From the Fan-KKM principle to extended real-valued equilibria and to variational-hemivariational inequalities with application to nonmonotone contact problems

This paper starts off by the celebrated Knaster–Kuratowski–Mazurkiewicz principle in the formulation by Ky Fan. We provide a novel variant of this principle and build an existence theory for extended real-valued equilibrium problems with general, then monotone and pseudomonotone bifunctions. We develop our existence theory first in general topological vector spaces, then in reflexive Banach spaces, where we investigate the issue of coerciveness for existence on unbounded sets. Thereafter we use the Clarke generalized differential calculus for locally Lipschitz functions and derive existence results for nonlinear variational-hemivariational inequalities and hemivariational quasivariational inequalities. As application, we treat a unilateral contact problem in solid mechanics with nonmonotone friction.


Introduction
This paper starts off by the celebrated Knaster-Kuratowski-Mazurkiewicz (KKM) principle in the formulation by Ky Fan [22,23], which is known to be equivalent to topological fixed point theorems [28]. First, we provide a new refined version of the Fan-KKM principle, which relies on a general coercivity condition and unifies some results on this topic in [17,21]. Based on this refined version of the Fan-KKM principle, we build a broad existence theory for extended real-valued equilibrium problems with general bifunctions, then with bifunctions of monotone type, including monotone and (Brèzis or topologically) pseudomonotone bifunctions. We show how by this equilibrium theory existence results for variational inequalities of monotone type with set-valued operators can be derived. The setting of our exposition is first in general topological vector spaces, then in reflexive Banach spaces, where we discuss the issue of coerciveness for existence on unbounded sets in depth. Thereafter we use the Clarke generalized differential calculus [18,Sect. 2.1] for locally Lipschitz functions and obtain existence results for nonlinear variational-hemivariational inequalities and hemivariational quasivariational inequalities. For this latter class of nonlinear variational problems, we present a direct approach to existence results based on equilibrium theory without additional resorting to Banach's contraction theorem as exposed in the monograph of Sofonea and Migorski on variationalhemivariational inequalities [57].
We emphasize that extended real-valued equilibrium problems -a further novel feature of the present paper -cannot be reduced to classical equilibrium problems as pioneered by Blum and Oettli [7]. Although a reduction is trivial for equilibrium problems with convex, lower semicontinuous real-valued functions, the standard trick for extended real-valued convex, lower semicontinuous functions that works with the epigraph of such functions fails, since the transfer to a product space does not preserve compactness and coerciveness properties.
For existence on noncompact sets, we avoid, similarly to [40], coercivity conditions involving interior points, since the topological interior and the algebraic interior of convex sets in infinite-dimensional function spaces may be empty.
As application, we treat a unilateral contact problem in continuum mechanics with nonmonotone friction in the framework of nonlinear variational-hemivariational inequalities. The theory of hemivariational inequalities has been introduced and studied by Panagiotopoulos [50] in the 1980s and 1990s as a generalization of variational inequalities with an aim to model many problems coming from mechanics when the energy functionals are nonconvex; see [25,44,51,56].
The paper is organized as follows. In the next section, we establish our novel variant of the Fan-KKM principle and a general existence theorem for extended real-valued equilibria. In Sect. 3, we are concerned with monotone and pseudomonotone bifunctions, and develop further the existence theory for extended real-valued equilibria. Section 4 provides an application of the equilibrium theory to variational inequalities of monotone type with set-valued operators. Then in Sect. 5, we switch from topological vector spaces to reflexive Banach spaces, and in this setting, we discuss the issue of coerciveness in depth. Next, in Sect. 6, we use the Clarke generalized differential calculus for locally Lipschitz functions and obtain existence results for nonlinear variational-hemivariational inequalities and hemivariational quasivariational inequalities. As a model application, we study a scalar variational problem with the p-Laplacian and nonmonotone, possibly set-valued boundary conditions. In Sect. 7, we present an application to a full vectorial contact problem. Here we treat a unilateral contact problems with material nonlinearity and nonmonotone friction. The paper ends in Sect. 8 with some concluding remarks giving an outlook to some related directions of research.

A variant of the Fan-KKM principle and a general existence theorem for extended real-valued equilibria
Let us give some notations and terminology used in the following. In a topological vector space E, F(E) is the set of all finite-dimensional subspaces of E, ordered by inclusion. Conv X and cl X or X denote the convex hull and the topological closure of a subset X Theorem 2.2 Let X be a nonempty convex subset of a topological vector space, and let M : X → 2 X be a KKM covering. Suppose (i) there exists a convex compact subset C 0 of X such that cl X ( x∈C 0 M(x)) is compact, (ii) for any N ∈ N (X) and any x ∈ Conv N , (iii) for any compact K ⊂ X and any N ∈ N (X), we have, for D = Conv N and Then x∈X M(x) = ∅.
Then G ⊂ X is compact, since with N = {x 1 , . . . , x n } ⊂ X and the standard simplex S n in R n+1 , G is the continuous image of the compact C 0 × S n under the map χ given by χ(x 0 , λ 0 , λ 1 , . . . , λ n ) = λ 0 x 0 + n j=1 λ j x j for Since Y H is contained in the compact set G ⊂ X and the family {Y H : H ∈ N (G)} has the finite intersection property, there exists some y ∈ G ∩ {cl X Y H : H ∈ N (G)}. Now let y 1 ∈ G be arbitrary. Consider H 1 = {y 1 , y} ∈ N (G). Then by construction and by ( * ) By (iii) conclude for K := G, D := Conv H 1 , y ∈ {M(y) : y ∈ Conv H 1 } ⊂ M(y 1 ), and therefore y ∈ G lies in {M(y) : y ∈ G}, as desired.
3) Introduce the family which is ordered by inclusion. According to the preceding step, for any G ∈ G, there exists some Then Z G ⊂ L, and the family {cl X Z G : G ∈ G} has the finite intersection property. Hence there exists z ∈ L such that z ∈ G∈G cl X Z G . Finally, we show that z ∈ {M(x) : x ∈ X}. Note that for all N ⊃ N with N ∈ N (X) and Now let x 0 ∈ X be arbitrary. Consider N 0 = {x 0 , z} and G 0 = Conv(K 0 ∪ N 0 ). Then by construction and by ( * * ) A discussion of our FKKM theorem in relation to the literature is in order. Clearly, the coercivity assumption (i) is implied by the assumption of compactness of cl X M(x 0 ) for some x 0 ∈ X, which is used in [17] and also by Ansari, Lin, and Yao [4] and Kalmoun [36]. Note that in this case, we choose K = C 0 = {x 0 } in (i), then G = Conv(C 0 ∪ N) = ConvÑ withÑ = N ∪ {x 0 }, and so (iii) simplifies to: For any compact K ⊂ X and any N ∈ N (X), which follows from the "transfer-closedness" of the sets M(x): which is used instead of (iii) by Ansari, Lin, and Yao [4].
Finally, (i) is more general than the assumption of compactness of cl X ( x∈N M(x)) for some N ∈ N (X), which is used in [21]. This comes out in the following example.
Example 2.1 Let X be an infinite-dimensional real Hilbert space endowed with norm x and scalar product x, y . For x ∈ X, define Then M(x) = X if x = 0, and M(x) is a closed convex halfspace of X if x = 0. Furthermore, for all x ∈ X and r = x , we have since by the Cauchy-Schwarz inequality, for all y ∈ K r , This shows that M is a KKM covering. Indeed, let N ∈ N (X). Define ρ = max{ x : x ∈ N}. Then Conv N ⊂ K ρ and K ρ ⊂ {M(x) : x ∈ N}, and the claim follows.
On the contrary, we cannot find some N ∈ N (X) such that {M(x) : x ∈ N} is compact. Indeed, consider the finite-dimensional subspace F = span N . Since X is of infinite dimension, there exists z = 0 such that z ⊥ F, and hence Rz ⊂ {M(x) : x ∈ N}.
Based on the FKKM Theorem 2.2, we establish the following general existence theorem for extended real-valued equilibria.

Theorem 2.3 Let C be a closed convex subset of a real Hausdorff topological vector space E.
Let the extended real-valued function f : C → (-∞, +∞], f ≡ ∞, and a real-valued bifunction ϕ : C × C → R, nonnegative on the diagonal of C × C, be given such that the following assumptions are satisfied: There exist a compact subsetK of E andz ∈ C ∩K such that for all x ∈ C \K , Then there exists a solutionx of the extended real-valued equilibrium problem P[ϕ, f ; C], that is, for all y ∈ C, we have Proof We define for any z ∈ C = X the set To obtain the conclusion, we have to show that M is a KKM covering that satisfies (i)-(iii). Take K 0 = {z}, which trivially is convex compact, and (i) holds by (A 2 ).
(iii) is implied by (A 4 ). Indeed, let N ∈ N (C). Then K 0 ∪ N = {z} ∪ N =: N ∈ N (C) and G = Conv(K 0 ∪ N) = Conv N . Further, let K be a compact in C, and let This means that x = lim x t for some net {x t } t∈T ⊂ K and x t satisfy for all t ∈ T and x ∈ G, Finally, we show that M is a KKM covering by an indirect argument. Assume that there exist z i ∈ C and λ i ≥ 0 (i = 1, , . . . , k) with λ i = 1 such that By quasiconvexity and (A 1 ) we arrive at f (z) < ∞ and which contradicts ϕ(z, z) ≥ 0. The theorem is proved.

Existence of extended real-valued equilibria with bifunctions of monotone type
As introduced by [ (shortly p.m.) on a subset X of E if for any net {x t } t∈T that converges to some x ∈ X and satisfies lim inf t∈T ϕ(x t , x) ≥ 0, we have for all y ∈ X, Note that ϕ(x, y) := s(y)s(x) with a lower semicontinuous (l.s.c.) function s : C → R is a simple example of a p.m. bifunction.
The following proposition shows that such bifunctions ϕ satisfy the assumption (A 4 ), provided that the function f is l.s.c. Proof Let N ∈ N (C), and let (x t ) t∈T be a net that converges to some x ∈ Conv N such that Choosing y = x ∈ Conv N , by the lower semicontinuity of f we obtain Hence, for all y ∈ Conv N ⊂ C, we get since ϕ is p.m. and f is l.s.c., and the proposition follows.
Thus we immediately obtain the following existence result for extended real-valued equilibria with p.m. bifunctions.

Theorem 3.1 Let C be a closed convex subset of a real Hausdorff topological vector space E.
Let an extended real-valued l.s.c. function f : C → (-∞, +∞], f ≡ ∞, and a real-valued p.m. bifunction ϕ : C × C → R, nonnegative on the diagonal of C × C, be given such that the following assumptions are satisfied: (B 3 ) For all y ∈ C and N ∈ N (C), ϕ(·, y) is upper semicontinuous on Conv N . Then there exists a solutionx ∈ C ∩K of the extended real-valued equilibrium problem P[ϕ, f ; C], that is, for all y ∈ C, Next, we sharpen some assumptions in Theorem 3.1 and derive the following corollary.
be given such that the following assumptions are satisfied: Note that in the case C = E, conditions (C3) and (B3) are in fact equivalent. Indeed, for all x 0 ∈ F and F ∈ F(E), we can construct some N ∈ N (F) such that x 0 is an interior point of Conv N . Hence the continuity on Conv N, N ∈ N (E), implies the continuity on F for all F ∈ F(E).
From now on we impose that ϕ vanishes on the diagonal of C × C and that for x ∈ C, ϕ(x, ·) is convex. Let us recall from [7,26] that a bifunction ϕ : Note that ϕ(x, y) := c(y)-c(x) with c : C → R is a simple example of a monotone bifunction. For a monotone bifunction ϕ, the following relaxed notion of continuity is important: is upper semicontinuous at t = 0. For a hemicontinuous monotone bifunction ϕ, the "Minty trick" [10, Lemma 1], [42, Lemma 1] applies: , f ≡ ∞, be convex and l.s.c., and let D be a convex subset of C. Consider the statements If ϕ is monotone on D, then (i) implies (ii); if ϕ is hemicontinuous on D, then (i) follows from (ii).
Since ϕ(z s , ·) and f are convex, Now we can show that a hemicontinuous monotone bifunction ϕ that is l.s.c. in the second argument together with a convex l.s.c. extended real-valued function f satisfies assumption (A 4 ) of the general existence Theorem 2.3: Let D be an arbitrary convex subset of C, and let {x t } ⊂ C be a net converging to some Thus Theorem 2.3 applies to conclude the existence of equilibria with monotone bifunctions. However, its assumption (A 3 ) imposes for any y ∈ C the upper semicontinuity of ϕ(·, y) on finite-dimensional convex parts Conv N for N ∈ N (C), which is slightly stronger than the hemicontinuity. To arrive at an existence result for extended real-valued equilibria with monotone bifunctions that are only hemicontinuous, we again appeal to the basic Theorem 2.2 and slightly modify the monotonicity argument of [7, Lemma 2]. , f ≡ ∞, and a real-valued hemicontinuous monotone bifunction ϕ : C × C → R that vanishes on the diagonal of C × C be given such that the following assumptions are satisfied:

Then there exists a solutionx of the extended real-valued equilibrium problem
Proof In virtue of Proposition 3.2, we have to show the existence ofx ∈ C ∩ K that satisfies for all y ∈ C. Thus we apply Theorem 2.2 to the subsets {y 1 , . . . , y k } be an arbitrary finite subset of C, and let {λ 1 , . . . , λ k } be arbitrary nonnegative This means that On the other hand, since ϕ(y i , ·) and f are convex, Hence altogether by linear combination with This results in which contradicts the monotonicity of ϕ. Hence there existsẑ ∈ C that is an extended real-valued equilibrium of P[ϕ, f ; C]. In view of (D 2 ),ẑ ∈K .

Variational inequalities of monotone type with set-valued operators. An equilibrium approach
In this section, we derive existence results for variational inequalities (VIs) of monotone type from our existence results for extended real-valued equilibria. Here we focus to variational inequalities of the first kind with set-valued operators ("multis") of monotone type in general topological vector spaces (t.v.s.). The crucial element in our approach to existence results for multis is the following extension lemma [29, Theorem 2.2], which is a refined version of the famous Fan-Glicksberg-Hoffman theorem of alternative, which can be derived from Kneser's minimax theorem [37] and from Simons' more general two-function minimax theorem [55,Theorem 5] or can be directly proved from the separation theorem [7,29]. Lemma 4.1 Let C be a convex set in a vector space. Llet K be a compact convex set in a topological vector space. Let L be a real-valued functional on C × K such that, for all y ∈ K , L(·, y) is convex and, for every x ∈ C, L(x, ·) is concave and upper semicontinuous. Then the following statements are equivalent.
1. For each x ∈ C, there exists y ∈ K such that L(x, y) ≥ 0.

Suppose that f is upper semicontinuous and S is upper semicontinuous with compact values. Thenf is upper semicontinuous.
Let E be real Hausdorff topological vector space with dual space E * and duality ·, · . Further, as before, let C be a closed convex nonempty subset of E. Let S : C → 2 E * be a set-valued operator with nonempty, convex, and We are interested in the variational inequality problem P(S; C): Find a pair (x,ξ ) ∈ E ×E * such thatx ∈ C,ξ ∈ S(x), and for all y ∈ C, To apply our theory for equilibria of the previous section, we consider the real-valued bifunction ϕ : which clearly vanishes on the diagonal of C × C and is convex and l.s.c. on C with respect to the second argument. Now we can derive from Theorem 3.2 the following existence result for VIs with monotone set-valued operators.

Theorem 4.1 Let C be a closed convex subset of a real Hausdorff topological vector space E.
is upper semicontinuous at t = 0 and is monotone, that is, for all x, y ∈ C, ξ ∈ S(x), and η ∈ S(y), we have ξη, xy ≥ 0. Suppose there exist a compact subsetK of E andz ∈ C ∩K such that for all x ∈ C \K , there exist ζ ∈ S(z) such that ζ , x -z < 0.
Then there exists a solutionx to the variational inequality problem P(S; C).
Proof Obviously, the bifunction ϕ given by (2) is monotone and hemicontinuous by Lemma 4.2. Since the coercivity condition (D 2 ) is satisfied, Theorem 3.2 yields the existence ofx ∈ C such that for each y ∈ C, there exists ξ ∈ S(x) such that ξ , yx ≥ 0. Finally, apply Lemma 4.1 to obtain the result. Next, let S : C → 2 E * be a set-valued operator with nonempty, convex, and σ (E * , E)compact values E(x) for x ∈ C that is u.s.c. from every finite-dimensional part C ∩ F (where F is a finite-dimensional subspace of E) into E * equipped with the weak topology σ (E * , E). Then S is called pseudomonotone (p.m.) (see, e.g., [52, Definition 2.10.1], going back to [8]) if the following condition holds: Here we focus on this notion of generalized monotonicity. For other notions of generalized monotonicity for multis and their relation to topological pseudomonotonicity, we refer to [52,Sect. 2.10].
Again, we use construction (2). It is evident that a p.m. set-valued operator gives a p.m. bifunction ϕ. So by a similar reasoning as before, in particular, using Lemmas 4.1 and 4.2, we can conclude from Theorem 3.1 the following existence result for VIs with p.m. set-valued operators.

Theorem 4.2 Let C be a closed convex subset of a real Hausdorff topological vector space E.
Let S : C → E * be a p.m. set-valued operator that is u.s.c. from every part C ∩ F into E * for any finite-dimensional subspace F of E. Suppose there exist a compact subsetK of E and z ∈ C ∩K such that for all x ∈ C \K , there exists ζ ∈ S(z) such that ζ , x -z < 0. Then there exists a solutionx to the variational inequality problem P(S; C).

Extended real-valued equilibrium problems of monotone type in a reflexive Banach space
Let V be a reflexive Banach space with dual V * . We denote by ·, · the duality pairing between V and V * and by · and · * the norm and the dual norm on V and V * , respectively. We denote by " " and "→" the weak and strong convergence in V , respectively. As before, C stands for a nonempty closed convex subset of V .
In the following, we use weak sequential convergence on V and, in particular, the compactness of the unit ball of V with respect to weak sequential convergence. Moreover, since from any weakly convergent net in a reflexive Banach space there can be extracted a weakly convergent subsequence (with the same limit) (see [11,Prop. 7.2]), we can simplify the definition of a p.m. bifunction ϕ as follows. Now ϕ : C × C → R is called Brèzis pseudomonotone or topologically pseudomonotone (p.m.) on C if for any sequence {x n } n∈N ⊂ C that converges weakly to some x ∈ V and satisfies lim inf n∈N ϕ(x n , x) ≥ 0, we have for all y ∈ C. The stability of monotonicity with respect to addition is evident, and it can be shown that the sum of two p.m. bifunctions is p.m., too; see [30,Proposition].
Also, for further use, we note that a hemicontinuous monotone operator T defined throughout V is continuous on finite-dimensional subspaces and pseudomonotone on V ; see [8,Propositions 6 and 9] and [59,Proposition 27.6]. Hence, in particular, the bifunction ϕ T (x, y) := Tx, yx associated with T is p.m. on V . Now we are in a position to apply our general existence theory of Sect. 3 and derive existence results for extended real-valued equilibrium problems of monotone type in the setting of a reflexive Banach space, first, on bounded sets and then on unbounded sets, where we discuss the issue of coercivity following [26].

Extended real-valued equilibrium problems of monotone type on bounded sets
Let V be a reflexive Banach space with norm · and dual V * . As before, C stands for a nonempty closed convex subset of V . In the following, let an extended real-valued function f : C → (-∞, +∞], f ≡ ∞, be convex and l.s.c., and let a real-valued bifunction ϕ : C × C → R vanish on the diagonal of C × C and be convex and l.s.c. with respect to the second argument. We are interested in the existence of a solutionx of the extended real-valued equilibrium problem P[ϕ, f ; C], that is, for all y ∈ C. From Theorem 3.1 we immediately obtain the following: When the bifunction ϕ is only defined on C × C, where C is a proper subset of V , we can appeal to Theorem 3.2 and directly obtain the following Hartman-Stampacchia-like result [34].

Extended real-valued equilibrium problems of monotone type on unbounded sets. Coercivity
Throughout this subsection, we assume that the equilibrium problem P[ϕ, f ; D] is solvable for any bounded closed convex subset D ⊂ C. For brevity, we write ψ(x, y) := ϕ(x, y)+f (y)f (x).
First, we consider the coercivity condition that goes back to Moré [43, Theorem 2] dealing with the solution of complementarity problems.
Next, we consider the coercivity condition that goes back to Stampacchia in [58, Theorem 2.5].
In addition, assume that ϕ is monotone. Then the coercivity condition of Proposition 5.2 implies the existence of x ρ with x ρ < ρ such that ψ(y, x ρ ) ≤ 0 for all y ∈ C ρ . Hence x ρ satisfies the coercivity condition of Proposition 5.1.
Next, we provide asymptotic coercivity conditions for equilibrium problems on unbounded sets.
Then there exists a solution to P[ϕ, f ; C].
Proof By assumption there exists a solution x n of P[ϕ, f ; C n ] on the bounded set C n := C ∩ K(0, n) for all n ∈ N. By a contradiction argument we can show that (3) implies that the norms x n are bounded. Hence there exists a weakly convergent subsequence {x k } k∈K , K ⊂ N, such that x k x for somex ∈ C andx ∈ C k for sufficiently large k ∈ K .
For a hemicontinous and monotone bifunction ϕ, we use the Minty trick as follows. By construction and Proposition 3.2 we have Since the sets C k give rise to an ascending set sequence and exhaust the set C, it follows for all y ∈ C, By Proposition 3.2x solves P[ϕ, f ; C].

The asymptotic coercivity condition (3) is in particular satisfied
A coercivity condition of this type can already be found in [10, Theorem 3].

Extended real-valued equilibrium problems of monotone type under asymptotic coercivity condition
Under the asymptotic condition (4), we formulate the following useful existence results. First, in the monotone case, we combine Theorem 5.1 and Corollary 5.2 to immediately obtain the following: In part (B), we introduce an applicable growth condition for the bifunction ϕ, which can be guaranteed in the context of variational-hemivariational inequalities we will study in the next section.
where the function G : R + = [0, +∞) → R + is strictly increasing with G(0) = 0. Further, suppose that ϕ satisfies, instead of (4), the following growth condition: there exists a positive constant c 1 such that Then there exists a solutionx of the extended real-valued equilibrium problem P[A, ϕ, f ; C], that is, for all y ∈ C, Proof (A) This is a consequence of Theorem 3.1 in combination with Theorem 5.1. (B) First, observe that the hemicontinuous and monotone operator A defined throughout the space V is pseudomonotone and upper semicontinuous on finite-dimensional subspaces. Hence the associated bifunction ϕ A (x, y) = Ax, yx is p.m., and ϕ A (·, y) is upper semicontinuous on the convex hull of any finite subset of C for all y ∈ C; see the proof of Corollary 3.1.
Since summation preserves pseudomonotonicity (see [30]), the bifunction ψ := ϕ A + ϕ is p.m., and ψ(·, y) is upper semicontinuous on the convex hull of any finite subset of C for all y ∈ C.
We claim that f (x)ψ(x, 0), x ∈ C, satisfies the asymptotic coercivity condition (4). Indeed, (5) and (6) entail the estimate and f is bounded below, with appropriate λ ∈ V * and c f ∈ R. This latter estimate can be obtained by strong separation (Hahn-Banach theorem) of the convex closed set epi f = {(x, r) ∈ C × R : r ≥ f (x)} from the compact set (x 1 , f (x 1 ) -1), where x 1 ∈ C is taken such that f (x 1 ) < ∞. Thus we can again apply the Theorem 3.1 in combination with Theorem 5.1 to obtain the conclusion.

A class of variational-hemivariational inequalities
Let V be a reflexive Banach space endowed with norm · . Further, suppose we have a bounded domain D ⊂ R d (with d = 1, 2, 3) and a linear compact operator χ : V → L p (D) (1 < p < ∞). Important particular cases of this framework are: , and χ is the canonical injection that is compact by the Rellich-Kondrachev theorem; see [1], [ [56].
In this setting, we treat a class of variational-hemivariational inequalities with the following main ingredients: -a convex closed subset C of V that can explicitly describe constraints -an extended real-valued convex l.s.c. function f : C → (-∞, +∞], f ≡ ∞, that can implicitly describe constraints -a monotone nonlinearity given by a monotone operator A : V → V * -a nonmonotone nonlinearity given by the generalized Clarke directional derivative of a locally Lipschitz function j (the so-called "superpotential") that gives rise to a p.m. bifunction To make the latter ingredient precise, let us consider a function j : D × R d → R such that j(·, ξ ) : D → R is measurable on D for all ξ ∈ R d and j(s, ·) : R d → R is locally Lipschitz on R d for almost all (a.a.) s ∈ D. Let j 0 (s, ·; ·) stand for the generalized Clarke directional derivative [18] of j(s, ·), and let ∂j(s, ξ ) := ∂j(s, ·)(ξ ) denote the generalized subdifferential of j(s, ·) at the point ξ in the sense of Clarke [18]. Then we define the bifunction ϕ on V × V by Finally, let l ∈ V * be a continuous linear form, which -similarly as C and f -will be specified further.
With this data, we consider the nonlinear variational-hemivariational inequality (HVI): Find u ∈ C such that From Theorem 5.3(B) we derive the following existence result, which extends [15, Theorem 3.1] from hemivariational inequalities with bilinear forms to nonlinear variationalhemivariational inequalities.
where c A > 0 and 1 < p < ∞. Suppose that there exist positive constants c 1 and c 2 such that for a.a. s ∈ D, all ξ ∈ R d , and all η ∈ ∂j(s, ξ ), the following inequalities hold: Then there exists a solution to the HVI (9).

Proof
First, note that conditions (i) and (ii) ensure that the integral in (8) is well defined. Indeed, it follows from (i) and (ii) that for a.a. s ∈ D, and Moreover, thanks to [49, Lemma 1] and [31, Lemma 4.1], the functional ϕ is p.m. and satisfies ϕ(u, 0) ≤ c 3 meas 1/2 (D) χ u ∀u ∈ V (13) for some positive constant c 3 . Hence the given bifunction ϕ satisfies the linear growth condition (6). Finally, relation (5) is satisfied with G(t) = c A t p-1 . Thus Theorem 5.3(B) applies to arrive at the conclusion.
An example is the following scalar variational problem that models nonmonotone contact problems and employs the p-Laplace operator. Hence for some positive constant c A , we have It follows that the operator A : V → V * is monotone and relation (5) is satisfied with G(t) = c A t p-1 . We prescribe nonmonotone, generally set-valued boundary conditions on c via the bifunction ϕ : V × V → R given by Here j 0 (ξ ; η) is the generalized Clarke derivative [18] of a locally Lipschitz function j : Further, γ stands for the linear compact trace operator from V ⊂ W 1,p ( ) into L p ( c ) with the norm γ . Moreover, the given right-hand side f 0 ∈ L 2 ( ), f 1 ∈ L p ( N ) defines the linear form Let C be a nonempty closed convex subset of V to be specified further.
According to [48,Lemma 1], the bifunction ϕ : V × V → R is well-defined, pseudomonotone, and upper-semicontinuous, and relation (6) is satisfied with Thus altogether this example fits to the frame described above, and Theorem 5.3(B) applies to conclude the existence of a solution to the HVI (15).
Finally, in this subsection, we comment on uniqueness of the solution to the HVI (15).
Remark 6.1 Introduce the upper Lipschitz condition on the bifunction ϕ: There exists a constant c ϕ > 0 such that

A more general class of variational-hemivariational inequalities
In this subsection, we turn to a more general class of variational-hemivariational inequalities on a reflexive Banach space V . Following [57,Sect. 5.4], there are the following ingredients: a convex closed subset C ⊂ V , an operator A : C → V * , a bifunction ψ : C × C → R, and a locally Lipschitz function J : C → R with the Clarke generalized directional derivative J 0 . In addition, we introduce an extended real-valued convex and l.s.c. function f : C → (-∞, +∞], f ≡ ∞, and consider the following extended real-valued HVI problem: Find an elementû ∈ C such that We require the following conditions on the data A, ψ, J: [C ψ ] ψ : C × C → R is such that (1) ψ(u, ·) : C → R is convex and l.s.c. on C for all u ∈ C; (2) there exists α ψ > 0 such that there holds the Lipschitz condition [C J ] J : C → R is such that (1) J is locally Lipschitz; (2) there exists α J > 0 such that Then the HVI (18) is uniquely solvable.
Proof Introduce Then for all u ∈ C, (u, ·) is convex and l.s.c. on C, and (u, u) = 0. The bifunction ϕ 2 is hemicontinuous, that is, for all u, v ∈ C, the function t ∈ [0, 1] → ϕ 2 (tv + (1t) Also, the bifunction ϕ 1 is hemicontinuous. To see this, use the shorthand w t := tv +(1t)u for fixed u, v ∈ C. Then w t → u as t → 0, and since ψ(u, ·) is l.s.c. Then write Thanks to the Lipschitz condition, the term in the curled brackets is bounded above by In virtue of (20), the hemicontinuity follows. Thus the bifunction is also hemicontinuous. Next, we estimate using [C A ](2), [C ψ ](2), and [C J ](2): Hence the smallness condition (19) implies that the bifunction is strongly monotone. Further, (21) and (19) imply with the lower bound (7) for the convex l.s.c. function f , shown in the proof of Theorem 5.3, that f (·) -(·, u 0 ) for some u 0 ∈ C satisfies the asymptotic coercivity condition (4). Thus Theorem 5.2 yields the existence of a solution to the HVI (18).
Remark 6.2 1) When we drop ϕ in (9) and when we drop ψ and J 0 in (18), we obtain a standard elliptic variational inequality with nonlinear monotone operator A. On the other hand, dropping the operator A in (9) and in (18) leads in view of the ϕ and J 0 terms, respectively, to a hemivariational inequality. Therefore, following [57], (9) and (18) are called elliptic variational-hemivariational inequalities. In contrast, further in (18), there is a convex l.s.c. function ψ(, ·) on C depending on v ∈ C. So problem (18) can also be considered as a quasi-hemivariational inequality (of the second kind), and thus Theorem 6. 3) It is worth noting that because of the restrictive feature of the smallness condition (19), in the application to static contact problems in [57,Chap. 8] when (see [57, (4.9), p.126]), analogously to (14), the norm of the trace operator γ : V → L 2 ( c ; R d ) enters the smallness condition; see [57,Lemma 8,p.126], but the compactness of γ is not required. In contrast, Theorem 6.1 and Example 6.1 demand the compactness of γ .

A frictional unilateral contact problem
In this section, we apply the main result of Sect. 5 to study the existence of solutions for a broad class of variational-hemivariational inequalities that model unilateral contact problems with nonmonotone, generally set-valued friction laws in solid mechanics. Similar nonmonotone friction phenomena occur with adhesion/cohesion and delamination problems in material science; see, for instance, [20,35,45] and the references therein. Let us consider a deformable body which occupies the Lipschitz domain ⊂ R d (d = 2, 3). The boundary = ∂ is decomposed into three disjoint open parts such that = D ∪ N ∪ C with meas( D ) > 0 and meas( C ) > 0. Suppose that the process is static and, in addition, the body is subjected to volume forces of density f 0 in and to surface tractions of density f 1 on N . On C the body is in frictional unilateral contact with a rigid obstacle (foundation). We model the friction by a boundary condition in the tangential direction involving Clarke's generalized gradient and leading to the study of a nonlinear variational-hemivariational inequality problem.
Let S d be the space of second-order symmetric tensors on R d . For u = (u i ), v = (v i ) ∈ R d , and σ , τ ∈ S d , the inner product and the Euclidean norm on S d and R d are, respectively, u · v = u i v i , u = (u · u) 1/2 , σ · τ = σ ij τ ij , σ = (σ · σ ) 1/2 . We also use the usual notation for the normal components and the tangential parts of vectors and tensors, respectively, given by u N = u · n, u T = uu N n, σ N = σ ij n i n j , and σ T = σ nσ N n, ∂ N u = ∇u · n, where n = (n i ) represents the outward unit normal vector to the Lipschitz boundary ∂ , which is defined almost everywhere.
We consider the following contact problem for a nonlinear elastic body with unilateral constraints and nonmonotone friction condition, where the friction coefficient is slip dependent.
Problem (P 1 ): Find a displacement field u = (u i ) : → R d and a stress field σ = (σ ij ) : Div σ (u) Equation (22) represents the constitutive law for nonlinear elastic materials. The contact is assumed to be static, and hence we use the equilibrium equation (23), f 0 being the body force density. Equations (24) and (25) are the classical displacement and traction boundary conditions: the body is fixed on D , and surface tractions of density f N are applied on N . Assuming that the foundation is perfectly rigid, we have the unilateral Signorini condition (26) on the surface C , where g describes the gap between body and foundation. Relation (27) exhibits the friction law, where ∂j denotes the generalized gradient of the given locally Lipschitz function j, and μ is the friction coefficient, which is assumed to be a positive function on C . The function μ may depend on the slip, that is, on the tangential displacement. For details on mathematical description of static contact models, see, for instance, [41,56].
We require the following conditions on the nonlinear elastic operator A, the superpotential j, and the friction coefficient μ.
[C j ] j : C × R d → R is such that (1) j(x, ·) is locally Lipschitz on R d for a.a. x ∈ C ; (2) j(·, ξ ) is measurable on C for all ξ ∈ R d ; (3) |η| ≤ a(1 + |ξ |) for all η ∈ ∂j(x, ξ ), ξ ∈ R d and a.a. x ∈ C with a > 0; (4) η · (-ξ ) ≤ b|ξ | for all η ∈ ∂j(x, ξ ), ξ ∈ R d and a.a. x ∈ C with b > 0. [C μ ] μ : C × R + → R + is such that (1) μ(·, s) is measurable on C for all s ∈ R + ; (2) There exists L μ > 0 such that μ(x, s 1 )μ(x, s 2 ) ≤ L μ |s 1s 2 |, for all s 1 , s 2 ∈ R + and a.a. x ∈ C ; (3) There exists μ 0 > 0 such that μ(x, s) ≤ μ 0 for all s ∈ R + and a.a. x ∈ C . Moreover, we suppose that the following regularity conditions are satisfied by the gap function and the densities of the body forces and surface traction: To give the weak formulation of problem (P 1 ), we consider the following Hilbert spaces: The inner products over the spaces H and V are given respectively by Let γ : V → L 2 ( ; R d ) be the trace operator, which is continuous and compact, and where γ denotes the norm of γ . We define the load functional l ∈ V * by In the following, we omit the symbol γ and simply write v for the trace of an element v ∈ V when considered on a boundary part. The set of admissible displacement fields is given by which is a closed and convex subset of V . Let A : V → V be the nonlinear operator defined by A(u), v V := A(ε(u)), ε(v) H for u, v ∈ V . Then the weak formulation of problem (P 1 ) is the following.
Problem (P 2 ): Find a displacement fieldū ∈ C such that Conditions [C j ] and [C μ ] ensure that the integral in (29) is well defined. By conditions [C A ], the operator A : V → V * is well-defined, hemicontinuous and uniformly monotone. Introduce the bifunction ϕ : V × V → R by  (6). Thus, proceeding similarly as above in Example 6.1 and as in the proof of Theorem 6.1, we derive from Theorem 5.3(B) the existence of a solutionū ∈ C to problem (P 2 ).

Concluding remarks. An outlook
There is a vast body of literature on the KKM principle with a lot of extensions and ramifications. When these results generalize and as well rely on the classical KKM principle, then they are obviously equivalent in a broad sense. Nevertheless, further results that, similarly to Theorem 2.2, unify some ramifications and reveal the close relationship among them; see, for example, [53]. In our equilibrium approach to variational inequalities with set-valued operators, we focused on variational inequalities of the first kind. Extended real-valued set-valued variational inequalities of the second kind with a lower semicontinuous convex function as a further ingredient are also worth studying.
Further, we focused on a class of variational-hemivariational inequalities with application to static contact problems. Whereas these problems are of interest in their own right, they are also an important building block in the study of more complicated timedependent problems encountered with contact problems in viscoelasticy and viscoplasticity; here we refer, for example, to [19,33,54,56,57].
We also focused on coercive equilibrium problems and coercive variational inequalities. With loss of coerciveness, the existence can be guaranteed under extra conditions for the right-hand side (see the recent paper [32] and the references therein), and the Browder-Tikhonov regularization methods come into play; see, for example, [3,13,16,24,39].