# A New System of Multivalued Mixed Variational Inequality Problem

Authors: Xi Li; Xue-song Li
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider a new system of multivalued mixed variational inequality problem, which includes some known systems of variational inequalities as special cases. Under suitable conditions, the existence of solutions for the system of multivalued mixed variational inequality problem and the convergence of iterative sequences generated by the generalizedf-projection algorithm are proved. A perturbational algorithm for solving a special case of multivalued mixed variational inequality problem is formally constructed. The results concerned with the existence of solutions and the convergence of iterative sequences generated by the perturbational algorithm are also given. Some known results are improved and generalized.


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## Body

## 1. Introduction

Variational inequalities are known to be very useful tool to formulate and investigate various network equilibrium problems arising in economic, management, and engineering. An important and useful generalization of the variational inequality is called the mixed variational inequality. This problem was originally considered by Lescarret [1] and Browder [2] in connection with its numerous applications. Konnov and Volotskaya [3] considered rather broad classes of general economic equilibrium problems and oligopolistic equilibrium problems which can be formulated as mixed variational inequality problems.Recently, some interesting and important problems related to variational inequalities and mixed variational inequalities have been considered by many authors. Chang et al. [4] introduced a generalized system for relaxed cocoercive variational inequalities in Hilbert spaces and established some algorithms for the system. Petrot [5] studied a generalized system for relaxed cocoercive mixed variational inequality problem in Hilbert spaces and found the common solutions for the system using a resolvent operator technique. For more details related to variational inequalities and mixed variational inequalities, we refer to [6–12] and the references therein.It is well known that projection methods have represented an important tool for solving variational inequalities. In 1994, Alber [13] introduced the generalized projections in uniformly convex and uniformly smooth Banach spaces and studied their properties in detail. Recently, Wu and Huang [14] introduced a new generalized f-projection operator in a Banach space, which was a useful tool for solving mixed variational inequality problems. They extended the definition of the generalized projection operators introduced by Alber [13, 15] and proved some properties of the generalized f-projection operator. Fan et al. [16] presented some basic results for the generalized f-projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces by using iterative schemes. Very recently, Li et al. [17] proved some stability results for the generalized f-projection operators with perturbations of constraint sets in Banach spaces.Inspired and motivated by the previous works mentioned above, in this paper, we introduce a new system of multivalued mixed variational inequality problem in Hilbert spaces. This class of systems includes some known systems of variational inequalities as special cases. We construct a new generalizedf-projection algorithm for solving the system of multivalued mixed variational inequality problem. The existence of solutions and the convergence of iterative sequences generated by the algorithm are presented in this paper. We also construct a perturbational algorithm for solving a special case of multivalued mixed variational inequality problem and give the existence of solutions and the convergence of iterative sequences generated by the perturbational algorithm. Our results improve and generalize some known corresponding ones.


## 2. Preliminaries

LetH be a real Hilbert space with scalar product and norm denoted by 〈·,·〉 and ∥·∥, respectively. We recall the concept of the generalized f-projector operator, together with its properties. Let R=(-∞,+∞), let K⊂H be a closed convex subset, and let f:K→R∪{+∞} be a proper convex and lower semicontinuous functional. Let G:H×K→R∪{+∞} be a functional defined as follows:
(1)G(x,ξ)=∥x∥2-2〈x,ξ〉+∥ξ∥2+2ρf(ξ),
where ξ∈K,x∈H, and ρ is a positive number.Definition 1 (see [14]).
LetH be a real Hilbert space, and let K be a nonempty closed and convex subset of H. Let f:K→R∪{+∞} be a proper, convex, and lower semicontinuous functional. One says that PKf,ρ:H→2K is a generalized f-projection operator if
(2)PKf,ρx={u∈K:G(x,u)=inf⁡ξ∈KG(x,ξ)},∀x∈H.Lemma 2 (see [14, 16]).
LetH be a real Hilbert space, and let K be a nonempty closed and convex subset of H. Let f:K→R∪{+∞} be a proper, convex, and lower semicontinuous functional. Then the following statements hold: (i)
PKf,ρ is a single-valued mapping with nonempty values;(ii)
for allx∈H,x*=PKf,ρx if and only if
(3)〈x*-x,y-x*〉+ρf(y)-ρf(x*)≥0,∀y∈K;(iii)
PKf,ρ is continuous.Lemma 3 (see [18]).
LetH be a real Hilbert space, and let K be a nonempty closed and convex subset of H. Let f:K→R∪{+∞} be a proper, convex, and lower semicontinuous functional. Then
(4)∥PKf,ρv-PKf,ρu∥≤∥v-u∥,∀v,u∈H.LetK1 and K2 be closed convex subsets in H. It is known that the Hausdorff distance between K1 and K2 is defined as follows:
(5)H(K1,K2)=max⁡{sup⁡x∈K1inf⁡y∈K2∥x-y∥,sup⁡y∈K2inf⁡x∈K1∥x-y∥}.Lemma 4 (see [17]).
LetH be a real Hilbert space and let K be a nonempty closed convex subset of H. Let f:H→R∪{+∞} be a convex and uniformly continuous mapping. Let {Kn}n=0+∞⊂H be a family of nonempty closed convex subsets such that f is proper on each Kn and H(Kn,K)→0 as n→+∞. Then, for any x0∈H,
(6)lim⁡n→+∞PKnf,ρx0=PKf,ρx0.Lemma 5 (see [19]).
Let{an} and {bn} be two nonnegative real sequences satisfying
(7)an+1≤kan+bn,
with 0<k<1 and bn→0. Then lim⁡n→0⁡an=0.In order to obtain our results, the following definitions are crucial to us.Definition 6.
LetT:H×H→H be a mapping. T is said to be (i)
γ-Lipschitz continuous with respect to the first argument, if there exists a constant γ>0 such that
(8)∥T(x,z)-T(y,z)∥≤γ∥x-y∥,∀x,y,z∈H;(ii)
μ-strongly monotone with respect to the first argument, if there exists a constant μ>0 such that
(9)〈T(x,z)-T(y,z),x-y〉≥μ∥x-y∥2,∀x,y,z∈H.Definition 7 (see [20]).
LetF:H→2H be a multivalued mapping. F is said to be ξ-H-Lipchitz continuous, if there exists a constant ξ>0 such that
(10)H(F(x),F(y))≤ξ∥x-y∥,∀x,y∈H.


## 3. The Multivalued Mixed Variational Inequality System

In this section, we will introduce a new system of multivalued mixed variational inequality in a Hilbert space. LetH be a real Hilbert space, and let K be a nonempty closed convex subset of H. Let S,T:H×H→H be two single-valued mappings, let f:K→R∪{+∞} be a proper, convex, and lower semicontinuous mapping, and let E,F:H→2H be two set-valued mappings. We consider the following system of multivalued mixed variational inequality problem: find (x,y)∈K×K, u∈E(x), and v∈F(y) such that
(11)〈S(x,v),z-x〉+f(z)-f(x)≥0,∀z∈K,〈T(y,u),z-y〉+f(z)-f(y)≥0,∀z∈K.IfE=F≡I, where I is the identity mapping, then the problem (11) reduces immediately to the following mixed variational inequality system: find (x,y)∈K such that
(12)〈S(x,y),z-x〉+f(z)-f(x)≥0,∀z∈K,〈T(y,x),z-y〉+f(z)-f(y)≥0,∀z∈K.IfK=H, E=F≡I, S(x,y)=sA(y,x)+x-y, and T(y,x)=rA(x,y)+x-y for all x,y∈H, where A:H×H→H, then the problem (11) is equivalent to finding (x,y)∈H such that
(13)〈sA(y,x)+x-y,z-x〉+f(z)-f(x)≥0,fffffffffffffffffffffffffffffffffffffff∀z∈H,s>0,〈rA(x,y)+y-x,z-y〉+f(z)-f(y)≥0,fffffffffffffffffffffffffffffffffffffff∀z∈H,r>0,
which was studied by Petrot [5].IfE=F≡I, f≡0, S(x,y)=sA(y,x), and T(y,x)=rA(x,y) for all x,y∈H, where A:H×H→H and s,r>0, then the problem (11) is equivalent to finding (x,y)∈K such that
(14)〈sA(y,x)+x-y,z-x〉≥0,∀z∈K,s>0,〈rA(x,y)+y-x,z-y〉≥0,∀z∈K,r>0.
The system (14) was studied by Chang et al. [4].In brief, the system (11) of multivalued mixed variational inequality is more general and includes many systems of variational inequalities as special cases (see [6–9, 21]).The property of generalizedf-projection operator plays an important role in solving the system of multivalued mixed variational inequality problem. From Lemma 2, it is easy to see that the following lemma holds trivially.Lemma 8.
The problem (11) is equivalent to finding (x,y)∈K×K, u∈E(x), and v∈F(y) such that
(15)x=PKf,ρ1(x-ρ1S(x,v)),y=PKf,ρ2(y-ρ2T(y,u)),
where ρ1,ρ2>0.For any given(x0,y0)∈K×K, we choose u0∈E(x0) and v0∈F(y0). By Nadler Jr. [22], for any (x1,y1)∈K×K and ɛ>0, there exist u1∈E(x1) and v1∈F(y1) such that
(16)∥u0-u1∥≤(1+ɛ)H(E(x0),E(x1)),∥v0-v1∥≤(1+ɛ)H(F(y0),F(y1)).Based on (15) and (16), we can construct the following algorithm for solving the problem (11).Algorithm 9.
Assume thatK,S,T,E,F, and f are the same as in the problem (11). For any given (x0,y0)∈K×K, u0∈E(x0), and v0∈F(y0), we compute xn+1, yn+1, un+1, and vn+1 as follows:
(17)xn+1=(1-αn)xn+αnPKf,ρ1(xn-ρ1S(xn,vn)),yn+1=(1-βn)yn+βnPKf,ρ2(yn-ρ2T(yn,un)),vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvun+1∈E(xn+1),∥un+1-un∥≤(1+(n+1)-1)H(E(xn+1),E(xn)),vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvn+1∈F(yn+1),∥vn+1-vn∥≤(1+(n+1)-1)H(F(yn+1),F(yn)),mmmmmmmmmmmmmmmmmmn=0,1,2,…,
where 0<αn,βn≤1 are constants.Let{Kn}n=0+∞ be a family of nonempty closed convex subsets in H. Now we construct the following perturbational algorithm for the problem (12).Algorithm 10.
Assume thatK,S,T, and f are the same as in the problem (11). For any given (x0,y0)∈K×K, we compute xn+1 and yn+1 as follows:
(18)xn+1=PKnf,ρ1(xn-ρ1S(xn,yn)),yn+1=PKnf,ρ2(yn-ρ2T(yn,xn)),mmmmfmmmmn=0,1,2,….


## 4. Existence and Convergence Theorems

In this section, we will prove the existence of solutions for the problem (11) and the unique existence of solutions for the problem (12), respectively. In addition, we will provide the convergence results of Algorithms 9 and 10, respectively.Theorem 11.
LetH be a real Hilbert space, let K be a nonempty closed convex subset of H, and let f:K→R∪{+∞} be a proper, convex, and lower semicontinuous mapping. Let S:H×H→H be μ1-strongly monotone and r1-Lipschitz continuous with respect to the first variable and τ1-Lipschitz continuous with respect to the second variable. Let T:H×H→H be μ2-strongly monotone and r2-Lipschitz continuous with respect to the first variable and τ2-Lipschitz continuous with respect to the second variable. Let E:H→C(H) be ξ1-H-Lipschitz continuous; let F:H→C(H) be ξ2-H-Lipschitz continuous, where C(H) denotes the collection of all closed subsets of H. Suppose that 0<α,β≤1, and ρ1,ρ2>0 satisfy
(19)θ1=1-2ρ1μ1+ρ12r12<1,θ2=1-2ρ2μ2+ρ22r22<1,lim⁡n→+∞αn=α,lim⁡n→+∞βn=β,βρ2τ2ξ1<α(1-θ1),αρ1τ1ξ2<β(1-θ2).
Then (xn,yn,un,vn) generated by Algorithm 9 converges strongly to (x,y,u,v) as n→+∞; moreover, (x,y,u,v) is a solution of system (11).Proof.
From Algorithm9 and Lemma 3, we have
(20)∥xn+1-xn∥≤(1-αn)∥xn-xn-1∥+αn∥PKf,ρ1(xn-ρ1S(xn,vn))mmmmmmm-PKf,ρ1(xn-1-ρ1S(xn-1,vn-1))∥≤(1-αn)∥xn-xn-1∥+αn∥xn-xn-1-ρ1(S(xn,vn)-S(xn-1,vn-1))∥≤(1-αn)∥xn-xn-1∥+αnρ1∥S(xn-1,vn)-S(xn-1,vn-1)∥+αn∥xn-xn-1-ρ1(S(xn,vn)-S(xn-1,vn))∥.
Since S is μ1-strongly monotone and r1-Lipschitz continuous with respect to the first variable, we obtain
(21)∥xn-xn-1-ρ1(S(xn,vn)-S(xn-1,vn))∥2≤∥xn-xn-1∥2-2ρ1〈S(xn,vn)-S(xn-1,vn),xn-xn-1〉+ρ12∥S(xn,vn)-S(xn-1,vn)∥2≤∥xn-xn-1∥2-2μ1ρ1∥xn-xn-1∥2+ρ12r12∥xn-xn-1∥2.
The ξ2-H-Lipschitz continuity of F yields that
(22)∥S(xn-1,vn)-S(xn-1,vn-1)∥≤τ1∥vn-vn-1∥≤τ1(1+1n)H(F(yn),F(yn-1))≤τ1(1+1n)ξ2∥yn-yn-1∥.
It follows from inequalities (20)–(22) that
(23)∥xn+1-xn∥≤(1-αn)∥xn-xn-1∥+αnρ1τ1ξ2(1+1n)∥yn-yn-1∥+αn1-2ρ1μ1+ρ12r12∥xn-xn-1∥=(1-αn+αnθ1)∥xn-xn-1∥+an∥yn-yn-1∥,
where an=αnρ1τ1ξ2(1+1/n) and θ1=1-2ρ1μ1+ρ12r12.
By the assumptions ofT and E, following very similar arguments from (20)–(23), we have
(24)∥T(yn-1,un)-T(yn-1,un-1)∥≤τ2∥un-un-1∥≤τ2(1+1n)H(E(xn),E(xn-1))≤τ2(1+1n)ξ1∥xn-xn-1∥,(25)∥yn-yn-1-ρ2(T(yn,un)-T(yn-1,un))∥2≤(1-2ρ2μ2+ρ22r22)∥yn-yn-1∥2.
Hence,
(26)∥yn+1-yn∥≤(1-βn)∥yn-yn-1∥+βnρ2τ2ξ1(1+1n)×∥xn-xn-1∥+βn1-2ρ2μ2+ρ22r22∥yn-yn-1∥=(1-βn+βnθ2)∥yn-yn-1∥+bn∥xn-xn-1∥,
where bn=βnρ2τ2ξ1(1+1/n) and θ2=1-2ρ2μ2+ρ22r22.
Now (23) and (26) imply that
(27)∥xn+1-xn∥+∥yn+1-yn∥≤(1-αn+αnθ1+bn)∥xn-xn-1∥+(1-βn+βnθ2+an)∥yn-yn-1∥≤λn(∥xn-xn-1∥+∥yn-yn-1∥),
where
(28)λn=max⁡{1-αn+αnθ1+bn,1-βn+βnθ2+an},hhhhhhhhhhhhhhhhhhhhhhhhhh∀n=1,2,….
Since lim⁡n→+∞⁡αn=α and lim⁡n→+∞⁡βn=β, we know that lim⁡n→+∞⁡λn=λ, where
(29)λ=max⁡{1-α(1-θ1)+βρ2τ2ξ1,hhhhh1-β(1-θ2)+αρ1τ1ξ2}.
It is clear that 0<λ<1 due to (19). Consequently, there exist N>0 and λ<λ^<1 such that λn≤λ^ for all n≥N. It follows from (27) that
(30)∥xm-xn∥+∥ym-yn∥≤∑k=nm-1‍(∥xk+1-xk∥+∥yk+1-yk∥)≤∑k=nm-1‍λk(∥xk-xk-1∥+∥yk-yk-1∥)≤∑k=nm-1‍λ^k-N(∥xN+1-xN∥+∥yN+1-yN∥),
for all m>n≥N. Letting n→+∞ in (30), we obtain lim⁡n→+∞∥xm-xn∥=0 and lim⁡n→+∞∥ym-yn∥=0, and so {xn} and {yn} are Cauchy sequences in K. Therefore, there exist x,y∈K such that lim⁡n→+∞xn=x and lim⁡n→+∞yn=y.
From inequalities (22) and (24), we know that {un} and {vn} are both Cauchy sequences in H, and so there exist u,v∈H such that lim⁡n→+∞un=u and lim⁡n→+∞vn=v. Since un∈E(xn) and vn∈F(yn), we have
(31)d(u,E(x))=inf⁡{∥u-z∥,z∈E(x)}≤∥u-un∥+d(un,E(x))≤∥u-un∥+H(E(xn),E(x))≤∥u-un∥+ξ1∥xn-x∥⟶0,d(v,F(y))=inf⁡{∥v-z∥,z∈F(y)}≤∥v-vn∥+d(vn,F(y))≤∥v-vn∥+H(F(yn),F(y))≤∥v-vn∥+ξ2∥yn-y∥⟶0.
Thus, u∈E(x) and v∈F(y). By the continuity of PKf, S, and T, it follows from (17) that
(32)x=PKf,ρ1(x-ρ1S(x,v)),y=PKf,ρ2(y-ρ2T(y,u)).
Lemma 8 shows that (x,y,u,v) is a solution of system (11). This completes the proof.Theorem 12.
LetH be a real Hilbert space and let K be a nonempty closed convex subsets of H. Let f:H→R∪{+∞} be a convex and uniformly continuous mapping and let {Kn}n=0+∞⊂H be a family of nonempty closed convex subsets such that f is proper on each Kn and H(Kn,K)→0 as n→+∞. Let S,T be the same as in Theorem 11. Suppose that ρ1,ρ2>0 are such that
(33)1-2ρ1μ1+ρ12r12+ρ2τ2<1,1-2ρ2μ2+ρ22r22+ρ1τ1<1.
Then (xn,yn) generated by Algorithm 10 converges strongly to (x,y) as n→+∞; moreover, (x,y) is the unique solution of system (12).Proof.
Define the norm∥·∥1 on product space H×H by
(34)∥(u,v)∥1=∥u∥+∥v∥,∀(u,v)∈H×H.
It is easy to see that (H×H,∥·∥1) is a Banach space. Let G:K×K→K×K be defined by
(35)G(u,v)=(PKf,ρ1(u-ρ1S(u,v)),PKf,ρ2(v-ρ2T(v,u))),mmmmmmmmmmmmmmm∀(u,v)∈H×H.
For any(u1,v1),(u2,v2)∈H×H, it follows from Lemma 3 that
(36)∥G(u1,v1)-G(u2,v2)∥=∥PKf,ρ1(u1-ρ1S(u1,v1))-PKf,ρ1(u2-ρ1S(u2,v2))∥+∥PKf,ρ2(v1-ρ2T(v1,u1))-PKf,ρ2(v2-ρ2T(v2,u2))∥≤∥u1-u2-ρ1S(u1,v1)+ρ1S(u2,v2)∥+∥v1-v2-ρ2T(v1,u1)+ρ2T(v2,u2)∥≤∥u1-u2-ρ1S(u1,v1)+ρ1S(u2,v1)∥+ρ1∥S(u2,v1)-S(u2,v2)∥+∥v1-v2-ρ2T(v1,u1)+ρ2T(v2,u1)∥+ρ2∥T(v2,u1)-T(v2,u2)∥≤∥u1-u2-ρ1S(u1,v1)+ρ1S(u2,v1)∥+ρ1τ1∥v1-v2∥+∥v1-v2-ρ2T(v1,u1)+ρ2T(v2,u1)∥+ρ2τ2∥u1-u2∥.
By the assumptions imposed on S and T, we obtain(37)∥u1-u2-ρ1S(u1,v1)+ρ1S(u2,v1)∥2≤∥u1-u2∥2-2ρ1〈S(u1,v1)-S(u2,v1),u1-u2〉+ρ12∥S(u1,v1)-S(u2,v1)∥2≤(1-2μ1ρ1+ρ12r12)∥u1-u2∥2,∥v1-v2-ρ2T(v1,u1)+ρ2T(v2,u1)∥2≤∥v1-v2∥2-2ρ2〈T(v1,u1)-T(v2,u1),v1-v2〉+ρ22∥T(v1,u1)-T(v2,u1)∥2≤(1-2μ2ρ2+ρ22r22)∥v1-v2∥2.
From (36)-(37), we have
(38)∥G(u1,v1)-G(u2,v2)∥≤(1-2ρ1μ1+ρ12r12+ρ2τ2)∥u1-u2∥+(1-2ρ2μ2+ρ22r22+ρ1τ1)∥v1-v2∥≤k∥(u1,v1)-(u2,v2)∥1,
where k=max⁡{1-2ρ1μ1+ρ12r12+ρ2τ2,1-2ρ2μ2+ρ22r22+ρ1τ1}. It follows assumption (33) that 0<k<1. This shows that G:K×K→K×K is a contractive operator, and so there exists a unique (x,y)∈K×K such that G(x,y)=(x,y). Thus, (x,y) is the unique solution of system (12).
Now we prove thatxn→x and yn→y as n→+∞. In fact, it follows from (18) that
(39)∥xn+1-x∥=∥PKnf,ρ1(xn-ρ1S(xn,yn))-PKf,ρ1(x-ρ1S(x,y))∥≤∥PKnf,ρ1(xn-ρ1S(xn,yn))-PKnf,ρ1(x-ρ1S(x,y))∥+∥PKnf,ρ1(x-ρ1S(x,y))-PKf,ρ1(x-ρ1S(x,y))∥,∥yn+1-y∥=∥PKnf,ρ2(yn-ρ2T(yn,xn))-PKf,ρ2(y-ρ2T(y,x))∥=∥PKnf,ρ2(yn-ρ2T(yn,xn))-PKnf,ρ2(y-ρ2T(y,x))∥+∥PKnf,ρ2(y-ρ2T(y,x))-PKf,ρ2(y-ρ2T(y,x))∥.
Following very similar arguments from (36)-(38), we have
(40)∥PKnf,ρ1(xn-ρ1S(xn,yn))-PKnf,ρ1(x-ρ1S(x,y))∥+∥PKnf,ρ2(yn-ρ2T(yn,xn))-PKnf,ρ2(y-ρ2T(y,x))∥≤k(∥xn-x∥+∥yn-y∥).
It follows from (39)-(40) that
(41)∥xn+1-x∥+∥yn+1-y∥≤k(∥xn-x∥+∥yn-y∥)+bn,
where
(42)bn=∥PKnf,ρ1(x-ρ1S(x,y))-PKf,ρ1(x-ρ1S(x,y))∥+∥PKnf,ρ2(y-ρ2T(y,x))-PKf,ρ2(y-ρ2T(y,x))∥.
An application of Lemma 4 yields that bn→0, as n→+∞. Now Lemma 5 implies that ∥xn+1-x∥+∥yn+1-y∥→0, and so xn→x and yn→y as n→+∞. This completes the proof.Remark 13.
Theorems11 and 12 improve and generalize some known corresponding results. (i)
IfK=H, E=F≡I, S(x,y)=sA(y,x)+x-y, and T(y,x)=rA(x,y)+x-y, for all x,y∈H, then the problem (11) is equivalent to the problem (13) studied by Petrot [5]. We can get the existence and convergence results of solutions for the problem (13) from Theorems 11 and 12.(ii)
IfE=F≡I, f≡0, S(x,y)=sA(y,x), and T(y,x)=rA(x,y), for all x,y∈H, then the problem (11) is equivalent to the problem (14) studied by Chang et al. [4]. We can get the existence and convergence results of solutions for the problem (14) from Theorems 11 and 12.



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