# Solving Generalized Mixed Equilibria, Variational Inequalities, and Constrained Convex Minimization

Authors: A. E. Al-Mazrooei; A. Latif; J. C. Yao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We propose implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings in a real Hilbert space. We prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets under very mild conditions.


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## Body

## 1. Introduction and Problems Formulation

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, let C be a nonempty closed convex subset of H, and let PC be the metric projection of H onto C. Let S:C→C be a self-mapping on C. We denote by Fix(S) the set of fixed points of S and by R the set of all real numbers. Recall that a mapping A:C→H is said to be L-Lipschitz continuous if there exists a constant L≥0 such that
(1)∥Ax-Ay∥≤L∥x-y∥,∀x,y∈C.
In particular, if L=1, then A is called a nonexpansive mapping [1], and if L∈[0,1), then A is called a contraction.Recall that a mappingA:C→H is called(i)
monotone if(2)〈Ax-Ay,x-y〉≥0,∀x,y∈C;(ii)
η-strongly monotone if there exists a constant η>0 such that
(3)〈Ax-Ay,x-y〉≥η∥x-y∥2,∀x,y∈C;(iii)
α-inverse strongly monotone if there exists a constant α>0 such that
(4)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.It is obvious that ifA is α-inverse strongly monotone, then A is monotone and (1/α)-Lipschitz continuous.LetA:C→H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point x∈C such that
(5)〈Ax,y-x〉≥0,∀y∈C.
The solution set of VIP (5) is denoted by VI(C,A).The VIP (5) was first discussed by Lions [2] and is now well known. The VIP (5) has many potential applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and so on; see, for example, [3–5] and the references therein.In 1976, Korpelevich [6] proposed an iterative algorithm for solving the VIP (5) in Euclidean space Rn:
(6)yn=PC(xn-τAxn),xn+1=PC(xn-τAyn),∀n≥0,
with τ>0, a given number which is known as the extragradient method. The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many researchers. See, for example, [7–16] and the references therein. In particular, motivated by the idea of Korpelevich’s extragradient method [6], Nadezhkina and Takahashi [17] introduced an extragradient iterative scheme:
(7)x0=x∈Cchosenarbitrary,yn=PC(xn-λnAxn),xn+1=αnxn+(1-αn)SPC(xn-λnAyn),αnxn+(1-αn)SPC(xn-λnAyn)∀n≥0,
where A:C→H is a monotone, L-Lipschitz continuous mapping, S:C→C is a nonexpansive mapping, {λn}⊂[a,b] for some a,b∈(0,1/L), and {αn}⊂[c,d] for some c,d∈(0,1). They proved the weak convergence of {xn} to an element of Fix(S)∩VI(C,A).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 [18] introduced the following generalized mixed equilibrium problem (GMEP) of finding x∈C such that
(8)Θ(x,y)+φ(y)-φ(x)+〈Ax,y-x〉≥0,∀y∈C.
We denote the set of solutions of GMEP (8) by GMEP(Θ,φ,A). The GMEP (8) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems in noncooperative games. The GMEP is further considered and studied. See, for example, [19, 20]. Some special cases of GMEP (8) are as follows.Ifφ=0, then GMEP (8) reduces to the generalized equilibrium problem (GEP) which is to find x∈C such that
(9)Θ(x,y)+〈Ax,y-x〉≥0,∀y∈C.
It is introduced and studied by S. Takahashi and W. Takahashi [21]. The set of solutions of GEP is denoted by GEP(Θ,A).IfA=0, then GMEP (8) reduces to the mixed equilibrium problem (MEP) which is to find x∈C such that
(10)Θ(x,y)+φ(y)-φ(x)≥0,∀y∈C.
It is considered and studied in [22]. The set of solutions of MEP is denoted by MEP(Θ,φ).Ifφ=0 and A=0, then GMEP (8) reduces to the equilibrium problem (EP) which is to find x∈C such that
(11)Θ(x,y)≥0,∀y∈C.
It is considered and studied in [23]. The set of solutions of EP is denoted by EP(Θ).Throughout this paper, it is assumed as in [18] 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,
(12)limsupt→0+Θ(tz+(1-t)x,y)≤Θ(x,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,
(13)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0;(B2)
C is a bounded set.Next we list some known results for the MEP as follows.Proposition 1 (see [22]).
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:
(14)Tr(Θ,φ)(x)={1rz∈C:Θ(z,y)+φ(y)-φ(z){1r+1r〈y-z,z-x〉≥0,∀y∈C},
for all x∈H. Then the following conditions hold:(i)
for eachx∈H, Tr(Θ,φ)(x)is nonempty and single-valued;(ii)
Tr(Θ,φ)is firmly nonexpansive; that is, for anyx,y∈H,
(15)∥Tr(Θ,φ)x-Tr(Θ,φ)y∥2≤〈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)x∥2≤((s-t)/s)〈Ts(Θ,φ)x-Tt(Θ,φ)x,Ts(Θ,φ)x-x〉,for alls,t>0andx∈H.Letλn,1,λn,2,…,λn,N∈(0,1], n≥1. Given the nonexpansive mappings S1,S2,…,SN on H, for each n≥1, the mappings Un,1,Un,2,…,Un,N are defined by
(16)Un,1=λn,1S1+(1-λn,1)I,Un,2=λn,2SnUn,1+(1-λn,2)I,Un,n-1=λn-1Tn-1Un,n+(1-λn-1)I,⋮Un,N-1=λn,N-1SN-1Un,N-2+(1-λn,N-1)I,Wn:=Un,N=λn,NSNUn,N-1+(1-λn,N)I.TheWn is called the W-mapping generated by S1,…,SN and λn,1,λn,2,…,λn,N. Note that the nonexpansivity of Si implies the nonexpansivity of Wn.In 2012, combining the hybrid steepest-descent method in [24] and hybrid viscosity approximation method in [25], Ceng et al. [20] proposed and analyzed the following hybrid iterative method for finding a common element of the set of solutions of GMEP (8) and the set of fixed points of a finite family of nonexpansive mappings {Si}i=1N.Theorem CGY (see [20, Theorem 3.1]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ:C×C→R be a bifunction satisfying assumptions (A1)–(A4) and let φ:C→R be a lower semicontinuous and convex function with restriction (B1) or (B2). Let the mapping A:H→H be δ-inverse strongly monotone, and let {Si}i=1N be a finite family of nonexpansive mappings on H such that ⋂i=1NFix(Si)∩GMEP(Θ,φ,A)≠∅. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with constants κ,η>0 and V:H→H a ρ-Lipschitzian mapping with constant ρ≥0. Let 0<μ<2η/κ2 and 0≤γρ<τ, where τ=1-1-μ(2η-μκ2). Suppose {αn} and {βn} are two sequences in (0,1), {γn} is a sequence in (0,2δ], and {λn,i}i=1N is a sequence in [a,b] with 0<a≤b<1. For every n≥1, let Wn be the W-mapping generated by S1,…,SN and λn,1,λn,2,…,λn,N. Given x1∈H arbitrarily, suppose the sequences {xn} and {un} are generated iteratively by
(17)Θ(un,y)+φ(y)-φ(un)+〈Axn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=αnγVxn+βnxn+((1-βn)I-αnμF)Wnun,∀n≥1,
where the sequences {αn}, {βn}, and {rn} and the finite family of sequences {λn,i}i=1N satisfy the following conditions:(i)
limn→∞αn=0and∑n=1∞‍αn=∞;(ii)
0<liminfn→∞βn≤limsupn→∞βn<1;(iii)
0<liminfn→∞rn≤limsupn→∞rn<2δandlimn→∞(rn+1-rn)=0;(iv)
limn→∞(λn+1,i-λn,i)=0,for alli=1,2,…,N.Then both {xn} and {un} converge strongly to x*∈⋂i=1NFix(Si)∩GMEP(Θ,φ,A), where x*=P⋂i=1NFix(Si)∩GMEP(Θ,φ,A)(I-μF+γf)x* is a unique solution of the variational inequality problem (VIP):
(18)∀x∈∀〈(μF-γV)x*,x*-x〉≤0,∀x∈⋂i=1NFix(Si)∩GMEP(Θ,φ,A).Letf:C→R be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing f over the constraint set C:
(19)minx∈Cf(x)
(assuming the existence of minimizers). We denote by Γ the set of minimizers of CMP (19). It is well known that the gradient-projection algorithm (GPA) generates a sequence {xn} determined by the gradient ∇f and the metric projection PC:
(20)xn+1:=PC(xn-λ∇f(xn)),∀n≥0,
or more generally,
(21)xn+1:=PC(xn-λn∇f(xn)),∀n≥0,
where, in both (20) and (21), the initial guess x0 is taken from C arbitrarily and the parameters λ or λn are positive real numbers. The convergence of algorithms (20) and (21) depends on the behavior of the gradient ∇f. As a matter of fact, it is known that, if ∇f is α-strongly monotone and L-Lipschitz continuous, then, for 0<λ<2α/L2, the operator
(22)S:=PC(I-λ∇f)
is a contraction. Hence, the sequence {xn} defined by the GPA (20) converges in norm to the unique solution of CMP (19). More generally, if the sequence {λn} is chosen to satisfy the property
(23)0<liminfn→∞λn≤limsupn→∞λn<2αL2,
then the sequence {xn} defined by the GPA (21) converges in norm to the unique minimizer of CMP (19). If the gradient ∇f is only assumed to be Lipschitz continuous, then {xn} can only be weakly convergent if H is infinite dimensional (a counterexample is given in Section 5 of Xu [26]).Since the Lipschitz continuity of the gradient∇f implies that it is actually (1/L)-inverse strongly monotone (ism) [27], its complement can be an averaged mapping (i.e., it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping). Consequently, the GPA can be rewritten as the composite of a projection and an averaged mapping, which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Recently, Xu [26] used averaged mappings to study the convergence analysis of the GPA, which is hence an operator-oriented approach.In 2011, combining the hybrid steepest-descent method in [24], viscosity approximation method, and averaged mapping approach to the GPA in [26], Ceng et al. [28] introduced and analyzed the following implicit and explicit iterative algorithms:
(24)xλ=PC[sγVxλ+(I-sμF)Tλxλ],λ∈(0,2L),(25)xn+1=PC[snγVxn+(I-snμF)Tnxn],∀n≥0,
where V:C→H is l-Lipschitzian mapping with constant l≥0 and F:C→H is a κ-Lipschitzian and η-strongly monotone operator with constants κ,η>0. Assume that 0<μ<2η/κ2, 0≤γl<τ=1-1-μ(2η-μκ2), s:=s(λ)=(2-λL)/4 for each λ∈(0,2/L), PC(I-λ∇f)=sI+(1-s)Tλ for each λ∈(0,2/L), sn:=sn(λn)=(2-λnL)/4 with {λn}⊂(0,2/L) and λn→2/L, and PC(I-λn∇f)=snI+(1-sn)Tn. The authors proved that the net {xλ} defined by (24) converges strongly to some q∈Γ, which is a unique solution of the variational inequality problem (VIP):
(26)〈(μF-γV)q,p-q〉≥0,∀p∈Γ.
Furthermore, utilizing control conditions (i) sn→0, (ii) ∑n=0∞‍sn=∞, and (iii) either ∑n=0∞‍|sn+1-sn|<∞ or limn→∞sn+1/sn=1, the authors also proved that the sequence {xn} generated by (25) converges strongly to some q∈Γ, which is a unique solution of the VIP (26).Motivated and inspired by the above facts, in this paper we introduce implicit and explicit iterative algorithms for finding a common element of the set of solutions of the CMP (19) for a convex functional f:C→R with L-Lipschitz continuous gradient ∇f, the set of solutions of a finite family of GMEPs, and the set of solutions of a finite family of VIPs for inverse strong monotone mappings in a real Hilbert space. Under very mild control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets. Our iterative algorithms are based on Korpelevich’s extragradient method, hybrid steepest-descent method in [24], viscosity approximation method, and averaged mapping approach to the GPA in [26]. The results obtained in this paper improve and extend the corresponding results announced by many others.


## 2. Preliminaries

Throughout this paper, we assume thatH is a real Hilbert space with inner product and norm 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,
(27)ωw(xn)≔{{xn}x∈H:xni⇀xforsomesubsequence{xni}of{xn}}.The metric 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
(28)∥x-PCx∥=infy∈C∥x-y∥=:d(x,C).Some important properties of projections are listed in the following proposition.Proposition 2.
For givenx∈H and z∈C,(i)
z=PCx⇔〈x-z,y-z〉≤0,for ally∈C;(ii)
z=PCx⇔∥x-z∥2≤∥x-y∥2-∥y-z∥2,for ally∈C;(iii)
〈PCx-PCy,x-y〉≥∥PCx-PCy∥2,for ally∈H.Consequently,PC is nonexpansive and monotone. If A is an α-inverse strongly monotone mapping of C into H, then it is obvious that A is (1/α)-Lipschitz continuous. We also have that, for all u,v∈C and λ>0,
(29)∥(I-λA)u-(I-λA)v∥2=∥(u-v)-λ(Au-Av)∥2=∥u-v∥2-2λ〈Au-Av,u-v〉+λ2∥Au-Av∥2≤∥u-v∥2+λ(λ-2α)∥Au-Av∥2.
So, if λ≤2α, then I-λA is a nonexpansive mapping from C to H.Definition 3.
A mappingT:H→H is said to be(a)
nonexpansive [1] if
(30)∥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),
(31)〈x-y,Tx-Ty〉≥∥Tx-Ty∥2,∀x,y∈H;
alternatively, T is firmly nonexpansive if and only if T can be expressed as
(32)T=12(I+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.Definition 4.
A mappingT:H→H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping; that is,
(33)T≡(1-α)I+αS,
where α∈(0,1) and S:H→H is nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus firmly nonexpansive mappings (in particular, projections) are (1/2)-averaged mappings.Proposition 5 (see [29]).
LetT:H→H be a given mapping.(i)
Tis nonexpansive if and only if the complementI-Tis(1/2)-ism.(ii)
IfTisν-ism, then, forγ>0,γTis(ν/γ)-ism.(iii)
Tis averaged if and only if the complementI-Tisν-ism for someν>1/2. Indeed, forα∈(0,1),Tisα-averaged if and only ifI-Tis(1/2α)-ism.Proposition 6 (see [29]).
LetS,T,V:H→H be given operators.(i)
IfT=(1-α)S+αVfor someα∈(0,1)and ifSis averaged andVis nonexpansive, thenTis averaged.(ii)
Tis firmly nonexpansive if and only if the complementI-Tis firmly nonexpansive.(iii)
IfT=(1-α)S+αVfor someα∈(0,1)and ifSis firmly nonexpansive andVis nonexpansive, thenTis averaged.(iv)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings{Ti}i=1Nis averaged, then so is the compositeT1⋯TN. In particular, ifT1isα1-averaged andT2isα2-averaged, whereα1,α2∈(0,1), then the compositeT1T2isα-averaged, whereα=α1+α2-α1α2.(v)
If the mappings{Ti}i=1Nare averaged and have a common fixed point, then(34)⋂i=1N‍Fix(Ti)=Fix(T1···TN).
The notationFix(T) denotes the set of all fixed points of the mapping T; that is, Fix(T)={x∈H:Tx=x}.We need some facts and tools in a real Hilbert spaceH which are listed as lemmas below.Lemma 7.
LetX be a real inner product space. Then there holds the following inequality:
(35)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈X.Lemma 8.
LetA:C→H be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 2(i)) implies
(36)u∈VI(C,A)⟺u=PC(u-λAu),forsomeλ>0.Lemma 9 (see [30, Demiclosedness principle]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C with Fix(T)≠∅. 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.Lemma 10 (see [31]).
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):=limn→∞∏k=1n‍(1-αk)=0;(ii)
limsupn→∞βn≤0, or ∑n=1∞‍|αnβn|<∞. Then limn→∞sn=0.Lemma 11 (see [32]).
Let{xn} and {zn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with
(39)0<liminfn→∞βn≤limsupn→∞βn<1.
Suppose that xn+1=(1-βn)zn+βnxn for each n≥1 and
(40)limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0.
Then limn→∞∥zn-xn∥=0.The following lemma can be easily proven and, therefore, we omit the proof.Lemma 12.
LetV:H→H be an l-Lipschitzian mapping with constant l≥0, and let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Then for 0≤γl<μη,
(41)〈(μF-γV)x-(μF-γV)y,x-y〉≥(μη-γl)∥x-y∥2,〈(μF-γV)x-(μF-γV)y,x-y〉≥(μη-γ)∀x,y∈H.
That is, μF-γV is strongly monotone with constant μη-γl.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
(42)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 following conditions:
(43)∥Fx-Fy∥≤κ∥x-y∥,〈Fx-Fy,x-y〉≥η∥x-y∥2,
for all x,y∈H.Lemma 13 (see [31, Lemma 3.1]).
T
λ is a contraction provided 0<μ<2η/κ2; that is,
(44)∥Tλx-Tλy∥≤(1-λτ)∥x-y∥,∀x,y∈C,
where τ=1-1-μ(2η-μκ2)∈(0,1].Remark 14.
(i) SinceF is κ-Lipschitzian and η-strongly monotone on H, we get 0<η≤κ. Hence, whenever 0<μ<2η/κ2, we have
(45)0≤(1-μη)2=1-2μη+μ2η2≤1-2μη+μ2κ2<1-2μη+2ηκ2μκ2=1,
which implies
(46)0<1-1-2μη+μ2κ2≤1.
So, τ=1-1-μ(2η-μκ2)∈(0,1].
(ii) In Lemma13, put F=(1/2)I and μ=2. Then we know that κ=η=1/2,0<μ=2<2η/κ2=4, and
(47)τ=1-1-μ(2η-μκ2)=1-1-2(2×12-2×(12)2)=1.
Finally, recall that a set-valued mappingT:H→2H is called monotone if, for all x,y∈H, f∈Tx and g∈Ty imply 〈x-y,f-g〉≥0. A monotone mapping T:H→2H is maximal if its graph G(T) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if, for (x,f)∈H×H,〈x-y,f-g〉≥0 for all (y,g)∈G(T) implies f∈Tx. Let A:C→H be a monotone, L-Lipschitz continuous mapping and let NCv be the normal cone to C at v∈C; that is, NCv={w∈H:〈v-u,w〉≥0, for all u∈C}. Define
(48)Tv={Av+NCv,ifv∈C,∅,ifv∉C.
It is known that in this case T is maximal monotone, and 0∈Tv if and only if v∈Ω; see [33].


## 3. Implicit Iterative Algorithm and Its Convergence Criteria

We now state and prove the first main result of this paper.Theorem 15.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let M and 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 Bk:H→H and Ai:C→H be μk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2,…,M} and i∈{1,2,…,N}. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let V:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=⋂k=1MGMEP(Θk,φk,Bk)∩⋂i=1NVI(C,Ai)∩Γ≠∅ and that either (B1) or (B2) holds. Let {xn} be a sequence generated by
(49)un=TrM,n(ΘM,φM)(I-rM,nBM)TrM-1,n(ΘM-1,φM-1)(I-rM-1,nBM-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn,vn=PC(I-λN,nAN)PC(I-λN-1,nAN-1)⋯PC(I-λ2,nA2)PC(I-λ1,nA1)un,xn=snγVxn+(I-snμF)Tnvn,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive and sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L)). Assume that the following conditions hold:(i)
sn∈(0,1/2) for each λn∈(0,2/L), limn→∞sn=0  (⇔limn→∞λn=2/L);(ii)
{λi,n}⊂[ai,bi]⊂(0,2ηi), for all i∈{1,2,…,N};(iii)
{rk,n}⊂[ek,fk]⊂(0,2μk), for all k∈{1,2,…,M}.
Then{xn} converges strongly as λn→2/L (⇔sn→0) to a point q∈Ω, which is a unique solution of the VIP:
(50)〈(μF-γV)q,p-q〉≥0,∀p∈Ω.
Equivalently, q=PΩ(I-μF+γV)q.Proof.
First of all, let us show that the sequence{xn} is well defined. Indeed, since ∇f is L-Lipschitzian, it follows that ∇f is 1/L-ism; see [34]. By Proposition 5(ii) we know that, for λ>0,λ∇f is (1/λL)-ism. So by Proposition 5(iii) we deduce that I-λ∇f is (λL/2)-averaged. Now since the projection PC is (1/2)-averaged, it is easy to see from Proposition 6(iv) that the composite PC(I-λ∇f) is ((2+λL)/4)-averaged for λ∈(0,2/L). Hence we obtain that for each n≥1, PC(I-λn∇f) is ((2+λnL)/4)-averaged for each λn∈(0,2/L). Therefore, we can write
(51)PC(I-λn∇f)=2-λnL4I+2+λnL4Tn=snI+(1-sn)Tn,
where Tn is nonexpansive and sn:=sn(λn)=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L). It is clear that
(52)λn⟶2L⟺sn⟶0.
Put(53)Δnk=Trk,n(Θk,φk)(I-rk,nBk)Trk-1,n(Θk-1,φk-1)(I-rk-1,nBk-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn,
for all k∈{1,2,…,M} and n≥1,
(54)Λni=PC(I-λi,nAi)PC(I-λi-1,nAi-1)⋯PC(I-λ2,nA2)PC(I-λ1,nA1),
for all i∈{1,2,…,N} and n≥1, and Δn0=Λn0=I, where I is the identity mapping on H. Then we have that un=ΔnMxn and vn=ΛnNun.
Consider the following mappingGn on H defined by
(55)Gnx=snγVx+(I-snμF)TnΛnNΔnMx,(I-snμF)111TnΛnNΔnM∀x∈H,n≥1,
where sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L). By Proposition 1(ii) and Lemma 13 we obtain from (29) that, for all x,y∈H,
(56)∥Gnx-Gny∥≤snγ∥Vx-Vy∥+∥(I-snμF)TnΛnNΔnMx-(I-snμF)TnΛnNΔnMy∥≤snγl∥x-y∥+(1-snτ)∥ΛnNΔnMx-ΛnNΔnMy∥≤snγl∥x-y∥+(1-snτ)∥(I-λN,nAN)ΛnN-1ΔnMx-(I-λN,nAN)ΛnN-1ΔnMy∥≤snγl∥x-y∥+(1-snτ)∥ΛnN-1ΔnMx-ΛnN-1ΔnMy∥⋮≤snγl∥x-y∥+(1-snτ)∥Λn0ΔnMx-Λn0ΔnMy∥=snγl∥x-y∥+(1-snτ)∥ΔnMx-ΔnMy∥≤snγl∥x-y∥+(1-snτ)∥(I-rM,nBM)ΔnM-1x-(I-rM,nBM)ΔnM-1y∥≤snγl∥x-y∥+(1-snτ)∥ΔnM-1x-ΔnM-1y∥⋮≤snγl∥x-y∥+(1-snτ)∥Δn0x-Δn0y∥=snγl∥x-y∥+(1-snτ)∥x-y∥=(1-sn(τ-γl))∥x-y∥.
Since 0<1-sn(τ-γl)<1, Gn:H→H is a contraction. Therefore, by the Banach contraction principle, Gn has a unique fixed point xn∈H, which uniquely solves the fixed point equation
(57)xn=snγVxn+(I-snμF)TnΛnNΔnMxn.
This shows that the sequence {xn} is defined well.
Note that0≤γl<τ and μη≥τ⇔κ≥η. Hence by Lemma 12 we know that
(58)〈(μF-γV)x-(μF-γV)y,x-y〉≥(μη-γl)∥x-y∥2,∀x,y∈H.
That is, μF-γV is strongly monotone for 0≤γl<τ≤μη. Moreover, it is clear that μF-γV is Lipschitz continuous. So the VIP (50) has only one solution. Below we use q∈Ω to denote the unique solution of the VIP (50).
Now, let us show that{xn} is bounded. In fact, take p∈Ω arbitrarily. Then from (29) and Proposition 1(ii) we have
(59)∥un-p∥=∥TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn-TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1p∥≤∥(I-rM,nBM)ΔnM-1xn-(I-rM,nBM)ΔnM-1p∥≤∥ΔnM-1xn-ΔnM-1p∥⋮≤∥Δn0xn-Δn0p∥=∥xn-p∥.
Similarly, we have
(60)∥vn-p∥=∥PC(I-λN,nAN)ΛnN-1un-PC(I-λN,nAN)ΛnN-1p∥≤∥(I-λN,nAN)ΛnN-1un-(I-λN,nAN)ΛnN-1p∥≤∥ΛnN-1un-ΛnN-1p∥⋮≤∥Λn0un-Λn0p∥=∥un-p∥.
Combining (59) and (60), we have
(61)∥vn-p∥≤∥xn-p∥.
Since
(62)p=PC(I-λn∇f)p=snp+(1-sn)Tnp,p=PC(I-λn∇f)p=snp+∀λn∈(0,2L),
where sn:=sn(λn)=(2-λnL)/4∈(0,1/2). It is clear that Tnp=p for each λn∈(0,2/L). Thus, utilizing Lemma 13 and the nonexpansivity of Tn, we obtain from (61) that
(63)∥xn-p∥=∥sn(γVxn-μFp)+(I-snμF)Tnvn-(I-snμF)Tnp∥≤∥(I-snμF)Tnvn-(I-snμF)Tnp∥+sn∥γVxn-μFp∥≤(1-snτ)∥vn-p∥+sn(γ∥Vxn-Vp∥+∥γVp-μFp∥)≤(1-snτ)∥xn-p∥+sn(γl∥xn-p∥+∥γVp-μFp∥)=(1-sn(τ-γl))∥xn-p∥+sn∥γVp-μFp∥.
This implies that ∥xn-p∥≤∥γVp-μFp∥/(τ-γl). Hence {xn} is bounded. So, according to (59) and (61) we know that {un},{vn},{Tnvn},{Vxn}, and {FTnvn} are bounded.
Next let us show that∥un-xn∥→0, ∥vn-un∥→0, and ∥xn-Tnxn∥→0 as n→∞.
Indeed, from (29) it follows that, for all i∈{1,2,…,N} and k∈{1,2,…,M},
(64)∥vn-p∥2=∥ΛnNun-p∥2≤∥Λniun-p∥2=∥PC(I-λi,nAi)Λni-1un-PC(I-λi,nAi)p∥2≤∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2≤∥Λni-1un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2≤∥un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2≤∥xn-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2,∥un-p∥2=∥ΔnMxn-p∥2≤∥Δnkxn-p∥2=∥Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nBk)p∥2≤∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2≤∥Δnk-1xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2≤∥xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2.
Thus, utilizing Lemma 7, from (49) and (64) we have
(65)∥xn-p∥2=∥sn(γVxn-μFp)+(I-snμF)Tnvn-(I-snμF)Tnp∥2≤∥(I-snμF)Tnvn-(I-snμF)Tnp∥2+2sn〈γVxn-μFp,xn-p〉≤(1-snτ)2∥vn-p∥2+2sn〈γVxn-μFp,xn-p〉=(1-snτ)2∥vn-p∥2+2snγ〈Vxn-Vp,xn-p〉+2sn〈γVp-μFp,xn-p〉≤(1-snτ)2∥vn-p∥2+2snγl∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥≤(1-snτ)2[×∥AiΛni-1un-Aip∥2∥un-p∥2+λi,n(λi,n-2ηi)×∥AiΛni-1un-Aip∥2]+2snγl∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥≤(1-snτ)2[∥AiΛni-1un-Aip∥2∥xn-p∥2+rk,n(rk,n-2μk)×∥BkΔnk-1xn-Bkp∥2+λi,n(λi,n-2ηi)×∥AiΛni-1un-Aip∥2]+2snγl∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥=[1-2sn(τ-γl)+sn2τ2]∥xn-p∥2-(1-snτ)2rk,n(2μk-rk,n)×∥BkΔnk-1xn-Bkp∥2+λi,n(2ηi-λi,n)∥AiΛni-1un-Aip∥2+2sn∥γVp-μFp∥∥xn-p∥≤∥xn-p∥2+sn2τ2∥xn-p∥2-(1-snτ)2[∥AiΛni-1un-Aip∥2rk,n(2μk-rk,n)×∥BkΔnk-1xn-Bkp∥2+λi,n(2ηi-λi,n)×∥AiΛni-1un-Aip∥2]+2sn∥γVp-μFp∥∥xn-p∥,
which implies that
(66)(1-snτ)2[∥AiΛni-1un-Aip∥2rk,n(2μk-rk,n)∥BkΔnk-1xn-Bkp∥2+λi,n(2ηi-λi,n)∥AiΛni-1un-Aip∥2∥AiΛni-1un-Aip∥2]≤sn2τ2∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥.
Since {λi,n}⊂[ai,bi]⊂(0,2ηi) and {rk,n}⊂[ek,fk]⊂(0,2μk), for all i∈{1,2,…,N} and k∈{1,2,…,M}, from sn→0 we conclude immediately that
(67)limn→∞∥AiΛni-1un-Aip∥=0,limn→∞∥BkΔnk-1xn-Bkp∥=0,
for all i∈{1,2,…,N} and k∈{1,2,…,M}.
Furthermore, by Proposition1(ii) we obtain that for each k∈{1,2,…,M}(68)∥Δnkxn-p∥2=∥Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nBk)p∥2≤〈(I-rk,nBk-(I-rk,nBk)p,Δnkxn-p)Δnk-1xn〉=12(-(Δnkxn-p)∥2∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2+∥Δnkxn-p∥2-∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p-(Δnkxn-p)∥2)≤12(-(Δnkxn-p)∥2∥Δnk-1xn-p∥2+∥Δnkxn-p∥2-∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2),
which implies that
(69)∥Δnkxn-p∥2≤∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2=∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn∥2-rk,n2∥BkΔnk-1xn-Bkp∥2+2rk,n〈Δnk-1xn-Δnkxn,BkΔnk-1xn-Bkp〉≤∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥×∥BkΔnk-1xn-Bkp∥≤∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥×∥BkΔnk-1xn-Bkp∥.
Also, by Proposition 2(iii), we obtain that for each i∈{1,2,…,N}(70)∥Λniun-p∥2=∥PC(I-λi,nAi)Λni-1un-PC(I-λi,nAi)∥2≤〈(I-λi,nAi)Λni-1un-(I-λi,nAi)p,Λniun-p〉=12(-(I-λi,nAi)p-(Λniun-p)∥2∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2+∥Λniun-p∥2-∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p-(Λniun-p)∥2)≤12(-(I-λi,nAi)p-(Λniun-p)∥2∥Λni-1un-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2)≤12(∥un-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2-(I-λi,nAi)p-(Λniun-p)∥2),
which implies
(71)∥Λniun-p∥2≤∥un-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2=∥un-p∥2-∥Λni-1un-Λniun∥2-λi,n2∥AiΛni-1un-Aip∥2+2λi,n〈Λni-1un-Λniun,AiΛni-1un-Aip〉≤∥un-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥×∥AiΛni-1un-Aip∥.
Thus, utilizing Lemma 7, from (49), (69), and (71) we have
(72)∥xn-p∥2=∥sn(γVxn-μFp)+(I-snμF)Tnvn-(I-snμF)Tnp∥2≤∥(I-snμF)Tnvn-(I-snμF)Tnp∥2+2sn〈γVxn-μFp,xn-p〉≤(1-snτ)2∥vn-p∥2+2snγl∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥=(1-snτ)2∥ΛnNun-p∥2+2snγl∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥≤(1-snτ)2∥Λniun-p∥2+2snγl∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥≤(1-snτ)2[∥AiΛni-1un-Aip∥∥un-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥×∥AiΛni-1un-Aip∥∥un-p∥2]+2snγl∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥=(1-snτ)2[∥AiΛni-1un-Aip∥∥ΔnMxn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥×∥AiΛni-1un-Aip∥∥ΔnMxn-p∥2]+2snγl∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥≤(1-snτ)2[∥AiΛni-1un-Aip∥∥Δnkxn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥×∥AiΛni-1un-Aip∥∥Δnkxn-p∥2]+2snγl∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥≤(1-snτ)2[∥AiΛni-1un-Aip∥∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥×∥BkΔnk-1xn-Bkp∥-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥×∥AiΛni-1un-Aip∥]+2snγl∥xn-p∥2+2sn∥γVp-μFp∥∥xn-p∥≤(1-2sn(τ-γl)+sn2τ2)∥xn-p∥2-(1-snτ)2×(∥Δnk-1xn-Δnkxn∥2+∥Λni-1un-Λniun∥2)+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2sn∥γVp-μFp∥∥xn-p∥≤∥xn-p∥2+sn2τ2∥xn-p∥2-(1-snτ)2×(∥Δnk-1xn-Δnkxn∥2+∥Λni-1un-Λniun∥2)+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2sn∥γVp-μFp∥∥xn-p∥.
It immediately follows that
(73)(1-snτ)2(∥Δnk-1xn-Δnkxn∥2+∥Λni-1un-Λniun∥2)≤sn2τ2∥xn-p∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2sn∥γVp-μFp∥∥xn-p∥.
Since {λi,n}⊂[ai,bi]⊂(0,2ηi) and {rk,n}⊂[ek,fk]⊂(0,2μk), for all i∈{1,2,…,N} and k∈{1,2,…,M}, from (67) and sn→0 we deduce that
(74)limn→∞∥Δnk-1xn-Δnkxn∥=0,limn→∞∥Λni-1un-Λniun∥=0,
for all i∈{1,2,…,N} and k∈{1,2,…,M}. Hence we get
(75)∥xn-un∥=∥Δn0xn-ΔnMxn∥≤∥Δn0xn-Δn1xn∥+∥Δn1xn-Δn2xn∥+⋯+∥ΔnM-1xn-ΔnMxn∥⟶0asn⟶∞,(76)∥un-vn∥=∥Λn0un-ΛnNun∥≤∥Λn0un-Λn1un∥+∥Λn1un-Λn2un∥+⋯+∥ΛnN-1un-ΛnNun∥⟶0asn⟶∞.
So, taking into account that ∥xn-vn∥≤∥xn-un∥+∥un-vn∥, we have
(77)limn→∞∥xn-vn∥=0.
Thus, from (77) and sn→0 we have
(78)∥vn-Tnvn∥≤∥vn-xn∥+∥xn-Tnvn∥=∥vn-xn∥+sn∥γVxn-μFTnvn∥⟶0asn⟶∞.
Now we show that ∥xn-Tnxn∥→0 as n→∞. In fact, from the nonexpansivity of Tn, we have
(79)∥xn-Tnxn∥≤∥xn-vn∥+∥vn-Tnvn∥+∥Tnvn-Tnxn∥≤2∥xn-vn∥+∥vn-Tnvn∥.
By (77) and (78), we get
(80)limn→∞∥xn-Tnxn∥=0.
From (78) it is easy to see that
(81)limn→∞∥xn-Tnvn∥=0.
Observe that
(82)∥PC(I-λn∇f)vn-vn∥=∥snvn+(1-sn)Tnvn-vn∥=(1-sn)∥Tnvn-vn∥≤∥Tnvn-vn∥,
where sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L). Hence we have
(83)∥PC(I-2L∇f)vn-vn∥≤∥PC(I-2L∇f)vn-PC(I-λn∇f)vn∥+∥PC(I-λn∇f)vn-vn∥≤∥(I-2L∇f)vn-(I-λn∇f)vn∥+∥PC(I-λn∇f)vn-vn∥≤(2L-λn)∥∇f(vn)∥+∥Tnvn-vn∥.
From the boundedness of {vn},sn→0  (⇔λn→2/L) and ∥Tnvn-vn∥→0  (due to (78)), it follows that
(84)limn→∞∥vn-PC(I-2L∇f)vn∥=0.
Further, we show thatωw(xn)⊂Ω. Indeed, since {xn} is bounded, there exists a subsequence {xni} of {xn} which converges weakly to some w. Note that limn→∞∥xn-un∥=0  (due to (75)). Hence xni⇀w. Since C is closed and convex, C is weakly closed. So, we have w∈C. From (74)-(75), we have that Δnikxni⇀w, Λnimuni⇀w, uni⇀w, and vni⇀w, where k∈{1,2,…,M} and m∈{1,2,…,N}. First, we prove that w∈⋂m=1NVI(C,Am). Let
(85)Tmv={Amv+NCv,v∈C,∅,v∉C,
where m∈{1,2,…,N}. Let (v,u)∈G(Tm). Since u-Amv∈NCv and Λnmun∈C, we have
(86)〈v-Λnmun,u-Amv〉≥0.
On the other hand, from Λnmun=PC(I-λm,nAm)Λnm-1un and v∈C, we have
(87)〈v-Λnmun,Λnmun-(Λnm-1un-λm,nAmΛnm-1un)〉≥0,
and hence
(88)〈v-Λnmun,Λnmun-Λnm-1unλm,n+AmΛnm-1un〉≥0.
Therefore we have
(89)〈v-Λnimuni,u〉≥〈v-Λnimuni,Amv〉≥〈v-Λnimuni,Amv〉-〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni+AmΛnim-1uni〉=〈v-Λnimuni,Amv-AmΛnimuni〉+〈v-Λnimuni,AmΛnimuni-AmΛnim-1uni〉-〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni〉≥〈v-Λnimuni,AmΛnimuni-AmΛnim-1uni〉-〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni〉.
From (74) and since Am is Lipschitz continuous, we obtain that limn→∞∥AmΛnmun-AmΛnm-1un∥=0. From Λnimuni⇀w,{λi,n}⊂[ai,bi]⊂(0,2ηi), for all i∈{1,2,…,N} and (74), we have
(90)〈v-w,u〉≥0.
Since Tm is maximal monotone, we have w∈Tm-10 and hence w∈VI(C,Am), m=1,2,…,N, which implies w∈⋂m=1NVI(C,Am). Next we prove that w∈⋂k=1MGMEP(Θk,φk,Bk). Since Δnkxn=Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn, n≥1, k∈{1,2,…,M}, we have
(91)Θk(Δnkxn,y)+φk(y)-φk(Δnkxn)+〈BkΔnk-1xn,y-Δnkxn〉+1rk,n〈y-Δnkxn,Δnkxn-Δnk-1xn〉≥0.
By (A2), we have
(92)φk(y)-φk(Δnkxn)+〈BkΔnk-1xn,y-Δnkxn〉+1rk,n〈y-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
(93)〈zt-Δnkxn,Bkzt〉≥φk(Δnkxn)-φk(zt)+〈zt-Δnkxn,Bkzt〉-〈zt-Δnkxn,BkΔnk-1xn〉-〈zt-Δnkxn,Δnkxn-Δnk-1xnrk,n〉+Θk(zt,Δnkxn)=φk(Δnkxn)-φk(zt)+〈zt-Δnkxn,Bkzt-BkΔnkxn〉+〈zt-Δnkxn,BkΔnkxn-BkΔnk-1xn〉-〈zt-Δnkxn,Δnkxn-Δnk-1xnrk,n〉+Θk(zt,Δnkxn).
By (74), we have ∥BkΔnkxn-BkΔnk-1xn∥→0 as n→∞. Furthermore, by the monotonicity of Bk, we obtain 〈zt-Δnkxn,Bkzt-BkΔnkxn〉≥0. Then, by (A4) we obtain
(94)〈zt-w,Bkzt〉≥φk(w)-φk(zt)+Θk(zt,w).
Utilizing (A1), (A4), and (94), we obtain
(95)0=Θk(zt,zt)+φk(zt)-φk(zt)≤tΘk(zt,y)+(1-t)Θk(zt,w)+tφk(y)+(1-t)φk(w)-φk(zt)≤t[Θk(zt,y)+φk(y)-φk(zt)]+(1-t)〈zt-w,Bkzt〉=t[Θk(zt,y)+φk(y)-φk(zt)]+(1-t)t〈y-w,Bkzt〉,
and hence
(96)0≤Θk(zt,y)+φk(y)-φk(zt)+(1-t)〈y-w,Bkzt〉.
Letting t→0, we have, for each y∈C,
(97)0≤Θk(w,y)+φk(y)-φk(w)+〈y-w,Bkw〉.
This implies that w∈GMEP(Θk,φk,Bk) and hence w∈⋂k=1MGMEP(Θk,φk,Bk). Further, let us show that w∈Γ. As a matter of fact, from (84), vni⇀w, and Lemma 9, we conclude that
(98)w=PC(I-2L∇f)w.
So, w∈VI(C,∇f)=Γ. Therefore, w∈⋂i=1NVI(C,Ai)∩⋂k=1MGMEP(Θk,φk,Bk)∩Γ=:Ω. This shows that ωw(xn)⊂Ω.
Finally, let us show thatxn→q as n→∞, where q is the unique solution of the VIP (50). Indeed, we note that, for w∈Ω with xni⇀w,
(99)xn-w=sn(γVxn-μFw)+(I-snμF)Tnvn-(I-snμF)w.
By (61) and Lemma 13, we obtain that
(100)∥xn-w∥2=sn〈γVxn-μFw,xn-w〉+〈(I-snμF)Tnvn-(I-snμF)w,xn-w〉=sn〈γVxn-μFw,xn-w〉+∥(I-snμF)Tnvn-(I-snμF)w∥∥xn-w∥≤sn〈γVxn-μFw,xn-w〉+(1-snτ)∥vn-w∥∥xn-w∥≤sn〈γVxn-μFw,xn-w〉+(1-snτ)∥xn-w∥2.
Hence it follows that
(101)∥xn-w∥2≤1τ〈γVxn-μFw,xn-w〉=1τ(γ〈Vxn-Vw,xn-w〉+〈γVw-μFw,xn-w〉)≤1τ(γl∥xn-w∥2+〈γVw-μFw,xn-w〉),
which hence leads to
(102)∥xn-w∥2≤〈γVw-μFw,xn-w〉τ-γl.
In particular, we have
(103)∥xni-w∥2≤〈γVw-μFw,xni-w〉τ-γl.
Since xni⇀w, it follows from (103) that xni→w as i→∞.
Now we show thatw solves the VIP (50). Since xn=snγVxn+(I-snμF)Tnvn, we have
(104)(μF-γV)xn=-1sn((I-snμF)xn-(I-snμF)Tnvn).
It follows that, for each p∈Ω,
(105)〈(μF-γV)xn,xn-p〉=-1sn〈(I-snμF)xn-(I-snμF)Tnvn,xn-p〉=-1sn〈(I-snμF)xn-(I-snμF)Tnvn,xn-p〉=-1sn〈(I-snμF)xn-(I-snμF)TnΛnNΔnMxn,xn-p〉=-1sn〈(I-TnΛnNΔnM)xn-(I-TnΛnNΔnM)p,xn-p〉+〈μFxn-μFTnΛnNΔnMxn,xn-p〉≤〈μFxn-μFTnΛnNΔnMxn,xn-p〉
since I-TnΛnNΔnM is monotone (i.e., 〈(I-TnΛnNΔnM)x-(I-TnΛnNΔnM)y,x-y〉≥0, for all x,y∈H. This is due to the nonexpansivity of TnΛnNΔnM). Since ∥xn-Tnvn∥=∥(I-TnΛnNΔnM)xn∥→0 as n→∞, by replacing n in (105) with ni and letting i→∞, we get
(106)〈(μF-γV)w,w-p〉=limi→∞〈(μF-γV)xni,xni-p〉≤limi→∞〈μFxni-μFTnivni,xni-p〉=0.
That is, w∈Ω is a solution of VIP (50).
Finally we show that the sequence{xn} converges strongly to q. To this end, let {xnk} be another subsequence of {xn} such that xnk→w^. By the same arguments as above, we have w^∈Ω. Moreover, it follows from (106) that
(107)〈(μF-γV)w,w-w^〉≤0.
Interchanging w and w^, we obtain
(108)〈(μF-γV)w^,w^-w〉≤0.
Utilizing Lemma 12 and adding the two inequalities (107) and (108), we have
(109)(μη-γl)∥w-w^∥2≤〈(μF-γV)w-(μF-γV)w^,w-w^〉≤0.
Hence w=w^. Therefore we conclude that xn→w as n→∞. Taking into account the uniqueness of solutions of VIP (50), we have w=q. The VIP (50) can be rewritten as
(110)〈(I-μF+γV)q-q,q-p〉≥0,∀p∈Ω.
By Proposition 2(i), this is equivalent to the fixed point equation
(111)PΩ(I-μF+γV)q=q.
This completes the proof.Corollary 16.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θ be a bifunction from C×C to R satisfying (A1)–(A4) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Let B:H→H and Ai:C→H be ζ-inverse strongly monotone and ηi-inverse strongly monotone, respectively, for i=1,2. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let V:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP(Θ,φ,B)∩VI(C,A1)∩VI(C,A2)∩Γ≠∅ and that either (B1) or (B2) holds. Let {xn} be a sequence generated by
(112)Θ(un,y)+φ(y)-φ(un)+〈Bxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,vn=PC(I-λ2,nA2)PC(I-λ1,nA1)un,xn=snγVxn+(I-snμF)Tnvn,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive and sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L)). Assume that the following conditions hold:(i)
sn∈(0,1/2) for each λn∈(0,2/L), limn→∞sn=0  (⇔limn→∞λn=2/L);(ii)
{λi,n}⊂[ai,bi]⊂(0,2ηi) for i=1,2;(iii)
{rn}⊂[e,f]⊂(0,2ζ).
Then{xn} converges strongly as λn→2/L (⇔sn→0) to a point q∈Ω, which is a unique solution of the VIP:
(113)〈(μF-γV)q,p-q〉≥0,∀p∈Ω.Corollary 17.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θ be a bifunction from C×C to R satisfying (A1)–(A4) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Let B:H→H and A:C→H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let V:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP(Θ,φ,B)∩VI(C,A)∩Γ≠∅ and that either (B1) or (B2) holds. Let {xn} be a sequence generated by
(114)Θ(un,y)+φ(y)-φ(un)+〈Bxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,vn=PC(I-ρnA)un,xn=snγVxn+(I-snμF)Tnvn,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive and sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L)). Assume that the following conditions hold:(i)
sn∈(0,1/2) for each λn∈(0,2/L), limn→∞sn=0  (⇔limn→∞λn=2/L);(ii)
{ρn}⊂[a,b]⊂(0,2ξ);(iii)
{rn}⊂[e,f]⊂(0,2ζ).
Then{xn} converges strongly as λn→2/L (⇔sn→0) to a point q∈Ω, which is a unique solution of the VIP:
(115)〈(μF-γV)q,p-q〉≥0,∀p∈Ω.


## 4. Explicit Iterative Algorithm and Its Convergence Criteria

We next state and prove the second main result of this paper.Theorem 18.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let M and 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 Bk:H→H and Ai:C→H be μk-inverse strongly monotone and ηi-inverse strongly monotone, respectively, where k∈{1,2,…,M} and i∈{1,2,…,N}. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let V:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=⋂k=1MGMEP(Θk,φk,Bk)∩⋂i=1NVI(C,Ai)∩Γ≠∅ and that either (B1) or (B2) holds. For arbitrarily given x1∈H, let {xn} be a sequence generated by
(116)un=TrM,n(ΘM,φM)(I-rM,nBM)TrM-1,n(ΘM-1,φM-1)(I-rM-1,nBM-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn,vn=PC(I-λN,nAN)PC(I-λN-1,nAN-1)⋯PC(I-λ2,nA2)PC(I-λ1,nA1)un,xn+1=snγVxn+βnxn+((1-βn)I-snμF)Tnvn,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive and sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L)). Assume that the following conditions hold:(i)
sn∈(0,1/2) for each λn∈(0,2/L), and limn→∞sn=0  (⇔limn→∞λn=2/L);(ii)
{βn}⊂(0,1) and 0<liminfn→∞βn≤limsupn→∞βn<1;(iii)
{λi,n}⊂[ai,bi]⊂(0,2ηi) and limn→∞|λi,n+1-λi,n|=0, for all i∈{1,2,…,N};(iv)
{rk,n}⊂[ek,fk]⊂(0,2μk) and limn→∞|rk,n+1-rk,n|=0, for all k∈{1,2,…,M}.
Then{xn} converges strongly as λn→2/L (⇔sn→0) to a point q∈Ω, which is a unique solution of VIP (50).Proof.
First of all, repeating the same arguments as in Theorem15, we can write
(117)PC(I-λn∇f)=2-λnL4I+2+λnL4Tn=snI+(1-sn)Tn,
where Tn is nonexpansive and sn:=sn(λn)=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L). It is clear that
(118)λn⟶2L⟺sn⟶0.
Put(119)Δnk=Trk,n(Θk,φk)(I-rk,nBk)Trk-1,n(Θk-1,φk-1)(I-rk-1,nBk-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn,
for all k∈{1,2,…,M} and n≥1,
(120)Λni=PC(I-λi,nAi)PC(I-λi-1,nAi-1)⋯PC(I-λ2,nA2)PC(I-λ1,nA1),
for all i∈{1,2,…,N} and n≥1, and Δn0=Λn0=I, where I is the identity mapping on H. Then we have that un=ΔnMxn and vn=ΛnNun. In addition, taking into consideration conditions (i) and (ii), we may assume, without loss of generality, that sn≤1-βn, for all n≥1.
We divide the remainder of the proof into several steps.
Step
1. Let us show that ∥xn-p∥≤max{∥x1-p∥,∥γVp-μFp∥/(τ-γl)}, for all n≥1 and p∈Ω. Indeed, take p∈Ω arbitrarily. Repeating the same arguments as those of (59)–(61) in the proof of Theorem 15, we obtain
(121)∥un-p∥≤∥xn-p∥,∥vn-p∥≤∥un-p∥,∥vn-p∥≤∥xn-p∥.
Taking into account conditions (i) and (ii), we may assume, without loss of generality, that sn≤1-βn, for all n≥1. Then from (121), Tnp=p, and Lemma 13, we have
(122)∥xn+1-p∥=∥sn(γVxn-μFp)+βn(xn-p)+((1-βn)I-snμF)Tnvn-((1-βn)I-snμF)Tnp∥≤sn∥γVxn-μFp∥+βn∥xn-p∥+(1-βn)×∥(I-sn1-βnμF)Tnvn-(I-sn1-βnμF)Tnp∥≤sn(∥γVxn-γVp∥+∥γVp-μFp∥)+βn∥xn-p∥+(1-βn)(1-snτ1-βn)∥vn-p∥≤sn(∥γVxn-γVp∥+∥γVp-μFp∥)+βn∥xn-p∥+(1-βn-snτ)∥xn-p∥≤snγl∥xn-p∥+sn∥γVp-μFp∥+(1-snτ)∥xn-p∥=(1-sn(τ-γl))∥xn-p∥+sn(τ-γl)∥γVp-μFp∥τ-γl≤max{∥xn-p∥,∥γVp-μFp∥τ-γl}.
By induction, we have
(123)∥xn-p∥≤max{∥x1-p∥,∥γVp-μFp∥τ-γl},≤max1{∥x1-p∥,∥γVp-μFp∥τ-γl}1∀n≥1.
Hence {xn} is bounded. According to (121), {un}, {vn}, {Tnvn}, {Vxn}, and {FTnvn} are also bounded.
Step
2. Let us show that ∥xn+1-xn∥→0 as n→∞. To this end, define
(124)xn+1=βnxn+(1-βn)zn,∀n≥1.
Observe that, from the definition of zn,
(125)zn+1-zn=xn+2-βn+1xn+11-βn+1-xn+1-βnxn1-βn=sn+1γVxn+1+((1-βn+1)I-sn+1μF)Tn+1vn+11-βn+1-snγVxn+((1-βn)I-snμF)Tnvn1-βn=sn+11-βn+1γVxn+1-sn1-βnγVxn+Tn+1vn+1-Tnvn+sn1-βnμFTnvn-sn+11-βn+1μFTn+1vn+1=sn+11-βn+1(γVxn+1-μFTn+1vn+1)+sn1-βn(μFTnvn-γVxn)+Tn+1vn+1-Tnvn.
Thus, it follows that
(126)∥zn+1-zn∥≤sn+11-βn+1(γ∥Vxn+1∥+μ∥FTn+1vn+1∥)+sn1-βn(μ∥FTnvn∥+γ∥Vxn∥)+∥Tn+1vn+1-Tnvn∥.
On the other hand, since ∇f is (1/L)-ism, PC(I-λn∇f) is nonexpansive for λn∈(0,2/L). So, it follows that, for any given p∈Ω,
(127)∥PC(I-λn+1∇f)vn∥≤∥PC(I-λn+1∇f)vn-p∥+∥p∥=∥PC(I-λn+1∇f)vn-PC(I-λn+1∇f)p∥+∥p∥≤∥vn-p∥+∥p∥≤∥vn∥+2∥p∥.
This together with the boundedness of {vn} implies that {PC(I-λn+1∇f)vn} is bounded. Also, observe that
(128)∥Tn+1vn-Tnvn∥=∥4PC(I-λn+1∇f)-(2-λn+1L)I2+λn+1Lvn-4PC(I-λn∇f)-(2-λnL)I2+λnLvn∥≤∥4PC(I-λn+1∇f)2+λn+1Lvn-4PC(I-λn∇f)2+λnLvn∥+∥2-λnL2+λnLvn-2-λn+1L2+λn+1Lvn∥=∥((2+λn+1L)(2+λnL))-1(4(2+λnL)PC(I-λn+1∇f)vn-4(2+λn+1L)PC(I-λn∇f)vn)×((2+λn+1L)(2+λnL))-1∥+4L|λn+1-λn|(2+λn+1L)(2+λnL)∥vn∥=∥×((2+λn+1L)(2+λnL))-1(×(PC(I-λn+1∇f)vn-PC(I-λn∇f)vn)4L(λn-λn+1)PC(I-λn+1∇f)vn+4(2+λn+1L)×(PC(I-λn+1∇f)vn-PC(I-λn∇f)vn))×((2+λn+1L)(2+λnL))-1∥+4L|λn+1-λn|(2+λn+1L)(2+λnL)∥vn∥≤4L|λn-λn+1|∥PC(I-λn+1∇f)vn∥(2+λn+1L)(2+λnL)+(((2+λn+1L)(2+λnL))-1(4(2+λn+1L)×∥PC(I-λn+1∇f)vn-PC(I-λn∇f)vn∥)×((2+λn+1L)(2+λnL))-1)+4L|λn+1-λn|(2+λn+1L)(2+λnL)∥vn∥≤|λn+1-λn|×[L∥PC(I-λn+1∇f)vn∥+4∥∇f(vn)∥+L∥vn∥]≤M~|λn+1-λn|,
where supn≥1{L∥PC(I-λn+1∇f)vn∥+4∥∇f(vn)∥+L∥vn∥}≤M~ for some M~>0. So, by (128), we have that
(129)∥Tn+1vn+1-Tnvn∥≤∥Tn+1vn+1-Tn+1vn∥+∥Tn+1vn-Tnvn∥≤∥vn+1-vn∥+M~|λn+1-λn|≤∥vn+1-vn∥+4M~L(sn+1+sn).
Note that
(130)∥vn+1-vn∥=∥Λn+1Nun+1-ΛnNun∥=∥PC(I-λN,n+1AN)Λn+1N-1un+1-PC(I-λN,nAN)ΛnN-1un∥≤∥PC(I-λN,n+1AN)Λn+1N-1un+1-PC(I-λN,nAN)Λn+1N-1un+1∥+∥PC(I-λN,nAN)Λn+1N-1un+1-PC(I-λN,nAN)ΛnN-1un∥≤∥(I-λN,n+1AN)Λn+1N-1un+1-(I-λN,nAN)Λn+1N-1un+1∥+∥(I-λN,nAN)Λn+1N-1un+1-(I-λN,nAN)ΛnN-1un∥≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥+∥Λn+1N-1un+1-ΛnN-1un∥≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥+|λN-1,n+1-λN-1,n|∥AN-1Λn+1N-2un+1∥+∥Λn+1N-2un+1-ΛnN-2un∥⋮≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥+|λN-1,n+1-λN-1,n|∥AN-1Λn+1N-2un+1∥+⋯+|λ1,n+1-λ1,n|∥A1Λn+10un+1∥+∥Λn+10un+1-Λn0un∥≤M~0∑i=1N‍|λi,n+1-λi,n|+∥un+1-un∥,
where supn≥1{∑i=1N‍∥AiΛn+1i-1un+1∥}≤M~0 for some M~0>0. Also, utilizing Proposition 1(ii), (v) we deduce that
(131)∥un+1-un∥=∥Δn+1Mxn+1-ΔnMxn∥=∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn∥≤∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1∥+∥TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1-TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn∥≤∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-TrM,n(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1∥+∥TrM,n(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1∥+∥(I-rM,nBM)Δn+1M-1xn+1-(I-rM,nBM)ΔnM-1xn∥≤|rM,n+1-rM,n|rM,n+1×∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-(I-rM,n+1BM)Δn+1M-1xn+1TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1∥+|rM,n+1-rM,n|∥BMΔn+1M-1xn+1∥+∥Δn+1M-1xn+1-ΔnM-1xn∥=|rM,n+1-rM,n|×[+1rM,n+1∥BMΔn+1M-1xn+1∥+1rM,n+1∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-(I-rM,n+1BM)Δn+1M-1xn+1TrM,n+1(ΘM,φM)∥1rM,n+1]+∥Δn+1M-1xn+1-ΔnM-1xn∥⋮≤|rM,n+1-rM,n|×[∥BMΔn+1M-1xn+1∥+1rM,n+1×∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-(I-rM,n+1BM)Δn+1M-1xn+1TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1∥+1rM,n+1]+⋯+|r1,n+1-r1,n|×[∥B1Δn+10xn+1∥+1r1,n+1×∥Tr1,n+1(Θ1,φ1)(I-r1,n+1B1)Δn+10xn+1-(I-r1,n+1B1)Δn+10xn+1Tr1,n+1(Θ1,φ1)(I-r1,n+1B1)Δn+10xn+1∥+1r1,n+1]+∥Δn+10xn+1-Δn0xn∥≤M~1∑k=1M‍|rk,n+1-rk,n|+∥xn+1-xn∥,
where M~1>0 is a constant such that, for each n≥1,
(132)∑k=1M‍[∥Trk,n+1(Θk,φk)(I-rk,n+1Bk)Δn+1k-1xn+1BkΔn+1k-1xn+1∥+1rk,n+1∥Trk,n+1(Θk,φk)(I-rk,n+1Bk)Δn+1k-1xn+1-(I-rk,n+1Bk)Δn+1k-1xn+1Trk,n+1(Θk,φk)(I-rk,n+1Bk)Δn+1k-1xn+1∥]≤M~1.
Combining (126)–(131), we get
(133)∥zn+1-zn∥-∥xn+1-xn∥≤sn+11-βn+1(γ∥Vxn+1∥+μ∥FTn+1vn+1∥)+sn1-βn(μ∥FTnvn∥+γ∥Vxn∥)+∥Tn+1vn+1-Tnvn∥-∥xn+1-xn∥≤sn+11-βn+1(γ∥Vxn+1∥+μ∥FTn+1vn+1∥)+sn1-βn(μ∥FTnvn∥+γ∥Vxn∥)+∥vn+1-vn∥+4M~L(sn+1+sn)-∥xn+1-xn∥≤sn+11-βn+1(γ∥Vxn+1∥+μ∥FTn+1vn+1∥)+sn1-βn(μ∥FTnvn∥+γ∥Vxn∥)+M~0∑i=1N‍|λi,n+1-λi,n|+∥un+1-un∥+4M~L(sn+1+sn)-∥xn+1-xn∥≤sn+11-βn+1(γ∥Vxn+1∥+μ∥FTn+1vn+1∥)+sn1-βn(μ∥FTnvn∥+γ∥Vxn∥)+M~0∑i=1N‍|λi,n+1-λi,n|+M~1∑k=1M‍|rk,n+1-rk,n|+∥xn+1-xn∥+4M~L(sn+1+sn)-∥xn+1-xn∥=sn+11-βn+1(γ∥Vxn+1∥+μ∥FTn+1vn+1∥)+sn1-βn(μ∥FTnvn∥+γ∥Vxn∥)+M~0∑i=1N‍|λi,n+1-λi,n|+M~1∑k=1M‍|rk,n+1-rk,n|+4M~L(sn+1+sn).
Thus, it follows from (133) and conditions (i)–(iv) that
(134)limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0.
Hence by Lemma 11 we have
(135)limn→∞∥zn-xn∥=0.
Consequently,
(136)limn→∞∥xn+1-xn∥=limn→∞(1-βn)∥zn-xn∥=0,
and by (129)–(131),
(137)limn→∞∥un+1-un∥=0,limn→∞∥vn+1-vn∥=0,limn→∞∥Tn+1vn+1-Tnvn∥=0.
Step
3. Let us show that ∥BkΔnk-1xn-Bkp∥→0 and ∥AiΛni-1un-Aip∥→0, for all k∈{1,2,…,M} and i∈{1,2,…,N}.
Indeed, since(138)xn+1=snγVxn+βnxn+((1-βn)I-snμF)Tnvn,
we have
(139)∥xn-Tnvn∥≤∥xn-xn+1∥+∥xn+1-Tnvn∥≤∥xn-xn+1∥+sn∥γVxn-μFTnvn∥+βn∥xn-Tnvn∥;
that is,
(140)∥xn-Tnvn∥≤11-βn∥xn-xn+1∥∥xn-Tnvn∥≤+sn1-βn(γ∥Vxn∥+μ∥FTnvn∥).
So, from sn→0, ∥xn+1-xn∥→0, and condition (ii), it follows that
(141)limn→∞∥xn-Tnvn∥=0.
Also, from (29) it follows that, for all i∈{1,2,…,N} and k∈{1,2,…,M},
(142)∥vn-p∥2=∥ΛnNun-p∥2≤∥Λniun-p∥2=∥PC(I-λi,nAi)Λni-1un-PC(I-λi,nAi)p∥2≤∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2≤∥Λni-1un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2≤∥un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2≤∥xn-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2,∥un-p∥2=∥ΔnMxn-p∥2≤∥Δnkxn-p∥2=∥Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nBk)p∥2≤∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2≤∥Δnk-1xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2≤∥xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2.
Furthermore, utilizing Lemma 7, we deduce from (116) that
(143)∥xn+1-p∥2=∥sn(γVxn-μFp)+βn(xn-Tnvn)+(I-snμF)Tnvn-(I-snμF)p∥2≤∥βn(xn-Tnvn)+(I-snμF)Tnvn-(I-snμF)p∥2+2sn〈γVxn-μFp,xn+1-p〉≤[βn∥xn-Tnvn∥+∥(I-snμF)Tnvn-(I-snμF)Tnp∥]2+2sn∥γVxn-μFp∥∥xn+1-p∥≤[βn∥xn-Tnvn∥+(1-snτ)∥vn-p∥]2+2sn∥γVxn-μFp∥∥xn+1-p∥=(1-snτ)2∥vn-p∥2+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥.
From (142)-(143), it follows that
(144)∥xn+1-p∥2≤∥vn-p∥2+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥≤∥un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥≤∥xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥,
and so
(145)rk,n(2μk-rk,n)∥BkΔnk-1xn-Bkp∥2+λi,n(2ηi-λi,n)∥AiΛni-1un-Aip∥2≤∥xn-p∥2-∥xn+1-p∥2+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥.
Since {λi,n}⊂[ai,bi]⊂(0,2ηi) and {rk,n}⊂[ek,fk]⊂(0,2μk), for all i∈{1,2,…,N} and k∈{1,2,…,M}, by (136), (141), and (145) we conclude immediately that
(146)limn→∞∥AiΛni-1un-Aip∥=0,limn→∞∥BkΔnk-1xn-Bkp∥=0,
for all i∈{1,2,…,N} and k∈{1,2,…,M}.
Step
4. Let us show that ∥xn-un∥→0, ∥un-vn∥→0, and ∥vn-Tnvn∥→0 as n→∞.
Indeed, by Proposition1(ii) we obtain that for each k∈{1,2,…,M}(147)∥Δnkxn-p∥2=∥Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nBk)p∥2≤〈(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p,Δnkxn-p〉=12(∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2+∥Δnkxn-p∥2-∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p-(Δnkxn-p)∥2)≤12(∥Δnk-1xn-p∥2+∥Δnkxn-p∥2-∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2),
which implies that
(148)∥Δnkxn-p∥2≤∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2=∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn∥2-rk,n2∥BkΔnk-1xn-Bkp∥2+2rk,n〈Δnk-1xn-Δnkxn,BkΔnk-1xn-Bkp〉≤∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥≤∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥.
Also, by Proposition 2(iii), we obtain that for each i∈{1,2,…,N}(149)∥Λniun-p∥2=∥PC(I-λi,nAi)Λni-1un-PC(I-λi,nAi)p∥2≤〈(I-λi,nAi)Λni-1un-(I-λi,nAi)p,Λniun-p〉=12(∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2+∥Λniun-p∥2-∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p-(Λniun-p)∥2)≤12(∥Λni-1un-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2)≤12(∥un-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2),
which implies
(150)∥Λniun-p∥2≤∥un-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2=∥un-p∥2-∥Λni-1un-Λniun∥2-λi,n2∥AiΛni-1un-Aip∥2+2λi,n〈Λni-1un-Λniun,AiΛni-1un-Aip〉≤∥un-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥.
Thus, from (143), (148), and (150), we have
(151)∥xn+1-p∥2≤(1-snτ)2∥vn-p∥2+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥=(1-snτ)2∥ΛnNun-p∥2+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥≤(1-snτ)2∥Λniun-p∥2+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥≤(1-snτ)2×[∥un-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥∥Λni-1un-Λniun∥2]+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥=(1-snτ)2×[∥ΔnMxn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥∥ΔnMxn-p∥2-∥Λni-1un-Λniun∥2]+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥≤(1-snτ)2×[∥Δnkxn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥∥Δnkxn-p∥2-∥Λni-1un-Λniun∥2]+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥≤(1-snτ)2×[∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2]+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥≤∥xn-p∥2-(1-snτ)2∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥-(1-snτ)2∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥;
that is,
(152)(1-snτ)2[∥Δnk-1xn-Δnkxn∥2+∥Λni-1un-Λniun∥2]≤∥xn-p∥2-∥xn+1-p∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+βn2∥xn-Tnvn∥2+2(1-snτ)βn∥vn-p∥∥xn-Tnvn∥+2sn∥γVxn-μFp∥∥xn+1-p∥.
So, from sn→0, (136), (141), and (146) we immediately get
(153)limn→∞∥Λni-1un-Λniun∥,limn→∞∥Δnk-1xn-Δnkxn∥=0,
for all i∈{1,2,…,N} and k∈{1,2,…,M}. Note that
(154)∥xn-un∥=∥Δn0xn-ΔnMxn∥≤∥Δn0xn-Δn1xn∥+∥Δn1xn-Δn2xn∥+⋯+∥ΔnM-1xn-ΔnMxn∥,∥un-vn∥=∥Λn0un-ΛnNun∥≤∥Λn0un-Λn1un∥+∥Λn1un-Λn2un∥+⋯+∥ΛnN-1un-ΛnNun∥.
Thus, from (153) we have
(155)limn→∞∥xn-un∥=0,limn→∞∥un-vn∥=0.
It is easy to see that as n→∞(156)∥xn-vn∥≤∥xn-un∥+∥un-vn∥⟶0.
Also, observe that
(157)∥Tnvn-vn∥≤∥Tnvn-xn∥+∥xn-vn∥.
Hence we have from (141)
(158)limn→∞∥Tnvn-vn∥=0.
Step
5. Let us show that limsupn→∞〈(μF-γV)q,q-xn〉≤0, where q∈Ω is the same as in Theorem 15; that is, q∈Ω is a unique solution of VIP (50). To show this inequality, we choose a subsequence {xni} of {xn} such that
(159)limsupn→∞〈(μF-γV)q,q-xn〉=limi→∞〈(μF-γV)q,q-xni〉.
Since {xn} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly to w. Without loss of generality, we may assume that xni⇀w. From Step 4, we have that Δnikxni⇀w, Λnimuni⇀w, uni⇀w, and vni⇀w, where k∈{1,2,…,M} and m∈{1,2,…,N}. Since vn-Tnvn→0 by Step 4, by the same arguments as in the proof of Theorem 15, we get w∈Ω. Since q=PΩ(I-μF+γV)q, it follows that
(160)limsupn→∞〈(μF-γV)q,q-xn〉=limi→∞〈(μF-γV)q,q-xni〉=〈(μF-γV)q,q-w〉≤0.
Step
6. Let us show that limn→∞∥xn-q∥=0, where q∈Ω is the same as in Theorem 15; that is, q∈Ω is a unique solution of VIP (50). From (116), we know that
(161)xn+1-q=sn(γVxn-μFq)+βn(Tnvn-q)+((1-βn)I-snμF)Tnvn-((1-βn)I-snμF)q.
Applying Lemmas 7 and 13 and noticing that Tnq=q and ∥vn-q∥≤∥xn-q∥, for all n≥1, we have
(162)∥xn+1-q∥2≤∥βn(Tnvn-q)+((1-βn)I-snμF)Tnvn-((1-βn)I-snμF)q∥2+2sn〈γVxn-μFq,xn+1-q〉≤[βn∥Tnvn-q∥+∥((1-βn)I-snμF)Tnvn-((1-βn)I-snμF)q∥]2+2sn〈γVxn-γVq,xn+1-q〉+2sn〈γVq-μFq,xn+1-q〉=[(I-sn1-βnμF)βn∥Tnvn-q∥+(1-βn)×∥(I-sn1-βnμF)Tnvn-(I-sn1-βnμF)Tnq∥]2+2sn〈γVxn-γVq,xn+1-q〉+2sn〈γVq-μFq,xn+1-q〉≤[βn∥vn-q∥+(1-βn)(1-snτ1-βn)∥vn-q∥]2+2sn〈γVxn-γVq,xn+1-q〉+2sn〈γVq-μFq,xn+1-q〉=[βn∥vn-q∥+(1-βn-snτ)∥vn-q∥]2+2sn〈γVxn-γVq,xn+1-q〉+2sn〈γVq-μFq,xn+1-q〉≤[βn∥xn-q∥+(1-βn-snτ)∥xn-q∥]2+2sn〈γVxn-γVq,xn+1-q〉+2sn〈γVq-μFq,xn+1-q〉≤(1-snτ)2∥xn-q∥2+2snγl∥xn-q∥∥xn+1-q∥+2sn〈γVq-μFq,xn+1-q〉≤(1-τsn)2∥xn-q∥2+snγl(∥xn-q∥2+∥xn+1-q∥2)+2sn〈γVq-μFq,xn+1-q〉.
This implies that
(163)∥xn+1-q∥2≤1-2τsn+τ2sn2+snγl1-snγl∥xn-q∥2+2sn1-snγl〈γVq-μFq,xn+1-q〉=(1-2(τ-γl)sn1-snγl)∥xn-q∥2+τ2sn21-snγl∥xn-q∥2+2sn1-snγl〈γVq-μFq,xn+1-q〉≤(1-2(τ-γl)1-snγlsn)∥xn-q∥2+2(τ-γl)sn1-snγl×(τ2sn2(τ-γl)M~2+1τ-γl〈μFq-γVq,q-xn+1〉)=(1-σn)∥xn-q∥2+σnδn,
where M~2=supn≥1∥xn-q∥2, σn=(2(τ-γl)/(1-snγl))sn, and
(164)δn=τ2sn2(τ-γl)M~2+1τ-γl〈μFq-γVq,q-xn+1〉.
From condition (i) and Step 5, it is easy to see that σn→0,∑n=0∞‍σn=∞ and limsupn→∞δn≤0. Hence, by Lemma 10, we conclude that xn→q as n→∞. This completes the proof.Corollary 19.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θ be a bifunction from C×C to R satisfying (A1)–(A4) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Let B:H→H and Ai:C→H be ζ-inverse strongly monotone and ηi-inverse strongly monotone, respectively, for i=1,2. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let V:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP(Θ,φ,B)∩VI(C,A1)∩VI(C,A2)∩Γ≠∅ and that either (B1) or (B2) holds. For arbitrarily given x1∈H, let {xn} be a sequence generated by
(165)Θ(un,y)+φ(y)-φ(un)+〈Bxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,vn=PC(I-λ2,nA2)PC(I-λ1,nA1)un,xn+1=snγVxn+βnxn+((1-βn)I-snμF)Tnvn,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive and sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L)). Assume that the following conditions hold:(i)
sn∈(0,1/2) for each λn∈(0,2/L), and limn→∞sn=0  (⇔limn→∞λn=2/L);(ii)
{βn}⊂(0,1) and 0<liminfn→∞βn≤limsupn→∞βn<1;(iii)
{λi,n}⊂[ai,bi]⊂(0,2ηi) and limn→∞|λi,n+1-λi,n|=0 for i=1,2;(iv)
{rn}⊂[e,f]⊂(0,2ζ) and limn→∞|rn+1-rn|=0.
Then{xn} converges strongly as λn→2/L(⇔sn→0) to a point q∈Ω, which is a unique solution of VIP (113).Corollary 20.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θ be a bifunction from C×C to R satisfying (A1)–(A4) and let φ:C→R∪{+∞} be a proper lower semicontinuous and convex function. Let B:H→H and A:C→H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let V:H→H be an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/κ2 and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that Ω:=GMEP(Θ,φ,B)∩VI(C,A)∩Γ≠∅ and that either (B1) or (B2) holds. For arbitrarily given x1∈H, let {xn} be a sequence generated by
(166)Θ(un,y)+φ(y)-φ(un)+〈Bxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,vn=PC(I-ρnA)un,xn+1=snγVxn+βnxn+((1-βn)I-snμF)Tnvn,∀n≥1,
where PC(I-λn∇f)=snI+(1-sn)Tn (here Tn is nonexpansive and sn=(2-λnL)/4∈(0,1/2) for each λn∈(0,2/L)). Assume that the following conditions hold:(i)
sn∈(0,1/2) for each λn∈(0,2/L), and limn→∞sn=0  (⇔limn→∞λn=2/L);(ii)
{βn}⊂(0,1) and 0<liminfn→∞βn≤limsupn→∞βn<1;(iii)
{ρn}⊂[a,b]⊂(0,2ξ) and limn→∞|ρn+1-ρn|=0;(iv)
{rn}⊂[e,f]⊂(0,2ζ) and limn→∞|rn+1-rn|=0.
Then{xn} converges strongly as λn→2/L(⇔sn→0) to a point q∈Ω, which is a unique solution of VIP (115).



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*Source: 587865-2014-01-08.xml*