∈Ω, we have(105)limsupn→∞D-γfq,q-xn≤0.
Step 6 (finally, we show that { x n } converges strongly to q). It is obvious that (106)yn-q=Tnun-q+αγndn≤Tnun-Tnq+αγndn≤un-q+αM1γn=un-q+M4γn≤xn-q+M4γn,where M4=αM1. And the first inequality is true because q∈Ω and Tnq=q according to the same reasoning to equality (50). The last inequality is obtained by q∈Ω and the same reasoning to inequality (47). Thus, from Lemma 1(i), we have (107)xn+1-q2=αnγfxn+βnxn+1-βnI-αnDyn-q2=αnγfxn-Dq+βnxn-q+1-βnI-αnDyn-q2≤βnxn-q+1-βnI-αnDyn-q2+2αnγfxn-Dq,xn+1-q≤βnxn-q+1-βn-αnγ¯yn-q)2+2αnγfxn-fq,xn+1-q+2αnγfq-Dq,xn+1-q≤1-αnγ¯xn-q+1-βn-αnγ¯M4γn2+2αnγηxn-qxn+1-q+2αnγfq-Dq,xn+1-q≤1-αnγ¯2xn-q2+M5γn+αnγηxn-q2+αnγηxn+1-q2+2αnγfq-Dq,xn+1-q,where M5=supn∈N21-αnγ¯1-βn-αnγ¯M4xn-q+1-βn-αnγ¯2M42γn.
As a result,(108)1-αnγηxn+1-q2≤1-2αnγ¯+αn2γ¯2+αnγηxn-q2+M5γn+2αnγfq-Dq,xn+1-q,which implies that(109)xn+1-q2≤1-2γ¯-γηαn1-αnγηxn-q2+2γ¯-γηαn1-αnγηαnγ¯2xn-q2+M5γn/αn2γ¯-γη+1γ¯-γηγfq-Dq,xn+1-q=1-tnxn-q2+tnδn,where tn=2(γ¯-γη)αn/1-αnγη, δn=αnγ¯2xn-q2+M5γn/αn/2(γ¯-γη) + 1/γ¯-γη〈γf(q)-Dq,xn+1-q〉.
According to (80), (C1),(C3), and ηγ<γ¯, we have ∑n=0∞tn=∞ and limsupn→∞δn≤0.
By Lemma7, xn→q, which completes the proof.


## 4. Consequently Results

In the above section, we discuss the iterative algorithm and prove the strong convergence theorem for finding a common solution of split generalized equilibrium and variational inequality problems. In this section, we give some corollaries, which can find a common solution of the special issues obtained from split generalized equilibrium and variational inequality problems.Ifh1=h2=0, then SGEP (5)-(6) reduces to the following split equilibrium problem (SEP).LetF1:C×C→R and F2:Q×Q→R be nonlinear bifunctions and A:H1→H2 be a bounded linear operator; then SEP is to find x∗∈C such that(110)F1x∗,x≥0,∀x∈C,and such that(111)y∗=Ax∗∈QsolvesF2y∗,y≥0,∀y∈Q.The solution set of SEP (110)-(111) is denoted by Γ1. And(112)TrF1x=z∈C:F1z,y+1ry-z,z-x≥0,∀y∈C,(113)TsF2w=d∈Q:F2d,e+1se-d,d-w≥0,∀e∈Q.According to Theorem 14, we can obtain the following corollary.Corollary 15.
LetH1 and H2 be two real Hilbert spaces and C⊂H1, Q⊂H2 be nonempty closed convex subsets. Let F1:C×C→R and F2:Q×Q→R satisfy Assumption 2 and F2 is upper semicontinuous in the first argument. Assume that Ω≔VI(C,B)∩Γ1≠∅, x0∈H1 and {un}, {yn}, and {xn} are generated by the following iterative scheme:(114)un=TrnF1xn+ξA∗TrnF2-IAxn,yn=un+αdn+1,xn+1=αnγfxn+βnxn+1-βnI-αnDyn,where dn+1=1/α(Tnun-un)+γndn, d0=1/α(T0u0-u0), α>0, and Tn is defined by (34). Suppose that {αn},{βn}⊂(0,1), {γn}⊂(0,1/2), {λn}⊂(0,2β), {rn}⊂(0,∞) satisfy conditions (C1)-(C6) in Theorem 14. Then sequence {xn} converges strongly to q∈Ω, where q=PΩ(I-D+γf)q, which is the unique solution of the variational inequality problem (115)D-γfq,x-q≥0,∀x∈Ω,or equivalently, q is the unique solution to the minimization problem (116)minx∈Q⁡12Dx,x-hx,where h is a potential function for γf such that h′(x)=γf(x) for x∈H1.Furthermore, ifF1=F2=F, H1=H2=H, and A=0, then SEP (110)-(111) reduce to the following equilibrium problem (EP).LetF:C×C→R be nonlinear bifunction; then EP is to find x∗∈C such that(117)Fx∗,x≥0,∀x∈C.The solution set of EP (117) is denoted by Γ2. And(118)TrFx=z∈C:Fz,y+1ry-z,z-x≥0,∀y∈C.According to Corollary 15, let D=I, and we can obtain the following corollary.Corollary 16.
LetH be real Hilbert space and C⊂H be nonempty closed convex subset. Let F:C×C→R satisfy Assumption 2. Assume that Ω≔VI(C,B)∩Γ2≠∅, x0∈H1, and {un}, {yn}, and {xn} are generated by the following iterative scheme:(119)un=TrnFxn,yn=un+αdn+1,xn+1=αnγfxn+βnxn+1-βn-αnyn,where dn+1=1/α(Tnun-un)+γndn, d0=1/α(T0u0-u0), α>0, and Tn is defined by (34). Suppose that {αn},{βn}⊂(0,1), {γn}⊂(0,1/2), {λn}⊂(0,2β), {rn}⊂(0,∞) satisfy conditions (C1)-(C6) in Theorem 14. Then sequence {xn} converges strongly to q∈Ω, where q=PΩγf(q).


## 5. Numerical Examples

In this section, we show some insight into the behavior of Algorithm13. The whole codes are written in Matlab 7.0. All the numerical results are carried out on a personal Lenovo Thinkpad computer with Intel(R) Core(TM) i7-6500U CPU 2.50GHz and RAM 8.00GB.Example 1.
In the variational inequality problem (3) as well as the split generalized equilibrium problem (5) and (6), let H1=H2=R2, C=x1,x2T∈R2∣1≤x1≤5,0.5≤x2≤4, Q=y1,y2T∈R2∣2≤y1≤10,3≤y2≤24, F1(x,y)=h1(x,y)=eT(x-y),∀x,y∈C, F2(x,y)=h2(x,y)=5eT(x-y),∀x,y∈Q, and A=2006, B=1/2001/2, where e=(1,1)T.It is easy to see that the optimal solution of Example1 is x∗=(5,4)T. Now, we use Algorithm 13 to compute the solution of the problem. Let α=1/2, ξ=1/7, D=9/10009/10, γ=1.2, rn=1/10+1/2n, λn=1/n, αn=1/n, βn=1/10+1/n2, γn=1/n2, and f(x)=x/2,∀x∈H1. The stopping criterion is xn+1-xn≤ϵ. For ϵ=10-7 and different initial points which are presented randomly, such as (120)x=rand2,1,x=10∗rand2,1,x=1000∗rand2,1,separately. Table 1 shows the initial value, the final value, and the cpu time for the above three cases. We denote Init., Fina., and Sec. the initial value, the final value, and the cpu time in seconds, respectively.Table 1
The final value and the cpu time for different initial points.
Init. Fina. Sec. x = ( 0.7060,0.0318 ) T x - = ( 4.9924,3.9939 ) T 1.14 x = ( 5.4722,1.3862 ) T x - = ( 4.9924,3.9939 ) T 1.35 x = ( 89.0903,95.9291 ) T x - = ( 4.9924,3.9939 ) T 1.36From Table1, we can see that the final value x¯ is not influenced by the initial value.To show the changing tendency of the final valuex¯ for different ϵ, Table 2 gives the different ϵ, the initial value, the final value, and the cpu time in seconds.Table 2
The final value and the cpu time for differentϵ.
ϵ Init. Fina. Sec. 1 0 - 5 x = ( 8.0028,1.4189 ) T x - = ( 4.9640,3.9712 ) T 0.08 1 0 - 6 x = ( 7.4313,3.9223 ) T x - = ( 4.9835,3.9868 ) T 0.30 1 0 - 7 x = ( 5.4722,1.3862 ) T x - = ( 4.9924,3.9939 ) T 1.35Now, we further express the status that the final valuex¯ tends to optimal solution x∗ through Figures 1 and 2. We carry out 100 experiments for different ϵ from 10-7 to 10-9. From Figures 1 and 2, we know that the final value x¯ tends to the optimal solution x∗ when ϵ tends to 0, which illustrates the efficiency of Algorithm 13.Figure 1
The behaviors of first coordinate ofx¯ for different ϵ.Figure 2
The behaviors of the second coordinate ofx¯ for different ϵ.


## 6. Conclusion

In this paper, we study the split generalized equilibrium problem and variational inequality problem. For finding their common solution, we propose a kind of conjugate gradient viscosity approximation algorithm. Under mild conditions, we prove that the sequence generated by the iterative algorithm converges strongly to the common solution. In comparison to [25], the authors introduce and study an effective iterative algorithm to approximate a common solution of a split generalized equilibrium problem and a fixed point problem. Under suitable conditions, they proved a strong convergence theorem for the sequence generated by the iterative scheme. Furthermore, we can study the iterative algorithm for finding a common solution of the split generalized equilibrium problem, variational inequality problem, and fixed point problem.



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