A New Class of History–Dependent Evolutionary Variational–Hemivariational Inequalities with Unilateral Constraints

In this paper we study a new abstract evolutionary variational–hemivariational inequality which involves unilateral constraints and history–dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected functions together with a fixed-point principle for history–dependent operators. Then, we apply the abstract result to show the unique weak solvability to a dynamic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity combined with the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.


Introduction
Mathematical models based on variational and hemivariational inequalities are particulary useful to determine and characterize the qualitative properties of systems in mechanics, economics, engineering sciences, etc. The theory of hemivariational and variational-hemivariational inequalities has been initiated in early 1980s with the pioneering works of Panagiotopoulos, see [35][36][37]. Since then, because of new and efficient approaches which combine convex and nonsmooth analysis, the theory has undergone a remarkable development in pure and applied mathematics, see monographs [12,16,20,21,33,34,37].
Hemivariational inequalities are variational descriptions of physical phenomena that include nonconvex, nondifferentiable and locally Lipschitz functions. They play an important role in a description of diverse mechanical problems arising in solid and fluid mechanics. We refer to [2][3][4]12,19,20,26,28,34,38,39] and the references therein for the recent results on the mathematical theory of contact mechanics and related issues.
In this paper we study the Cauchy problem for a new class of evolutionary first order variational-hemivariational inequalities in infinite dimensional spaces which involve history-dependent operators and a set of constraints. Given a reflexive Banach space V , and a nonempty, closed and convex set of constraints K ⊂ V , the problem reads as follows: find w : (0, T ) → V such that w(t) ∈ K for a.e. t ∈ (0, T ) and

(Sw)(t), Mw(t); Mv − Mw(t))
+ ϕ(t, (R 2 w)(t), Mv) − ϕ(t, (R 2 w)(t), Mw(t)) ≥ 0 (1.1) for all v ∈ K , a.e. t ∈ (0, T ). Here, the nonlinear operators A, R 1 , R 2 , S, and an affine operator M are prescribed, and (1.1) is supplemented by an initial condition. Since the inequality (1.1) contains two potentials, ϕ which is a convex function and j which is a locally Lipschitz function, it is called a variational-hemivariational inequality. Operators R 1 , R 2 and S are called history-dependent, for their properties and applications, see e.g. [18,25,29,30,32,40]. The main novelty of the paper is to establish a new result on existence, uniqueness and regularity of solution to the inequality (1.1). We extend the main theorem of [18,Theorem 9], where a particular case K = V (no constraints) has been studied. It is worth mentioning that in the study of (1.1), one can not proceed as in [18]. Specifically, since the set of constraints is nonempty, closed and convex, the indicator function I K of K is proper, convex and lower semicontinuous, so its subdifferential ∂ I K is maximal monotone. After inserting the term I K (v) − I K (w(t)) in (1.1) makes an equivalent problem with v in the whole space. However, it is clear that I K is not a locally Lipschitz function, so the existence results of [18,Theorems 6 and 9] are not applicable in this case. Note also that virtually in every mechanical system there is a situation in which unilateral effects occur, see [16]. We study the problem (1.1) in the framework of evolution triple of spaces, exploit recent result on the mixed equilibrium inequality, see [6,Theorem 3.4] and [7,Theorem 2.4], and a fixed-point principle for history-dependent operators. The second novelty is to obtain the unique weak solvability to a dynamic frictional contact problem for a viscoelastic material with long memory and unilateral constraints in velocity. The model of this contact problem is new and, to the best of our knowledge, it has not be solved in the literature yet. The contact condition involves a unilateral Signorini-type condition for the normal velocity combined with the nonmonotone normal damped response condition. The friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip. Moreover, a similar unique weak solvability result can be proved for other unilateral boundary condition and a nonmonotone frictional law governed by a nonconvex potential. Results on existence of a weak solution to dynamic viscoelastic unilateral contact problems with constraints in displacement can be found in [1,9,13,15,23] and with constraints in velocity in [17,31].
The outline of this paper is the following. In Sect. 2 we recall the notation and some preliminary materials. The main result of the paper on constrained history-dependent variational-hemivariational inequality is stated and proved in Sect. 3. Finally, in Sect. 4, we examine a dynamic nonsmooth contact problem of viscoelasticity with multivalued contact and friction conditions for which we prove existence and uniqueness of a weak solution.

Essential Tools
In this section we introduce the notation and recall some preliminary results.
Let (X , · X ) be a Banach space, X * denote its dual space and ·, · X * ×X be the duality pairing between X * and X . By J : X → 2 X * we denote the duality mapping defined by It is known, see e.g. [10,41], that if X is a reflexive Banach space, then an equivalent norm can be introduced so that X is a strictly convex space Banach space and, therefore, the duality map is a single-valued and continuous operator. The symbols "→" and " " denote the strong and the weak convergence, respectively. For a set D ⊂ X , conv(D) is the convex hull of D. The notation L(E, F) stands for the space of linear bounded operators from a Banach space E to a Banach space F, and it is well known that L(E, F) is a Banach space endowed with the usual norm · L(E,F) . For a set S ⊂ X , we write S X = sup{ u X | u ∈ S}. We recall some concepts for single-valued operators and bifunctions which can be found in [2,6,7,11,28].
maximal monotone, if it is monotone and the conditions (u, u * ) ∈ X × X * and u * − Av, u − v X * ×X ≥ 0 for all v ∈ X imply u * = Au, (iv) quasimonotone, if lim sup Au n , u n − u X * ×X ≥ 0 for any sequence {u n } ⊂ X with u n u in X , (v) pseudomonotone, if for any sequence {u n } ⊂ X such that u n u in X and lim sup Au n , u n − u X * ×X ≤ 0, we have lim inf Au n , u n − v X * ×X ≥ Au, u − v X * ×X for all v ∈ X , (vi) bounded, if it maps bounded subsets of X into bounded subsets of X * .
for all x ∈ X .

Definition 3
Let K be a nonempty, closed and convex subset of X . A real-valued function F : K × K → R, called later a bifunction, is said to be Definition 4 Let K be a nonempty, closed and convex subset of X . Let F : K ×K → R be a real-valued bifunction with F(u, u) = 0 for all u ∈ K . The bifunction F is said to be maximal monotone if for every u ∈ K and for every convex function ψ : K → R with ψ(u) = 0, we have The following properties collected from [6,7] will be useful in next sections. Below, K denotes a nonempty, closed and convex subset of X .

Remark 5 (i) If
A : X → X * is pseudomonotone (respectively, quasimonotone), then the bifunction F : is pseudomonotone (respectively, quasimonotone). (ii) If the bifunction F is usc with respect to the first argument for the weak topology, then it is pseudomonotone, and in particular, if the condition F(u, u) ≥ 0 for all u ∈ K is satisfied, then F is quasimonotone. (iii) If F, G : K × K → R are pseudomonotone bifunctions such that F(u, u) ≤ 0 and G(u, u) ≤ 0 for all u ∈ K , then F + G is also pseudomonotone. (iv) If A : X → X * is maximal monotone, then the bifunction F : K × K → R defined by F(u, v) = Au, v − u X * ×X for u, v ∈ K , is monotone and maximal monotone.
v ∈ K , then F is monotone and maximal monotone.
We recall an abstract result on existence of solution to the mixed equilibrium problem involving bifunctions of the following form. Let U be a subset of a reflexive Banach space X .
We need the following hypotheses on the data of Problem 6. H (U ) : U is a nonempty, closed and convex subset of X .
(1) F is monotone and maximal monotone, (2) F(u, ·) is convex and lsc for all u ∈ U , (H coer ) : There exists a nonempty weakly compact subset W such that for each λ > 0 small enough, there exists a weakly compact and convex subset B λ of U satisfying the following condition: We have the following existence result whose proof can be found in [ can be omitted in Theorem 7. (ii) If F(u, ·) is convex and lsc for all u ∈ U , and F(u, u) = 0 for all u ∈ U , then the condition (H coer ) is satisfied, if for some v 0 ∈ U , we have Next, we recall the generalized directional derivative and the generalized gradient for a locally Lipschitz function, see [8,10,28].

Definition 9
Let X be a Banach space and j : X → R be a locally Lipschitz function, that is, for each x ∈ X , there are a neighborhood N = N(x) and a constant k The generalized gradient of j at x, denoted by ∂ j(x), is given by Finally, we state the fixed-point result which is a consequence of the Banach contraction principle, see [22,Lemma 7] or [40,Corollary 27].

Existence and Uniqueness Result
The goal of this paper is to study the evolutionary variational-hemivariational inequality with constraints and history-dependent operators. Let (V , H , V * ) be an evolution triple of spaces, that is, V is a separable, reflexive Banach space, H is a separable Hilbert space, the embedding V ⊂ H is continuous and V is dense in H , see e.g. [28,Definition 1.52]. We also assume that the embedding V ⊂ H is compact. Given 0 < T < +∞, we introduce the following Bochner spaces where v denotes the distributional derivative of v. From the reflexivity of V , it follows that both V and V * are reflexive Banach spaces, see [41, p. 411]. Endowed with the graph norm, W is a separable, reflexive Banach space. It is well known that the embeddings V ⊂ L 2 (0, T ; H ) ⊂ V * are continuous and W ⊂ L 2 (0, T ; H ) is compact, see, e.g., [11,Theorem 3.4.13]. The duality pairing between V * and V is denoted by where ·, · V * ×V is the duality pairing between V * and V . We recall that the operator Let X , Y and Z be Banach spaces. The evolutionary history-dependent variational-hemivariational inequality is of the form.
The hypotheses for Problem 11 are following.
We show that the solution to Problem 14 is unique. For simplicity, we skip the subscripts ξ , η and ζ for this part of the proof. Let w i ∈ W, i = 1, 2 be solutions to Problem 14, i.e., w i (t) ∈ K for a.e. t ∈ (0, T ), w i (0) = 0 and From the above inequalities, we obtain Integrating the above inequality on (0, t), using the integration by parts formula, and the hypothesis H (A)(4) on the left hand side, and applying the hypothesis H ( j) (5) to the right hand side, we have for all t ∈ [0, T ]. Hence, due to condition (H 2 ), we have w 1 = w 2 . This completes the proof of uniqueness of the solution.
Step 2 We now prove that Problem 14 has a solution. To this end, we introduce and Consider the following problem.
Lemma 16 Problems 14 and 15 are equivalent.
Proof Let w ∈ W be a solution to Problem 14. This means that w ∈ K and w(0) = 0 which entails w ∈ K 1 . Let z ∈ K 1 . Then, we have Integrating the last inequality on (0, T ), since z ∈ K 1 is arbitrary, we deduce that w ∈ K 1 is a solution to Problem 15. Vice versa, assume that w ∈ K 1 solves Problem 15. Since 0 ∈ K and K is a convex set, by using [24, Theorem 9.1, p.270], we deduce that the condition (9.18) in [24, p.269] holds, which, in particular, implies that Exploiting the density (3.4), we obtain that w ∈ K 1 is a solution to the following problem.
Indeed, let z ∈ K = L 2 (0, T ; K ). From (3.4), there exists a sequence z n ∈ D(L) ∩ K such that z n → z in V. By Problem 15, for all n ∈ N, we have We will pass to the limit in (3.5), as n → ∞. By passing to a subsequence if necessary, we deduce, see e.g. [28,Theorem 2.39], that z n (t) → z(t) in V , for a.e. t ∈ (0, T ) and there is g ∈ L 2 (0, T ) such that z n (t) V ≤ g(t) for a.e. t ∈ (0, T ). for a.e. t ∈ (0, T ). The latter together with H ( j)(4) and the following estimate for a.e. t ∈ (0, T ) with h ∈ L 1 (0, T ), allows to apply the Fatou lemma to get Next, since the function ϕ(t, η(t), ·) is continuous on X for a.e. t ∈ (0, T ), see e.g.
In the final part of the proof, we show that Problems 14 and 17 are equivalent. It is obvious that Problem 14 implies Problem 17. The converse implication follows from [5, Lemma 2.3] (applied with ϕ : is monotone-convex with a suitable growth condition). We conclude that Problems 14 and 15 are equivalent, which completes the proof of the lemma.
It follows from Step 1 and Lemma 16 that Problems 14 and 15 have the same unique solution. Therefore, to show the existence of a solution to Problem 14, it is enough to prove that there exists a solution to Problem 15. In what follows, we prove that Problem 15 has at least one solution. This can be done in several ways. For our purpose, it is sufficient to apply Theorem 7. We introduce the following additional notation which allow to rewrite Problem 15 in the form of the mixed equilibrium inequality as stated in Problem 6. Let the operators A : V → V * , M : V → L 2 (0, T ; X ), and the functions : V → R, J : V → R be defined by Next, we introduce the bifunctions F, G, H : for w, z ∈ K 1 . Under these notation, Problem 15 can be equivalently formulated as follows.

Problem 18
The existence of a solution to Problem 18 is demonstrated by using Theorem 7. We will verify the hypotheses of this theorem for U = K 1 .

Claim 1 The bifunction F defined by (3.8) satisfies the condition H (F).
First, we prove H (F) (1). Since F(w, z)+ F(z, w) = − Lw − Lz, w− z V * ×V ≤ 0 for all w, z ∈ K 1 , it is clear that F is monotone. We show that F is maximal monotone. Let F 1 (w, z) = Lw, z − w V * ×V for w, z ∈ K 1 . Assume that for every w ∈ K 1 and every convex function ψ : (3.12) By the maximal monotonicity of the operator L : V → V * , it follows that F 1 is maximal monotone. Thus, the condition (3.12) implies Thus

η(t), Mz(t)) dt = (z).
Thus for w ∈ V , we have which implies that F(w, ·) is lsc for all w ∈ K 1 . The condition H (F)(3) is obvious. Hence, we deduce that Claim 1 holds.

Claim 2 The bifunction G defined by (3.9) satisfies the condition H (G).
We first show that, for each finite subset D of K 1 , G(·, z) is usc on conv(D) for all z ∈ K 1 . Let {w n } ⊂ conv(D) be such that w n → w in V. Since conv(D) is a closed and convex set, we deduce that w ∈ conv(D). From hypothesis H (A) (2) and [41, Proposition 27.7(b)], it follows that A : V → V * is demicontinuous. Thus, we have Aw n Aw in V * and Hence G(·, z) is usc on conv(D) for all z ∈ K 1 , i.e., the condition H (G) (2) holds. It follows from hypothesis H (A) (2),(4) that G is pseudomonotone on K 1 . The proof is analogous to the one in [7,Lemma 3.4]
Step 4 In this part of the proof we use a fixed point argument. We define the operator : where w ξηζ ∈ W denotes the unique solution to Problem 14 corresponding to (ξ, η, ζ ).
Finally, we use these relations in Problem 14, and conclude that w ξ * η * ζ * is the unique solution to Problem 11. This completes the proof of the theorem.
Next, we consider the constrained variational-hemivariational inequality with the nonhomogeneous initial condition w 0 ∈ V , w 0 = 0.

Problem 19
Find w ∈ W such that w(t) ∈ K for a.e. t ∈ (0, T ) and (3.17) With this notation, Problem 19 can be reformulated as follows.

Problem 21
Find w ∈ W such that w(t) ∈ K for a.e. t ∈ (0, T ) and We will apply Theorem 13 to obtain that Problem 21 has a unique solution w ∈ W. It is sufficient to show that the operators A and M defined by (3.16)

and (3.17) satisfy the hypotheses H (A) and H (M).
We first verify that A satisfies H (A). The conditions Hence, H (A)(3) holds with a 0 (t) = a 0 (t) + a 1 w 0 V and a 1 = a 1 . Moreover, for v 1 , v 2 ∈ V , a.e. t ∈ (0, T ), we have Hence for v ∈ V and (3.17), we have which means that M is an affine operator. Furthermore, for v ∈ V , a.e. t ∈ (0, T ), we get It follows from the compactness of M that M is compact. Therefore, M satisfies H (M).
It follows from Theorem 13 that Problem 21 has a unique solution w ∈ W. Therefore, w ∈ W given by w(t) = w(t) + w 0 is the unique solution to Problem 19. This completes the proof.

Unilateral Frictional Contact Problem
In this section, we study a unilateral viscoelastic frictional contact problem to which our main results of Sect. 3 can be applied. We give the classical formulation of the contact problem, provide its variational formulation, and prove a result on its unique weak solvability.
Consider the following physical setting. A viscoelastic body occupies a bounded domain ⊂ R d , d = 1, 2, 3. The boundary of , denoted by , is assumed to be Lipschitz continuous. We denote by ν the outward unit normal at . We suppose that consists of three mutually disjoint and measurable parts D , N and C such that meas ( D ) > 0. The symbol S d denotes the space of d × d symmetric matrices. The standard inner products and norms on R d and S d are given by respectively. For a vector field v, v ν and v τ denote its normal and tangential components on the boundary defined by Given a tensor σ , the symbols σ ν and σ τ denote its normal and tangential components on the boundary, i.e., σ ν = (σ ν) · ν and σ τ = σ ν − σ ν ν.
The classical model for the contact process on the finite time interval is the following.

Problem 22
Find a displacement field u : × (0, T ) → R d and a stress field σ : × (0, T ) → S d such that for all t ∈ (0, T ), and In this problem, equation (4.1) represents the constitutive law for viscoelastic materials with long memory in which A is the viscosity operators, B represents the elasticity operator and C is the relaxation tensor, and ε(u) denotes the linearized strain tensor defined by Equation (4.2) is the equation of motion in which Divσ = (σ i j, j ) and f 0 denotes the density of the body forces. The displacement homogeneous boundary condition (4.3) means that the body is fixed on D , while (4.4) is the traction boundary condition with surface tractions of density f N acting on N . Condition (4.5) is the Signorini unilateral contact boundary condition for the normal velocity in which g > 0 and ∂ j ν denotes the Clarke subgradient of a prescribed function j ν . Condition η(t) ∈ k(u ν (t)) ∂ j ν (u ν (t)) on C represents the normal damped response condition where k is a given damper coefficient depending on the normal displacement. Condition (4.6) is a version of the Coulomb law of dry friction in which F b denotes the friction bound. The latter may depend on the total (accumulated slip) represented by Examples and details on mechanical interpretation of conditions (4.5) and (4.6) can be found in [18,28] and references therein. Finally, conditions as in (4.7) are the initial conditions in which u 0 and w 0 represent the initial displacement and the initial velocity, respectively.
(1) k(·, r ) is measurable on C for all r ∈ R.
Further, we introduce the set of admissible velocity fields U defined by and an element f ∈ V * by We now turn to the weak formulation of Problem 22. Let (u, σ ) be a smooth solution to this problem which means that the data are smooth functions such that all the derivatives and all the conditions are satisfied in the usual sense at each point. Let v ∈ U and t ∈ (0, T ). We multiply (4.2) by v − u (t), use an the integration by parts formula, and apply the boundary conditions (4.3) and (4.4) to obtain From (4.5) and the definition of the Clarke subgradient, we have On the other hand, the friction law (4.6) can be equivalently written as (4.10) Combining (4.9), (4.10) and the decomposition formula, see [28, (6.33)], we obtain The latter and the definition (4.8) imply Finally, exploiting (4.1), we are lead to the following variational formulation of Problem 22.
The following result concerns the unique solvability and regularity of solution to Problem 23. Then Problem 23 has a unique solution u ∈ C([0, T ]; V ), u ∈ W with u (t) ∈ U for a.e. t ∈ (0, T ).
Proof We will apply Theorem 20 with the following functional framework: X = Y = Z = L 2 ( C ), K = U , and M = γ . Let the operator A : V → V * , and functions ϕ : Y × X → R and j : Z × X → R be defined by Note that A, ϕ and j are now independent of t. Moreover, we introduce operators Let w(t) = u (t) for all t ∈ (0, T ). Then, with the above notation, Problem 23 can be formulated as follows. can be verified as in the proof in [18,Theorem 13]. We deduce from Theorem 20 that Problem 25 has a solution w ∈ W such that w(t) ∈ U for a.e. t ∈ (0, T ). From the relation u(t) = t 0 w(s) ds + u 0 for all t ∈ (0, T ), we conclude that u ∈ C([0, T ]; V ), u ∈ W with u (t) ∈ U for a.e. t ∈ (0, T ). This completes the proof.
A couple of functions (u, σ ) which satisfies Problem 23 and (4.1) is called a weak solution to Problem 22. Under the assumptions of Theorem 24, Problem 22 has a unique weak solution. Further, this solution has the regularity u ∈ C([0, T ]; V ), u ∈ W with u (t) ∈ U for a.e. t ∈ (0, T ), σ ∈ L 2 (0, T ; H), and Div σ ∈ V * .
Finally, we note that the main contribution of this paper consists in extending [18,Theorem 9] from the case of no constraints (i.e., K = V ) to the case of a nonempty, closed and convex set of constraints K ⊂ V . Moreover, we have used the recent existence result for an equilibrium problem involving bifunctions which is a novelty for variational-hemivariational inequalities. It is reasonable to conjecture that Problem 11 has solutions without the smallness assumption in (H 2 ), however, the uniqueness is likely to be lost.