# A New System of Generalized Mixed Quasivariational Inclusions with Relaxed Cocoercive Operators and Applications

Authors: Zhongping Wan; Jia-Wei Chen; Hai Sun; Liuyang Yuan
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

A new system of generalized mixed quasivariational inclusions
(for short, SGMQVI) with relaxed cocoercive operators, which develop some preexisting
variational inequalities, is introduced and investigated in Banach spaces. Next, the existence and uniqueness of solutions to the problem (SGMQVI) are established in real Banach
spaces. From fixed point perspective, we propose some new iterative algorithms for solving
the system of generalized mixed quasivariational inclusion problem (SGMQVI). Moreover,
strong convergence theorems of these iterative sequences generated by the corresponding
algorithms are proved under suitable conditions. As an application, the strong convergence
theorem for a class of bilevel variational inequalities is derived in Hilbert space. The main
results in this paper develop, improve, and unify some well-known results in the literature.


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## Body

## 1. Introduction

Generalized mixed quasivariational inclusion problems, which are extensions of variational inequalities introduced by Stampacchia [1] in the early sixties, are among the most interesting and extensively investigated classes of mathematics problems and have many applications in the fields of optimization and control, abstract economics, electrical networks, game theory, auction, engineering science, and transportation equilibria (see, e.g., [2–5] and the references therein). For the past few decades, existence results and iterative algorithms for variational inequality and variational inclusion problems have been obtained (see, e.g., [6–14] and the references cited therein). Recently, some new problems, which are called to be the system of variational inequality and equilibrium problems, received many attentions. Ansari et al. [2] considered a system of vector variational inequalities and obtained its existence results. In [3], Pang stated that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled as a system of variational inequalities. Verma [15] and J. K. Kim and D. S. Kim [16] investigated a system of nonlinear variational inequalities. Cho et al. [17] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces and obtained the existence and uniqueness properties of solutions for the system of nonlinear variational inequalities. In [18], Peng and Zhu introduced a new system of generalized mixed quasivariational inclusions involving (H,η)-monotone operators. Very recently, Qin et al. [19] studied the approximation of solutions to a system of variational inclusions in Banach spaces and established a strong convergence theorem in uniformly convex and 2 uniformly smooth Banach spaces. In [20], Kamraksa and Wangkeeree gave a general iterative method for a general system of variational inclusions and proved a strong convergence theorem in strictly convex and 2 uniformly smooth Banach spaces. Further, Wangkeeree and Kamraksa [21] introduced an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general system of variational inequalities and then obtained the strong convergence of the iterative in Hilbert spaces. Petrot [22] applied the resolvent operator technique to find the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point problems for Lipschitz mappings in Hilbert spaces. Zhao et al. [23] obtained some existence results for a system of variational inequalities by Brouwer’s fixed point theory and proved the convergence of an iterative algorithm in finite Euclidean spaces. Chen and Wan [24] also proved the existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces, proposed some iterative methods for finding the common element of the solutions set for the system of quasivariational inclusions and the fixed point set for Lipschitz mapping, and obtained the convergent rates of corresponding iterative sequences. On the other hand, various bilevel programming problems, bilevel decision problems, and mathematical program problems with equilibrium (variational inequalities) constraints have been wildly investigated (see, e.g., [25, 26]). To the best of our knowledge, there is few results concerning the algorithms and convergence analysis of solutions to bilevel variational inequalities in Hilbert spaces.The aim of this paper is to introduce and study a new system of generalized mixed quasivariational inclusion problem (SGMQVI) in uniformly smooth Banach spaces which includes some previous variational inequalities as special cases. Furthermore, the existence and uniqueness theorems of solutions for the problem (SGMQVI) are established by using resolvent techniques. Thirdly, we also propose some new iterative algorithms for solving the problem (SGMQVI). Strong convergence of the iterative sequences generated by the corresponding iterative algorithms are proved under suitable conditions. As an application, we study the properties for the lower-level variational inequalities of a class of bilevel variational inequalities (for short, (BVI)) in Hilbert spaces and then suggest a reasonable iterative algorithm for (BVI). Finally, the strong convergence theorem for (BVI) are derived under appropriate assumptions. The results presented in this paper improve, develop, and extend the results of [8, 23, 24, 27].


## 2. Preliminaries

Throughout this paper, letE be a real q-uniformly Banach space with its dual E*,q>1, denote the duality between E and E* by 〈·,·〉 and the norm of E by ∥·∥, and let T:E→E be a nonlinear mapping. If {xn} is a sequence in E, we denote strong convergence of {xn} to x∈E by xn→x. A Banach space E is called smooth if limt→0(∥x+ty∥-∥x∥)/t exists for all x,y∈E with ∥x∥=∥y∥=1. It is called uniformly smooth if the limit is attained uniformly for ∥x∥=∥y∥=1. The function (2.1)ρE(t)=sup⁡{‖x+y‖+‖x-y‖2-1:‖x‖=1,‖y‖≤t}
is called the modulus of smoothness of E. E is called q-uniformly smooth if there exists a constant cq>0 such that ρE(t)≤cqtq.Example 2.1 (see [4]).
All Hilbert spaces,Lp(orlp), and the Sobolev spaces Wmp, (p≥2) are 2-uniformly smooth, while Lp(orlp) and Wmp spaces (1<p≤2) are p-uniformly smooth.
The generalized duality mappingJq:E→2E* defined as follows:
(2.2)Jq(x)={f*∈E*:〈f*,x〉=‖f*‖‖x‖=‖x‖q,‖f*‖=‖x‖q-1},
for all x∈E. As we know that J=J2 is the usual normalized duality mapping, and Jq(x)=∥x∥q-2J(x) for x≠0, Jq(tx)=tq-1Jq(x), and Jq(-x)=-Jq(x) for all x∈E and t∈[0,+∞), and Jq is single-valued if E is smooth (see, e.g., [28]). If E is a Hilbert space, then J=I, where I is the identity operator. Let gj:E→E, let Aj:E×E→E be single-valued mappings, and let Mj:E→2E be set-valued mappings for all j∈{1,2,…,n}. We consider the system of generalized mixed quasivariational inclusions problem (for short, (SGMQVI)) as follows: find (x1*,x2*,…,xn-1*,xn*)∈En such that
(2.3)0∈x1*-g1(x2*)+ρ1(A1(x2*,x1*)+M1(x1*)),0∈x2*-g2(x3*)+ρ2(A2(x3*,x2*)+M2(x2*)),⋮0∈xn-1*-gn-1(xn*)+ρn-1(An-1(xn*,xn-1*)+Mn-1(xn-1*)),0∈xn*-gn(x1*)+ρn(An(x1*,xn*)+Mn(xn*)),
where ρi(i=1,2,…,n) are positive constants. Denote the set of solutions to (SGMQVI) by Ξ.
Special cases are as follows:
(I) Ifgj(x)=x and Aj(x,y)=Tj(x)+Sj(x) for all x,y∈E and j=1,2,…,n, where Tj,Sj:E→E are single-valued mappings, then the problem (SGMQVI) is equivalent to find (x1*,x2*,…,xn-1*,xn*)∈En such that
(2.4)0∈x1*-x2*+ρ1(T1(x2*)+S1(x2*)+M1(x1*)),0∈x2*-x3*+ρ2(T2(x3*)+S2(x3*)+M2(x2*)),⋮0∈xn-1*-xn*+ρn-1(Tn-1(xn*)+Sn-1(xn*)+Mn-1(xn-1*)),0∈xn*-x1*+ρn(Tn(x1*)+Sn(x1*)+Mn(xn*)),
where ρi(i=1,2,…,n) are positive constants, which is called the system of generalized nonlinear mixed variational inclusions problem [8].
(II) Ifn=2,A1=A2=A,E=H is a Hilbert space, g1(x)=g2(x)=x and M1(x)=M2(x)=∂ϕ(x) for all x∈E, where ϕ:E→R∪{+∞} is a proper, convex, and lower semicontinuous functional, and ∂ϕ denotes the subdifferential operator of ϕ, then the problem (SGMQVI) is equivalent to find (x*,y*)∈E×E such that
(2.5)〈ρ1A(y*,x*)+x*-y*,x-x*〉+ϕ(x)-ϕ(x*)≥0,∀x∈E,〈ρ2A(x*,y*)+y*-x*,x-y*〉+ϕ(x)-ϕ(y*)≥0,∀x∈E,
where ρi(i=1,2) are positive constants, which is called the generalized system of relaxed cocoercive mixed variational inequality problem [29].
(III) Ifn=2,E=H is a Hilbert space, and K is a closed convex subset of E, and ϕ(x)=δK(x) for all x∈K, where δK is the indicator function of K defined by
(2.6)ϕ(x)=δK(x)={0,ifx∈K,+∞,otherwise,
then the problem (SGMQVI) is equivalent to find (x*,y*)∈K×K such that
(2.7)〈ρ1A1(y*,x*)+x*-g1(y*),g1(x)-x*〉≥0,∀x∈K,〈ρ2A2(x*,y*)+y*-g2(x*),g2(x)-y*〉≥0,∀x∈K,
where ρi(i=1,2) are positive constants, which is called the system of general variational inequalities problem [27].
(IV) Ifn=2,A1=A2=A,E=H is a Hilbert space, and K is a closed convex subset of E, g1(y)=g2(y)=y, and ϕ(x)=δK(x) for all x∈K,y∈E, where δK is the indicator function of K defined by
(2.8)ϕ(x)=δK(x)={0,ifx∈K,+∞,otherwise,
then the problem (SGMQVI) is equivalent to find (x*,y*)∈K×K such that
(2.9)〈ρ1A(y*,x*)+x*-y*,x-x*〉≥0,∀x∈K,〈ρ2A(x*,y*)+y*-x*,x-y*〉≥0,∀x∈K,
where ρi(i=1,2) are positive constants, which is called the generalized system of relaxed cocoercive variational inequality problem [30].
(V) If for eachi∈{1,2},z∈E,Ai(x,z)=Ψi(x), and gi(x)=x for all x∈E, where Ψi:E→E, then the problem (SGMQVI) is equivalent to find (x*,y*)∈E×E such that
(2.10)0∈x*-y*+ρ1(Ψ1(y*)+M1(x*)),0∈y*-x*+ρ2(Ψ2(x*)+M2(y*)),
where ρi  (i=1,2) are positive constants, which is called the system of quasivariational inclusion [19, 20].
(VI) Ifn=2, for each i∈{1,2},z∈E,Ai(x,z)=Ψ(x), and gi(x)=x for all x∈E, where Ψ:E→E and M1(x)=M2(x)=M, then the problem (SGMQVI) is equivalent to find (x*,y*)∈E×E such that
(2.11)0∈x*-y*+ρ1(Ψ(y*)+M(x*)),0∈y*-x*+ρ2(Ψ(x*)+M(y*)),
where ρi(i=1,2) are positive constants, which is called the system of quasivariational inclusion [20].
We first recall some definitions and lemmas which are needed in our main results.Definition 2.2.
LetM:dom(M)⊂E→2E be a set-valued mapping, where dom(M) is the effective domain of the mapping M. M is said to be (i)
accretive if, for anyx,y∈dom(M),u∈M(x) and v∈M(y), there exists jq(x-y)∈Jq(x-y) such that
(2.12)〈u-v,jq(x-y)〉≥0,(ii)
m-accretive (maximal-accretive) if M is accretive and (I+ρM)dom(M)=E holds for every ρ>0, where I is the identity operator on E.Remark 2.3.
IfE is a Hilbert space, then accretive operator and m-accretive operator are reduced to monotone operator and maximal monotone operator, respectively.Definition 2.4 (see [24, 31]).
LetT:E→E be a single-valued mapping. T is said to be(i)
γ-Lipschitz continuous mapping if there exists a constant γ>0 such that
(2.13)‖Tx-Ty‖≤γ‖x-y‖,∀x,y∈E,(ii)
(a,b)-relaxed cocoercive if there exist two constants a≥0 and b>0 such that(2.14)〈T(x)-T(x̃),Jq(x-x̃)〉≥(-a)‖T(x)-T(x̃)‖q+b‖x-x̃‖q,∀x,x̃∈E.Remark 2.5 (see [24]).
(1) Ifγ=1, then a γ-Lipschitz continuous mapping reduces to a nonexpansive mapping.
(2) Ifγ∈(0,1), then a γ-Lipschitz continuous mapping reduces to a contractive mapping.
(3) It is easy to see that the identity operatorI:E→E is (0,1) relaxed cocoercive, where I(x)=x for all x∈E.Definition 2.6 (see [24]).
LetA:E×E→E be a mapping. A is said to be (i)
τ-Lipschitz continuous in the first variable if there exists a constant τ>0 such that, for x,x̃∈E,
(2.15)‖A(x,y)-A(x̃,ỹ)‖≤τ‖x-x̃‖,∀y,ỹ∈E(ii)
α-strongly accretive if there exists a constant α>0 such that
(2.16)〈A(x,y)-A(x̃,ỹ),Jq(x-x̃)〉≥α‖x-x̃‖q,∀(x,y),(x̃,ỹ)∈E×E,
or equivalently,
(2.17)〈A(x,y)-A(x̃,ỹ),J(x-x̃)〉≥α‖x-x̃‖,∀(x,y),(x̃,ỹ)∈E×E,(iii)
(μ,ν) relaxed cocoercive if there exist two constants μ≥0 and ν>0 such that(2.18)〈A(x,y)-A(x̃,ỹ),Jq(x-x̃)〉≥(-μ)‖A(x,y)-A(x̃,ỹ)‖q+ν‖x-x̃‖q,∀(x,y),(x̃,ỹ)∈E×E.Remark 2.7.
(1) Everyα-strongly accretive mapping is a (μ,α) relaxed cocoercive for any positive constant μ. But the converse is not true in general.
(2) The conception of the cocoercivity is applied in several directions, especially for solving variational inequality problems by using the auxiliary problem principle and projection methods [14]. Several classes of relaxed cocoercive variational inequalities have been investigated in [5, 22, 28, 30].Definition 2.8 (see [9, 32]).
Let the set-valued mappingM:dom(M)⊂E→2E be m-accretive. For any positive number ρ>0, the mapping RρM:E→dom(M) defined by
(2.19)RρM(x)=(I+ρM)-1(x),x∈E,
is called the resolvent operator associated with M and ρ, where I is the identity operator on E.Remark 2.9.
LetC⊂E be a nonempty closed convex set. If E is a Hilbert space and M=∂ϕ, the subdifferential of the indicator function ϕ, that is,
(2.20)ϕ(x)=δC(x)={0,ifx∈C,+∞,otherwise,
then R(ρ,M)=PC, the metric projection operator from E onto C.Lemma 2.10 (see [9, 32]).
Let the set-valued mappingM:dom(M)⊂E→2E be m-accretive. Then the resolvent operator RρM is single-valued and nonexpansive for all ρ>0.Lemma 2.11 (see [33]).
Let{Bn},{Cn}, and {Dn} be three nonnegative real sequences satisfying the following conditions:
(2.21)Bn+1≤(1-λn)Bn+Cn+Dn,∀n≥n0,
for some n0∈N,{λn}⊂(0,1) with ∑n=0∞λn=∞, Cn=0(λn) and ∑n=0∞Dn<+∞. Then limn→∞Bn=0.Lemma 2.12 (see [34]).
LetE be a real q-uniformly Banach space. Then there exists a constant cq>0 such that
(2.22)‖x+y‖q≤‖x‖q+q〈y,Jq(x)〉+cq‖y‖q,∀x,y∈E.


## 3. Existence Theorems

In this section, we will investigate the existence and uniqueness of solutions for the problem (SGMQVI) inq-uniformly smooth Banach space under some suitable conditions.Theorem 3.1.
Let(x1*,x2*,…,xn-1*,xn*)∈En, Mi:E→2E(i=1,2,…,n) be maximal accretive. Then (x1*,x2*,…,xn-1*,xn*) is a solution of the problem (SGMQVI) if and only if
(3.1)x1*=Rρ1M1(g1(x2*)-ρ1A1(x2*,x1*)),x2*=Rρ2M2(g2(x3*)-ρ2A2(x3*,x2*)),⋮xn-1*=Rρn-1Mn-1(gn-1(xn*)-ρn-1An-1(xn*,xn-1*)),xn*=RρnMn(gn(x1*)-ρnAn(x1*,xn*)),
where ρi(i=1,2,…,n) are positive constants.Proof.
Let(x1*,x2*,…,xn-1*,xn*)∈En be a solution of the problem (SGMQVI). Then, for any given positive constants ρi(i=1,2,…,n), the problem (SGMQVI) is equivalent to
(3.2)g1(x2*)-ρ1A1(x2*,x1*)∈x1*+ρ1M1(x1*),g2(x3*)-ρ2A2(x3*,x2*)∈x2*+ρ2M2(x2*),⋮gn-1(xn*)-ρn-1An-1(xn*,xn-1*)∈xn-1*+ρn-1Mn-1(xn-1*),gn(x1*)-ρnAn(x1*,xn*)∈xn*+ρnMn(xn*).
From Definition 2.8 and Lemma 2.10, it yields that (3.2) is equivalent to (3.1). This completes the proof.Theorem 3.2.
LetE be a real q-uniformly smooth Banach space. Let j∈{1,2,…,n}, Mj:E→2E be m-accretive mapping, Let Aj:E×E→E be (μj,νj)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τj, and Let gj:E→E be (aj,bj)-relaxed cocoercive and Lipschitz continuous with constant ιj. Then, for each j∈{1,2,…,n},x∈E, the mapping RρjMj(gj(x)-ρjAj(x,·)):E→E has at most one fixed point. If
(3.3)min⁡{1+qajιjq+cqιjq-qbj,1+qρjμjτjq+cqρjqτjq-qρjνj}≥0,
then the implicit function yj(x) determined by
(3.4)yj(x)=RρjMj(gj(x)-ρjAj(x,yj(x)))
is continuous on E.Proof.
Letx∈E. We show by contradiction that RρjMj(gj(x)-ρjAj(x,·)):E→E has at most one fixed point. Suppose to the contrary that y,ỹ∈E and y≠ỹ such that
(3.5)y=RρjMj(gj(x)-ρjAj(x,y)),ỹ=RρjMj(gj(x)-ρjAj(x,ỹ)).
Since Aj is Lipschitz continuous in the first variable with constant τj, then
(3.6)‖y-ỹ‖=‖RρjMj(gj(x)-ρjAj(x,y))-RρjMj(gj(x)-ρjAj(x,ỹ))‖≤‖gj(x)-ρjAj(x,y)-(gj(x)-ρjAj(x,ỹ))‖=ρj‖Aj(x,y)-Aj(x,ỹ)‖≤ρjτj‖x-x‖=0,
which is a contradiction. Therefore,  the mapping RρjMj(gj(x)-ρjAj(x,·)):E→E has at most one fixed point.
Next, we show that the implicit functionyj(x) is continuous on E. For any sequence, {xn}⊂E,x0∈E,xn→x0 as n→∞. Since Aj:E×E→E is (μj,νj)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τj and gj:E→E is (aj,bj)-relaxed cocoercive and Lipschitz continuous with constant ιj, one has
(3.7)L̃Aj=‖xn-x0-ρj[Aj(xn,y(xn))-Aj(x0,y(x0))]‖q≤‖xn-x0‖q-qρj〈Aj(xn,y(xn))-Aj(x0,y(x0)),Jq(xn-x0)〉+cqρjq‖Aj(xn,y(xn))-Aj(x0,y(x0))‖q≤qρj(μj‖Aj(xn,y(xn))-Aj(x0,y(x0))‖q-νj‖xn-x0‖q)+(1+cqρjqτjq)‖xn-x0‖q≤qρj(μjτjq-νj)‖xn-x0‖q+(1+cqρjqτjq)‖xn-x0‖q=(1+qρjμjτjq+cqρjqτjq-qρjνj)‖xn-x0‖q,L̃gj=‖xn-x0-(gj(xn)-gj(x0))‖q≤‖xn-x0‖q-q〈gj(xn)-gj(x0),Jq(xn-x0)〉+cq‖gj(xn)-gj(x0)‖q≤(1+cqιjq)‖xn-x0‖q+q(aj‖gj(xn)-gj(x0)‖q-bj‖xn-x0‖q)≤(1+cqιjq)‖xn-x0‖q+q(ajιjq-bj)‖xn-x0‖q=(1+qajιjq+cqιjq-qbj)‖xn-x0‖q.
Therefore, from Lemma 2.10, we get
(3.8)‖yj(xn)-yj(x0)‖=‖RρjMj(gj(xn)-ρjAj(xn,yj(xn)))-RρjMj(gj(x0)-ρjAj(x0,yj(x0)))‖≤‖gj(xn)-ρjAj(xn,yj(xn))-(gj(x0)-ρjAj(x0,yj(x0)))‖=‖(gj(xn)-gj(x0))-ρj(Aj(xn,yj(xn))-Aj(x0,yj(x0)))‖≤‖xn-x0-(gj(xn)-gj(x0))‖+‖xn-x0-ρj(Aj(xn,yj(xn))-Aj(x0,yj(x0)))‖≤L̃Ajq+L̃gjq≤(1+qρjμjτjq+cqρjqτjq-qρjνjq+1+qajιjq+cqιjq-qbjq)‖xn-x0‖.
From (3.3), it follows that the implicit function yj(x) is continuous on E. This completes the proof.Ifj=2 and g(x)=x for all x∈E, then Theorem 3.2 is reduced to the following result.Corollary 3.3 (see [24]).
LetE be a real q-uniformly smooth Banach space. Let M2:E→2E be m-accretive mapping; Let A2:E×E→E be (μ2,ν2)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ2. Then, for each x∈E, the mapping Rρ2M2(x-ρ2A2(x,·)):E→E has at most one fixed point. If
(3.9)1+qρ2μ2τ2q+cqρ2qτ2q-qρ2ν2≥0,
then the implicit function y(x) determined by
(3.10)y(x)=Rρ2M2(x-ρ2A2(x,y2(x)))
is continuous on E.Theorem 3.4.
LetE be a real q-uniformly smooth Banach space. Let Mj:E→2E be m-accretive mapping, Let Aj:E×E→E be (μj,νj)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τj, and let gj:E→E be (aj,bj)-relaxed cocoercive and Lipschitz continuous with constant ιj for j∈{1,2,…,n}. Assume that
(3.11)min⁡{1+qajιjq+cqιjq-qbj,1+qρjμjτjq+cqρjqτjq-qρjνj}≥0,j=1,2,…,n,(3.12)0≤∏j=1n(1+qajιjq+cqιjq-qbjq+1+qρjμjτjq+cqρjqτjq-qρjνjq)<1.
Then the problem (SGMQVI) has a solution. Moreover, the solutions set Ξ of (SGMQVI) is a singleton.Proof.
By Theorem3.1, the problem (SGMQVI) has a solution if and only if (3.1) holds. For the convenience, we define a mapping F:E→E by
(3.13)F(x)=Rρ1M1(g1(y2(x))-ρ1A1(y2(x),x)),y2(x)=Rρ2M2(g2(y3(x))-ρ2A2(y3(x),y2(x))),⋮yn-1(x)=Rρn-1Mn-1(gn-1(yn(x))-ρn-1An-1(yn(x),yn-1(x))),yn(x)=RρnMn(gn(x)-ρnAn(x,yn(x))),x∈E.
Since Aj:E×E→E are (μj,νj)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τj and gj:E→E are (aj,bj)-relaxed cocoercive and Lipschitz continuous with constant ιj for j∈{1,2,…,n}, by Theorem 3.2, we know that F(x) and yi(x)(i=2,3,…,n) are continuous on E. For any x,z∈E,
(3.14)LAj=‖yj+1(x)-yj+1(z)-ρj[Aj(yj+1(x),yj(x))-Aj(yj+1(z),yj(z))]‖q≤-qρj〈Aj(yj+1(x),yj(x))-Aj(yj+1(z),yj(z)),Jq(yj+1(x)-yj+1(z))〉+‖yj+1(x)-yj+1(z)‖q+cqρjq‖Aj(yj+1(x),yj(x))-Aj(yj+1(z),yj(z))‖q≤qρj(μj‖Aj(yj+1(x),yj(x))-Aj(yj+1(z),yj(z))‖q-νj‖yj+1(x)-yj+1(z)‖q)+(1+cqρjqτjq)‖yj+1(x)-yj+1(z)‖q≤(1+qρjμjτjq+cqρjqτjq-qρjνj)‖yj+1(x)-yj+1(z)‖q,j=2,3,…,n-1,Lgj=‖yj+1(x)-yj+1(z)-(gj(yj+1(x))-gj(yj+1(z)))‖q≤‖yj+1(x)-yj+1(z)‖q-q〈gj(yj+1(x))-gj(yj+1(z)),Jq(yj+1(x)-yj+1(z))〉+cq‖gj(yj+1(x))-gj(yj+1(z))‖q≤(1+qajιjq+cqιjq-qbj)‖yj+1(x)-yj+1(z)‖q,j=2,3,…,n-1,LAn=‖x-z-ρn[An(x,yn(x))-An(z,yn(z))]‖q≤‖x-z‖q-qρn〈An(x,yn(x))-An(z,yn(z)),Jq(x-z)〉+cqρnq‖An(x,yn(x))-An(z,yn(z))‖q≤(1+qρnμnτnq+cqρnqτnq-qρnνn)‖x-z‖q,Lgn=‖x-z-(gn(x)-gn(z))‖q≤‖x-z‖q-q〈gn(x)-gn(z),Jq(x-z)〉+cq‖gn(x)-gn(z)‖q≤(1+qanιnq+cqιnq-qbn)‖x-z‖q,LA1=‖y2(x)-y2(z)-ρ1[A1(y2(x),x)-A1(y2(z),z)]‖q≤(1+qρ1μ1τ1q+cqρ1qτ1q-qρ1ν1)‖y2(x)-y2(z)‖q,Lg1=‖y2(x)-y2(z)-(g1(y2(x))-g1(y2(z)))‖q≤(1+qa1ι1q+cqι1q-qb1)‖y2(x)-y2(z)‖q.
From Lemma 2.10, it yields that
(3.15)‖F(x)-F(z)‖=‖Rρ1M1(g1(y2(x))-ρ1A1(y2(x),x))-Rρ1M1(g1(y2(z))-ρ1A1(y2(z),z))‖≤‖(g1(y2(x))-ρ1A1(y2(x),x))-(g1(y2(z))-ρ1A1(y2(z),z))‖=‖(g1(y2(x))-g1(y2(z)))-ρ1(A1(y2(x),x)-A1(y2(z),z))‖≤‖(y2(x)-y2(z))-ρ1(A1(y2(x),x)-A1(y2(z),z))‖+‖(y2(x)-y2(z))-(g1(y2(x))-g1(y2(z)))‖≤(1+qa1ι1q+cqι1q-qb1q+1+qρ1μ1τ1q+cqρ1qτ1q-qρ1ν1q)‖y2(x)-y2(z)‖.
Note that, for each j∈{2,3,…,n-1},
(3.16)‖yj(x)-yj(z)‖=‖RρjMj(gj(yj+1(x))-ρjAj(yj+1(x),yj(x)))-RρjMj(gj(yj+1(z))-ρjAj(yj+1(z),yj(z)))‖≤‖(gj(yj+1(x))-ρjAj(yj+1(x),yj(x)))-(gj(yj+1(z))-ρjAj(yj+1(z),yj(z)))‖=‖(gj(yj+1(x))-gj(yj+1(z)))-ρj(Aj(yj+1(x),yj(x))-Aj(yj+1(z),yj(z)))‖≤‖(yj+1(x)-yj+1(z))-ρj(Aj(yj+1(x),yj(x))-Aj(yj+1(z),yj(z)))‖+‖(yj+1(x)-yj+1(z))-(gj(yj+1(x))-gj(yj+1(z)))‖≤(1+qρjμjτjq+cqρjqτjq-qρjνjq+1+qajιjq+cqιjq-qbjq)‖yj+1(x)-yj+1(z)‖,‖yn(x)-yn(z)‖=‖RρnMn(gn(x)-ρnAn(x,yn(x)))-RρnMn(gn(z)-ρnAn(z,yn(z)))‖≤‖(gn(x)-gn(z))-ρn(An(x,yn(x))-An(z,yn(z)))‖≤‖x-z-ρn(An(x,yn(x))-An(z,yn(z)))‖+‖x-z-(gn(x)-gn(z))‖≤(1+qρnμnτnq+cqρnqτnq-qρnνnq+1+qanιnq+cqιnq-qbnq)‖x-z‖.
Therefore, we obtain
(3.17)‖F(x)-F(z)‖≤∏j=1n(1+qρjμjτjq+cqρjqτjq-qρjνjq+1+qajιjq+cqιjq-qbjq)‖x-z‖.
It follows from (3.12) that the mapping F is contractive. By banach’s contraction principle, there exists a unique x1*∈E such that F(x1*)=x1*. Therefore, by Theorem 3.2, there exists an unique (x1*,x2*,…,xn*)∈En such that (x1*,x2*,…,xn*) is a solution of the problem (SGMQVI), where xi*=yi(x1*) for i=2,3,…,n, that is, Ξ={(x1*,x2*,…,xn*)}. This completes the proof.


## 4. Convergence Analysis

In this section, we introduce several implicit algorithms with errors and explicit algorithms without errors for solving the system of generalized mixed quasivariational inclusions problem (SGMQVI) and then explore the convergence analysis of the iterative sequences generated by the corresponding algorithms.From Section3, we know that the system of generalized mixed quasivariational inclusions problem (SGMQVI) is equivalent to the fixed point problem (3.1). This equivalent formulation is crucial from the numerical analysis point of view. As we know, this fixed point formulation has been used to suggest and analyze some iterative methods for solving variational inequalities and related optimization problems. By using the relations between the problem (SGMQVI) and the fixed point problem (3.1), we construct the following iterative algorithms for solving the system of generalized mixed quasivariational inclusions problem (2.3).Algorithm 4.1.
Letρj be positive constants for all j=1,2,…,n. For any given points x1,0∈E, define sequences {xj,k}j=1,2,…,n in E by the following implicit algorithm:
(4.1)xn,k=RρnMn(gn(x1,k)-ρnAn(x1,k,xn,k)),xn-1,k=Rρn-1Mn-1(gn-1(xn,k)-ρn-1An-1(xn,k,xn-1,k)),⋮x2,k=Rρ2M2(g2(x3,k)-ρ2A2(x3,k,x2,k)),x1,k+1=(1-αk)x1,k+αkRρ1M1(g1(x2,k)-ρ1A1(x2,k,x1,k))+ek,k=0,1,2,…,
where {ek}⊂E and {αk} is a real sequence in [0,1].Ifek≡0 for all k≥0, then Algorithm 4.1 is reduced to the following result.Algorithm 4.2.
Letρj be positive constants for all j=1,2,…,n. For any given points x1,0∈E, define sequences {xj,k}j=1,2,…,n in E by the following implicit algorithm
(4.2)xn,k=RρnMn(gn(x1,k)-ρnAn(x1,k,xn,k)),xn-1,k=Rρn-1Mn-1(gn-1(xn,k)-ρn-1An-1(xn,k,xn-1,k)),⋮x2,k=Rρ2M2(g2(x3,k)-ρ2A2(x3,k,x2,k)),x1,k+1=(1-αk)x1,k+αkRρ1M1(g1(x2,k)-ρ1A1(x2,k,x1,k)),k=0,1,2,...,
where {αk} is a real sequence in [0,1].Now we construct an explicit algorithms for solving the system of generalized mixed quasivariational inclusions problem (SGMQVI).Algorithm 4.3.
Letρj be positive constants for all j=1,2,…,n. For any given points (x1,0,x2,0,…,xn-1,0,xn,0)∈En, define sequences {xj,k}(j=1,2,…,n) in E by the following explicit algorithm
(4.3)xn,k+1=RρnMn(gn(x1,k)-ρnAn(x1,k,xn,k)),xn-1,k+1=Rρn-1Mn-1(gn-1(xn,k+1)-ρn-1An-1(xn,k+1,xn-1,k)),⋮x2,k+1=Rρ2M2(g2(x3,k+1)-ρ2A2(x3,k+1,x2,k)),x1,k+1=(1-αk)x1,k+αkRρ1M1(g1(x2,k+1)-ρ1A1(x2,k+1,x1,k)),k=0,1,2,…,
where {αk} is a real sequence in [0,1].Remark 4.4.
Ifn=2,E=H is a Hilbert space, and K is a closed convex subset of E, ϕ(x)=δK(x) for all x∈K, and M1(x)=M2(x)=∂ϕ(x) for all x∈E, where δK is the indicator function of K, and ∂ϕ denotes the subdifferential operator of ϕ, then, from Remark 2.9, Algorithms 4.1 and 4.3 are reduced to the Algorithms 4.5 and 4.6 for solving the system of general variational inequalities problem (2.7).Algorithm 4.5.
Letρj be positive constants for all j=1,2. For any given points x1,0∈E, define sequences {xj,k}j=1,2 in E by the following implicit algorithm:
(4.4)x2,k=PK(g2(x1,k)-ρ2A2(x1,k,x2,k)),x1,k+1=(1-αk)x1,k+αkPK(g1(x2,k)-ρ1A1(x2,k,x1,k))+ek,k=0,1,2,…,
where {ek}⊂E and {αk} is a real sequence in [0,1].Algorithm 4.6.
Letρj be positive constants for all j=1,2. For any given points (x1,0,x2,0)∈E2, define sequences {xj,k}(j=1,2) in E by the following explicit algorithm:
(4.5)x2,k+1=PK(g2(x1,k)-ρ2A2(x1,k,x2,k)),x1,k+1=(1-αk)x1,k+αkPK(g1(x2,k+1)-ρ1A1(x2,k+1,x1,k)),k=0,1,2,…,
where {αk} is a real sequence in [0,1].Theorem 4.7.
LetE be a real q-uniformly smooth Banach space, and let Mj,Aj, and gj(j=1,2,…,n) be the same as in Theorem 3.4. Assume that {αn} is a real sequence in (0,1] and satisfy the following conditions:(i)
∑k=0∞αk=∞;(ii)
∑k=0∞∥ek∥<+∞;(iii)
min{1+qajιjq+cqιjq-qbj,1+qρjμjτjq+cqρjqτjq-qρjνj}≥0,j=1,2,…,n;(iv)
0<∏j=1n(1+qajιjq+cqιjq-qbjq+1+qρjμjτjq+cqρjqτjq-qρjνjq)<1.Then the sequences {xj,k}(j=1,2,…,n) generated by Algorithm 4.1 converge strongly to xj*(j=1,2,…,n), respectively, such that (x1*,x2*,…,xn*)∈Ξ.Proof.
By Theorem3.4, we know that there exist an unique point (x1*,x2*,…,xn*)∈En such that (x1*,x2*,…,xn*)∈Ξ. Then, from Theorem 3.1, one has
(4.6)x1*=Rρ1M1(g1(x2*)-ρ1A1(x2*,x1*)),x2*=Rρ2M2(g2(x3*)-ρ2A2(x3*,x2*)),⋮xn-1*=Rρn-1Mn-1(gn-1(xn*)-ρn-1An-1(xn*,xn-1*)),xn*=RρnMn(gn(x1*)-ρnAn(x1*,xn*)).
Therefore, from both (4.1) and (4.6), we have
(4.7)‖x1,k+1-x1*‖=‖(1-αk)x1,k+αkRρ1M1(g1(x2,k)-ρ1A1(x2,k,x1,k))+ek-x1*‖=‖(1-αk)(x1,k-x1*)+αk(Rρ1M1(g1(x2,k)-ρ1A1(x2,k,x1,k))-x1*)+ek‖=‖αk(Rρ1M1(g1(x2,k)-ρ1A1(x2,k,x1,k))-Rρ1M1(g1(x2*)-ρ1A1(x2*,x1*)))+(1-αk)(x1,k-x1*)+ek(Rρ1M1(g1(x2,k)-ρ1A1(x2,k,x1,k))-Rρ1M1(g1(x2*)-ρ1A1(x2*,x1*)))‖≤αk‖Rρ1M1(g1(x2,k)-ρ1A1(x2,k,x1,k))-Rρ1M1(g1(x2*)-ρ1A1(x2*,x1*))‖+(1-αk)‖x1,k-x1*‖+‖ek‖≤αk‖(g1(x2,k)-ρ1A1(x2,k,x1,k))-(g1(x2*)-ρ1A1(x2*,x1*))‖+(1-αk)‖x1,k-x1*‖+‖ek‖=αk‖(g1(x2,k)-g1(x2*))-(ρ1A1(x2,k,x1,k)-ρ1A1(x2*,x1*))‖+(1-αk)‖x1,k-x1*‖+‖ek‖≤(1-αk)‖x1,k-x1*‖+‖ek‖+αk‖(x2,k-x2*)-(g1(x2,k)-g1(x2*))‖+αk‖(x2,k-x2*)-ρ1(A1(x2,k,x1,k)-A1(x2*,x1*))‖.
Since Aj:E×E→E are (μj,νj)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τj and gj:E→E are (aj,bj)-relaxed cocoercive and Lipschitz continuous with constant ιj for j∈{1,2,…,n}, we can conclude
(4.8)LAi*=‖xi+1,k-xi+1*-ρi[Ai(xi+1,k,xi,k)-Ai(xi+1*,xi*)]‖q≤-qρi〈Ai(xi+1,k,xi,k)-Ai(xi+1*,xi*),Jq(xi+1,k-xi+1*)〉+‖xi+1,k-xi+1*‖q+cqρiq‖Ai(xi+1,k,xi,k)-Ai(xi+1*,xi*)‖q≤qρi(μi‖Ai(xi+1,k,xi,k)-Ai(xi+1*,xi*)‖q-νi‖xi+1,k-xi+1*‖q)+(1+cqρiqτiq)‖xi+1,k-xi+1*‖q≤(1+qρiμiτiq+cqρiqτiq-qρiνi)‖xi+1,k-xi+1*‖q,i=1,2,…,n-1,Lgi*=‖xi+1,k-xi+1*-(gi(xi+1,k)-gi(xi+1*))‖q≤-q〈gi(xi+1,k)-gi(xi+1*),Jq(xi+1,k-xi+1*)〉+‖xi+1,k-xi+1*‖q+cq‖gi(xi+1,k)-gi(xi+1*)‖q≤(1+qaiιiq+cqιiq-qbi)‖xi+1,k-xi+1*‖q,i=1,2,…,n-1,LAn*=‖x1,k-x1*-ρn[An(x1,k,xn,k)-An(x1*,xn*)]‖q≤‖x1,k-x1*‖q-qρn〈An(x1,k,xn,k)-An(x1*,xn*),Jq(x1,k-x1*)〉+cqρnq‖An(x1,k,xn,k)-An(x1*,xn*)‖q≤(1+qρnμnτnq+cqρnqτnq-qρnνn)‖x1,k-x1*‖q,Lgn*=‖x1,k-x1*-(gn(x1,k)-gn(x1*))‖q≤‖x1,k-x1*‖q-q〈gn(x1,k)-gn(x1*),Jq(x1,k-x1*)〉+cq‖gn(x1,k)-gn(x1*)‖q≤(1+qanιnq+cqιnq-qbn)‖x1,k-x1*‖q.
Noticing that, for each i∈{1,2,…,n-2},
(4.9)‖xi+1,k-xi+1*‖=‖Rρi+1Mi+1(gi+1(xi+2,k)-ρi+1Ai+1(xi+2,k,xi+1,k))-Rρi+1Mi+1(gi+1(xi+2*)-ρi+1Ai+1(xi+2*,xi+1*))‖≤‖(gi+1(xi+2,k)-ρi+1Ai+1(xi+2,k,xi+1,k))-(gi+1(xi+2*)-ρi+1Ai+1(xi+2*,xi+1*))‖=‖(gi+1(xi+2,k)-gi+1(xi+2*))-ρi+1(Ai+1(xi+2,k,xi+1,k)-Ai+1(xi+2*,xi+1*))‖≤‖(xi+2,k-xi+2*)-ρi+1(Ai+1(xi+2,k,xi+1,k)-Ai+1(xi+2*,xi+1*))‖+‖(xi+2,k-xi+2*)-(gi+1(xi+2,k)-gi+1(xi+2*))‖≤(1+qρi+1μi+1τi+1q+cqρi+1qτi+1q-qρi+1νi+1q+1+qai+1ιi+1q+cqιi+1q-qbi+1q)‖xi+2,k-xi+2*‖,‖xn,k-xn*‖=‖RρnMn(gn(x1,k)-ρnAn(x1,k,xn,k))-RρnMn(gn(x1*)-ρnAn(x1*,xn*))‖≤‖(gn(x1,k)-ρnAn(x1,k,xn,k))-(gn(x1*)-ρnAn(x1*,xn*))‖=‖(gn(x1,k)-gn(x1*))-ρn(An(x1,k,xn,k)-An(x1*,xn*))‖≤‖(x1,k-x1*)-(ρnAn(x1,k,xn,k)-ρnAn(x1*,xn*))‖+‖(x1,k-x1*)-(gn(x1,k)-gn(x1*))‖≤(1+qρnμnτnq+cqρnqτnq-qρnνnq+1+qanιnq+cqιnq-qbnq)×‖x1,k-x1*‖.
As a consequence, we have
(4.10)‖x1,k+1-x1*‖≤αk∏j=1n(1+qρjμjτjq+cqρjqτjq-qρjνjq+1+qajιjq+cqιjq-qbjq)‖x1,k-x1*‖+(1-αk)‖x1,k-x1*‖+‖ek‖=[1-αk(1-∏j=1n(1+qρjμjτjq+cqρjqτjq-qρjνjq+1+qajιjq+cqιjq-qbjq))]×‖x1,k-x1*‖+‖ek‖.
Putting λk=αk(1-∏j=1n(1+qρjμjτjq+cqρjqτjq-qρjνjq+1+qajιjq+cqιjq-qbjq)), Ck=0,Bk=∥x1,k-x1*∥, and Dk=∥ek∥. Then Bk+1≤(1-λk)Bk+Ck+Dk. From the conditions (i)–(iv), it follows that
(4.11)∑k=0∞λk=∞,Ck=0(λk),∑k=0∞Dk<∞,0<λk<1,∀k∈N.
Therefore, by Lemma 2.11 and (4.11), one has
(4.12)lim⁡k→∞Bk=0,
that is, limk→∞x1,k=x1*. Again from (iii), this shows that
(4.13)1+qρjμjτjq+cqρjqτjq-qρjνjq+1+qajιjq+cqιjq-qbjq≥0,j=2,3,…,n,
and so,
(4.14)lim⁡k→∞‖xj,k-xj*‖=0,
That is, xj,k→xj* as k→∞ for j=2,3,…,n. Thus, (x1,k,x2,k,…,xn,k) converges strongly to (x1*,x2*,…,xn*). This completes the proof.Theorem 4.8.
LetE be a real q-uniformly smooth Banach space, and let Mj,Aj and gj(j=1,2,…,n) be the same as in Theorem 3.4. Assume that {αn} is a real sequence in (0,1] and satisfies the following conditions:(i)
∑k=0∞αk=∞;(ii)
min{1+qajιjq+cqιjq-qbj,1+qρjμjτjq+cqρjqτjq-qρjνj}≥0, j=1,2,…,n;(iii)
0<∏j=1n(1+qajιjq+cqιjq-qbjq+1+qρjμjτjq+cqρjqτjq-qρjνjq)<1.Then the sequences {xj,k}(j=1,2,…,n) generated by Algorithm 4.2 converge strongly to xj*(j=1,2,…,n), respectively, such that (x1*,x2*,…,xn*)∈Ξ.Proof.
It directly follows from Theorem4.7, and so the proof is omitted. This completes the proof.Theorem 4.9.
LetE be a real q-uniformly smooth Banach space, and let Mj,Aj and gj(j=1,2,…,n) be the same as in Theorem 3.4. Assume that {αn} is a real sequence in (0,1] and satisfy the following conditions:(i)
∑k=0∞αk=∞;(ii)
min{1+qajιjq+cqιjq-qbj,1+qρjμjτjq+cqρjqτjq-qρjνj}≥0,j=1,2,…,n;(iii)
0<∏j=1n(1+qajιjq+cqιjq-qbjq+1+qρjμjτjq+cqρjqτjq-qρjνjq)<1.Then the sequences {xj,k}(j=1,2,…,n) generated by Algorithm 4.3 converge strongly to xj*(j=1,2,…,n), respectively, such that (x1*,x2*,…,xn*)∈Ξ.Proof.
As in the proof of Theorem4.7, we know that there exists an unique point (x1*,x2*,…,xn*)∈En such that (x1*,x2*,…,xn*)∈Ξ, and so
(4.15)x1*=Rρ1M1(g1(x2*)-ρ1A1(x2*,x1*)),x2*=Rρ2M2(g2(x3*)-ρ2A2(x3*,x2*)),⋮xn-1*=Rρn-1Mn-1(gn-1(xn*)-ρn-1An-1(xn*,xn-1*)),xn*=RρnMn(gn(x1*)-ρnAn(x1*,xn*)),
Note that
(4.16)‖x1,k+1-x1*‖=‖(1-αk)x1,k+αkRρ1M1(g1(x2,k+1)-ρ1A1(x2,k+1,x1,k))-x1*‖=‖(1-αk)(x1,k-x1*)+αk[Rρ1M1(g1(x2,k+1)-ρ1A1(x2,k+1,x1,k))-x1*]‖=‖αk[Rρ1M1(g1(x2,k+1)-ρ1A1(x2,k+1,x1,k))-Rρ1M1(g1(x2*)-ρ1A1(x2*,x1*))]+(1-αk)(x1,k-x1*)[Rρ1M1(g1(x2,k+1)-ρ1A1(x2,k+1,x1,k))-Rρ1M1(g1(x2*)-ρ1A1(x2*,x1*))]‖≤αk‖Rρ1M1(g1(x2,k+1)-ρ1A1(x2,k+1,x1,k))-Rρ1M1(g1(x2*)-ρ1A1(x2*,x1*))‖+(1-αk)‖x1,k-x1*‖≤αk‖(g1(x2,k+1)-ρ1A1(x2,k+1,x1,k))-(g1(x2*)-ρ1A1(x2*,x1*))‖+(1-αk)‖x1,k-x1*‖=αk‖(g1(x2,k+1)-g1(x2*))-ρ1(A1(x2,k+1,x1,k)-A1(x2*,x1*))‖+(1-αk)‖x1,k-x1*‖≤(1-αk)‖x1,k-x1*‖+αk‖(x2,k+1-x2*)-(g1(x2,k)-g1(x2*))‖+αk‖(x2,k+1-x2*)-ρ1(A1(x2,k+1,x1,k)-A1(x2*,x1*))‖.
Since Aj:E×E→E are (μj,νj)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τj and gj:E→E are (aj,bj)-relaxed cocoercive and Lipschitz continuous with constant ιj for j∈{1,2,…,n}, we can conclude that
(4.17)L̂Ai*=‖xi+1,k+1-xi+1*-ρi[Ai(xi+1,k+1,xi,k)-Ai(xi+1*,xi*)]‖q≤-qρi〈Ai(xi+1,k+1,xi,k)-Ai(xi+1*,xi*),Jq(xi+1,k+1-xi+1*)〉+‖xi+1,k+1-xi+1*‖q+cqρiq‖Ai(xi+1,k+1,xi,k)-Ai(xi+1*,xi*)‖q≤qρi(μi‖Ai(xi+1,k+1,xi,k)-Ai(xi+1*,xi*)‖q-νi‖xi+1,k+1-xi+1*‖q)+(1+cqρiqτiq)‖xi+1,k+1-xi+1*‖q≤(1+qρiμiτiq+cqρiqτiq-qρiνi)‖xi+1,k+1-xi+1*‖q,i=1,2,…,n-1,L̂gi*=‖xi+1,k+1-xi+1*-(gi(xi+1,k+1)-gi(xi+1*))‖q≤-q〈gi(xi+1,k+1)-gi(xi+1*),Jq(xi+1,k+1-xi+1*)〉+‖xi+1,k+1-xi+1*‖q+cq‖gi(xi+1,k+1)-gi(xi+1*)‖q≤(1+qaiιiq+cqιiq-qbi)‖xi+1,k+1-xi+1*‖q,i=1,2,…,n-1,L̂An*=‖x1,k-x1*-ρn[An(x1,k,xn,k)-An(x1*,xn*)]‖q≤‖x1,k-x1*‖q-qρn〈An(x1,k,xn,k)-An(x1*,xn*),Jq(x1,k-x1*)〉+cqρnq‖An(x1,k,xn,k)-An(x1*,xn*)‖q≤(1+qρnμnτnq+cqρnqτnq-qρnνn)‖x1,k-x1*‖q,L̂gn*=‖x1,k-x1*-(gn(x1,k)-gn(x1*))‖q≤‖x1,k-x1*‖q-q〈gn(x1,k)-gn(x1*),Jq(x1,k-x1*)〉+cq‖gn(x1,k)-gn(x1*)‖q≤(1+qanιnq+cqιnq-qbn)‖x1,k-x1*‖q.
Noticing that, for each i∈{1,2,…,n-2},
(4.18)‖xi+1,k+1-xi+1*‖=‖Rρi+1Mi+1(gi+1(xi+2,k+1)-ρi+1Ai+1(xi+2,k+1,xi+1,k))-Rρi+1Mi+1(gi+1(xi+2*)-ρi+1Ai+1(xi+2*,xi+1*))‖≤‖(gi+1(xi+2,k+1)-ρi+1Ai+1(xi+2,k+1,xi+1,k))-(gi+1(xi+2*)-ρi+1Ai+1(xi+2*,xi+1*))(gi+1(xi+2,k+1)-ρi+1Ai+1(xi+2,k+1,xi+1,k))‖=‖(gi+1(xi+2,k+1)-gi+1(xi+2*))-ρi+1×(Ai+1(xi+2,k+1,xi+1,k)-Ai+1(xi+2*,xi+1*))‖≤‖(xi+2,k+1-xi+2*)-ρi+1(Ai+1(xi+2,k+1,xi+1,k)-Ai+1(xi+2*,xi+1*))‖+‖(xi+2,k+1-xi+2*)-(gi+1(xi+2,k+1)-gi+1(xi+2*))‖≤(1+qρi+1μi+1τi+1q+cqρi+1qτi+1q-qρi+1νi+1q+1+qai+1ιi+1q+cqιi+1q-qbi+1q)‖xi+2,k+1-xi+2*‖,‖xn,k+1-xn*‖=‖RρnMn(gn(x1,k)-ρnAn(x1,k,xn,k))-RρnMn(gn(x1*)-ρnAn(x1*,xn*))‖≤‖(gn(x1,k)-ρnAn(x1,k,xn,k))-(gn(x1*)-ρnAn(x1*,xn*))‖=‖(gn(x1,k)-gn(x1*))-ρn(An(x1,k,xn,k)-An(x1*,xn*))‖≤‖(x1,k-x1*)-(ρnAn(x1,k,xn,k)-ρnAn(x1*,xn*))‖+‖(x1,k-x1*)-(gn(x1,k)-gn(x1*))‖≤(1+qρnμnτnq+cqρnqτnq-qρnνnq+1+qanιnq+cqιnq-qbnq)×‖x1,k-x1*‖.
Consequently, we have
(4.19)‖x1,k+1-x*‖≤αk∏j=1n(1+qρjμjτjq+cqρjqτjq-qρjνjq+1+qajιjq+cqιjq-qbjq)‖x1,k-x1*‖+(1-αk)‖x1,k-x*‖=[1-αk(1-∏j=1n(1+qρjμjτjq+cqρjqτjq-qρjνjq+1+qajιjq+cqιjq-qbjq))]‖x1,k-x1*‖.
Taking λk=αk(1-∏j=1n(1+qρjμjτjq+cqρjqτjq-qρjνjq+1+qajιjq+cqιjq-qbjq)), Ck=Dk=0, and Bk=∥x1,k-x1*∥, then Bk+1≤(1-λk)Bk+Ck+Dk. It follows from the conditions (i)–(iii) that
(4.20)∑k=0∞λk=∞,Ck=0(λk),∑k=0∞Dk<∞,0<λk<1,∀k∈N.
Therefore, by Lemma 2.11 and (4.20), one has limk→∞Bk=0. By the same argument of Theorem 4.7, we get
(4.21)lim⁡k→∞‖xj,k-xj*‖=0,j=1,2,…,n,
that is, xj,k→xj* as k→∞ for j=1,2,…,n. Thus, (x1,k,x2,k,…,xn,k) converges strongly to (x1*,x2*,…,xn*). This completes the proof.By Remark4.4, we have the following strong convergence theorems for the system of general variational inequalities problem (2.7).Corollary 4.10.
LetK be a closed convex subset of a real Hilbert space E, and let Aj and gj  (j=1,2) be the same as in Theorem 3.4. Assume that {αn} is a real sequence in (0,1] and satisfies the following conditions:(i)
∑k=0∞αk=∞;(ii)
∑k=0∞∥ek∥<+∞;(iii)
min{1+2ajιj2+ιj2-2bj,1+2ρjμjτj2+ρj2τj2-2ρjνj}≥0,j=1,2;(iv)
0<∏j=1n(1+2ajιj2+ιj2-2bj+1+2ρjμjτj2+ρj2τj2-2ρjνj)<1.Then the sequences {xj,k}(j=1,2) generated by Algorithm 4.5 converge strongly to xj*(j=1,2), respectively, such that (x1*,x2*) is the unique solution of the system of general variational inequalities problem (2.7).Corollary 4.11.
LetK be a closed convex subset of a real Hilbert space E, and let Aj and gj(j=1,2) be the same as in Theorem 3.4. Assume that {αn} is a real sequence in (0,1] and satisfies the following conditions:(i)
∑k=0∞αk=∞;(ii)
min{1+2ajιj2+ιj2-2bj,1+2ρjμjτj2+ρj2τj2-2ρjνj}≥0, j=1,2;(iii)
0<∏j=1n(1+2ajιj2+ιj2-2bj+1+2ρjμjτj2+ρj2τj2-2ρjνj)<1.Then the sequences {xj,k}(j=1,2) generated by Algorithm 4.6 converge strongly to xj*(j=1,2), respectively, such that (x1*,x2*) is the unique solution of the system of general variational inequalities problem (2.7).


## 5. An Application

In this section, we applied the obtained results to study a class of bilevel variational inequalities in Hilbert space, which includes some bilevel programming as special cases and widely used in many practical problems. Moreover, an iterative algorithm and convergence theorem for solutions to the bilevel variational inequalities are given in Hilbert space.LetK1 and K2 be nonempty closed convex subsets of a Hilbert space E, and let g,h:E→E and A:E×E→E be single-valued mappings. We consider the following bilevel variational inequalities (for short, (BVI)): find (x*,y*)∈K1×K2 such that (5.1)〈x*+h(y*),h(x)-x*〉≥0,∀x∈K1,y*∈Ψ(x*),
where Ψ(x*) is the solutions set of the following parametric variational inequalities with respect to the parametric variable x*: (5.2)〈ρA(x*,y*)+y*-g(x*),g(y)-y*〉≥0,∀y∈K2,
where ρ is a positive constant. Equations (5.1) and (5.2) are called the upper-level variation inequality (for short, (UVI)) and the lower-level variation inequality (for short, LVI), respectively. Denote the set of solutions to the (BVI) by Θ. An important question for the (BVI) is how to solve the bilevel variational inequalities.From Remark2.9 and Theorem 3.1, one can easily conclude the following result.Lemma 5.1.
Let(x*,y*)∈K1×K2. Then (x*,y*) is a solution of the problem (BVI) if and only if x*=PK1(-h(y*)), where y*=PK2(g(x*)-ρA(x*,y*)) and ρ is a positive constant.Lemma 5.2.
LetK1 and K2 be nonempty closed convex subsets of a Hilbert space E. Let A:E×E→E be (μ,ν)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ, and let g:E→E be (a2,b2)-relaxed cocoercive and Lipschitz continuous with constant c2. Assume that {1+2a2c22+c22-2b2,1+2ρμτ2+ρ2τ2-2ρν}≥0 and
(5.3)0≤1+2a2c22+c22-2b2+1+2ρμτ2+ρ2τ2-2ρν<1.
Then, for each x∈K1, the parametric variational inequalities (5.2) have a uniquely solution. Further, the solution mapping y(x) of the parametric variational inequalities (5.2) is continuous on K1.Proof.
It directly follows from Theorems3.2 and 3.4. This completes the proof.Theorem 5.3.
LetK1 and K2 be nonempty closed convex subsets of a Hilbert space E. Let A:E×E→E be (μ,ν)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ, h:E→E a (a1,b1)-relaxed cocoercive and Lipschitz continuous with constant c1, and let g:E→E be a (a2,b2)-relaxed cocoercive and Lipschitz continuous with constant c2. Assume that {αk} is a real sequence in (0,1] and satisfies the following conditions:(i)
∑k=0∞αk=∞;(ii)
min{1+2ajcj2+cj2-2bj,1+2ρμτ2+ρ2τ2-2ρν,j=1,2}≥0;(iii)
0<∏j=1n(1+qajιjq+cqιjq-qbjq+1+qρjμjτjq+cqρjqτjq-qρjνjq)<1.The sequences {xk} and {yk} generated by the following algorithm:
(5.4)x0∈K1,yk=PK2(g(xk)-ρA(xk,yk)),xk+1=(1-αk)xk+αkPK1(-h(yk)),k=0,1,2,…,
where ρ is a positive constant. Then the sequences {xk} and {yk} converge strongly to x* and y*, respectively, such that (x*,y*) is a solution of the (BVI).Proof.
The proof is similar to Remark2.9 and Theorem 4.7, and so the proof is omitted. This completes the proof.



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*Source: 961038-2011-09-11.xml*