# A System of Generalized Variational-Hemivariational Inequalities with Set-Valued Mappings

Authors: Zhi-bin Liu; Jian-hong Gou; Yi-bin Xiao; Xue-song Li
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

By using surjectivity theorem of pseudomonotone and coercive operators rather than the KKM theorem and fixed point theorem used in recent literatures, we obtain some conditions under which a system of generalized variational-hemivariational inequalities concerning set-valued mappings, which includes as special cases many problems of hemivariational inequalities studied in recent literatures, is solvable. As an application, we prove an existence theorem of solutions for a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives.


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## Body

## 1. Introduction

LetV1,V2,…,Vn be real, separable, reflexive Banach spaces with dual spaces V1*,V2*,…,Vn*, and let X1,X2,…,Xn be real reflexive Banach spaces with dual spaces X1*,X2*,…,Xn* such that there exist linear continuous and compact operators Ti:Vi→Xi for i=1,2,…,n. We denote by 〈·,·〉E*×E the duality pairing between Banach space E and its dual E* and by ∥·∥E, ∥·∥E* the norms on the space E and its dual space E*, respectively, where E={Vi,Xi,i=1,2,…,n}. Assume that Ai:∏k=1n‍Vk→2Vi*, i=1,2,…,n are set-valued mappings on the product space ∏k=1n‍Vk of Banach spaces V1,…,Vn and that J:∏k=1n‍Xk→R is a functional on the product space ∏k=1n‍Xk of Banach spaces X1,…,Xn, which is locally Lipschitz with respect to each component; that is, for all i=1,2,…,n, the functionals J(x1,x2,…,xi-1,·,xi+1,…,xn):Xi→R are locally Lipschitz for all fixed xj, j≠i, and Gi:Vi→R∪{+∞},  i=1,2,…,n are proper, convex, and lower semicontinuous functionals. In this paper, we study a system of generalized variational-hemivariational inequalities concerning set-valued mappings, which is specified as follows.For alli=1,2,…,n, find ui∈Vi and μi∈Ai(u) such that
(1)(P)〈μi,vi-ui〉Vi*×Vi+Ji∘(u^;v^i-u^i)+Gi(vi)-Gi(ui)≥0,∀vi∈Vi,
where u=(u1,u2,…,un)∈∏k=1n‍Vk,u^=(u^1,u^2,…,u^n)=(T1u1,T2u2,…,Tnun), v^i=Tivi and Ji∘(u^;v^i-u^i) is the partial generalized directional derivative (in the sense of Clarke) of the functional J, which is locally Lipschitz for each component, with respect to the ith component at the point u^i∈Xi in the direction v^i-u^i for all given u^j∈Xj, j≠i, which can be defined by
(2)Ji∘(u^;v^i-u^i)=limsupx→u^i,λ↓0J(u^1,…,u^i-1,x+λ(v^i-u^i),u^i+1,…,u^n)λ-J(u^1,…,u^i-1,x,u^i+1,…,u^n)λ.In the last few years, there are many researchers who dedicated themselves to the study of various types of hemivariational inequalities and systems of hemivariational inequalities, which are a generalization of the variational inequalities, and related problems such as equilibrium problems. In these papers, based on Clarke’s generalized directional derivative and Clarke's generalized gradient for locally Lipchitz functions, the researchers study the existence and uniqueness of solution by mainly using KKM theorems, surjectivity theorems for pseudomonotone and coercive operators, fixed point theorems, critical point theory, and so on. We refer readers for the study of hemivariational inequalities to monographs of Carl et al. [1], Migórski et al. [2], Naniewicz and Panagiotopoulos [3], and Panagiotopoulos [4]. For the system of hemivariational inequalities, Denkowski and Migórski [5] studied a dynamic thermoviscoelastic frictional contact problem which was modeled by a system of evolution hemivariational inequalities. They proved the existence and uniqueness of the weak solution for the problem by using a surjectivity result for operators of pseudomonotone type. In 2011, Repovš and Varga [6] studied the Nash equilibrium point by using the Ky Fan version of the KKM theorem and the Tarafdar fixed point theorem for a class of hemivariational inequality system. It is obvious that some problems studied in literatures are special cases of our system of generalized variational-hemivariational inequalities under some special conditions, such as n=1, Ai are single-valued, or Gi are indicators of some convex subsets Ki for i=1,2,…,n.  Although it seems that our problem (P) cannot include the problem studied in [6] as a special case, we remark here that, in essence, the problem (P) is a generalization of the problem in [6] since, when Ai (i=1,2,…,n) are single-valued and Gi are the indicators of the convex subsets Ki, the problem (P) reduces to the problem studied by Repovš and Varga [6] with Ti=Si and Ai∘(Tu;Tivi-Tiui) being incorporated into Ji∘(Su;Sivi-Siui) under the regularity condition. For more information on the research of hemivariational inequalities and systems of hemivariational inequalities, we can refer to [7–16] and references therein.It is well known that, by surjectivity theorem of pseudomonotone and coercive operators, there exists solutionui to each variational-hemivariational inequality in the system (1) for all uj∈Vj, j≠i under some suitable conditions on the operators Ai, J, and Gi. A natural question is whether these conditions are sufficient for the existence of solutions to the system (1) which is combined by solvable variational-hemivariational inequalities. If not, what other stronger conditions do we need to guarantee the solvability of the system (1)? In this paper, we are devoted to these questions by using surjectivity theorem of pseudomonotone and coercive operators rather than the KKM theorem, and the fixed point theorem used by Repovš and Varga in [6] to obtain the existence of the solutions to the problem (P) of a system of generalized variational-hemivariational inequalities concerning set-valued mappings.As will be seen in the proof of our main theorem in Section3, the case where n=k for any finite positive integer k>2 is a natural generalization of the case where n=2. Therefore, in what follows, We will focus on the problem of a system of two generalized variational-hemivariational inequalities, which can be reformulated as follows. Consider u1∈V1,  u2∈V2, μ1∈A1(u) and μ2∈A2(u) such that
(3)(P′){〈μ1,v1-u1〉V1*×V1+J1∘(u^;v^1-u^1)+G1(v1)-G1(u1)≥0,∀v1∈V1,〈μ2,v2-u2〉V2*×V2+J2∘(u^;v^2-u^2)+G2(v2)-G2(u2)≥0,∀v2∈V2,
where u=(u1,u2), u^=(u^1,u^2)=(T1u1,T2u2) and v^i=Tivi for i=1,2.The paper is structured as follows. In Section2, we recall some preliminary material. Section 3 gives conditions under which the problem (P) of a system of generalized variational-hemivariational inequalities concerning set-valued mapping is solvable by considering the simple case, the problem (P′) of a system of two generalized variational-hemivariational inequalities. At last, in Section 4, we are concerned with an application of our results to a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives.


## 2. Preliminaries

In this section, we recall some important notations and useful results on nonlinear analysis, nonsmooth analysis, and operators of monotone type, which can be found in [2, 3, 17, 18].Without confusion of symbols, we suppose, just in this section, thatX is a Banach space with its dual X* and duality paring 〈·,·〉 between X* and X, G:X→R∪{+∞} is a proper and convex functional, and J:X→R is a locally Lipschitz functional with Clarke's generalized directional derivative J∘(u,v). We denote by ∂^G(u):X→2X*∖{∅} and ∂J(u):X→2X*∖{∅} the subgradient of the convex functional G in the sense of convex analysis and Clarke's generalized gradient of the locally Lipschitz functional J, respectively. Then,
(4)∂^G(u)={u*∈X*:G(v)-G(u)≥〈u*,v-u〉,∀v∈Xu*∈X*:G(v)-G(u)},∂J(u)={ω∈X*:J∘(u,v)≥〈ω,v〉,∀v∈X≥〈ω,v〉}.We have the following basic properties on Clarke's generalized directional derivative and Clarke's generalized gradient (see, e.g., [2, 17]).Proposition 1.
LetX be Banach space, and let u,v∈X, and J be locally Lipschitz functional defined on X. Then one has the following.  (1)
The functionv↦J∘(u,v)  is finite, positively homogeneous, subadditive, and then convex onX;(2)
J∘(u,v)  is upper semicontinuous as a function of(u,v), but as a function ofv  alone, it is Lipschitz continuous onX.(3)
∂J(u)  is a nonempty, convex, bounded, and weak*-compact subset ofX*.(4)
For everyv∈X, one has(5)J∘(u,v)=max{〈ξ,v〉:ξ∈∂J(u)}.(5)
The graph of the Clarke generalized gradient  ∂J(u)  is closed in  X×(w*-X*)  topology.(6)
The multifunction  X∋u→∂J(u)⊆X*  is upper semicontinuous from  X  intow*-X*.Definition 2.
A locally Lipschitz functionalJ:X→R is said to be regular (in the sense of Clarke) at u∈X if (i)
for allv∈X the directional derivative J′(u,v) exists;(ii)
for allv∈X, J′(u,v)=J∘(u,v), where J′(u,v) is directional derivative of J at u∈X in the direction v∈X, which is defined by
(6)J′(u,v)=limλ↓0J(u+λv)-J(u)λ,
whenever this limit exists. The functional J is regular (in the sense of Clarke) on X if it is regular at every point u∈X.Proposition 3.
LetX1 and X2 be two Banach spaces. If J:X1×X2→R is locally Lipschitz and either J or -J is regular at u=(u1,u2)∈X1×X2, then
(7)∂J(u1,u2)⊆∂1J(u1,u2)×∂2J(u1,u2),
or equivalently one has
(8)J∘(u1,u2;v1,v2)≤J1∘(u1,u2;v1)+J2∘(u1,u2;v2),hhhhhhhhhhhhhhhhhhhhh∀(v1,v2)∈X1×X2,
where ∂1(u1,u2) (resp., ∂2(u1,u2)) represents the partial generalized subgradient of J(·,u2) (resp., J(u1,·)) and J1∘(u1,u2;v1) (resp., J2∘(u1,u2;v2)) denotes the partial generalized directional derivative of J(·,u2) (resp., J(u1,·)) at the point u1 (resp., u2) in the direction v1 (resp., v2), but the converse of inclusion (7) and inequality (8) is not true in general.Definition 4.
LetX be real reflexive Banach space with dual X*. A mapping T from X into 2X* is said to be pseudomonotone if  (1)
the setTu is nonempty, bounded, closed, and convex for all u∈X;(2)
T is upper semicontinuous from each finite dimensional subspace of X to X* endowed with the weak topology;(3)
{ui} is a sequence in X converging weakly to u, and ui*∈Tui is such that
(9)limsup〈ui*,ui-u〉≤0,
then for each element v∈X there exists u*(v)∈Tu such that
(10)liminf〈ui*,ui-v〉≥〈u*(v),u-v〉.Definition 5.
LetX be real reflexive Banach space with dual X*. A mapping T from X into 2X* is said to be generalized pseudomonotone if for any sequences {ui}⊂X, {ui*}⊂X* with ui*∈Tui, ui→u weakly in X, ui*→u* weakly in X* and
(11)limsup〈ui*,ui-u〉≤0;
then one has u*∈Tu and 〈ui*,ui〉→〈u*,u〉.Proposition 6.
LetX be real reflexive Banach space with dual X* and let T1, T2 be two pseudomonotone mappings from X into 2X*. Then T1+T2 is pseudomonotone.Proposition 7.
LetX be real reflexive Banach space with dual X* and let T:X→2X* be a pseudomonotone mapping from X into 2X*. Then T is a generalized pseudomonotone.Proposition 8.
LetX be real reflexive Banach space with dual X* and let T be a bounded, generalized pseudomonotone mapping from X into 2X*. Assume that, for each u∈X, Tu is a nonempty closed convex subset of X*. Then T is pseudomonotone.Definition 9.
LetX be real reflexive Banach space with dual X*. The operator T:X→2X* is said to be as follows:  (1)
monotone if for all(u,u*), (v,v*) lying in the graph G(T) of T, one has
(12)〈u*-v*,u-v〉≥0.(2)
maximal monotone if it is monotone and if(u,u*)∈X×X* is such that
(13)〈u*-v*,u-v〉≥0,∀(v,v*)∈G(T);
then (u,u*)∈G(T).(3)
quasibounded if for eachM>0 there exists K(M)>0 such that, whenever (u,u*)∈G(T), 〈u*,u〉≤M∥u∥, and ∥u∥≤M; then
(14)∥u*∥≤K(M).(4)
strongly quasibounded if for eachM>0 there exists K(M)>0 such that for all (u,u*)∈G(T) with 〈u*,u〉≤M and ∥u∥≤M, one has
(15)∥u*∥≤K(M).Definition 10.
LetX be real reflexive Banach space with dual X*. A mapping T from X into 2X* is said to be as follows:  (1)
coercive if there exists a real-valued functionc on R+ with limr→∞c(r)=∞ such that for all (u,u*)∈G(T), one has
(16)〈u*,u〉≥c(∥u∥)∥u∥,(2)
coercive with constantα>0 if
(17)〈u*,u〉≥α∥u∥2,(3)
u0-coercive if there exists a real-valued function c on R+ with limr→∞c(r)=∞ such that for some u0∈X and for all (u,u*)∈G(T), one has
(18)〈u*,u-u0〉≥c(∥u∥)∥u∥.The following theorem is a surjectivity theorem for the sum of a pseudomonotone, coercive operator, and a maximal monotone operator, which is important to the proof of our main results.Theorem 11 (see [3]).
LetX be a real reflexive Banach space with dual X*, let T^ be a maximal monotone mapping from X into 2X* with u0∈D(T)={x:Tx≠∅}, and let T be a u0-coercive, pseudomonotone operator from X into 2X*. Suppose further that either Tu0:X→2X* is quasibounded or T^u0:X→2X* is strongly quasibounded, where Tu0(v)=T(u0+v) and the same for T^u0. Then R(T+T^)=X*.


## 3. Main Results

In this section, we first give an existence theorem for the solution to the problem (P′) of a system of two generalized variational-hemivariational inequalities. And then, as a natural generalization, an existence theorem for the solution to the problem (P), a system of generalized variational-hemivariational inequalities concerning set-valued mappings is also obtained.Before we present the main existence theorem, for the simplicity of writing, we define some useful symbols and give a crucial lemma in advance, which establishes the relationship between the problem (P′) of a system of two variational-hemivariational inequalities and a generalized vector variational-hemivariational inequality. Let V=V1×V2. Endowed with the norm defined by
(19)∥u∥V=∥u1∥V1+∥u2∥V2,∀u=(u1,u2)∈V,V is a reflexive Banach space with dual V*. The duality pairing between V and V* is given by
(20)〈u*,u〉V*×V=〈u1*,u1〉V1*×V1+〈u2*,u2〉V2*×V2,hhhhhhh∀u*=(u1*,u2*)∈V*,u=(u1,u2)∈V.On the Banach spaceV defined above, we further define a set-valued mapping A:V→2V*, an operator T:V→X1×X2, and a functional G:V→R∪{+∞}, which are specified as follows. For all u=(u1,u2)∈V, one has
(21)A(u)=(A1(u),A2(u)),T(u)=(T1u1,T2u2),G(u)=G1(u1)+G2(u2).Lemma 12.
Assume thatGi:Vi→R∪{+∞},  i=1,2 are proper, convex, and lower semicontinuous functionals. The functional G defined above is also a proper, convex, and lower semicontinuous functional on V. Moreover, ∂^G(u)=∂^G1(u1)×∂^G2(u2).Proof.
Since the functionalsGi:Vi→R∪{+∞}, i=1,2 are proper and convex, it is easy to show that G is also proper and convex by the inequality G(λu+(1-λv))=G1(λu1+(1-λ)v1)+G2(λu2+(1-λ)v2)≤λG(u)+(1-λ)G(v) for all u,v∈V and λ∈[0,1]. As for the lower semicontinuity of the functional G, by assuming that un→u in V, which implies u1n→u1 in V1 and u2n→u2 in V2, we can get from the lower semicontinuity of G1 and G2 that
(22)liminfG(un)=liminf(G1(u1n)+G2(u2n))≥liminfG1(u1n)+liminfG2(u2n)≥G1(u1)+G2(u2)=G(u),
which means that G is lower semicontinuous on V.
Now, we prove the equality∂^G(u)=∂^G1(u1)×∂^G2(u2). Assume that μ∈∂^G(u)⊂V*, which says that
(23)G(v)-G(u)≥〈μ,v-u〉V*×V,∀v∈V.
In particular, for anyv1∈V1, let v=(v1,u2) in (23), and then we can get that
(24)G1(v1)-G1(u1)≥〈μ1,v1-u1〉V1*×V1,∀v1∈V1.
Similarly, by lettingv=(u1,v2) in (23) for any v2∈V2, we can obtain
(25)G2(v2)-G2(u2)≥〈μ2,v2-u2〉V2*×V2,∀v2∈V2,
which together with (24) implies that μ∈∂^G1(u1)×∂^G2(u2); that is, ∂^G(u)⊆∂^G1(u1)×∂^G2(u2).
Conversely, letμ=(μ1,μ2)∈∂^G1(u1)×∂^G2(u2). For all vi∈Vi,  i=1,2, it follows from μi∈∂^Gi(ui) that
(26)G1(v1)-G1(u1)≥〈μ1,v1-u1〉V1*×V1,∀v1∈V1,G2(v2)-G2(u2)≥〈μ2,v2-u2〉V2*×V2,∀v2∈V2.By adding the two inequalities (26), we obtain
(27)G(v)-G(u)≥〈μ,v-u〉V*×V,∀v∈V,
which implies that μ∈∂^G(u); that is, ∂^G(u)⊇∂^G1(u1)×∂^G2(u2). This completes the proof of Lemma 12.Now, we consider the following generalized vector variational-hemivariational inequality. Findu=(u1,u2)∈V and μ=(μ1,μ2)∈A(u) such that
(28)(P′′)〈μ,v-u〉V*×V+J∘(Tu,Tv-Tu)+G(v)hhhhhhhhhhhhhhh-G(u)≥0,∀v∈V.We first give a crucial lemma which establishes the relationship between the problem (P′) of a system of two variational-hemivariational inequalities and the problem (P′′) of a generalized vector variational-hemivariational inequality.Lemma 13.
Assume that the locally Lipschitz functionalJ:X1×X2→R is regular on X1×X2. Then any solution u=(u1,u2)∈V to the problem (P′′) is always a solution to the problem (P′).Proof.
Assume thatu=(u1,u2) solves the problem (P′′), which says that there exists an μ=(μ1,μ2)∈A(u) such that for all v∈V, one has
(29)〈μ,v-u〉V*×V+J∘(Tu,Tv-Tu)+G(v)-G(u)≥0.
Specially, for anyv1∈V1, let v=(v1,u2)∈V in (29), and then we can get from Proposition 3 that
(30)0≤〈μ1,v1-u1〉V1*×V1+J∘(Tu,(T1v1-T1u1,0))+G1(v1)-G1(u1)≤〈μ1,v1-u1〉V1*×V1+J1∘(Tu,T1v1-T1u1)+G1(v1)-G1(u1),∀v1∈V1.
Similarly, by lettingv=(u1,v2)∈V in (29) for any v2∈V2, we can obtain that
(31)〈μ2,v2-u2〉V2*×V2+J2∘(Tu,T2v2-T2u2)+G2(v2)-G2(u2)≥0,∀v2∈V2,
which together with the inequality (30) implies that u=(u1,u2) is a solution to the problem (P′). This completes the proof of Lemma 13.Remark 14.
It follows from Proposition3 that, just under regularity condition of the functional J, J∘(u,v)=J1∘(u,v1)+J2∘(u,v2) does not hold in general, while the inequality J∘(u,v)≤J1∘(u,v1)+J2∘(u,v2) is true. Therefore, without other much stronger conditions on functional J, the inverse of the Lemma 13 is not true in general.We give some assumptions on the operatorsAi and J in the system (3) of two generalized variational-hemivariational inequalities.  The assumption (HA) is as follows.
(1)
A1:V1×V2→2V1* is bounded on V1×V2 and pseudomonotone with respect to the first argument; that is, for all u2∈V2, the operator A1(·,u2):V1→2V1* is pseudomonotone on V1.(2)
A2:V1×V2→2V2* is bounded on V1×V2 and pseudomonotone with respect to the second argument; that is, for all u1∈V1, the operator A2(u1,·):V2→2V2* is pseudomonotone on V2.(3)
For allu2∈V2, there exist an element w1∈D(∂^G1)⊂V1 and a constant α1>0 such that
(32)〈u1*,u1-w1〉V1*×V1≥α1∥u1∥V12,hhhh∀u1∈V1,u1*∈A1(u1,u2).(4)
For allu1∈V1, there exist an element w2∈D(∂^G2)⊂V2 and a constant α2>0 such that
(33)〈u2*,u2-w2〉V2*×V2≥α2∥u2∥V22,hhhh∀u2∈V2,u2*∈A2(u1,u2).Remark 15.
It is clear that the hypotheses (1) and (2) in the assumption (HA) imply that the operatorA defined in (21) is also bounded on V. The hypotheses (3) on the operator A1 and (4) on the operator A2 in the assumption (HA) imply the w1-coercivity of A1 with respect to the first argument and w2-coercivity of A2 with respect to the second argument, respectively. Moreover, for w=(w1,w2)∈D(∂^G), the operator A defined in (21) is also w-coercive with constant β=min{α1,α2}/2. In fact, for all μ∈A(u), one has
(34)〈μ,u-w〉V*×V=〈μ1,u1-w1〉V1*×V1+〈μ2,u2-w2〉V2*×V2≥α1∥u1∥V12+α2∥u2∥V22≥min{α1,α2}2∥u∥V2,=β∥u∥V2,
which implies the w-coercivity with constant β of operator A on V. The assumption (HJ) is as follows.
(1)
For allx2∈X2, there exist constants c1,d1≥0 such that
(35)∥η1∥X1*≤c1+d1∥x1∥X1,∀η1∈∂1J(x1,x2).(2)
For allx1∈X1, there exist constants c2,d2≥0 such that
(36)∥η2∥X2*≤c2+d2∥x2∥X2,∀η2∈∂2J(x1,x2).Remark 16.
It is clear that the hypotheses in assumption (HJ) imply that∂1J(·,x2) and ∂2J(x1,·) are bounded on X1 and X2, respectively. Moreover, if J is regular on X, then ∂J is also bounded on X. (In the following, let X=X1×X2 and X*=X1*×X2* for simplicity of writing.) In fact, since J is regular on X, the inclusion ∂J(x1,x2)⊆∂1J(x1,x2)×∂2J(x1,x2) holds. It follows from (35) and (36) that
(37)∥η∥X*≤c+d∥x∥X,∀η∈∂J(x),
with c=c1+c2≥0 and d=max{d1,d2}≥0. This also means that ∂J is bounded on X.We are now in a position to give our main result on the existence of solution to the problem (P′), a system of two generalized variational-hemivariational inequalities.Theorem 17.
Suppose that the set-valued mappingsAi:V1×V2→2Vi*, i=1,2, which satisfy the assumption (HA), is such that the operator A defined in (21) is pseudomonotone on V. Let Ti:Vi→Xi be linear continuous and compact operators, let J:X→R be a regular, locally Lipschitz functional which satisfies the hypothesis (HJ), let and Gi:Vi→R∪{+∞}, i=1,2 be proper, convex, and lower semicontinuous functionals. Then the problem (P′) admits at least one solution under the condition
(38)β>d∥T∥2,
where ∥T∥ is the norm of the operator T defined by (21).Proof.
By Lemma13, the existence of solution to the problem (P′) of a system of two generalized variational-hemivariational inequalities can be proved as long as the problem (P′′) of a generalized vector variational-hemivariational inequality is solvable. Therefore, we consider the following inclusion problem. Find u∈V such that
(39)F(u)+∂^G(u)∋0,
where F:V→2V* with F(u)=A(u)+T*∘∂J∘T(u) for all u∈V. We will prove the existence of solution to the inclusion problem (39) by the surjectivity theorem (Theorem 11), which implies that the problem (P′′) is solvable.
Claim  1(Fis bounded onV). Since the operator A is bounded on V under assumption (HA) by Remark 15, ∂J is also bounded on X under the assumption (HJ) by Remark 16, and T is linear continuous by the linearity and continuity of the operators Ti,i=1,2, and it is easy to check that F is bounded on V, which implies that Fu0:V→2V* with Fu0(u)=F(u0+u) is quasibounded for any u0∈V.
Claim  2   (Fis pseudomonotone onV). Since F=A+T*∘∂J∘T:V→2V* and the operator A is pseudomonotone, we only need to prove that T*∘∂J∘T:V→2V* is pseudomonotone. To this end, firstly, we prove that T*∘∂J∘T is generalized pseudomonotone. Let un→u weakly in V,ξn∈T*(∂J(Tun)) with ξn→ξ weakly in V* and limsup〈ξn,un-u〉V*×V≤0. There exist ηn∈∂J(Tun) such that ξn=T*ηn. Since ∂J is bounded on X by Remark 16, Tun→Tu in X by the compactness of the operators Ti, i=1,2, and ηn∈∂J(Tun), we have the fact that ηn is bounded in X*. Thus there exists a subsequence, which is also denoted by ηn, such that ηn→η weakly in X* with some η∈X*. By using the equality ξn=T*ηn, it is easy to get that ξ=T*η. Since ηn∈∂J(Tun) with ηn→η weakly in X* and Tun→Tu in X, we get by the closedness of ∂J with X×(w*-X*) topology and the reflexivity of X that η∈∂J(Tu), and thus ξ=T*η∈T*(∂J(Tu)). Moreover, it follows from ηn→η weakly in X* and Tun→Tu in X that
(40)〈ξn,un〉V*×V=〈ηn,Tun〉X*×X⟶〈η,Tu〉X*×X=〈T*η,u〉V*×V=〈ξ,u〉V*×V,
which together with ξ∈T*(∂J(Tu)) implies that T*∘∂J∘T is generalized pseudomonotone on V. Secondly, it is easy to check that T*(∂J(Tu)) is nonempty, convex, and closed in V* for all u∈V since ∂J(x) is a nonempty, convex, and closed subset in X* for all x∈X and T is linear and continuous on V. Thirdly, the operator T*∘∂J∘T is bounded on V, which has been proved in Claim 1. Consequently, it follows from the Proposition 8 that T*∘∂J∘T is pseudomonotone on V.
Claim  3  (Fisw-coercive onV). Let u∈V and τ∈F(u), and then there exist μ∈A(u) and η∈∂J(Tu) such that τ=μ+T*η. By Remarks 15 and 16, we have
(41)〈τ,u-w〉V*×V=〈μ,u-w〉V*×V+〈T*η,u-w〉V*×V≥β∥u∥V2+〈η,T(u-w)〉X*×X≥β∥u∥V2-∥η∥X*∥T∥(∥u∥V+∥w∥V)≥β∥u∥V2-(c+d∥Tu∥X)∥T∥(∥u∥V+∥w∥V)≥(β-d∥T∥2)∥u∥V2-∥T∥(c+d∥T∥∥w∥V)∥u∥V-c∥T∥∥w∥V,
which together with the condition β>d∥T∥2 means that F is w-coercive on V with function c(t)=(β-d∥T∥2)t-∥T∥(c+d∥T∥∥w∥V)-c∥T∥∥w∥V/t.
It is well known that∂^G is a maximal monotone operator on V since, by Lemma 12, the functional G is proper, convex, and lower semicontinuous on V (see [18]). We are now in a position to apply Theorem 11 to the set-valued operators F and ∂^G. We deduce that F+∂^G is surjective, which implies that there exist u∈V such that
(42)0∈F(u)+∂^G(u).
By the definition of the operatorF, there exist μ∈A(u), η∈∂J(Tu), and ξ∈∂^G(u) such that
(43)μ+T*η+ξ=0.
By multiplying the equality (43) by v-u for all v∈V, we obtain from the definition of Clarke's generalized subgradient of the functional J and subgradient in the sense of convex analysis of the functional G that
(44)0=〈μ,v-u〉V*×V+〈T*η,v-u〉V*×V+〈ξ,v-u〉V*×V=〈μ,v-u〉V*×V+〈η,Tv-Tu〉X*×X+〈ξ,v-u〉V*×V≤〈μ,v-u〉V*×V+J∘(Tu,Tv-Tu)+G(v)-G(u),
which implies that u solves the problem (P′′) of a generalized vector variational-hemivariational inequality. As stated at the beginning of our proof, u is also a solution to the problem (P′) of a system of two generalized variational-hemivariational inequalities by Lemma 13. This completes the proof of Theorem 17.Remark 18.
The pseudomonotonicity of the operatorA defined in (21) is necessary for the proof of the existence of solution to the problem (P′) by the surjectivity theorem since the pseudomonotonicity of operator A1 with respect to the first argument and the pseudomonotonicity of operator A2 with respect to the second argument, which are necessary to prove the existence of solution to each generalized variational-hemivariational inequality in problem (P′), cannot guarantee the pseudomonotonicity of the operator A defined in (21) in general. However, some special cases in which the pseudomonotonicity of operator A1 with respect to the first argument and the pseudomonotonicity of operator A2 with respect to the second argument imply the pseudomonotonicity of the operator A defined in (21) can be given under some stronger conditions (see [5]).It is obvious that, by similar arguments as proof of Theorem17, we have the following results for the existence of solution to each generalized variational-hemivariational inequality in the system (3).Theorem 19.
Suppose that, fori=1,2, Ai:V1×V2→2Vi* are set-valued mappings satisfying the assumption (HA) and that Ti:Vi→Xi are linear continuous and compact operators. Let J:X→R be a regular, locally Lipschitz functional on X, which satisfies the hypothesis (HJ), and let Gi:Vi→R∪{+∞}, i=1,2 be proper, convex, and lower semicontinuous functionals. Then, the ith i=1,2 generalized variational-hemivariational inequality in the system (3) admits at least one solution ui∈Vi for all uj∈Vj,  j≠i under the condition
(45)αi>di∥Ti∥2,
where ∥Ti∥ is the norm of the operator Ti.Remark 20.
By comparing Theorems17 with 19, we remark here that, in addition to the pseudomonotonicity of the operator A defined in (21), we need strongly condition (38) than (45) to obtain the existence of solution to the problem (P′) of a system of two generalized variational-hemivariational inequalities.As a natural generalization of Theorem17 for the existence of solution to the problem (P′) of a system of two generalized variational-hemivariational inequalities, we can obtain the following theorem for the existence of solution to the problem (P) of a system of generalized variational-hemivariational inequalities concerning set-valued mappings.Theorem 21.
Suppose that the following assumptions on the operators in the problem(P) of a system of generalized variational-hemivariational inequalities hold. (1)
Fori=1,2,…,n, Ai:∏k=1n‍Vk→2Vi* are set-valued mappings satisfying the following.
(a)
Ai are bounded on ∏k=1n‍Vk and pseudomonotone with respect to the ith argument.(b)
the operatorA:V=∏k=1n‍Vk→2V*, which is defined by A(u)=(A1(u),…,An(u)), is pseudomonotone on V.(c)
For alluj∈Vj, j≠i, there exist an element wi∈D(∂^Gi)⊂Vi and a constant αi>0 such that
(46)〈ui*,ui-wi〉Vi*×Vi≥αi∥ui∥Vi2,∀ui∈Vi,ui*∈Ai(u1,…,un).(2)
J:∏k=1n‍Xk→R is a regular and locally Lipschitz functional which satisfies that for all i=1,2,…,n and xj∈Xj, j≠i, there exist constants ci,di≥0 such that
(47)∥ηi∥Xi*≤ci+di∥xi∥Xi,∀xi∈Xi,ηi∈∂iJ(x1,…,xn).(3)
Fori=1,2,…,n, Ti:Vi→Xi are linear continuous and compact operators and Gi:Vi→R∪{+∞} are proper, convex, and lower semicontinuous functionals. Then the problem (P) admits at least one solution under the condition
(48)β~>d~∥T~∥2,
where β~=min{αi}/n, d~=max{di} and ∥T~∥ is the norm of the operator T~:∏k=1n‍Vk→∏k=1n‍Xk defined by T~(u1,…,un)=(T1u1,…,Tnun).


## 4. An Application

In this section, we are concerned with an application of our results to a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives.LetΩ⊂Rn be a bounded and open set in Rn, let V1,V2,…,Vn be real, separable, and reflexive Banach spaces with dual spaces V1*,V2*,…,Vn*. For i=1,2,…,n, Ai:∏k=1n‍Vk→2Vi* are set-valued mappings, Ti:Vi→L2(Ω) are linear continuous and compact operators on Vi, and Gi:Vi→R∪{+∞} are proper, convex, and lower semicontinuous functionals. We consider the following system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives. For all i=1,2,…,n, find ui∈Vi and μi∈Ai(u) such that
(49)〈μi,vi-ui〉Vi*×Vi+∫Ω‍ji∘(x,u^(x);v^i(x)-u^i(x))dx+Gi(vi)-Gi(ui)≥0,∀vi∈Vi,
where u=(u1,u2,…,un)∈∏k=1n‍Vk,u^=(u^1,u^2,…,u^n)=(T1u1,T2u2,…,Tnun)∈L2(Ω;Rn), v^i=Tivi and j(x,y):Ω×Rn→R is a function satisfying the following assumption:  (Hj) is as follows.
(1)
j(x,·) is locally Lipschitz on Rn for a.e.x∈Ω.(2)
j(x,y):Ω×Rn→R is a Carathe´odory function.(3)
Eitherj(x,·) or -j(x,·) is regular on Rn for a.e. x∈Ω.(4)
For alli=1,2,…,n and a.e. x∈Ω, there exists constant ci,di≥0 such that
(50)|s|≤ci+di|yi|,∀y∈Rn,s∈∂ij(x,y).Remark 22.
The problem (49) we considered in this section includes the problem studied by Panagiotopoulos et al. [19] by using Brouwer's fixed point theorem as a special case where n=1, Ai is single-valued and Gi is an indicator of a convex subset K.We define a functionalJ on L2(Ω,Rn) as follows:
(51)J(u)=∫Ω‍j(x,u(x))dx,∀u∈L2(Ω,Rn).It follows from Theorem 3.47 in [2] that, under the assumption (Hj) on the function j, J defined by (51) is a locally Lipschitz functional on L2(Ω,Rn), which satisfies
(52)Ji∘(u,vi)=∫Ω‍ji∘(x,u(x),vi(x))dx,∀u∈L2(Ω,Rn),vi∈L2(Ω),∥η∥L2(Ω)≤c~i+d~i∥ui∥L2(Ω),∀u∈L2(Ω,Rn),η∈∂iJ(u),
where c~i=2ci|Ω|≥0 and d~i=2di≥0.Now, under the conditions (52), we are in a position to apply our result, Theorem 21, to the problem (49), a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives. We conclude this section with the following theorem, which gives the existence of solution to the problem (49).Theorem 23.
For the problem (49), a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives, one assumes the following.  (1)
Fori=1,2,…,n, Ai:∏k=1n‍Vk→2Vi* are set-valued mappings satisfying the following.
(a)
Ai are bounded on ∏k=1n‍Vk and pseudomonotone with respect to the ith argument.(b)
the operatorA:V=∏k=1n‍Vk→2V*, which is defined by A(u)=(A1(u),…,An(u)), is pseudomonotone on V.(c)
For alluj∈Vj, j≠i, there exist an element wi∈D(∂^Gi)⊂Vi and a constant αi>0 such that
(53)〈ui*,ui-wi〉Vi*×Vi≥αi∥ui∥Vi2,∀ui∈Vi,ui*∈Ai(u1,…,un).(2)
j(x,y):Ω×Rn→R is a function satisfying the assumption  (Hj).(3)
Fori=1,2,…,n, Ti:Vi→L2(Ω) are linear continuous and compact operators and Gi:Vi→R∪{+∞} are proper, convex, and lower semicontinuous functionals. Then the problem (49) admits at least one solution under the condition
(54)β~>d~∥T~∥2,
where β~=min{αi}/n, d~=2max{di}, and ∥T~∥ is the norm of the operator T~:∏k=1n‍Vk→L2(Ω;Rn) defined by T~(u1,…,un)=(T1u1,…,Tnun).



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