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

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.

1. Introduction

Let 𝑉₁,𝑉₂,…,𝑉_𝑛 be real, separable, reflexive Banach spaces with dual spaces 𝑉^∗₁,𝑉^∗₂,…,𝑉^∗_𝑛, and let 𝑋₁,𝑋₂,…,𝑋_𝑛 be real reflexive Banach spaces with dual spaces 𝑋^∗₁,𝑋^∗₂,…,𝑋^∗_𝑛 such that there exist linear continuous and compact operators 𝑇_𝑖∶𝑉_𝑖→𝑋_𝑖 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={𝑉_𝑖,𝑋_𝑖,𝑖=1,2,…,𝑛}. Assume that 𝐴_𝑖∶∏^𝑛_𝑘=1𝑉_𝑘→2^𝑉∗𝑖, 𝑖=1,2,…,𝑛 are set-valued mappings on the product space ∏^𝑛_𝑘=1𝑉_𝑘 of Banach spaces 𝑉₁,…,𝑉_𝑛 and that 𝐽∶∏^𝑛_𝑘=1𝑋_𝑘→𝑅 is a functional on the product space ∏^𝑛_𝑘=1𝑋_𝑘 of Banach spaces 𝑋₁,…,𝑋_𝑛, which is locally Lipschitz with respect to each component; that is, for all 𝑖=1,2,…,𝑛, the functionals 𝐽(𝑥₁,𝑥₂,…,𝑥_𝑖−1,⋅,𝑥_𝑖+1,…,𝑥_𝑛)∶𝑋_𝑖→𝑅 are locally Lipschitz for all fixed 𝑥_𝑗, 𝑗≠𝑖, and 𝐺_𝑖∶𝑉_𝑖→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 all 𝑖=1,2,…,𝑛, find 𝑢_𝑖∈𝑉_𝑖 and 𝜇_𝑖∈𝐴_𝑖(𝐮) such that (P)𝜇_𝑖,𝑣_𝑖−𝑢_{𝑖𝑉𝑖∗×𝑉𝑖}+𝐽^∘_𝑖𝐮;𝑣_𝑖̂−𝑢_𝑖+𝐺_𝑖𝑣_𝑖−𝐺_𝑖𝑢_𝑖≥0,∀𝑣_𝑖∈𝑉_𝑖, (1) where 𝐮=(𝑢₁,𝑢₂,…,𝑢_𝑛)∈∏^𝑛_𝑘=1𝑉_𝑘,𝐮=(𝑢₁,𝑢₂,…,𝑢_𝑛)=(𝑇₁𝑢₁,𝑇₂𝑢₂,…,𝑇_𝑛𝑢_𝑛), 𝑣_𝑖̂=𝑇_𝑖𝑣_𝑖 and 𝐽^∘_𝑖(𝐮;𝑣_𝑖̂−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∈𝑋_𝑖 in the direction 𝑣_𝑖̂−𝑢_𝑖 for all given 𝑢_𝑗∈𝑋_𝑗, 𝑗≠𝑖, which can be defined by 𝐽^∘_𝑖𝐮;𝑣_𝑖̂−𝑢_𝑖=limsup_𝑥→𝑢_𝑖^,𝜆↓0𝐽𝑢₁,…,𝑢_𝑖−1,𝑥+𝜆𝑣_𝑖̂−𝑢_𝑖,𝑢_𝑖+1,…,𝑢_𝑛𝜆−𝐽𝑢₁,…,𝑢_𝑖−1,𝑥,𝑢_𝑖+1,…,𝑢_𝑛𝜆. (2)

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, 𝐴_𝑖 are single-valued, or 𝐺_𝑖 are indicators of some convex subsets 𝐾_𝑖 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 𝐴_𝑖 (𝑖=1,2,…,𝑛) are single-valued and 𝐺_𝑖 are the indicators of the convex subsets 𝐾_𝑖, the problem (P) reduces to the problem studied by Repovš and Varga [6] with 𝑇_𝑖=𝑆_𝑖 and 𝐴^∘_𝑖(𝑇𝑢;𝑇_𝑖𝑣_𝑖−𝑇_𝑖𝑢_𝑖) being incorporated into 𝐽^∘_𝑖(𝑆𝑢;𝑆_𝑖𝑣_𝑖−𝑆_𝑖𝑢_𝑖) 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 solution 𝑢_𝑖 to each variational-hemivariational inequality in the system (1) for all 𝑢_𝑗∈𝑉_𝑗, 𝑗≠𝑖 under some suitable conditions on the operators 𝐴_𝑖, 𝐽, and 𝐺_𝑖. 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 Section 3, 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 P^′𝜇₁,𝑣₁−𝑢_{1𝑉∗1×𝑉1}+𝐽^∘₁𝑢;𝑣₁̂−𝑢₁+𝐺₁𝑣₁−𝐺₁𝑢₁≥0,∀𝑣₁∈𝑉₁,𝜇₂,𝑣₂−𝑢_{2𝑉∗2×𝑉2}+𝐽^∘₂𝑢;𝑣₂̂−𝑢₂+𝐺₂𝑣₂−𝐺₂𝑢₂≥0,∀𝑣₂∈𝑉₂, (3) where 𝑢=(𝑢₁,𝑢₂), 𝑢=(𝑢₁,𝑢₂)=(𝑇₁𝑢₁,𝑇₂𝑢₂) and 𝑣_𝑖̂=𝑇_𝑖𝑣_𝑖 for 𝑖=1,2.

The paper is structured as follows. In Section 2, 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, that 𝑋 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→2^𝑋∗⧵{∅} and 𝜕𝐽(𝑢)∶𝑋→2^𝑋∗⧵{∅} 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, 𝜕𝐺(𝑢)=𝑢^∗∈𝑋^∗(𝑣)(𝑢)≥𝑢∶𝐺−𝐺_∗,(𝑢)=,𝑣−𝑢,∀𝑣∈𝑋𝜕𝐽𝜔∈𝑋^∗∶𝐽^∘(𝑢,𝑣)≥⟨𝜔,𝑣⟩,∀𝑣∈𝑋. (4)

We have the following basic properties on Clarke's generalized directional derivative and Clarke's generalized gradient (see, e.g., [2, 17]).

Proposition 1. Let 𝑋 be Banach space, and let 𝑢,𝑣∈𝑋, and 𝐽 be locally Lipschitz functional defined on 𝑋. Then one has the following. (1)The function 𝑣↦𝐽^∘(𝑢,𝑣)  is finite, positively homogeneous, subadditive, and then convex on 𝑋;(2)𝐽^∘(𝑢,𝑣)  is upper semicontinuous as a function of (𝑢,𝑣), but as a function of 𝑣  alone, it is Lipschitz continuous on 𝑋.(3)𝜕𝐽(𝑢)  is a nonempty, convex, bounded, and weak*-compact subset of 𝑋^∗.(4)For every 𝑣∈𝑋, one has 𝐽^∘(𝑢,𝑣)=max𝜉,𝑣∶𝜉∈𝜕𝐽(𝑢). (5)(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  into 𝑤^∗−𝑋^∗.

Definition 2. A locally Lipschitz functional 𝐽∶𝑋→𝑅 is said to be regular (in the sense of Clarke) at 𝑢∈𝑋 if (i)for all 𝑣∈𝑋 the directional derivative 𝐽^′(𝑢,𝑣) exists;(ii)for all 𝑣∈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 𝐽^′(𝑢,𝑣)=lim_𝜆↓0𝐽(𝑢+𝜆𝑣)−𝐽(𝑢)𝜆, (6) 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. Let 𝑋₁ 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 𝜕𝐽𝑢₁,𝑢₂⊆𝜕₁𝐽𝑢₁,𝑢₂×𝜕₂𝐽𝑢₁,𝑢₂, (7) or equivalently one has 𝐽^∘𝑢₁,𝑢₂;𝑣₁,𝑣₂≤𝐽^∘₁𝑢₁,𝑢₂;𝑣₁+𝐽^∘₂𝑢₁,𝑢₂;𝑣₂,∀𝑣₁,𝑣₂∈𝑋₁×𝑋₂, (8) where 𝜕₁(𝑢₁,𝑢₂) (resp., 𝜕₂(𝑢₁,𝑢₂)) represents the partial generalized subgradient of 𝐽(⋅,𝑢₂) (resp., 𝐽(𝑢₁,⋅)) and 𝐽^∘₁(𝑢₁,𝑢₂;𝑣₁) (resp., 𝐽^∘₂(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. Let 𝑋 be real reflexive Banach space with dual 𝑋^∗. A mapping 𝑇 from 𝑋 into 2^𝑋∗ is said to be pseudomonotone if (1)the set 𝑇𝑢 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){𝑢_𝑖} is a sequence in 𝑋 converging weakly to 𝑢, and 𝑢^∗_𝑖∈𝑇𝑢_𝑖 is such that limsup𝑢^∗_𝑖,𝑢_𝑖−𝑢≤0, (9) then for each element 𝑣∈𝑋 there exists 𝑢^∗(𝑣)∈𝑇𝑢 such that liminf𝑢^∗_𝑖,𝑢_𝑖−𝑣≥𝑢^∗(𝑣),𝑢−𝑣. (10)

Definition 5. Let 𝑋 be real reflexive Banach space with dual 𝑋^∗. A mapping 𝑇 from 𝑋 into 2^𝑋∗ is said to be generalized pseudomonotone if for any sequences {𝑢_𝑖}⊂𝑋, {𝑢^∗_𝑖}⊂𝑋^∗ with 𝑢^∗_𝑖∈𝑇𝑢_𝑖, 𝑢_𝑖→𝑢 weakly in 𝑋, 𝑢^∗_𝑖→𝑢^∗ weakly in 𝑋^∗ and limsup𝑢^∗_𝑖,𝑢_𝑖−𝑢≤0; (11) then one has 𝑢^∗∈𝑇𝑢 and ⟨𝑢^∗_𝑖,𝑢_𝑖⟩→⟨𝑢^∗,𝑢⟩.

Proposition 6. Let 𝑋 be real reflexive Banach space with dual 𝑋^∗ and let 𝑇₁, 𝑇₂ be two pseudomonotone mappings from 𝑋 into 2^𝑋∗. Then 𝑇₁+𝑇₂ is pseudomonotone.

Proposition 7. Let 𝑋 be real reflexive Banach space with dual 𝑋^∗ and let 𝑇∶𝑋→2^𝑋∗ be a pseudomonotone mapping from 𝑋 into 2^𝑋∗. Then 𝑇 is a generalized pseudomonotone.

Proposition 8. Let 𝑋 be real reflexive Banach space with dual 𝑋^∗ and let 𝑇 be a bounded, generalized pseudomonotone mapping from 𝑋 into 2^𝑋∗. Assume that, for each 𝑢∈𝑋, 𝑇𝑢 is a nonempty closed convex subset of 𝑋^∗. Then 𝑇 is pseudomonotone.

Definition 9. Let 𝑋 be real reflexive Banach space with dual 𝑋^∗. The operator 𝑇∶𝑋→2^𝑋∗ 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 𝑢^∗−𝑣^∗,𝑢−𝑣≥0. (12)(2)maximal monotone if it is monotone and if (𝑢,𝑢^∗)∈𝑋×𝑋^∗ is such that 𝑢^∗−𝑣^∗,𝑢−𝑣≥0,∀𝑣,𝑣^∗∈𝐺(𝑇); (13) then (𝑢,𝑢^∗)∈𝐺(𝑇). (3)quasibounded if for each 𝑀>0 there exists 𝐾(𝑀)>0 such that, whenever (𝑢,𝑢^∗)∈𝐺(𝑇), ⟨𝑢^∗,𝑢⟩≤𝑀‖𝑢‖, and ‖𝑢‖≤𝑀; then 𝑢^∗≤𝐾(𝑀). (14)(4)strongly quasibounded if for each 𝑀>0 there exists 𝐾(𝑀)>0 such that for all (𝑢,𝑢^∗)∈𝐺(𝑇) with ⟨𝑢^∗,𝑢⟩≤𝑀 and ‖𝑢‖≤𝑀, one has 𝑢^∗≤𝐾(𝑀). (15)

Definition 10. Let 𝑋 be real reflexive Banach space with dual 𝑋^∗. A mapping 𝑇 from 𝑋 into 2^𝑋∗ is said to be as follows: (1)coercive if there exists a real-valued function 𝑐 on 𝑅⁺ with lim_𝑟→∞𝑐(𝑟)=∞ such that for all (𝑢,𝑢^∗)∈𝐺(𝑇), one has 𝑢^∗,𝑢≥𝑐(‖𝑢‖)‖𝑢‖, (16)(2)coercive with constant 𝛼>0 if 𝑢^∗,𝑢≥𝛼‖𝑢‖², (17)(3)𝑢₀-coercive if there exists a real-valued function 𝑐 on 𝑅⁺ with lim_𝑟→∞𝑐(𝑟)=∞ such that for some 𝑢₀∈𝑋 and for all (𝑢,𝑢^∗)∈𝐺(𝑇), one has 𝑢^∗,𝑢−𝑢₀≥𝑐(‖𝑢‖)‖𝑢‖. (18)

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]). Let 𝑋 be a real reflexive Banach space with dual 𝑋^∗, let 𝑇 be a maximal monotone mapping from 𝑋 into 2^𝑋∗ with 𝑢₀∈𝐷(𝑇)={𝑥∶𝑇𝑥≠∅}, and let 𝑇 be a 𝑢₀-coercive, pseudomonotone operator from 𝑋 into 2^𝑋∗. Suppose further that either 𝑇_𝑢0∶𝑋→2^𝑋∗ is quasibounded or 𝑇_𝑢0∶𝑋→2^𝑋∗ is strongly quasibounded, where 𝑇_𝑢0(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 ‖𝑢‖_𝑉=𝑢_1𝑉1+𝑢_2𝑉2,∀𝑢=𝑢₁,𝑢₂∈𝑉, (19)𝑉 is a reflexive Banach space with dual 𝑉^∗. The duality pairing between 𝑉 and 𝑉^∗ is given by 𝑢^∗,𝑢_{𝑉∗×𝑉}=𝑢^∗₁,𝑢_{1𝑉∗1×𝑉1}+𝑢^∗₂,𝑢_{2𝑉∗2×𝑉2},∀𝑢^∗=𝑢^∗₁,𝑢^∗₂∈𝑉^∗,𝑢=𝑢₁,𝑢₂∈𝑉. (20)

On the Banach space 𝑉 defined above, we further define a set-valued mapping 𝐴∶𝑉→2^𝑉∗, 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 𝐴(𝑢)=𝐴₁(𝑢),𝐴₂(𝑢),𝑇(𝑢)=𝑇₁𝑢₁,𝑇₂𝑢₂,𝐺(𝑢)=𝐺₁𝑢₁+𝐺₂𝑢₂. (21)

Lemma 12. Assume that 𝐺_𝑖∶𝑉_𝑖→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 functionals 𝐺_𝑖∶𝑉_𝑖→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 𝑢^𝑛→𝑢 in 𝑉, which implies 𝑢^𝑛₁→𝑢₁ in 𝑉₁ and 𝑢^𝑛₂→𝑢₂ in 𝑉₂, we can get from the lower semicontinuity of 𝐺₁ and 𝐺₂ that liminf𝐺𝑢^𝑛=liminf𝐺₁𝑢^𝑛₁+𝐺₂𝑢^𝑛₂≥liminf𝐺₁𝑢^𝑛₁+liminf𝐺₂𝑢^𝑛₂≥𝐺₁𝑢₁+𝐺₂𝑢₂=𝐺(𝑢), (22) 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 𝐺(𝑣)−𝐺(𝑢)≥𝜇,𝑣−𝑢_{𝑉∗×𝑉},∀𝑣∈𝑉. (23)
In particular, for any 𝑣₁∈𝑉₁, let 𝑣=(𝑣₁,𝑢₂) in (23), and then we can get that 𝐺₁𝑣₁−𝐺₁𝑢₁≥𝜇₁,𝑣₁−𝑢_{1𝑉∗1×𝑉1},∀𝑣₁∈𝑉₁. (24)
Similarly, by letting 𝑣=(𝑢₁,𝑣₂) in (23) for any 𝑣₂∈𝑉₂, we can obtain 𝐺₂𝑣₂−𝐺₂𝑢₂≥𝜇₂,𝑣₂−𝑢_{2𝑉∗2×𝑉2},∀𝑣₂∈𝑉₂, (25) which together with (24) implies that 𝜇∈𝜕𝐺₁(𝑢₁)×𝜕𝐺₂(𝑢₂); that is, 𝜕𝐺(𝑢)⊆𝜕𝐺₁(𝑢₁)×𝜕𝐺₂(𝑢₂).
Conversely, let 𝜇=(𝜇₁,𝜇₂)∈𝜕𝐺₁(𝑢₁)×𝜕𝐺₂(𝑢₂). For all 𝑣_𝑖∈𝑉_𝑖,  𝑖=1,2, it follows from 𝜇_𝑖∈𝜕𝐺_𝑖(𝑢_𝑖) that 𝐺₁𝑣₁−𝐺₁𝑢₁≥𝜇₁,𝑣₁−𝑢_{1𝑉∗1×𝑉1},∀𝑣₁∈𝑉₁,𝐺₂𝑣₂−𝐺₂𝑢₂≥𝜇₂,𝑣₂−𝑢_{2𝑉∗2×𝑉2},∀𝑣₂∈𝑉₂. (26)By adding the two inequalities (26), we obtain 𝐺(𝑣)−𝐺(𝑢)≥𝜇,𝑣−𝑢_{𝑉∗×𝑉},∀𝑣∈𝑉, (27) 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. Find 𝑢=(𝑢₁,𝑢₂)∈𝑉 and 𝜇=(𝜇₁,𝜇₂)∈𝐴(𝑢) such that P^′′𝜇,𝑣−𝑢_{𝑉∗×𝑉}+𝐽^∘()(𝑣)(𝑢)𝑇𝑢,𝑇𝑣−𝑇𝑢+𝐺−𝐺≥0,∀𝑣∈𝑉. (28)

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 functional 𝐽∶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 that 𝑢=(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 𝜇,𝑣−𝑢_{𝑉∗×𝑉}+𝐽^∘(𝑇𝑢,𝑇𝑣−𝑇𝑢)+𝐺(𝑣)−𝐺(𝑢)≥0. (29)
Specially, for any 𝑣₁∈𝑉₁, let 𝑣=(𝑣₁,𝑢₂)∈𝑉 in (29), and then we can get from Proposition 3 that 0≤𝜇₁,𝑣₁−𝑢_{1𝑉∗1×𝑉1}+𝐽^∘𝑇𝑢,𝑇₁𝑣₁−𝑇₁𝑢₁,0+𝐺₁𝑣₁−𝐺₁𝑢₁≤𝜇₁,𝑣₁−𝑢_{1𝑉∗1×𝑉1}+𝐽^∘₁𝑇𝑢,𝑇₁𝑣₁−𝑇₁𝑢₁+𝐺₁𝑣₁−𝐺₁𝑢₁,∀𝑣₁∈𝑉₁. (30)
Similarly, by letting 𝑣=(𝑢₁,𝑣₂)∈𝑉 in (29) for any 𝑣₂∈𝑉₂, we can obtain that 𝜇₂,𝑣₂−𝑢_{2𝑉∗2×𝑉2}+𝐽^∘₂𝑇𝑢,𝑇₂𝑣₂−𝑇₂𝑢₂+𝐺₂𝑣₂−𝐺₂𝑢₂≥0,∀𝑣₂∈𝑉₂, (31) 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 Proposition 3 that, just under regularity condition of the functional 𝐽, 𝐽^∘(𝑢,𝑣)=𝐽^∘₁(𝑢,𝑣₁)+𝐽^∘₂(𝑢,𝑣₂) does not hold in general, while the inequality 𝐽^∘(𝑢,𝑣)≤𝐽^∘₁(𝑢,𝑣₁)+𝐽^∘₂(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 operators 𝐴_𝑖 and 𝐽 in the system (3) of two generalized variational-hemivariational inequalities.  The assumption (HA) is as follows.(1)𝐴₁∶𝑉₁×𝑉₂→2^𝑉∗1 is bounded on 𝑉₁×𝑉₂ and pseudomonotone with respect to the first argument; that is, for all 𝑢₂∈𝑉₂, the operator 𝐴₁(⋅,𝑢₂)∶𝑉₁→2^𝑉∗1 is pseudomonotone on 𝑉₁.(2)𝐴₂∶𝑉₁×𝑉₂→2^𝑉∗2 is bounded on 𝑉₁×𝑉₂ and pseudomonotone with respect to the second argument; that is, for all 𝑢₁∈𝑉₁, the operator 𝐴₂(𝑢₁,⋅)∶𝑉₂→2^𝑉∗2 is pseudomonotone on 𝑉₂.(3) For all 𝑢₂∈𝑉₂, there exist an element 𝑤₁∈𝐷(𝜕𝐺₁)⊂𝑉₁ and a constant 𝛼₁>0 such that 𝑢^∗₁,𝑢₁−𝑤_{1𝑉∗1×𝑉1}≥𝛼₁𝑢₁²_𝑉1,∀𝑢₁∈𝑉₁,𝑢^∗₁∈𝐴₁𝑢₁,𝑢₂. (32)(4) For all 𝑢₁∈𝑉₁, there exist an element 𝑤₂∈𝐷(𝜕𝐺₂)⊂𝑉₂ and a constant 𝛼₂>0 such that 𝑢^∗₂,𝑢₂−𝑤_{2𝑉∗2×𝑉2}≥𝛼₂𝑢₂²_𝑉2,∀𝑢₂∈𝑉₂,𝑢^∗₂∈𝐴₂𝑢₁,𝑢₂. (33)

Remark 15. It is clear that the hypotheses (1) and (2) in the assumption (HA) imply that the operator 𝐴 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 𝜇,𝑢−𝑤_{𝑉∗×𝑉}=𝜇₁,𝑢₁−𝑤_{1𝑉∗1×𝑉1}+𝜇₂,𝑢₂−𝑤_{2𝑉∗2×𝑉2}≥𝛼₁𝑢₁²_𝑉1+𝛼₂𝑢₂²_𝑉2≥min𝛼₁,𝛼₂2‖𝑢‖²_𝑉,=𝛽‖𝑢‖²_𝑉, (34) which implies the 𝑤-coercivity with constant 𝛽 of operator 𝐴 on 𝑉.   The assumption (HJ) is as follows.(1) For all 𝑥₂∈𝑋₂, there exist constants 𝑐₁,𝑑₁≥0 such that 𝜂_1𝑋∗1≤𝑐₁+𝑑₁𝑥_1𝑋1,∀𝜂₁∈𝜕₁𝐽𝑥₁,𝑥₂. (35)(2) For all 𝑥₁∈𝑋₁, there exist constants 𝑐₂,𝑑₂≥0 such that 𝜂_2𝑋∗2≤𝑐₂+𝑑₂𝑥_2𝑋2,∀𝜂₂∈𝜕₂𝐽𝑥₁,𝑥₂. (36)

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^∗=𝑋^∗₁×𝑋^∗₂ 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 𝜂_𝑋∗≤𝑐+𝑑‖𝑥‖_𝑋,∀𝜂∈𝜕𝐽(𝑥), (37) 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 mappings 𝐴_𝑖∶𝑉₁×𝑉₂→2^𝑉∗𝑖, 𝑖=1,2, which satisfy the assumption (HA), is such that the operator 𝐴 defined in (21) is pseudomonotone on 𝑉. Let 𝑇_𝑖∶𝑉_𝑖→𝑋_𝑖 be linear continuous and compact operators, let 𝐽∶𝑋→𝑅 be a regular, locally Lipschitz functional which satisfies the hypothesis (HJ), let and 𝐺_𝑖∶𝑉_𝑖→𝑅∪{+∞}, 𝑖=1,2 be proper, convex, and lower semicontinuous functionals. Then the problem (P^′) admits at least one solution under the condition 𝛽>𝑑‖𝑇‖², (38) where ‖𝑇‖ is the norm of the operator 𝑇 defined by (21).

Proof. By Lemma 13, 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 𝐹(𝑢)+𝜕𝐺(𝑢)∋0, (39) where 𝐹∶𝑉→2^𝑉∗ 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 (𝐹 is bounded on 𝑉). 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 𝑇_𝑖,𝑖=1,2, and it is easy to check that 𝐹 is bounded on 𝑉, which implies that 𝐹_𝑢0∶𝑉→2^𝑉∗ with 𝐹_𝑢0(𝑢)=𝐹(𝑢₀+𝑢) is quasibounded for any 𝑢₀∈𝑉.
Claim  2   (𝐹 is pseudomonotone on 𝑉). Since 𝐹=𝐴+𝑇^∗∘𝜕𝐽∘𝑇∶𝑉→2^𝑉∗ and the operator 𝐴 is pseudomonotone, we only need to prove that 𝑇^∗∘𝜕𝐽∘𝑇∶𝑉→2^𝑉∗ is pseudomonotone. To this end, firstly, we prove that 𝑇^∗∘𝜕𝐽∘𝑇 is generalized pseudomonotone. Let 𝑢^𝑛→𝑢 weakly in 𝑉,𝜉^𝑛∈𝑇^∗(𝜕𝐽(𝑇𝑢^𝑛)) with 𝜉^𝑛→𝜉 weakly in 𝑉^∗ and limsup⟨𝜉^𝑛,𝑢^𝑛−𝑢⟩_𝑉^∗_×𝑉≤0. There exist 𝜂^𝑛∈𝜕𝐽(𝑇𝑢^𝑛) such that 𝜉^𝑛=𝑇^∗𝜂^𝑛. Since 𝜕𝐽 is bounded on 𝑋 by Remark 16, 𝑇𝑢^𝑛→𝑇𝑢 in 𝑋 by the compactness of the operators 𝑇_𝑖, 𝑖=1,2, and 𝜂^𝑛∈𝜕𝐽(𝑇𝑢^𝑛), we have the fact that 𝜂^𝑛 is bounded in 𝑋^∗. Thus there exists a subsequence, which is also denoted by 𝜂^𝑛, such that 𝜂^𝑛→𝜂 weakly in 𝑋^∗ with some 𝜂∈𝑋^∗. By using the equality 𝜉^𝑛=𝑇^∗𝜂^𝑛, it is easy to get that 𝜉=𝑇^∗𝜂. Since 𝜂^𝑛∈𝜕𝐽(𝑇𝑢^𝑛) with 𝜂^𝑛→𝜂 weakly in 𝑋^∗ and 𝑇𝑢^𝑛→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 𝜂^𝑛→𝜂 weakly in 𝑋^∗ and 𝑇𝑢^𝑛→𝑇𝑢 in 𝑋 that 𝜉^𝑛,𝑢^𝑛_{𝑉∗×𝑉}=𝜂^𝑛,𝑇𝑢^𝑛_{𝑋∗×𝑋}⟶𝜂,𝑇𝑢_{𝑋∗×𝑋}=𝑇^∗𝜂,𝑢_{𝑉∗×𝑉}=𝜉,𝑢_{𝑉∗×𝑉}, (40) 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  (𝐹 is 𝑤-coercive on 𝑉). 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 ⟨𝜏,𝑢−𝑤⟩_𝑉^∗_×𝑉=𝜇,𝑢−𝑤_{𝑉∗×𝑉}+𝑇^∗𝜂,𝑢−𝑤_{𝑉∗×𝑉}≥𝛽‖𝑢‖²_𝑉+𝜂,𝑇(𝑢−𝑤)_{𝑋∗×𝑋}≥𝛽‖𝑢‖²_𝑉−𝜂_𝑋∗‖𝑇‖‖𝑢‖_𝑉+‖𝑤‖_𝑉≥𝛽‖𝑢‖²_𝑉−𝑐+𝑑‖𝑇𝑢‖_𝑋‖𝑇‖‖𝑢‖_𝑉+‖𝑤‖_𝑉≥𝛽−𝑑‖𝑇‖²‖𝑢‖²_𝑉−‖𝑇‖𝑐+𝑑‖𝑇‖‖𝑤‖_𝑉‖𝑢‖_𝑉−𝑐‖𝑇‖‖𝑤‖_𝑉, (41) 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 0∈𝐹(𝑢)+𝜕𝐺(𝑢). (42)
By the definition of the operator 𝐹, there exist 𝜇∈𝐴(𝑢), 𝜂∈𝜕𝐽(𝑇𝑢), and 𝜉∈𝜕𝐺(𝑢) such that 𝜇+𝑇^∗𝜂+𝜉=0. (43)
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 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), (44) 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 operator 𝐴 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 Theorem 17, we have the following results for the existence of solution to each generalized variational-hemivariational inequality in the system (3).

Theorem 19. Suppose that, for 𝑖=1,2, 𝐴_𝑖∶𝑉₁×𝑉₂→2^𝑉∗𝑖 are set-valued mappings satisfying the assumption (HA) and that 𝑇_𝑖∶𝑉_𝑖→𝑋_𝑖 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 𝐺_𝑖∶𝑉_𝑖→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 𝑢_𝑖∈𝑉_𝑖 for all 𝑢_𝑗∈𝑉_𝑗,  𝑗≠𝑖 under the condition 𝛼_𝑖>𝑑_𝑖𝑇_𝑖², (45) where ‖𝑇_𝑖‖ is the norm of the operator 𝑇_𝑖.

Remark 20. By comparing Theorems 17 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 Theorem 17 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)For 𝑖=1,2,…,𝑛, 𝐴_𝑖∶∏^𝑛_𝑘=1𝑉_𝑘→2^𝑉∗𝑖 are set-valued mappings satisfying the following. (a)𝐴_𝑖 are bounded on ∏^𝑛_𝑘=1𝑉_𝑘 and pseudomonotone with respect to the 𝑖th argument. (b)the operator 𝐴∶𝑉=∏^𝑛_𝑘=1𝑉_𝑘→2^𝑉∗, which is defined by 𝐴(𝑢)=(𝐴₁(𝑢),…,𝐴_𝑛(𝑢)), is pseudomonotone on 𝑉. (c)For all 𝑢_𝑗∈𝑉_𝑗, 𝑗≠𝑖, there exist an element 𝑤_𝑖∈𝐷(𝜕𝐺_𝑖)⊂𝑉_𝑖 and a constant 𝛼_𝑖>0 such that 𝑢^∗_𝑖,𝑢_𝑖−𝑤_{𝑖𝑉∗𝑖×𝑉𝑖}≥𝛼_𝑖𝑢_𝑖²_𝑉𝑖,∀𝑢_𝑖∈𝑉_𝑖,𝑢^∗_𝑖∈𝐴_𝑖𝑢₁,…,𝑢_𝑛. (46)(2)𝐽∶∏^𝑛_𝑘=1𝑋_𝑘→𝑅 is a regular and locally Lipschitz functional which satisfies that for all 𝑖=1,2,…,𝑛 and 𝑥_𝑗∈𝑋_𝑗, 𝑗≠𝑖, there exist constants 𝑐_𝑖,𝑑_𝑖≥0 such that 𝜂_{𝑖𝑋∗𝑖}≤𝑐_𝑖+𝑑_𝑖𝑥_𝑖𝑋𝑖,∀𝑥_𝑖∈𝑋_𝑖,𝜂_𝑖∈𝜕_𝑖𝐽𝑥₁,…,𝑥_𝑛. (47)(3)For 𝑖=1,2,…,𝑛, 𝑇_𝑖∶𝑉_𝑖→𝑋_𝑖 are linear continuous and compact operators and 𝐺_𝑖∶𝑉_𝑖→𝑅∪{+∞} are proper, convex, and lower semicontinuous functionals. Then the problem (P) admits at least one solution under the condition 𝛽>𝑑𝑇², (48) where 𝛽=min{𝛼_𝑖}/𝑛, 𝑑=max{𝑑_𝑖} and ‖𝑇‖ is the norm of the operator 𝑇∶∏^𝑛_𝑘=1𝑉_𝑘→∏^𝑛_𝑘=1𝑋_𝑘 defined by 𝑇(𝑢₁,…,𝑢_𝑛)=(𝑇₁𝑢₁,…,𝑇_𝑛𝑢_𝑛).

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.

Let Ω⊂𝑅^𝑛 be a bounded and open set in 𝑅^𝑛, let 𝑉₁,𝑉₂,…,𝑉_𝑛 be real, separable, and reflexive Banach spaces with dual spaces 𝑉^∗₁,𝑉^∗₂,…,𝑉^∗_𝑛. For 𝑖=1,2,…,𝑛, 𝐴_𝑖∶∏^𝑛_𝑘=1𝑉_𝑘→2^𝑉∗𝑖 are set-valued mappings, 𝑇_𝑖∶𝑉_𝑖→𝐿²(Ω) are linear continuous and compact operators on 𝑉_𝑖, and 𝐺_𝑖∶𝑉_𝑖→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 𝑢_𝑖∈𝑉_𝑖 and 𝜇_𝑖∈𝐴_𝑖(𝑢) such that 𝜇_𝑖,𝑣_𝑖−𝑢_{𝑖𝑉∗𝑖×𝑉𝑖}+_Ω𝑗^∘_𝑖𝑥,𝐮(𝑥);𝑣_𝑖̂(𝑥)−𝑢_𝑖(𝑥)𝑑𝑥+𝐺_𝑖𝑣_𝑖−𝐺_𝑖𝑢_𝑖≥0,∀𝑣_𝑖∈𝑉_𝑖, (49) where 𝐮=(𝑢₁,𝑢₂,…,𝑢_𝑛)∈∏^𝑛_𝑘=1𝑉_𝑘,𝐮=(𝑢₁,𝑢₂,…,𝑢_𝑛)=(𝑇₁𝑢₁,𝑇₂𝑢₂,…,𝑇_𝑛𝑢_𝑛)∈𝐿²(Ω;𝑅^𝑛), 𝑣_𝑖̂=𝑇_𝑖𝑣_𝑖 and 𝑗(𝑥,𝑦)∶Ω×𝑅^𝑛→𝑅 is a function satisfying the following assumption:  (Hj) is as follows.(1)𝑗(𝑥,⋅) is locally Lipschitz on 𝑅^𝑛 for a.e.𝑥∈Ω.(2)𝑗(𝑥,𝑦)∶Ω×𝑅^𝑛→𝑅 is a Carathéodory function.(3) Either 𝑗(𝑥,⋅) or −𝑗(𝑥,⋅) is regular on 𝑅^𝑛 for a.e. 𝑥∈Ω.(4) For all 𝑖=1,2,…,𝑛 and a.e. 𝑥∈Ω, there exists constant 𝑐_𝑖,𝑑_𝑖≥0 such that |𝑠|≤𝑐_𝑖+𝑑_𝑖𝑦_𝑖,∀𝑦∈𝑅^𝑛,𝑠∈𝜕_𝑖𝑗𝑥,𝑦. (50)

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, 𝐴_𝑖 is single-valued and 𝐺_𝑖 is an indicator of a convex subset 𝐾.

We define a functional 𝐽 on 𝐿²(Ω,𝑅^𝑛) as follows: 𝐽(𝑢)=_Ω𝑗(𝑥,𝑢(𝑥))𝑑𝑥,∀𝑢∈𝐿²Ω,𝑅^𝑛. (51)

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,𝑅^𝑛), which satisfies 𝐽^∘_𝑖𝑢,𝑣_𝑖=_Ω𝑗^∘_𝑖𝑥,𝑢(𝑥),𝑣_𝑖(𝑥)𝑑𝑥,∀𝑢∈𝐿²Ω,𝑅^𝑛,𝑣_𝑖∈𝐿²(Ω),𝜂_𝐿2(Ω)≤𝑐_𝑖̃+𝑑_𝑖𝑢_{𝑖𝐿2(Ω)},∀𝑢∈𝐿²Ω,𝑅^𝑛,𝜂∈𝜕_𝑖𝐽(𝑢), (52) where 𝑐_𝑖̃=√2𝑐_𝑖|Ω|≥0 and 𝑑_𝑖=√2𝑑_𝑖≥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) For 𝑖=1,2,…,𝑛, 𝐴_𝑖∶∏^𝑛_𝑘=1𝑉_𝑘→2^𝑉∗𝑖 are set-valued mappings satisfying the following. (a)𝐴_𝑖 are bounded on ∏^𝑛_𝑘=1𝑉_𝑘 and pseudomonotone with respect to the 𝑖th argument.(b) the operator 𝐴∶𝑉=∏^𝑛_𝑘=1𝑉_𝑘→2^𝑉∗, which is defined by 𝐴(𝑢)=(𝐴₁(𝑢),…,𝐴_𝑛(𝑢)), is pseudomonotone on 𝑉.(c) For all 𝑢_𝑗∈𝑉_𝑗, 𝑗≠𝑖, there exist an element 𝑤_𝑖∈𝐷(𝜕𝐺_𝑖)⊂𝑉_𝑖 and a constant 𝛼_𝑖>0 such that 𝑢^∗_𝑖,𝑢_𝑖−𝑤_{𝑖𝑉∗𝑖×𝑉𝑖}≥𝛼_𝑖𝑢_𝑖²_𝑉𝑖,∀𝑢_𝑖∈𝑉_𝑖,𝑢^∗_𝑖∈𝐴_𝑖𝑢₁,…,𝑢_𝑛. (53)(2)𝑗(𝑥,𝑦)∶Ω×𝑅^𝑛→𝑅 is a function satisfying the assumption (Hj).(3) For 𝑖=1,2,…,𝑛, 𝑇_𝑖∶𝑉_𝑖→𝐿²(Ω) are linear continuous and compact operators and 𝐺_𝑖∶𝑉_𝑖→𝑅∪{+∞} are proper, convex, and lower semicontinuous functionals. Then the problem (49) admits at least one solution under the condition 𝛽>𝑑𝑇², (54) where 𝛽=min{𝛼_𝑖}/𝑛, 𝑑=√2max{𝑑_𝑖}, and ‖𝑇‖ is the norm of the operator 𝑇∶∏^𝑛_𝑘=1𝑉_𝑘→𝐿²(Ω;𝑅^𝑛) defined by 𝑇(𝑢₁,…,𝑢_𝑛)=(𝑇₁𝑢₁,…,𝑇_𝑛𝑢_𝑛).

Acknowledgments

This work was supposed by the National Natural Science Foundation of China (11101069) and the Open Fund (PLN1104, PLN1102) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).