# Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

Authors: L. C. Ceng; A. Latif; C. F. Wen; A. E. Al-Mazrooei
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions, and the solution set of general system of variational inequalities (GSVI), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and the GSVI. The results obtained in this paper improve and extend the corresponding results announced by many others.


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## Body

## 1. Introduction

LetC be a nonempty closed convex subset of a real Hilbert space H and let PC be the metric projection of H onto C. Let S:C→H be a nonlinear mapping on C. We denote by Fix⁡(S) the set of fixed points of S and by R the set of all real numbers. A mapping S:C→H is called L-Lipschitz continuous if there exists a constant L>0 such that
(1)Sx-Sy≤Lx-y,∀x,y∈C.
In particular, if L=1, then S is called a nonexpansive mapping; if L∈(0,1), then S is called a contraction.LetA:C→H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point x∈C such that
(2)Ax,y-x≥0,∀y∈C.
The solution set of VIP 2 is denoted by VI⁡(C,A).The VIP2 was first discussed by Lions [1]. In 1976, Korpelevich [2] proposed an iterative algorithm for solving the VIP 2 in Euclidean space Rn:
(3)yn=PCxn-τAxn,xn+1=PCxn-τAyn,∀n≥0,
with τ>0 a given number, which is known as the extragradient method.Letφ:C→R be a real-valued function, let A:H→H be a nonlinear mapping, and let Θ:C×C→R be a bifunction. In 2008, Peng and Yao [3] introduced the following generalized mixed equilibrium problem (GMEP) of finding x∈C such that
(4)Θx,y+φy-φx+Ax,y-x≥0,∀y∈C.
We denote the set of solutions of GMEP 4 by GMEP⁡(Θ,φ,A).It was assumed in [3] that Θ:C×C→R is a bifunction satisfying conditions (A1)–(A4) and φ:C→R is a lower semicontinuous and convex function with restriction (B1) or (B2), where(A1)
Θ(x,x)=0 for all x∈C;(A2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0 for any x,y∈C;(A3)
Θ is upper-hemicontinuous; that is, for each x,y,z∈C,
(5)limsup⁡t→0+Θtz+1-tx,y≤Θx,y;(A4)
Θ(x,·) is convex and lower semicontinuous for each x∈C;(B1)
for eachx∈H and r>0, there exists a bounded subset Dx⊂C and yx∈C such that, for any z∈C∖Dx,
(6)Θ(z,yx)+φ(yx)-φ(z)+1ryx-z,z-x<0;(B2)
C is a bounded set.For a given positive numberr, let Tr(Θ,φ):H→C be the solution set of the auxiliary mixed equilibrium problem; that is, for each x∈H,
(7)Tr(Θ,φ)x≔y∈C:Θy,z+φz-φ(y)1r+1ry-x,z-y≥0,∀z∈C.On the other hand, letB be a single-valued mapping of C into H and let R be a multivalued mapping with D(R)=C. Consider the following variational inclusion: find a point x∈C such that
(8)0∈Bx+Rx.
We denote by I(B,R) the solution set of the variational inclusion 8. It is known that problem 8 provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory. Let R:D(R)⊂H→2H be a maximal monotone set-valued mapping. We define the resolvent operator JR,λ:H→D(R)¯ associated with R and λ as follows:
(9)JR,λ=I+λR-1,∀x∈H,
where λ is a positive number.LetF1,F2:C→H be two mappings. Consider the following general system of variational inequalities (GSVI) of finding (x*,y*)∈C×C such that
(10)〈ν1F1y*+x*-y*,x-x*〉≥0,∀x∈C,〈ν2F2x*+y*-x*,x-y*〉≥0,∀x∈C,
where ν1>0 and ν2>0 are two constants. It was considered and studied in [4–6]. In particular, if F1=F2=A, then the GSVI 10 reduces to the following problem of finding (x*,y*)∈C×C such that
(11)ν1Ay*+x*-y*,x-x*≥0,∀x∈C,ν2Ax*+y*-x*,x-y*≥0,∀x∈C,
which is defined by Verma [7] and it is called a new system of variational inequalities (NSVI). Further, if x*=y*, additionally, then the NSVI reduces to the classical VIP 2. In 2008, Ceng et al. [6] transformed the GSVI 10 into a fixed point problem in the following way.Proposition CWY (see [6]).For givenx¯,y¯∈C,(x¯,y¯)is a solution of the GSVI10 if and only ifx¯is a fixed point of the mappingG:C→Cdefined by(12)Gx=PCI-ν1F1PCI-ν2F2x,∀x∈C,wherey¯=PC(I-ν2F2)x¯.In particular, if the mappingFj:C→H is ζj-inverse strongly monotone for j=1,2, then the mapping G is nonexpansive provided νj∈(0,2ζj] for j=1,2. We denote by GSVI(G) the fixed point set of the mapping G.LetS and T be two nonexpansive mappings. In 2009, Yao et al. [8] considered the following hierarchical VIP: find hierarchically a fixed point of T, which is a solution to the VIP for monotone mapping I-S; namely, find x~∈Fix⁡(T), such that
(13)〈(I-S)x~,p-x~〉≥0,∀p∈Fix⁡(T).
The solution set of the hierarchical VIP 13 is denoted by Λ. It is not hard to check that solving the hierarchical VIP 13 is equivalent to the fixed point problem of the composite mapping PFix⁡(T)S; that is, find x~∈C, such that x~=PFix⁡(T)Sx~. The authors [8] introduced and analyzed the following iterative algorithm for solving the hierarchical VIP 13:
(14)yn=βnSxn+1-βnxn,xn+1=αnVxn+1-αnTyn,∀n≥0.Theorem YLM (see [8, Theorem 3.2]).LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetSandTbe two nonexpansive mappings ofCinto itself. LetV:C→Cbe a fixed contraction withα∈(0,1). Let{αn}and{βn}be two sequences in(0,1). For any givenx0∈C, let{xn}be the sequence generated by14. Assume that the sequence{xn}is bounded and that(i)
∑n=0∞αn=∞;(ii)
lim⁡n→∞1/αn1/βn-1/βn-1=0,lim⁡n→∞1/βn1-αn-1/αn=0;(iii)
lim⁡n→∞βn=0,lim⁡n→∞(αn/βn)=0andlim⁡n→∞(βn2/αn)=0;(iv)
Fix⁡(T)∩int⁡C≠∅;(v)
there exists a constantk>0such thatx-Tx≥kDist(x,Fix⁡(T))for eachx∈C, whereDist(x,Fix⁡(T))=inf⁡y∈Fix⁡(T)x-y. Then,{xn}converges strongly tox~=PΛVx~which solves the hierarchical VIP(15)I-Sx~,p-x~≤0,∀p∈Fix⁡(T).Very recently, Iiduka [9, 10] considered a variational inequality with a variational inequality constraint over the set of fixed points of a nonexpansive mapping. Since this problem has a triple structure in contrast with bilevel programming problems or hierarchical constrained optimization problems or hierarchical fixed point problem, it is refereed as triple hierarchical constrained optimization problem (THCOP). He presented some examples of THCOP and developed iterative algorithms to find the solution of such a problem. The convergence analysis of the proposed algorithms was also studied in [9, 10]. Since the original problem is a variational inequality, in this paper, we call it a triple hierarchical variational inequality (THVI). Subsequently, Zeng et al. [11] introduced and considered the following triple hierarchical variational inequality (THVI).Problem I. Assume that(i)
eachTi:H→H is a nonexpansive mapping with ⋂i=1NFix⁡(Ti)≠∅;(ii)
A1:H→H is α-inverse strongly monotone;(iii)
A2:H→H is β-strongly monotone and L-Lipschitz continuous;(iv)
IV(⋂i=1NFix⁡(Ti),A1)≠∅.Then, the objective is to
(16)findx*∈VI⁡VI⁡⋂i=1NFix⁡Ti,A1,A2≔x*∈VI⁡⋂i=1NFix⁡Ti,A1:A2x*,v-x*≥0,∀v∈VI⁡⋂i=1NFix⁡Ti,A1.In [11], the authors proposed the following algorithm for solving Problem I.Algorithm ZWY ([11, Algorithm  3.2]). Let Ti:H→H(i=1,2,…,N) and Ai:H→H(i=1,2) satisfy assumptions (i)–(iv) in Problem I. The following steps are presented for solving Problem I.Step  0. Take {αn}n=0∞⊂(0,1], {ρn}n=0∞⊂(0,2α], and μ∈(0,2β/L2), choose x0∈H arbitrarily, and let n≔0.Step  1. Given xn∈H, compute xn+1∈H as
(17)yn=T[n+1](xn-ρnA1xn),xn+1=yn-μαnA2yn,
where T[k]≔Tkmod⁡N, for integer k≥1, with the mod function taking values in the set {1,2,…,N}; that is, if k=jN+q for some integers j≥0 and 0≤q<N, then T[k]=TN if q=0 and T[k]=Tq if 1≤q<N.The following convergence analysis was presented in [11].Theorem ZWY ([11, Theorem 3.2]).Letμ∈(0,2β/L2),{αn}n=0∞⊂(0,1], and{ρn}n=0∞⊂(0,2α], such that (i)lim⁡n→∞αn=0, (ii)∑n=0∞αn=∞, (iii)lim⁡n→∞(αn-αn+N)/αn+N=0or∑n=0∞‍αn+N-αn<∞, (iv)lim⁡n→∞(ρn-ρn+N)/ρn+N=0or∑n=0∞‍ρn+N-ρn<∞, and (v)ρn≤αnfor alln≥0. Assume in addition that(18)⋂i=1NFix⁡Ti=Fix⁡T1T2⋯TN=Fix⁡TNT1⋯TN-1=⋯=Fix⁡(T2T3⋯TNT1).
Then, the sequence {xn}n=0∞ generated by Algorithm ZWY satisfies the following properties:(a)
{xn}n=0∞ is bounded;(b)
lim⁡n→∞xn+N-xn=0 and lim⁡n→∞xn-Tn+N⋯Tn+1xn=0;(c)
{xn}n=0∞ converges strongly to the unique solution of Problem I provided xn-yn=o(ρn).In this paper, we introduce and study the following triple hierarchical variational inequality (THVI) with constraints of finitely many GMEPs, finitely many variational inclusions, and general system of variational inequalities.Problem II. Let M,N be two integers. Assume that(i)
{Tn}n=1∞ is a sequence of nonexpansive self-mappings on C and Fj:C→H is ζj-inverse strongly monotone for j=1,2;(ii)
A~1:H→H is α-inverse strongly monotone and A~2:H→H is β-strongly monotone and L-Lipschitz continuous;(iii)
Θk is a bifunction from C×C to R satisfying (A1)–(A4) and φk:C→R∪{+∞} is a proper lower semicontinuous and convex function, where k∈{1,2,…,M};(iv)
Ri:C→2H is a maximal monotone mapping, and Ak:H→H and Bi:C→H are μk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2,…,M} and i∈{1,2,…,N};(v)
VI⁡(Ω,A~1)≠∅, where Ω≔⋂n=1∞Fix⁡(Tn)∩⋂k=1MGMEP⁡(Θk,φk,Ak)∩⋂i=1NI(Bi,Ri)∩GSVI(G). Then, the objective is to
(19)findx*∈VI⁡VI⁡Ω,A~1,A~2≔x*∈VI⁡(Ω,A~1):〈A~2x*,v-x*〉≥0,∀v∈VI⁡(Ω,A~1).Motivated and inspired by the above facts, we introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove that under mild conditions, the proposed algorithm converges strongly to a common elementx*∈Ω≔⋂n=1∞Fix⁡(Tn)∩⋂k=1MGMEP⁡(Θk,φk,Ak)∩⋂i=1NI(Bi,Ri)∩GSVI(G) of the fixed point set of infinitely many nonexpansive mappings {Tn}n=1∞, the solution set of finitely many GMEPs, the solution set of finitely many variational inclusions, and the solution set of GSVI 10, which is just a unique solution of the THVI 19. In addition, we also consider the application of the proposed algorithm to solve a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and GSVI 10. That is, under appropriate conditions, we prove that the proposed algorithm converges strongly to a unique solution u*∈Ω of the VIP: 〈A~2u*,p-u*〉≥0, ∀p∈Ω; equivalently, PΩ(I-A~2)u*=u*. The results obtained in this paper improve and extend the corresponding results announced by many others including [12, 13]. A comprehensive survey on triple hierarchical variational inequalities can be found in [14].


## 2. Preliminaries

Throughout this paper, we assume thatH is a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ·, respectively. Let C be a nonempty closed convex subset of H. We write xn⇀x to indicate that the sequence {xn} converges weakly to x and xn→x to indicate that the sequence {xn} converges strongly to x. Moreover, we use ωw(xn) to denote the weak ω-limit set of the sequence {xn}; that is,
(20)ωwxn≔x∈H:xni⇀xxniforsomesubsequencexniofxn.Recall that a mappingA:C→H is called(i)
monotone if(21)〈Ax-Ay,x-y〉≥0,∀x,y∈C;(ii)
η-strongly monotone if there exists a constant η>0 such that
(22)〈Ax-Ay,x-y〉≥ηx-y2,∀x,y∈C;(iii)
α-inverse strongly monotone if there exists a constant α>0 such that
(23)〈Ax-Ay,x-y〉≥αAx-Ay2,∀x,y∈C.It is obvious that ifA is α-inverse strongly monotone, then A is monotone and (1/α)-Lipschitz continuous. Moreover, we also have that, for all u,v∈C and λ>0,
(24)I-λAu-I-λAv2=(u-v)-λ(Au-Av)2=u-v2-2λ〈Au-Av,u-v〉+λ2Au-Av2≤u-v2+λ(λ-2α)Au-Av2.
So, if λ≤2α, then I-λA is a nonexpansive mapping from C to H.The metric (or nearest point) projection fromH onto C is the mapping PC:H→C which assigns to each point x∈H the unique point PCx∈C, satisfying the property
(25)x-PCx=inf⁡y∈Cx-y=:d(x,C).Some important properties of projections are gathered in the following proposition.Proposition 1.
For givenx∈H and z∈C:(i)
z=PCx⇔〈x-z,y-z〉≤0, ∀y∈C;(ii)
z=PCx⇔x-z2≤x-y2-y-z2, ∀y∈C;(iii)
〈PCx-PCy,x-y〉≥PCx-PCy2, ∀y∈H.Consequently, PC is nonexpansive and monotone.Definition 2.
A mappingT:H→H is said to be
(a) nonexpansive if(26)Tx-Ty≤x-y,∀x,y∈H;
(b) firmly nonexpansive if2T-I is nonexpansive or, equivalently, if T is 1-inverse strongly monotone (1-ism),
(27)〈x-y,Tx-Ty〉≥Tx-Ty2,∀x,y∈H;
alternatively, T is firmly nonexpansive if and only if T can be expressed as
(28)T=12I+S,
where S:H→H is nonexpansive; projections are firmly nonexpansive.It can be easily seen that ifT is nonexpansive, then I-T is monotone. It is also easy to see that a projection PC is 1-ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.Next, we list some elementary conclusions for the mixed equilibrium problem (MEP) whereMEP⁡(Θ,φ) is the solution set.Proposition 3 (see [15]).
Assume thatΘ:C×C→R satisfies (A1)–(A4) and let φ:C→R be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and x∈H, define a mapping Tr(Θ,φ):H→C as follows:
(29)Tr(Θ,φ)x=z∈C:Θ(z,y)+φ(y)-φ(z)1r+1r〈y-z,z-x〉≥0,∀y∈C
for all x∈H. Then, the following hold:(i)
for eachx∈H,Tr(Θ,φ)(x) is nonempty and single-valued;(ii)
Tr(Θ,φ) is firmly nonexpansive; that is, for any x,y∈H,
(30)Tr(Θ,φ)x-Tr(Θ,φ)y2≤〈Tr(Θ,φ)x-Tr(Θ,φ)y,x-y〉;(iii)
Fix⁡(Tr(Θ,φ))=MEP(Θ,φ);(iv)
MEP(Θ,φ) is closed and convex;(v)
Ts(Θ,φ)x-Tt(Θ,φ)x2≤(s-t/s)〈Ts(Θ,φ)x-Tt(Θ,φ)x, Ts(Θ,φ)x-x〉 for all s,t>0 and x∈H.We need some facts and tools in a real Hilbert spaceH which are listed as lemmas below.Lemma 4.
LetX be a real inner product space. Then, there holds the following inequality:
(31)x+y2≤x2+2〈y,x+y〉,∀x,y∈X.Lemma 5.
LetA:C→H be a monotone mapping. In the context of the variational inequality problem, the characterization of the projection (see Proposition 1(i)) implies
(32)u∈VI⁡C,A⟺u=PC(u-λAu),λ>0.Lemma 6 (see [16, demiclosedness principle]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C. Then, I-T is demiclosed. That is, whenever {xn} is a sequence in C weakly converging to some x∈C and the sequence {(I-T)xn} strongly converges to some y, it follows that (I-T)x=y. Here, I is the identity operator of H.Let{Tn}n=1∞ be an infinite family of nonexpansive self-mappings on C and let {λn}n=1∞ be a sequence of nonnegative numbers in [0,1]. For any n≥0, define a mapping Wn on C as follows:
(33)Un,n+1=I,Un,n=λnTnUn,n+1+(1-λn)I,Un,n-1=λn-1Tn-1Un,n+(1-λn-1)I,⋮Un,k=λkTkUn,k+1+(1-λk)I,Un,k-1=λk-1Tk-1Un,k+(1-λk-1)I,⋮Un,2=λ2T2Un,3+(1-λ2)I,Wn=Un,1=λ1T1Un,2+(1-λ1)I.
Such a mapping Wn is called the W-mapping generated by Tn,Tn-1,…,T1 and λn,λn-1,…,λ1.Lemma 7 (see [17, Lemma 3.2]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}n=1∞ be a sequence of nonexpansive self-mappings on C such that ⋂n=1∞Fix⁡(Tn)≠∅ and let {λn}n=1∞ be a sequence in (0,b] for some b∈(0,1). Then, for every x∈C and k≥1, the limit lim⁡n→∞Un,kx exists where Un,k is defined as in 33.Lemma 8 (see [17, Lemma 3.3]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}n=1∞ be a sequence of nonexpansive self-mappings on C such that ⋂n=1∞Fix⁡(Tn)≠∅, and let {λn}n=1∞ be a sequence in (0,b] for some b∈(0,1). Then, Fix⁡(W)=⋂n=1∞Fix⁡(Tn).LetC be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in (0,1] and let μ>0. Associating with a nonexpansive mapping T:C→H, we define the mapping Tλ:C→H by
(34)Tλx≔Tx-λμF(Tx),∀x∈C,
where F:H→H is an operator such that, for some positive constants κ,η>0, F is κ-Lipschitzian and η-strongly monotone on H; that is, F satisfies the conditions:
(35)Fx-Fy≤κx-y,Fx-Fy,x-y≥ηx-y2,
for all x,y∈H.Lemma 9 (see [18, Lemma 3.1]).
T
λ is a contraction provided 0<μ<2η/κ2; that is,
(36)Tλx-Tλy≤(1-λτ)x-y,∀x,y∈C,
where τ=1-1-μ(2η-μκ2)∈(0,1].Lemma 10 (see [18]).
Let{sn} be a sequence of nonnegative numbers satisfying the conditions
(37)sn+1≤(1-αn)sn+αnβn,∀n≥1,
where {αn} and {βn} are sequences of real numbers such that(i)
{αn}⊂[0,1] and ∑n=1∞‍αn=∞ or, equivalently,
(38)∏n=1∞‍(1-αn)≔lim⁡n→∞∏k=1n‍(1-αk)=0;(ii)
limsup⁡n→∞βn≤0, or ∑n=1∞‍αnβn<∞.Then, lim⁡n→∞sn=0.Lemma 11 (see [16]).
LetH be a real Hilbert space. Then, the following hold:(a)
x-y2=x2-y2-2x-y,y for all x,y∈H;(b)
λx+μy2=λx2+μy2-λμx-y2 for all x,y∈H and λ,μ∈[0,1] with λ+μ=1;(c)
If{xn} is a sequence in H such that xn⇀x, it follows that
(39)limsup⁡n→∞xn-y2=limsup⁡n→∞xn-x2+x-y2,∀y∈H.Finally, recall that a set-valued mappingT:D(T)⊂H→2H is called monotone if, for all x,y∈D(T), f∈Tx and g∈Ty imply
(40)〈f-g,x-y〉≥0.
A set-valued mapping T is called maximal monotone if T is monotone and (I+λT)D(T)=H for each λ>0, where I is the identity mapping of H. We denote by G(T) the graph of T. It is known that a monotone mapping T is maximal if and only if, for (x,f)∈H×H,〈f-g,x-y〉≥0 for every (y,g)∈G(T) implies f∈Tx. Let A:C→H be a monotone, k-Lipschitz-continuous mapping and let NCv be the normal cone to C at v∈C; that is,
(41)NCv=u∈H:v-p,u≥0,∀p∈C.
Define
(42)T~v=Av+NCv,ifv∈C,∅,ifv∉C.
Then, T~ is maximal monotone (see [19]) such that
(43)0∈T~v⟺v∈VI⁡(C,A).LetR:D(R)⊂H→2H be a maximal monotone mapping. Let λ,μ>0 be two positive numbers.Lemma 12 (see [20]).
There holds the resolvent identity(44)JR,λx=JR,μμλx+1-μλJR,λx,∀x∈H.Remark 13.
Forλ,μ>0, there holds the following relation:
(45)JR,λx-JR,μy≤x-y+λ-μ1λJR,λx-y+1μx-JR,μy,∀x,y∈H.In terms of Huang [21], there holds the following property for the resolvent operator JR,λ:H→D(R)¯.Lemma 14.
J
R
,
λ is single-valued and firmly nonexpansive; that is,
(46)〈JR,λx-JR,λy,x-y〉≥JR,λx-JR,λy2,∀x,y∈H.
Consequently, JR,λ is nonexpansive and monotone.Lemma 15 (see [22]).
LetR be a maximal monotone mapping with D(R)=C. Then, for any given λ>0, u∈C is a solution of problem 12 if and only if u∈C satisfies
(47)u=JR,λ(u-λBu).Lemma 16 (see [23]).
LetR be a maximal monotone mapping with D(R)=C and let B:C→H be a strongly monotone, continuous, and single-valued mapping. Then, for each z∈H, the equation z∈(B+λR)x has a unique solution xλ for λ>0.Lemma 17 (see [22]).
LetR be a maximal monotone mapping with D(R)=C and let B:C→H be a monotone, continuous, and single-valued mapping. Then, (I+λ(R+B))C=H for each λ>0. In this case, R+B is maximal monotone.


## 3. Main Results

In this section, we will introduce and analyze a relaxed iterative algorithm for finding a solution of the THVI19 with constraints of several problems: finitely many GMEPs, finitely many variational inclusions, and GSVI 10 in a real Hilbert space. This algorithm is based on Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove the strong convergence of the proposed algorithm to a unique solution of THVI 19 under suitable conditions. In addition, we also consider the application of the proposed algorithm to solve a hierarchical VIP with the same constraints.We are now in a position to state and prove our first main result.Theorem 18.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M, N be two integers. Let Θk be a bifunction from C×C to R satisfying (A1)–(A4) and let φk:C→R∪{+∞} be a proper lower semicontinuous and convex function, where k∈{1,2,…,M}. Let Ri:C→2H be a maximal monotone mapping and let Ak:H→H and Bi:C→H be μk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2,…,M}, i∈{1,2,…,N}. Let {Tn}n=1∞ be a sequence of nonexpansive self-mappings on C and {λn}n=1∞ be a sequence in (0,b] for some b∈(0,1). Let Fj:C→H be ζj-inverse strongly monotone for j=1,2. Let A~1:H→H be α-inverse strongly monotone with (I-A~1)C⊂C and let A~2:H→H be β-strongly monotone and L-Lipschitz continuous. Assume that either (B1) or (B2) holds and that VI⁡(Ω,A~1)≠∅ where Ω≔⋂n=1∞Fix⁡(Tn)∩⋂k=1MGMEP(Θk,φk,Ak)∩⋂i=1NI(Bi,Ri)∩GSVI(G). Let μ∈(0,2β/L2) and {αn}n=1∞ and {ρn}n=1∞ be sequences in (0,1]. For arbitrarily given x1∈H, let {xn} be a sequence generated by
(48)un=TrM,nΘM,φMI-rM,nAMTrM-1,nΘM-1,φM-1·(I-rM-1,nAM-1)⋯Tr1,n(Θ1,φ1)(I-r1,nA1)xn,vn=JRN,λN,nI-λN,nBNJRN-1,λN-1,n·(I-λN-1,nBN-1)⋯JR1,λ1,n(I-λ1,nB1)un,yn=WnG(vn-ρnA~1vn),xn+1=yn-μαnA~2yn,∀n≥1,
where G is defined as in 12 with νj∈(0,2ζj) for j=1,2, and Wn is the W-mapping defined by 33. Suppose that the following conditions are satisfied:(H1)
lim⁡n→∞αn=0, ∑n=1∞‍αn=∞ and lim⁡n→∞1/αn1-ρn-1/ρn=0;(H2)
lim⁡n→∞1/αn1/ρn-1/ρn-1=0 and lim⁡n→∞1/ρn1-αn-1/αn=0;(H3)
lim⁡n→∞(ρn/αn)=0 and lim⁡n→∞(bn/αnρn)=0;(H4)
{λi,n}⊂[ai,bi]⊂(0,2ηi) and lim⁡n→∞(λi,n-λi,n-1/αnρn)=0 for all i∈{1,2,…,N};(H5)
{rk,n}⊂[ek,fk]⊂(0,2μk) and lim⁡n→∞(rk,n-rk,n-1/ρnαn)=0 for all k∈{1,2,…,M}. Then, there hold the following:(i)
lim⁡n→∞(xn+1-xn/ρn)=0;(ii)
ωw(xn)⊂Ω;(iii)
ωw(xn)⊂VI⁡(Ω,A~1) provided xn-yn=o(ρn) additionally.Proof.
Let{x*}=VI⁡(VI⁡(Ω,A~1),A~2). Taking into account that lim⁡n→∞(ρn/αn)=0, we may assume, without loss of generality, that ρn≤αn and ρn≤2α for all n≥1. Since A~2 is L-Lipschitz continuous, we get
(49)A~2yn-A~2x*≤Lyn-x*,∀n≥1.
Putting v~n=vn-ρnA~1vn for all n≥1, we have
(50)xn+1=yn-μαnA~2yn=WnGv~n-μαnA~2WnGv~n=(I-μαnA~2)WnGv~n,∀n≥1.
Moreover, observe that
(51)v~n=vn-ρnA~1vn=(1-ρn)vn+ρn(I-A~1)vn.
(This is Mann’s iteration method.) Since (I-A~1)C⊂C and {vn}n=1∞⊂C, we know from the convexity of C that {v~n}n=1∞⊂C. Put
(52)Δnk=Trk,n(Θk,φk)(I-rk,nAk)Trk-1,n(Θk-1,φk-1)·(I-rk-1,nAk-1)⋯Tr1,n(Θ1,φ1)(I-r1,nA1)xn
for all k∈{1,2,…,M} and n≥1,
(53)Λni=JRi,λi,nI-λi,nBiJRi-1,λi-1,n·I-λi-1,nBi-1⋯JR1,λ1,n(I-λ1,nB1)
for all i∈{1,2,…,N}, Δn0=I, and Λn0=I, where I is the identity mapping on H. Then, we have that un=ΔnMxn and vn=ΛnNun.
We divide the rest of the proof into several steps.
Step  1. We prove that {xn} is bounded.
Indeed, utilizing24 and Proposition 3(ii), we have
(54)un-x*=TrM,nΘM,φMI-rM,nBMΔnM-1xn-TrM,nΘM,φMI-rM,nBMΔnM-1x*≤I-rM,nBMΔnM-1xn-I-rM,nBMΔnM-1x*≤ΔnM-1xn-ΔnM-1x*⋮≤Δn0xn-Δn0x*=xn-x*.
Utilizing 24 and Lemma 14, we have
(55)vn-x*=JRN,λN,n(I-λN,nAN)ΛnN-1un-JRN,λN,n(I-λN,nAN)ΛnN-1x*≤(I-λN,nAN)ΛnN-1un-(I-λN,nAN)ΛnN-1x*≤ΛnN-1un-ΛnN-1x*⋮≤Λn0un-Λn0x*=un-x*.
Combining 54 and 55, we have
(56)vn-x*≤xn-x*.
Since x*=Gx*=PC(I-ν1F1)PC(I-ν2F2)x*,Fj is ζj-inverse strongly monotone for j=1,2, and 0≤νj≤2ζj for j=1,2, we deduce that, for any n≥1,
(57)Gv~n-x*2=PCI-ν1F1PCI-ν2F2v~n-PCI-ν1F1PCI-ν2F2x*2≤(I-ν1F1)PC(I-ν2F2)v~n-I-ν1F1PCI-ν2F2x*2=PCI-ν2F2v~n-PCI-ν2F2x*-ν1F1PCI-ν2F2vn-F1PCI-ν2F2x*2≤PC(I-ν2F2)v~n-PC(I-ν2F2)x*2+ν1(ν1-2ζ1)·F1PC(I-ν2F2)vn-F1PC(I-ν2F2)p2≤PC(I-ν2F2)v~n-PC(I-ν2F2)x*2≤(I-ν2F2)v~n-(I-ν2F2)x*2=(v~n-p)-ν2(F2v~n-F2x*)2≤v~n-x*2+ν2(ν2-2ζ2)F2v~n-F2x*2≤v~n-x*2.
Since A~1 is α-inverse strongly monotone and {ρn}n=1∞⊂(0,2α], we have
(58)vn-x*-ρn(A~1vn-A~1x*)2=vn-x*2-2ρn〈A~1vn-A~1x*,xn-x*〉+ρn2A~1vn-A~1x*2≤vn-x*2-ρn(2α-ρn)A~1vn-A~1x*2≤vn-x*2.
Utilizing Lemma 9, the nonexpansivity of Wn and the one of G (due to Proposition CWY), we obtain from 24 and 56 that
(59)xn+1-x*=(I-μαnA~2)WnGv~n-x*≤(I-μαnA~2)WnGv~n-(I-μαnA~2)WnGx*+(I-μαnA~2)WnGx*-WnGx*≤(1-αnτ)Gv~n-Gx*+αnμA~2WnGx*≤(1-αnτ)v~n-x*+αnμA~2x*=(1-αnτ)vn-x*-ρnA~1vn+αnμA~2x*=(1-αnτ)vn-x*-ρnA~1vn-A~1x*-ρnA~1x*+αnμA~2x*≤(1-αnτ)vn-x*-ρnA~1vn-A~1x*+ρnA~1x*+αnμA~2x*≤1-αnτvn-x*+ρnA~1x*+αnμA~2x*≤1-αnτxn-x*+ρnA~1x*+αnμA~2x*≤(1-αnτ)xn-x*+ρnA~1x*+αnμA~2x*≤(1-αnτ)xn-x*+αn(A~1x*+μA~2x*)=(1-αnτ)xn-x*+αnτ·A~1x*+μA~2x*τ≤max⁡xn-x*,A~1x*+μA~2x*τ,
where τ:=1-1-μ(2β-μL2). By induction, we find that
(60)xn+1-x*≤max⁡x1-x*,A~1x*+μA~2x*τ,∀n≥1.
Thus, {xn}n=1∞ is bounded and so are the sequences {un}n=1∞, {vn}n=1∞, {v~n}n=1∞, and {yn}n=1∞.
Step  2. We prove that lim⁡n→∞(xn+1-xn/ρn)=0.
Indeed, utilizing24 and 45, we obtain that
(61)vn+1-vn=Λn+1Nun+1-ΛnNun=JRN,λN,n+1I-λN,n+1BNΛn+1N-1un+1-JRN,λN,nI-λN,nBNΛnN-1un≤JRN,λN,n+1I-λN,n+1BNΛn+1N-1un+1-JRN,λN,n+1I-λN,nBNΛn+1N-1un+1+JRN,λN,n+1I-λN,nBNΛn+1N-1un+1-JRN,λN,nI-λN,nBNΛnN-1un≤I-λN,n+1BNΛn+1N-1un+1-I-λN,nBNΛn+1N-1un+1+I-λN,nBNΛn+1N-1un+1-(I-λN,nBN)ΛnN-1un+λN,n+1-λN,n×1λN,n+1JRN,λN,n+1I-λN,nBNΛn+1N-1un+1JRN,λN,n+1I-λN,nBN-I-λN,nBNΛnN-1un+1λN,nI-λN,nBNΛn+1N-1un+1JRN,λN,nI-λN,nBN1λN,n+1-JRN,λN,nI-λN,nBNΛnN-1un≤λN,n+1-λN,nBNΛn+1N-1un+1+M~+Λn+1N-1un+1-ΛnN-1un≤λN,n+1-λN,nBNΛn+1N-1un+1+M~+λN-1,n+1-λN-1,nBN-1Λn+1N-2un+1+M~+Λn+1N-2un+1-ΛnN-2un⋮≤λN,n+1-λN,nBNΛn+1N-1un+1+M~+λN-1,n+1-λN-1,nBN-1Λn+1N-2un+1+M~+⋯+λ1,n+1-λ1,nB1Λn+10un+1+M~+Λn+10un+1-Λn0un≤M~0∑i=1N‍λi,n+1-λi,n+un+1-un,
where
(62)sup⁡n≥11λN,n+1JRN,λN,n+1(I-λN,nBN)Λn+1N-1un+1JRN,λN,n+1-(I-λN,nBN)ΛnN-1un+1λN,nI-λN,nBNΛn+1N-1un+11λN,n+1-JRN,λN,nI-λN,nBNΛnN-1un≤M~,
for some M~>0 and sup⁡n≥1{∑i=1N‍BiΛn+1i-1un+1+M~}≤M~0 for some M~0>0. Hence, it follows from 24 and {ρn}n=1∞⊂(0,2α] that
(63)v~n+1-v~n=vn+1-ρn+1A~1vn+1-vn-ρnA~1vn≤vn+1-ρn+1A~1vn+1-vn-ρn+1A~1vn+vn-ρn+1A~1vn-vn-ρnA~1vn≤vn+1-vn+ρn+1-ρnA~1vn≤M~0∑i=1N‍λi,n+1-λi,n+un+1-un+ρn+1-ρnA~1vn.
Also, utilizing Proposition 3(ii), (v), we deduce that
(64)un+1-un=Δn+1Mxn+1-ΔnMxn=TrM,n+1ΘM,φMI-rM,n+1AMΔn+1M-1xn+1-TrM,nΘM,φMI-rM,nAMΔnM-1xn≤TrM,n+1ΘM,φMI-rM,n+1AMΔn+1M-1xn+1-TrM,nΘM,φMI-rM,nAMΔn+1M-1xn+1+TrM,nΘM,φMI-rM,nAMΔn+1M-1xn+1-TrM,nΘM,φMI-rM,nAMΔnM-1xn≤TrM,n+1ΘM,φMI-rM,n+1AMΔn+1M-1xn+1-TrM,nΘM,φMI-rM,n+1AMΔn+1M-1xn+1+TrM,nΘM,φMI-rM,n+1AMΔn+1M-1xn+1-TrM,nΘM,φMI-rM,nAMΔn+1M-1xn+1+I-rM,nAMΔn+1M-1xn+1-I-rM,nAMΔnM-1xn≤rM,n+1-rM,nrM,n+1TrM,n+1ΘM,φMI-rM,n+1AMΔn+1M-1xn+1TrM,n+1ΘM,φMI-rM,n+1AM-I-rM,n+1AMΔn+1M-1xn+1+rM,n+1-rM,nAMΔn+1M-1xn+1+Δn+1M-1xn+1-ΔnM-1xn=rM,n+1-rM,n·AMΔn+1M-1xn+1TrM,n+1ΘM,φMI-rM,n+1AMΔn+1M-1xn+1+1rM,n+1TrM,n+1ΘM,φMI-rM,n+1AMΔn+1M-1xn+1TrM,n+1ΘM,φM-I-rM,n+1AMΔn+1M-1xn+1+Δn+1M-1xn+1-ΔnM-1xn⋮≤rM,n+1-rM,n·AMΔn+1M-1xn+1TrM,n+1ΘM,φMI-rM,n+1AMΔn+1M-1xn+1+1rM,n+1TrM,n+1ΘM,φMI-rM,n+1AMΔn+1M-1xn+1TrM,n+1ΘM,φMI-rM,n+1AM-I-rM,n+1AMΔn+1M-1xn+1+⋯+r1,n+1-r1,n·A1Δn+10xn+1Tr1,n+1Θ1,φ1-I-r1,n+1A1Δn+10xn+1+1r1,n+1Tr1,n+1Θ1,φ1I-r1,n+1A1Δn+10xn+1Tr1,n+1Θ1,φ1-I-r1,n+1A1Δn+10xn+1+Δn+10xn+1-Δn0xn≤M~1∑k=1M‍rk,n+1-rk,n+xn+1-xn,
where M~1>0 is a constant such that, for each n≥1,
(65)∑k=1M‍AkΔn+1k-1xn+1+1rk,n+1·Trk,n+1Θk,φkI-rk,n+1AkΔn+1k-1xn+11rk,n+1Trk,n+1Θk,φk-I-rk,n+1AkΔn+1k-1xn+1≤M~1.
In the meantime, from 33, since Wn, Tn, and Un,i are all nonexpansive, we have
(66)Wn+1Gv~n-WnGv~n=λ1T1Un+1,2Gv~n-λ1T1Un,2Gv~n≤λ1Un+1,2Gv~n-Un,2Gv~n=λ1λ2T2Un+1,3Gv~n-λ2T2Un,3Gv~n≤λ1λ2Un+1,3Gv~n-Un,3Gv~n⋮≤λ1λ2⋯λnUn+1,n+1Gv~n-Un,n+1Gv~n≤M~2∏i=1n‍λi,
where M~2 is a constant such that Un+1,n+1Gv~n+Un,n+1Gv~n≤M~2 for each n≥1. So, utilizing 24, 63–66, from {λn}⊂(0,b]⊂(0,1), {αn}⊂(0,1], {ρn}⊂(0,2α], and μ∈(0,2β/L2), we deduce that
(67)yn+1-yn=Wn+1Gv~n+1-WnGv~n≤Wn+1Gv~n+1-Wn+1Gv~n+Wn+1Gv~n-WnGv~n≤v~n+1-v~n+M~2∏i=1n‍λi≤v~n+1-v~n+M~2bn≤M~0∑i=1N‍λi,n+1-λi,n+un+1-un+ρn+1-ρnA~1vn+M~2bn≤M~0∑i=1N‍λi,n+1-λi,n+M~1∑k=1M‍rk,n+1-rk,n+xn+1-xn+ρn+1-ρnA~1vn+M~2bn,
and, by Lemma 9,
(68)xn+2-xn+1=I-μαn+1A~2Wn+1Gv~n+1-I-μαnA~2WnGv~n≤(I-μαn+1A~2)Wn+1Gv~n+1-(I-μαn+1A~2)WnGv~n+(I-μαn+1A~2)WnGv~n-(I-μαnA~2)WnGv~n≤1-αn+1τWn+1Gv~n+1-WnGv~n+αn+1-αnμA~2WnGv~n≤(1-αn+1τ)·M~0∑i=1N‍λi,n+1-λi,n+M~1∑k=1M‍rk,n+1-rk,n+xn+1-xn∑i=1N‍+ρn+1-ρnA~1vn+M~2bn+αn+1-αnμA~2WnGv~n≤1-αn+1τxn+1-xn+M~0∑i=1N‍λi,n+1-λi,n+M~1∑k=1M‍rk,n+1-rk,n+ρn+1-ρnA~1vn+M~2bn+αn+1-αnμA~2WnGv~n≤1-αn+1τxn+1-xn+M~3∑i=1N‍λi,n+1-λi,n+∑k=1M‍rk,n+1-rk,n∑i=1N‍+ρn+1-ρn+αn+1-αn+bn,
where sup⁡n≥1{M~0+M~1+M~2+A~1vn+μA~2WnGv~n}≤M~3 for some M~3>0. Consequently,
(69)xn+1-xnρn≤1-αnτxn-xn-1ρn+M~3∑i=1N‍λi,n-λi,n-1ρn+∑k=1M‍rk,n-rk,n-1ρn∑k=1M‍+ρn-ρn-1ρn+αn-αn-1ρn+bn-1ρn=(1-ταn)xn-xn-1ρn-1+1-ταnxn-xn-11ρn-1ρn-1+M~3∑i=1N‍λi,n-λi,n-1ρn+∑k=1M‍rk,n-rk,n-1ρn∑k=1M‍+ρn-ρn-1ρn+αn-αn-1ρn+bn-1ρn≤(1-ταn)xn-xn-1ρn-1+ταn·M~4τ1αn1ρn-1ρn-1bn-1αnρn+∑i=1N‍λi,n-λi,n-1αnρn+∑k=1M‍rk,n-rk,n-1αnρn+1αn1-ρn-1ρn+1ρn1-αn-1αn+bn-1αnρn,
where sup⁡n≥1{xn+1-xn+M~3}≤M~4 for some M~4>0. From (H1)–(H5), it follows that ∑n=1∞‍ταn=∞ and
(70)lim⁡n→∞M~4τ1αn1ρn-1ρn-1+∑i=1N‍λi,n-λi,n-1αnρn+∑k=1M‍rk,n-rk,n-1αnρn+1αn1-ρn-1ρn∑i=1N‍+1ρn1-αn-1αn+bn-1αnρn=0.
Thus, applying Lemma 10 to 69, we immediately conclude that
(71)lim⁡n→∞xn+1-xnρn=0.
So, from (H3), it follows that
(72)lim⁡n→∞xn+1-xn=0.
Step  3. We prove that lim⁡n→∞xn-un=0, lim⁡n→∞xn-vn=0,lim⁡n→∞v~n-Gv~n=0, and lim⁡n→∞v~n-Wv~n=0, where v~n=vn-ρnA~1vn.
Indeed, utilizing Lemma4, from 48, we get
(73)xn+1-x*2=(I-μαnA~2)WnGv~n-x*2=I-μαnA~2WnGv~n-I-μαnA~2WnGx*+I-μαnA~2WnGx*-WnGx*2≤I-μαnA~2WnGv~n-I-μαnA~2WnGx*2-2μαn〈A~2x*,xn+1-x*〉≤(1-αnτ)Gv~n-x*2-2μαn〈A~2x*,xn+1-x*〉≤(1-αnτ)v~n-x*2-2μαn〈A~2x*,xn+1-x*〉=(1-αnτ)vn-x*-ρnA~1vn2-2μαn〈A~2x*,xn+1-x*〉≤1-αnτvn-x*2-2ρnA~1vn,v~n-x*-2μαn〈A~2x*,xn+1-x*〉≤vn-x*2-2(1-αnτ)ρn〈A~1vn,v~n-x*〉-2μαn〈A~2x*,xn+1-x*〉.
On the other hand, observe that
(74)Δnkxn-x*2=Trk,n(Θk,φk)(I-rk,nAk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nAk)x*2≤(I-rk,nAk)Δnk-1xn-(I-rk,nAk)x*2≤Δnk-1xn-x*2+rk,n(rk,n-2μk)AkΔnk-1xn-Akx*2≤xn-x*2+rk,n(rk,n-2μk)AkΔnk-1xn-Akx*2,(75)Λniun-x*2=JRi,λi,n(I-λi,nBi)Λni-1un-JRi,λi,n(I-λi,nBi)x*2≤(I-λi,nBi)Λni-1un-(I-λi,nBi)x*2≤Λni-1un-x*2+λi,n(λi,n-2ηi)BiΛni-1un-Bix*2≤un-x*2+λi,n(λi,n-2ηi)BiΛni-1un-Bix*2≤xn-x*2+λi,n(λi,n-2ηi)BiΛni-1un-Bix*2.
Combining 73–75, we get
(76)xn+1-x*2≤vn-x*2-21-αnτρnA~1vn,v~n-x*-2μαn〈A~2x*,xn+1-x*〉≤Λniun-x*2+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*≤un-x*2+λi,nλi,n-2ηiBiΛni-1un-Bix*2+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*≤Δnkxn-x*2+λi,nλi,n-2ηiBiΛni-1un-Bix*2+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*≤xn-x*2+rk,nrk,n-2μkAkΔnk-1xn-Akx*2+λi,nλi,n-2ηiBiΛni-1un-Bix*2+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*,
which leads to
(77)rk,n2μk-rk,nAkΔnk-1xn-Akx*2+λi,n(2ηi-λi,n)BiΛni-1un-Bix*2≤xn-x*2-xn+1-x*2+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*≤xn-xn+1xn-x*+xn+1-x*+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*.
Since αn→0, ρn→0, xn+1-xn→0, and {xn},{vn},{v~} are bounded sequences, it follows from {λi,n}⊂[ai,bi]⊂(0,2ηi) and {rk,n}⊂[ek,fk]⊂(0,2μk) that
(78)lim⁡n→∞AkΔnk-1xn-Akx*=0,lim⁡n→∞BiΛni-1un-Bix*=0,
for all k∈{1,2,…,M} and i∈{1,2,…,N}.
Furthermore, by Proposition3(ii) and Lemma 11 (a), we have
(79)Δnkxn-x*2=Trk,n(Θk,φk)(I-rk,nAk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nAk)x*2≤I-rk,nAkΔnk-1xn-I-rk,nAkx*,Δnkxn-x*=12I-rk,nAkΔnk-1xn-I-rk,nAkx*2+Δnkxn-x*2-I-rk,nAkΔnk-1xn-I-rk,nAkx*I-rk,nAkΔnk-1-Δnkxn-x*2≤12Δnk-1xn-x*2+Δnkxn-x*2Δnk-1xn-Δnkxn-rk,nAkΔnk-1xn-Akx*2-Δnk-1xn-Δnkxn-rk,nAkΔnk-1xn-Akx*2,
which implies that
(80)Δnkxn-x*2≤Δnk-1xn-x*2-Δnk-1xn-Δnkxn-rk,n(AkΔnk-1xn-Akx*)2=Δnk-1xn-x*2-Δnk-1xn-Δnkxn2-rk,n2AkΔnk-1xn-Akx*2+2rk,n〈Δnk-1xn-Δnkxn,AkΔnk-1xn-Akx*〉≤Δnk-1xn-x*2-Δnk-1xn-Δnkxn2+2rk,nΔnk-1xn-ΔnkxnAkΔnk-1xn-Akx*≤xn-x*2-Δnk-1xn-Δnkxn2+2rk,nΔnk-1xn-ΔnkxnAkΔnk-1xn-Akx*.
By Lemma 11 (a) and Lemma 14, we obtain
(81)Λniun-x*2=JRi,λi,n(I-λi,nBi)Λni-1un-JRi,λi,n(I-λi,nBi)x*2≤I-λi,nBiΛni-1un-I-λi,nBix*,Λniun-x*=12I-λi,nBiΛni-1un-I-λi,nBix*2+Λniun-x*2-(I-λi,nBi)Λni-1unI-λi,nBiΛni-1un-I-λi,nBix*2-(I-λi,nBi)x*-(Λniun-x*)2≤12Λni-1un-x*2+Λniun-x*2-Λni-1un-Λniun-λi,n(BiΛni-1un-Bix*)2≤12un-x*2+Λniun-x*2-Λni-1un-Λniun-λi,n(BiΛni-1un-Bix*)2≤12xn-x*2+Λniun-x*2Λniun-x*2-Λni-1un-Λniun-λi,n(BiΛni-1un-Bix*)2,
which implies
(82)Λniun-x*2≤xn-x*2-Λni-1un-Λniun-λi,n(BiΛni-1un-Bix*)2=xn-x*2-Λni-1un-Λniun2-λi,n2BiΛni-1un-Bix*2+2λi,n〈Λni-1un-Λniun,BiΛni-1un-Bix*〉≤xn-x*2-Λni-1un-Λniun2+2λi,nΛni-1un-ΛniunBiΛni-1un-Bix*.
Combining 73 and 82, we conclude that
(83)xn+1-x*2≤vn-x*2-2(1-αnτ)ρn〈A~1vn,v~n-x*〉-2μαn〈A~2x*,xn+1-x*〉≤Λniun-x*2+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*≤xn-x*2-Λni-1un-Λniun2+2λi,nΛni-1un-ΛniunBiΛni-1un-Bix*+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*,
which yields
(84)Λni-1un-Λniun2≤xn-x*2-xn+1-x*2+2λi,nΛni-1un-ΛniunBiΛni-1un-Bix*+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*≤xn-xn+1xn-x*+xn+1-x*+2λi,nΛni-1un-ΛniunBiΛni-1un-Bix*+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*.
Since αn→0, ρn→0, and xn+1-xn→0 and xn, {vn}, and {v~} are bounded sequences, it follows from 78 and {λi,n}⊂[ai,bi]⊂(0,2ηi) that
(85)lim⁡n→∞Λni-1un-Λniun=0,∀i∈{1,2,…,N}.
Also, combining 55, 73, and 80, we deduce that
(86)xn+1-x*2≤vn-x*2-21-αnτρnA~1vn,v~n-x*-2μαn〈A~2x*,xn+1-x*〉≤un-x*2+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*≤Δnkxn-x*2+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*≤xn-x*2-Δnk-1xn-Δnkxn2+2rk,nΔnk-1xn-ΔnkxnAkΔnk-1xn-Akx*+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*,
which leads to
(87)Δnk-1xn-Δnkxn2≤xn-x*2-xn+1-x*2+2rk,nΔnk-1xn-ΔnkxnAkΔnk-1xn-Akx*+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*≤xn-xn+1xn-x*+xn+1-x*+2rk,nΔnk-1xn-ΔnkxnAkΔnk-1xn-Akx*+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*.
Since αn→0, ρn→0, and xn+1-xn→0 and {xn}, {vn} and, {v~} are bounded sequences, it follows from 78 and {rk,n}⊂[ek,fk]⊂(0,2μk) that
(88)lim⁡n→∞Δnk-1xn-Δnkxn=0,∀k∈{1,2,…,M}.
Hence, from 85 and 88, we get
(89)xn-un=Δn0xn-ΔnMxn≤Δn0xn-Δn1xn+Δn1xn-Δn2xn+⋯+ΔnM-1xn-ΔnMxn⟶0asn⟶∞,(90)un-vn=Λn0un-ΛnNun≤Λn0un-Λn1un+Λn1un-Λn2un+⋯+ΛnN-1un-ΛnNun⟶0asn⟶∞,
respectively. Thus, from 89 and 90, we obtain
(91)xn-vn≤xn-un+un-vn⟶0asn⟶∞.
Next, for simplicity, we writey*=PC(I-ν2F2)x*, wn=PC(I-ν2F2)v~n, and w~n=Gv~n=PC(I-ν1F1)wn for all n≥1. Then,
(92)x*=Gx*=PCI-ν1F1y*=PC(I-ν1F1)PC(I-ν2F2)x*.
We now show thatlim⁡n→∞Gv~n-v~n=0; that is, lim⁡n→∞w~n-v~n=0. As a matter of fact, from 56, 57, and 73 it follows that
(93)xn+1-x*2≤(1-αnτ)Gv~n-x*2-2μαn〈A~2x*,xn+1-x*〉≤w~n-x*2+2μαnA~2x*xn+1-x*≤wn-y*2+ν1(ν1-2ζ1)F1wn-F1y*2+2μαnA~2x*xn+1-x*≤v~n-x*2+ν2(ν2-2ζ2)F2v~n-F2x*2+ν1(ν1-2ζ1)F1wn-F1y*2+2μαnA~2x*xn+1-x*=vn-x*-ρnA~1vn2+ν2(ν2-2ζ2)F2v~n-F2x*2+ν1(ν1-2ζ1)F1wn-F1y*2+2μαnA~2x*xn+1-x*≤vn-x*2+2ρnA~1vnv~n-x*+ν2(ν2-2ζ2)F2v~n-F2x*2+ν1(ν1-2ζ1)F1wn-F1y*2+2μαnA~2x*xn+1-x*≤xn-x*2+2ρnA~1vnv~n-x*+ν2(ν2-2ζ2)F2v~n-F2x*2+ν1(ν1-2ζ1)F1wn-F1y*2+2μαnA~2x*xn+1-x*,
which immediately yields
(94)ν2(2ζ2-ν2)F2v~n-F2x*2+ν1(2ζ1-ν1)F1wn-F1y*2≤xn-x*2-xn+1-x*2+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*≤xn-xn+1xn-x*+xn+1-x*+2ρnA~1vnv~n-x*+2μαnA~2x*xn+1-x*.
Since αn→0, ρn→0, and xn+1-xn→0 and {xn}, {vn}, and {v~n} are bounded sequences, it follows from νj∈(0,2ζj), j=1,2, that
(95)lim⁡n→∞F2v~n-F2x*=0,lim⁡n→∞F1wn-F1y*=0.
In the meantime, in terms of the firm nonexpansivity of PC and the ζj-inverse strong monotonicity of Fj for j=1,2, we obtain from νj∈(0,2ζj), j=1,2, and 57 that
(96)wn-y*2=PC(I-ν2F2)v~n-PC(I-ν2F2)x*2≤〈(I-ν2F2)v~n-(I-ν2F2)x*,wn-y*〉=12(I-ν2F2)v~n-(I-ν2F2)x*2+wn-y*2-(I-ν2F2)v~n-(I-ν2F2)x*-(wn-y*)2≤12v~n-x*2+wn-y*2-(v~n-wn)-ν2(F2v~n-F2x*)-(x*-y*)2=12v~n-x*2+wn-y*2-(v~n-wn)-(x*-y*)2+2ν2v~n-wn-x*-y*,F2v~n-F2x*-ν22F2v~n-F2x*2≤12v~n-x*2+wn-y*2-v~n-wn-x*-y*2v~n-x*2+2ν2(v~n-wn)-(x*-y*)F2v~n-F2x*,w~n-x*2=PCI-ν1F1wn-PCI-ν1F1y*2≤I-ν1F1v~n-I-ν1F1p~,w~n-x*=12(I-ν1F1)wn-(I-ν1F1)y*2+w~n-x*2-(I-ν1F1)wn-(I-ν1F1)y*-(w~n-x*)2≤12wn-y*2+w~n-x*2-(wn-w~n)+(x*-y*)2+2ν1F1wn-F1y*,wn-w~n+x*-y*-ν12F1wn-F1y*2≤12v~n-x*2+w~n-x*2-wn-w~n+x*-y*2v~n-x*2+2ν1F1wn-F1y*,wn-w~n+x*-y*.
Thus, we have
(97)wn-y*2≤v~n-x*2-v~n-wn-x*-y*2+2ν2(v~n-wn)-(x*-y*)F2v~n-F2x*,(98)w~n-x*2≤v~n-x*2-wn-w~n+x*-y*2+2ν1F1wn-F1y*(wn-w~n)+(x*-y*),
respectively. Consequently, from 56, 73, and 98, it follows that
(99)xn+1-x*2≤1-αnτGv~n-x*2-2μαn〈A~2x*,xn+1-x*〉≤w~n-x*2+2μαnA~2x*xn+1-x*≤v~n-x*2-(wn-w~n)+(x*-y*)2+2ν1F1wn-F1y*wn-w~n+x*-y*+2μαnA~2x*xn+1-x*=vn-x*-ρnA~1vn2-(wn-w~n)+(x*-y*)2+2ν1F1wn-F1y*wn-w~n+x*-y*+2μαnA~2x*xn+1-x*≤vn-x*2+2ρnA~1vnv~n-x*-(wn-w~n)+(x*-y*)2+2ν1F1wn-F1y*wn-w~n+x*-y*+2μαnA~2x*xn+1-x*≤xn-x*2+2ρnA~1vnv~n-x*-(wn-w~n)+(x*-y*)2+2ν1F1wn-F1y*wn-w~n+x*-y*+2μαnA~2x*xn+1-x*,
which yields
(100)(wn-w~n)+(x*-y*)2≤xn-x*2-xn+1-x*2+2ρnA~1vnv~n-x*+2ν1F1wn-F1y*wn-w~n+x*-y*+2μαnA~2x*xn+1-x*≤xn-xn+1xn-x*+xn+1-x*+2ρnA~1vnv~n-x*+2ν1F1wn-F1y*wn-w~n+x*-y*+2μαnA~2x*xn+1-x*.
Since αn→0, ρn→0, and xn+1-xn→0 and {xn}, {vn}, {v~n}, {wn}, and {w~n} are bounded sequences, it follows from 95 that
(101)lim⁡n→∞wn-w~n+x*-y*=0.
Also, from 56, 57, 73, and 97, it follows that
(102)xn+1-x*2≤(1-αnτ)Gv~n-x*2-2μαn〈A~2x*,xn+1-x*〉≤w~n-x*2+2μαnA~2x*xn+1-x*≤wn-y*2+2μαnA~2x*xn+1-x*≤v~n-x*2-(v~n-wn)-(x*-y*)2+2ν2v~n-wn-x*-y*F2v~n-F2x*+2μαnA~2x*xn+1-x*=vn-x*-ρnA~1vn2-v~n-wn-x*-y*2+2ν2v~n-wn-x*-y*F2v~n-F2x*+2μαnA~2x*xn+1-x*≤vn-x*2+2ρnA~1vnv~n-x*-(v~n-wn)-(x*-y*)2+2ν2v~n-wn-x*-y*F2v~n-F2x*+2μαnA~2x*xn+1-x*≤xn-x*2+2ρnA~1vnv~n-x*-v~n-wn-x*-y*2+2ν2v~n-wn-x*-y*F2v~n-F2x*+2μαnA~2x*xn+1-x*,
which leads to
(103)v~n-wn-x*-y*2≤xn-x*2-xn+1-x*2+2ρnA~1vnv~n-x*+2ν2v~n-wn-x*-y*F2v~n-F2x*+2μαnA~2x*xn+1-x*≤xn-xn+1xn-x*+xn+1-x*+2ρnA~1vnv~n-x*+2ν2v~n-wn-x*-y*F2v~n-F2x*+2μαnA~2x*xn+1-x*.
Since αn→0, ρn→0, and xn+1-xn→0 and {xn}, {vn}, {v~n}, and {wn} are bounded sequences, it follows from 95 that
(104)lim⁡n→∞v~n-wn-x*-y*=0.
Note that
(105)v~n-w~n≤v~n-wn-x*-y*+(wn-w~n)+(x*-y*).
Hence, from 101 and 104, we get
(106)lim⁡n→∞v~n-Gv~n=lim⁡n→∞v~n-w~n=0.
Also, observe that v~n-xn≤vn-xn+ρnA~1vn and
(107)yn-xn≤xn+1-xn+μαnA~2yn.
Hence, from 91, αn→0, ρn→0, and xn+1-xn→0, we obtain that
(108)lim⁡n→∞v~n-xn=0,lim⁡n→∞yn-xn=0.
So, from 106 and 108, we deduce that
(109)Wnv~n-v~n≤Wnv~n-yn+yn-xn+xn-v~n=Wnv~n-WnGv~n+yn-xn+xn-v~n≤v~n-Gv~n+yn-xn+xn-v~n⟶0asn⟶∞.
In addition, it is clear that
(110)v~n-Wv~n≤v~n-Wnv~n+Wnv~n-Wv~n.
Thus, we conclude from [24, Remark 3.2], (3.28,) and the boundedness of {v~n} that
(111)lim⁡n→∞v~n-Wv~n=0.
Step  4. We prove that ωw(xn)⊂Ω.
Indeed, sinceH is reflexive and {xn} is bounded, there exists at least a weak convergence subsequence of {xn}. Hence, it is known that ωw(xn)≠∅. Now, take an arbitrary w∈ωw(xn). Then, there exists a subsequence {xni} of {xn} such that xni⇀w. From 85–89, 91, and 108, we have that uni⇀w, vni⇀w, v~ni⇀w, Λnimuni⇀w, and Δnikxni⇀w, where m∈{1,2,…,N} and k∈{1,2,…,M}. Utilizing Lemma 6, we deduce from v~ni⇀w, 106, and 111 that w∈GSVI(G) and w∈Fix⁡(W)=⋂n=1∞Fix⁡(Tn) (due to Lemma 8). Next, we prove that w∈⋂m=1NI(Bm,Rm). As a matter of fact, since Bm is ηm-inverse strongly monotone, Bm is a monotone and Lipschitz continuous mapping. It follows from Lemma 17 that Rm+Bm is maximal monotone. Let (v,g)∈G(Rm+Bm); that is, g-Bmv∈Rmv. Again, since Λnmun=JRm,λm,n(I-λm,nBm)Λnm-1un, n≥1, m∈{1,2,…,N}, we have
(112)Λnm-1un-λm,nBmΛnm-1un∈I+λm,nRmΛnmun.
That is,
(113)1λm,nΛnm-1un-Λnmun-λm,nBmΛnm-1un∈RmΛnmun.
In terms of the monotonicity of Rm, we get
(114)v-Λnmun,g-Bmv1λm,n-1λm,n(Λnm-1un-Λnmun-λm,nBmΛnm-1un)≥0
and hence
(115)〈v-Λnmun,g〉≥v-Λnmun,Bmv1λm,n+1λm,n(Λnm-1un-Λnmun-λm,nBmΛnm-1un)=v-Λnmun,Bmv-BmΛnmun+BmΛnmun1λm,n-BmΛnm-1un+1λm,n(Λnm-1un-Λnmun)≥v-Λnmun,BmΛnmun-BmΛnm-1un+v-Λnmun,1λm,n(Λnm-1un-Λnmun).
In particular,
(116)v-Λnimuni,g≥v-Λnimuni,BmΛnimuni-BmΛnim-1uni+v-Λnimuni,1λm,niΛnim-1uni-Λnimuni.
Since Λnmun-Λnm-1un→0 (due to 85) and BmΛnmun-BmΛnm-1un→0 (due to the Lipschitz continuity of Bm), we conclude from Λnimuni⇀w and {λi,n}⊂[ai,bi]⊂(0,2ηi) that
(117)lim⁡i→∞v-Λnimuni,g=〈v-w,g〉≥0.
It follows from the maximal monotonicity of Bm+Rm that 0∈(Rm+Bm)w; that is, w∈I(Bm,Rm). Therefore, w∈⋂m=1NI(Bm,Rm). Next, we prove that w∈⋂k=1MGMEP⁡(Θk,φk,Ak). Since Δnkxn=Trk,n(Θk,φk)(I-rk,nAk)Δnk-1xn, n≥1, k∈{1,2,…,M}, we have
(118)ΘkΔnkxn,y+φky-φkΔnkxn+AkΔnk-1xn,y-Δnkxn+1rk,ny-Δnkxn,Δnkxn-Δnk-1xn≥0.
By (A2), we have
(119)φky-φkΔnkxn+AkΔnk-1xn,y-Δnkxn+1rk,ny-Δnkxn,Δnkxn-Δnk-1xn≥Θk(y,Δnkxn).
Let zt=ty+(1-t)w for all t∈(0,1] and y∈C. This implies that zt∈C. Then, we have
(120)zt-Δnkxn,Akzt≥φkΔnkxn-φk(zt)+zt-Δnkxn,Akzt-zt-Δnkxn,AkΔnk-1xn-zt-Δnkxn,Δnkxn-Δnk-1xnrk,n+Θk(zt,Δnkxn)=φkΔnkxn-φk(zt)+zt-Δnkxn,Akzt-AkΔnkxn+zt-Δnkxn,AkΔnkxn-AkΔnk-1xn-zt-Δnkxn,Δnkxn-Δnk-1xnrk,n+Θkzt,Δnkxn.
By 88, we have AkΔnkxn-AkΔnk-1xn→0 as n→∞. Furthermore, by the monotonicity of Ak, we obtain 〈zt-Δnkxn,Akzt-AkΔnkxn〉≥0. Then, by (A4), we obtain
(121)〈zt-w,Akzt〉≥φk(w)-φk(zt)+Θk(zt,w).
Utilizing (A1), (A4), and 121, we obtain
(122)0=Θk(zt,zt)+φk(zt)-φk(zt)≤tΘk(zt,y)+(1-t)Θk(zt,w)+tφk(y)+(1-t)φk(w)-φk(zt)≤tΘkzt,y+φky-φkzt+1-tzt-w,Akzt=tΘkzt,y+φky-φkzt+1-tty-w,Akzt,
and hence
(123)0≤Θk(zt,y)+φk(y)-φk(zt)+1-ty-w,Akzt.
Letting t→0, we have, for each y∈C,
(124)0≤Θk(w,y)+φk(y)-φk(w)+y-w,Akw.
This implies that w∈GMEP⁡(Θk,φk,Ak) and hence w∈⋂k=1MGMEP⁡(Θk,φk,Ak). Thus, w∈Ω=⋂n=1∞Fix⁡(Tn)∩⋂k=1MGMEP⁡(Θk,φk,Ak)∩⋂m=1NI(Bm,Rm). Consequently, w∈⋂n=1∞Fix⁡(Tn)∩⋂k=1MGMEP⁡(Θk,φk,Ak)∩⋂m=1NI(Bm,Rm)∩GSVI(G)=:Ω. This shows that ωw(xn)⊂Ω.
Step  5. We prove that ωw(xn)⊂VI⁡(Ω,A~1) provided xn-yn=o(ρn) additionally.
Indeed, take an arbitraryw∈ωw(xn). Then, there exists a subsequence {xni} of {xn} such that xni⇀w. Utilizing the arguments similar to those of 56 and 57, we can readily obtain that for all p∈Ω(125)vn-p≤xn-p,Gv~n-p≤v~n-p.
Since A1 is α-inverse strongly monotone, from 48 and 125 we conclude that for all p∈Ω(126)yn-p2=WnGv~n-p2≤v~n-p2=vn-p-ρnA~1vn2≤vn-p2-2ρn〈A~1vn,v~n-p〉=vn-p2-2ρnA~1vn,vn-p+A~1vn,v~n-vn=vn-p2-2ρnA~1vn-A~1p,vn-p+A~1p,vn-p+A~1vn,v~n-vn≤xn-p2-2ρnA~1p,vn-p+2ρnA~1vnv~n-vn,
which implies that
(127)〈A~1p,vn-p〉≤12ρnxn-p2-yn-p2+A~1vnv~n-vn≤xn-yn2ρnxn-p+yn-p+ρnA~1vn2.
So, from ρn→0 and the assumption xn-yn=o(ρn), we get
(128)limsup⁡n→∞〈A~1p,vn-p〉≤0.
Thus, it follows from 91 that for all p∈Ω(129)A~1p,w-p=lim⁡i→∞A~1p,xni-p≤limsup⁡n→∞A~1p,xn-p=limsup⁡n→∞A~1p,vn-p+A~1p,xn-vn=limsup⁡n→∞A~1p,vn-p≤0.
That is,
(130)A~1p,p-w≥0,∀p∈Ω.
Since A~1 is α-inverse strongly monotone, by Minty’s Lemma [16], we know that 130 is equivalent to the VIP
(131)A~1w,p-w≥0,∀p∈Ω.
This shows that w∈VI⁡(Ω,A~1). Therefore, ωw(xn)⊂VI⁡(Ω,A~1).Theorem 19.
Assume that there hold all the conditions in Theorem18. Then, we have(i)
{xn} converges strongly to a point u*∈Ω, which is a unique solution of the VIP
(132)〈A~2u*,p-u*〉≥0,∀p∈Ω;(ii)
{xn} converges strongly to a unique solution of THVI 19 provided xn-yn=o(θn) additionally.Proof.
SinceA~2 is β-strongly monotone and L-Lipschitz continuous, there exists a unique solution u*∈Ω of the VIP
(133)〈A~2u*,p-u*〉≥0,∀p∈Ω.
Now, let us show that
(134)limsup⁡n→∞〈A~2u*,u*-xn〉≤0.
Since {xn} is bounded, we may assume, without loss of generality, that there exists a subsequence {xni} of {xn} such that xni⇀w and
(135)limsup⁡n→∞A~2u*,u*-xn=lim⁡i→∞〈A~2u*,u*-xni〉=〈A~2u*,u*-w〉.
In terms of Theorem 18 (ii), we know that w∈ωw(xn)⊂Ω. So, from 133, it follows that
(136)limsup⁡n→∞A~2u*,u*-xn=〈A~2u*,u*-w〉≤0.
Next, let us show thatlim⁡n→∞xn-u*=0. In fact, utilizing Lemma 4, from 48 and 125 with p=u*, we get
(137)xn+1-u*2=(I-μαnA~2)WnGv~n-u*2=I-μαnA~2WnGv~n-I-μαnA~2WnGu*+(I-μαnA~2)WnGu*-WnGu*2≤I-μαnA~2WnGv~n-I-μαnA~2WnGu*2-2μαn〈A~2u*,xn+1-u*〉≤1-αnτGv~n-u*2-2μαnA~2u*,xn+1-u*≤(1-αnτ)v~n-u*2-2μαnA~2u*,xn+1-u*=(1-αnτ)vn-u*-ρnA~1vn2-2μαn〈A~2u*,xn+1-u*〉≤1-αnτvn-u*2-2ρnA~1vn,v~n-u*-2μαnA~2u*,xn+1-u*≤(1-αnτ)xn-u*2+2ρnA~1vnv~n-u*+2μαn〈A~2u*,u*-xn+1〉=1-αnτxn-u*2+αnτ·2τρnαnA~1vnv~n-u*ρnαn+μA~2u*,u*-xn+1,
where τ=1-1-μ(2β-μL2). Since ∑n=1∞‍αn=∞, lim⁡n→∞(ρn/αn)=0, and limsup⁡n→∞〈A~2u*,u*-xn+1〉≤0 (due to 136), we deduce that ∑n=1∞‍αnτ=∞ and
(138)limsup⁡n→∞2τρnαnA~1vnv~n-u*+μA~2u*,u*-xn+1≤0.
Therefore, applying Lemma 10 to 137, we infer that lim⁡n→∞xn-u*=0.
Finally, we prove thatlim⁡n→∞xn-x*=0 provided xn-yn=o(αn) additionally, where {x*}=VI⁡(VI⁡(Ω,A~1),A~2).
Indeed, first of all, let us show thatωw(xn)={x*}. As a matter of fact, take an arbitrary w∈ωw(xn). Then, there exists a subsequence {xnj} of {xn} such that xnj⇀w. Moreover, by Theorem 18 (iii), we know that w∈ωw(xn)⊂VI⁡(Ω,A~1). Utilizing the arguments similar to those of 56 and 57, we can readily obtain that for all p∈VI⁡(Ω,A~1)(139)vn-p≤xn-p,Gv~n-p≤v~n-p.
Utilizing Lemma 4, from 48 and 139, we deduce that for all p∈VI⁡(Ω,A~1)(140)xn+1-p2=(I-μαnA~2)WnGv~n-p2=I-μαnA~2WnGv~n-I-μαnA~2WnGp+(I-μαnA~2)WnGp-WnGp2≤I-μαnA~2WnGv~n-I-μαnA~2WnGp2-2μαn〈A~2p,xn+1-p〉≤(1-αnτ)Gv~n-p2-2μαn〈A~2p,xn+1-p〉≤(1-αnτ)v~n-p2-2μαn〈A~2p,xn+1-p〉=(1-αnτ)vn-p-ρnA~1vn2-2μαn〈A~2p,xn+1-p〉≤1-αnτvn-p2-2ρnA~1vn,v~n-p-2μαn〈A~2p,xn+1-p〉≤vn-p2-2(1-αnτ)ρn〈A~1vn,v~n-p〉-2μαn〈A~2p,xn+1-p〉≤xn-p2-2(1-αnτ)ρn〈A~1vn,v~n-p〉-2μαn〈A~2p,xn+1-p〉,
where τ=1-1-μ(2β-μL2). So, it follows that
(141)〈A~2p,xn+1-p〉≤12μαnxn-p2-xn+1-p2-(1-αnτ)ρnμαn〈A~1vn,v~n-p〉≤xn-xn+12μαnxn-p+xn+1-p+(1-αnτ)ρnμαnA~1vnv~n-p.
Since αn→0,ρn=o(αn), and xn+1-xn=o(ρn), we find that
(142)lim⁡n→∞xn+1-xnαn=lim⁡n→∞xn+1-xnρn·ρnαn=0.
Hence, we conclude from 141 that for all p∈VI⁡(Ω,A~1)(143)〈A~2p,w-p〉=lim⁡j→∞A~2p,xnj-p≤limsup⁡n→∞A~2p,xn-p=limsup⁡n→∞A~2p,xn-xn+1+A~2p,xn+1-p=limsup⁡n→∞A~2p,xn+1-p≤0.
That is,
(144)〈A~2p,p-w〉≥0,∀p∈VI⁡(Ω,A~1).
Since A~2 is β-strongly monotone and L-Lipschitz continuous, by Minty’s Lemma [16], we know that 144 is equivalent to the VIP
(145)〈A~2w,p-w〉≥0,∀p∈VI⁡(Ω,A~1).
This shows that w∈VI⁡(VI⁡(Ω,A~1),A~2). Taking into account {x*}=VI⁡(VI⁡(Ω,A~1),A~2), we know that w=x*. Thus, ωw(xn)={x*}; that is, xn⇀x*.
Next we prove thatlim⁡n→∞xn-x*=0. As a matter of fact, utilizing 139 and 140 with p=x*, we get
(146)xn-1-x*2≤1-αnτvn-x*2-2ρnA~1vn,v~n-x*-2μαnA~2x*,xn+1-x*≤1-αnτxn-x*2-2ρnA~1vnv~n-x*+2μαnA~2x*,x*-xn+1=1-αnτxn-x*2+αnτ·2τρnαnA~1vnv~n-x*ρnαn+μA~2x*,x*-xn+1.
Since ∑n=1∞‍αn=∞, lim⁡n→∞(ρn/αn)=0, and lim⁡n→∞〈A~2u*,u*-xn+1〉=0 (due to xn⇀x*), we deduce that ∑n=1∞‍αnτ=∞ and
(147)lim⁡n→∞2τρnαnA~1vnv~n-u*+μA~2u*,u*-xn+1=0.
Therefore, applying Lemma 10 to 146, we infer that lim⁡n→∞xn-x*=0. This completes the proof.Remark 20.
It is obvious that iterative scheme48 is very different from Yao et al. iterative one 14 and Zeng et al. iterative one in Algorithm ZWY. Here, the two-step iterative scheme in [11, Algorithm 3.2] is extended to develop four-step iterative scheme 48 for the THVI 19 by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. It is worth pointing out that under the lack of the assumptions similar to those in [8, Theorem 3.2], for example, {xn} is bounded, Fix⁡(T)∩int⁡C≠∅, and x-Tx≥kDist(x,Fix⁡(T)), ∀x∈C, for some k>0, the sequence {xn} generated by 48 converges strongly to a point u*∈⋂n=1∞Fix⁡(Tn)∩⋂k=1MGMEP⁡(Θk,φk,Ak)∩⋂i=1NI(Bi,Ri)∩GSVI(G)=:Ω, which is a unique solution of the VIP: 〈A~2u*,p-u*〉≥0, ∀p∈Ω.Remark 21.
Theorems18 and 19 improve, extend, supplement, and develop Yao et al. [8, Theorems 3.1 and 3.2] and Zeng et al. [11, Theorem 3.2] in the following aspects.(a) THVI19 with the unique solution x*∈Ω satisfying
(148)x*=P⋂n=1∞Fix⁡(Tn)∩⋂k=1MGMEP⁡(Θk,φk,Ak)∩⋂i=1NI(Bi,Ri)∩GSVI(G)·(I-A~1)x*
is more general than the problem of finding a point x~∈C satisfying x~=PFix⁡(T)Sx~ in [8] and than the problem of finding a point x*∈⋂i=1NFix⁡(Ti) satisfying x*=P⋂i=1NFix⁡(Ti)(I-A1)x* in [11, Theorem 3.2]. It is worth pointing out that S is nonexpansive if and only if the complement I-S is (1/2)-inverse strongly monotone; see [25].(b) Four-step iterative scheme48 for THVI 19 is more flexible, more advantageous, and more subtle than Zeng et al. two-step iterative one in [11, Algorithm 3.2] and than Yao et al. two-step iterative one 14 because it can be used to solve several kinds of problems, for example, the THVI, the hierarchical VIP, and the problem of finding a common point of four sets: ⋂n=1∞Fix⁡(Tn), ⋂k=1MGMEP⁡(Θk,φk,Ak), ⋂i=1NI(Bi,Ri), and GSVI(G). In addition, it also drops the crucial requirements that Fix⁡(T)∩int⁡C≠∅ and x-Tx≥kDist(x,Fix⁡(T)), ∀x∈C, for some k>0 in [8, Theorem 3.2 (v)].(c) The argument techniques in Theorems18 and 19 are very different from the argument ones in [8, Theorems 3.1 and 3.2] and from the argument ones in [11, Theorem 3.2] because we use the W-mapping approach to fixed points of infinitely many nonexpansive mappings {Tn}n=1∞ (see Lemmas 7 and 8), the properties of resolvent operators and maximal monotone mappings, the fixed point equation x*=PC(I-ν1F1)PC(I-ν2F2)x* equivalent to the GSVI 10 (see Proposition CWY), and the contractive coefficient estimates for the contractions associated with nonexpansive mappings (see Lemma 9).(d) Compared with the restrictions on the parameter sequences of [8, Theorem 19] and [11, Theorem 3.2], respectively, the hypotheses (H3)–(H5) in Theorem 18 are additionally added because Theorem 18 involves the quite complex problem, that is, the THVI 19 (over the set ⋂n=1∞Fix⁡(Tn)) with constraints of several problems: finitely many GMEPs, finitely many variational inclusions, and GSVI 10.



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*Source: 980352-2015-05-19.xml*