Evolutionary variational–hemivariational inequalities with applications to dynamic viscoelastic contact mechanics

The purpose of this work is to introduce and investigate a complicated variational–hemivariational inequality of parabolic type with history-dependent operators. First, we establish an existence and uniqueness theorem for a first-order nonlinear evolution inclusion problem, which is driven by a convex subdifferential operator for a proper convex function and a generalized Clarke subdifferential operator for a locally Lipschitz superpotential. Then, we employ the fixed point principle for history-dependent operators to deliver the unique solvability of the parabolic variational–hemivariational inequality. Finally, a dynamic viscoelastic contact problem with the nonlinear constitutive law involving a convex subdifferential inclusion is considered as an illustrative application, where normal contact and friction are described, respectively, by two nonconvex and nonsmooth multi-valued terms.


Introduction
The contact processes between deformable bodies around in industry and our real-life and, for this reason, a considerable effort for modeling, mathematical analysis, numerical simulation and optimal control of various frictional contact problems are quite interesting and important.
The theory of variational inequalities can be used to describe the principles of virtual work and power which was initially proposed by Fourier in 1823. The prototypes, which lead to a class of variational inequalities, are the problems of Signorini-Fichera and frictional contact in elasticity. However, the first complete proof of unique solvability to Signorini Problem was provided by Signorini's student Fichera in 1964. The solution of the Signorini Problem coincides with the birth of the field of variational inequalities. For more on the initial developments of elasticity theory and variational inequalities, cf. e.g., [1]. With the gradual improvement of the theory of variational inequalities, there are numerous monographs dedicated to solving various complex phenomena in contact problems with different bodies and foundations, see for instance [7,8,12,32] and others. As the generalization of variational inequalities, the theory of hemivariational inequalities was first introduced and studied by Panagiotopoulos in [30]. The mathematical theory of hemivariational inequalities has been of great interest recently, which is due to the intensive development of applications of hemivariational inequalities in contact mechanics, control theory, games and so forth. Some comprehensive references are [4,13,[15][16][17][18][19]21,[24][25][26][27]29,31].
Before proving the main problem, it should be mentioned that all of the convex and Clarke subdifferentials which are appeared in the sequel of the present paper are always understood with respect to the last variable of the corresponding functions.
The abstract evolution inclusion problem of parabolic type under the consideration is formulated as follows.
where the function f and initial data w 0 are assumed to satisfy the following regularities To deliver the existence and uniqueness of solution to Problem 3.1, we make the following assumptions. The nonlinear function A : [0, T ] × V → V * satisfies the following conditions.
The main result of the section concerning the existence and uniqueness for Problem 3.1 is provided as follows. (3.2), and if, in addition, the inequality

Theorem 3.2. Under the assumptions of H(A), H(ϕ), H(ψ) and
holds, then Problem 3.1 admits a unique solution w ∈ W.
We shall employ the surjectivity result, Theorem 2.5, to obtain the desired conclusion in Theorem 3.2, by formulating Problem 3.1 to an abstract operator inclusion problem. To the end, we define an operator and introduce a convex function Φ : For any v ∈ V and t ∈ [0, T ] fixed, we may restate Problem 3.1 to the inequality problem, by multiplying the first equation of (3.1) with v(t) − w(t) and integrating the resulting In the meantime, consider the functions A w0 : Then, under the above definitions, it is easy to see that u ∈ W is a solution to problem (3.5), if and only if, z := u − w 0 ∈ D(L) solves the following operator inclusion problem find z ∈ D(L) such that Proof of Theorem 3.2. With respect to the existence of solutions to Problem 3.1, the proof will be based on Theorem 2.5.
Invoking [21,Lemma 3.64], it is well-known that the operator L defined in (3.7) is densely defined, linear, and maximal monotone. We assert that the mapping Q w0 : V → 2 V * defined by is coercive and bounded.
By virtue of hypotheses H(A)(iii), (iv), Hölder inequality and the element inequality On the other hand, H(ϕ)(iii) and Hölder inequality deduce we are now in a position to utilize the smallness condition (3.3) to conclude that Q w0 is coercive.
where the constants r 1 , r 2 , r 3 ≥ 0 are all independent of z. Therefore, Q w0 is a bounded mapping. Next, we shall demonstrate that Q w0 is L-pseudomonotone in the sense of Definition 2.1. To the end of this, we make the following three claims. Claim 1. The set Q w0 z is nonempty, bounded, closed and convex in V * for every z ∈ V.
Let z ∈ V be fixed. Proposition 2.4(i) implies that the set F w0 z is a nonempty and convex in V * , so does Q w0 z. However, the inequality (3.9) guarantees the boundedness of Q w0 z. To illustrate that the set of Q w0 z is closed, let {η n } ⊂ Q w0 z be such that η n → η in V * , as n → ∞. So, there exists a sequence {ζ n } ⊂ F w0 z such that η n = ζ n + A w0 z and ζ n → η − A w0 z in V * , as n → ∞. Then, passing to a subsequence if necessary, we assume that In accordance with Proposition 2.4(ii), it finds that the set of η − A w0 z ∈ F w0 z. Therefore, the set Q w0 z is also closed.
From [21,Proposition 3.8], it is enough to verify that for each weakly closed set C in V * , the set We now show that A w0 : V → V * is demicontinuous. Let {z n } ⊂ V be such that z n → z in V, as n → ∞. By passing to a subsequence if necessary, we may say (3.10) for all w ∈ V and a.e. t ∈ [0, T ]. The latter combined with hypothesis H(A)(ii) and Lebesgue-dominated convergence theorem implies

In view of the condition H(A)(ii), it reads
Invoking Proposition 2.4 and [21, Theorem 3.13] indicateŝ be a sequence such that w n → w in V, as n → ∞; thus, there isζ n = A w0 w n +ζ n withζ n ∈ F w0 w n . From the boundedness of F w0 , we may assume thatζ n →ζ weakly in V, as n → ∞, whereas by the demicontinuity of A w0 and the fact, Furthermore, recall that the subset C ⊂ V * is weakly closed, so it holds w ∈ C. Therefore, we have that We are going to showζ ∈ Q w0 z and ζ n , We now assert the convergence holds z n → z strongly in V. (3.14) Letζ n ∈ F w0 z n be such thatζ n =ζ n + A w0 z n . For anyζ ∈ F w0 z, H(ϕ)(iv) turns out for allζ ∈ Q w0 z and all z ∈ V. Then, if for the above inequality, passing to the upper limit, as n → ∞, and using (3.12), we derive But, the smallness condition (3.3) indicates that α − β > 0, namely (3.14) is valid. In the meanwhile, employing the demicontinuity of A w0 and the closedness of F w0 (see the proof of Claim 2), it yields ζ ∈ Q w0 z. This means that (3.13) is satisfied. Moreover, we also admit that Φ w0 : V → R is proper, convex and lower semicontinuous. The result Φ w0 ≡ +∞ is a direct consequence of hypothesis H(ψ)(i). Also, the convexity of Φ w0 can be obtained by applying the convexity of ψ. Let z n → z in V, as n → ∞. Passing to a subsequence, if necessary, one has However, from [28, Lemma 2.5(2)], we are able to find a function h Therefore, Φ w0 is lower semicontinuous on V. Invoking Proposition 2.3 indicates that ∂ c Φ w0 : V → 2 V * is maximal monotone.
Additionally, we shall demonstrate that ∂ c Φ w0 is strongly quasi-bounded on V with 0 ∈ ∂ c Φ w0 (0). For any M > 0 fixed, let z ∈ D(∂ c Φ w0 ) and ξ ∈ ∂ c Φ w0 (z) be such that Recall that w 0 ∈ intD(ψ), there exist an ε > 0 and K ε ∈ R such that ψ(y) ≤ K ε < +∞ for all y ∈ {x ∈ V | x − w 0 V < ε} (since ψ is locally Lipschitz continuous in intD(ψ)). Define the open neighborhood . Therefore, by using Proposition 2.2, we conclude that ∂ c Φ w0 is strongly quasi-bounded on V. On the other hand, the estimates To conclude, we have verified all conditions of Theorem 2.5. Using this theorem, we conclude that L+ Q w0 + ∂ c Φ w0 is onto; thus, the inclusion (3.8) has a solution z ∈ D(L). Consequently, w = z + w 0 ∈ W solves Problem 3.1. We illustrate that Problem 3.1 is unique solvability. Let w 1 , w 2 ∈ W be two solutions to Problem 3.1, i.e., for i = 1, 2, A simple calculation gives

History-dependent variational-hemivariational inequalities
In this section, we are interesting in the study of existence and uniqueness of solution to a generalized variational-hemivariational inequality involving history-dependent operators, in which the historydependent operators are, respectively, acted on the elastic operator and locally Lipschitz function. In what follows, let Y i for i = 1, 2, 3 be Banach spaces. The problem under investigation reads as follows.
(d) there are a 2 > 0 and a 1 ∈ L 2 (e) there exists a constant α > 0 such that (4.4) The main theorem of the section is delivered as follows.
We end the section by providing the following particular cases of Problem 4.1. Let K be a nonempty, closed and convex subset of V such that w 0 ∈int(K) = ∅, consider the function ψ : where ϕ : V → R is a convex and lower semicontinuous function, and I K : V → R ∪ {+∞} is the indicator function of K given by Obviously, we can see that the function ψ defined in (4.10) satisfies conditions H(ψ). In this case, we have the following corollary.
has a unique solution w ∈ W.
Indeed, under the suitable assumptions, this corollary, Corollary 4.3, can imply that problem (1.2) has a unique solution.

A dynamic viscoelastic contact problem
In the present section, we are concerned with the applicability of the results obtained in Sect. 4 to a new dynamic contact model for a viscoelastic material with the constitutive law involving a convex subdifferential inclusion, and multi-valued boundary conditions with nonconvex contact and friction potentials. The physical setting of the model is described as follows. Assume a viscoelastic body occupies a bounded and connected domain Ω in R d (d = 2, 3) such that its boundary Γ = ∂Ω is Lipschitz continuous. The boundary also is considered to be composed of three mutually disjoint and measurable parts Γ D , Γ N and Γ C with meas(Γ D ) > 0 (i.e., the measure of Γ D is positive). In the meanwhile, we adopt the standard notation and function spaces H, H and H 1 , which are mentioned in the end of Sect. 2. We set The classical formulation of the contact problem is stated as follows.
Problem 5.1. Find a displacement field u : Q → R d and a stress field σ : Q → S d such that We now provide a brief description on the equations, conditions and relations appeared in Problem 5.1. Inclusion (5.1) is a nonlinear viscoelastic constitutive law, where ϕ : S d → R ∪ {+∞} is a proper convex and lower semicontinuous function, and A : Q × S d × S d → S d presents a viscoelasticity operator (see for example, [34]), which is considered to read the following conditions.
(c) there exists α A > 0 such that As a special case, A can be specialized by the sum of a viscosity operator P and an elasticity operator B, i.e., A(x, t, ε, η) := P t, x, ε(u (t)) + B t, x, ε(u(t)) . In this moment, when ∂ c ϕ ≡ 0, the constitutive law (5.1) reduces to the nonlinear Kelvin-Voigt constitutive law, thus, x, ε(u(t)) for a.e. (x, t) ∈ Q, which has been frequently used to the study of various dynamic or quasi-static contact problems, see for instance, [23,35,36]. Condition (5.6) presents the initial displacement and velocity fields, which entail the following condition u 0 , w 0 ∈ V with w 0 ∈ intD(ψ) and 0 S d ∈ ∂ c ϕ(ε(w 0 (x))) for a.e. x ∈ Ω, (5.9) where ψ : V → R ∪ {+∞} is defined by and V is a closed subspace of H 1 given by Let V * be the dual space of V . Recall that meas(Γ D ) > 0, it follows from Korn's inequality that the space V is a real Hilbert space equipped with the inner product and the associated norm · V . However, by the Sobolev trace theorem, we have for some C 0 > 0, which only depends on the domain Ω, Γ D and Γ C .
To deliver the variational formulation of Problem 5.1, we now assume that there are the displacement field u and the stress field σ sufficiently smooth which satisfy (5.1)-(5.6). Denote w = u the velocity field. Also, we introduce the operator S : (5.14)
The existence and uniqueness theorem to Problem 5.2 is given as follows. for all z, q ∈ L 2 (Γ C ), all w ∈ V and a.e. t ∈ [0, T ]. Next, we shall prove that the problem: find w ∈ W such that for all v ∈ V and a.e. t ∈ [0, T ] with w(0) = w 0 , has a unique solution. The proof of the assertion is mainly based on the theoretical result, Theorem 4.2. Hence, the current goal is to illustrate that all of conditions presented in Theorem 5.4 are valid.
. From the formulations of S i , i = 1, 2, 3, we have the lemma.