Differential variational–hemivariational inequalities: existence, uniqueness, stability, and convergence

The goal of this paper is to study a comprehensive system called differential variational–hemivariational inequality which is composed of a nonlinear evolution equation and a time-dependent variational–hemivariational inequality in Banach spaces. Under the general functional framework, a generalized existence theorem for differential variational–hemivariational inequality is established by employing KKM principle, Minty’s technique, theory of multivalued analysis, the properties of Clarke’s subgradient. Furthermore, we explore a well-posedness result for the system, including the existence, uniqueness, and stability of the solution in mild sense. Finally, using penalty methods to the inequality, we consider a penalized problem-associated differential variational–hemivariational inequality, and examine the convergence result that the solution to the original problem can be approached, as a parameter converges to zero, by the solution of the penalized problem.


Introduction
The problems called differential variational inequalities (DVIs, for short) is a kind of dynamic systems which consist of a differential equation combined The main contributions of the paper are threefold. First, using KKM principle, Minty's approach, and the properties of Clarke's subgradient, we prove that the solution set of variational-hemivariational inequality (2) is nonempty, bounded, closed, and convex. As a result, the measurability and upper semicontinuity for variational-hemivariational inequality (2) with respect to the time variable and state variable are illustrated. Second, by applying a fixed point theorem for history-dependent operators, a well-posedness result for differential variational-hemivariational inequality (1), including the existence, uniqueness, and stability of the solution in mild sense, is established. Finally, the penalty methods are employed to differential variationalhemivariational inequality (1) to consider a penalized problem, problem (20) corresponding to original problem (1) (see Sect. 4), and examine the convergence result that the solution to the original problem can be approached, as a parameter converges to zero, by the solution of the penalized problem.
The outline of the paper is as follows. Basic notation and preliminary material needed in the sequel are recalled in Sect. 2. In Sects. 3 and 4, we deliver the main results of the paper which include existence, uniqueness, stability and convergence of the solution in mild sense for differential variationalhemivariational inequality (1). Section 5 gives a conclusion of the paper .

Mathematical background
In this section, we briefly review basic notation and some results which are needed in the sequel. For more details, we refer to monographs [3,4,25,36]. Throughout the paper, we denote by ·, · Y * ×Y the duality pairing between a Banach space Y and its dual Y * . The norm in a normed space Y is denoted by · Y . Given a subset D of Y , we write If no confusion arises, we often drop the subscripts. For any nonempty set X, we denote by P (X) the collection of its nonempty subsets. Besides, we denote by L(Y 1 , Y 2 ) the space of linear and bounded operators from a normed space Y 1 to a normed space Y 2 endowed with the usual norm · L(Y1, Y2) . In what follows, the symbols "→" and " " denote the strong and the weak convergence in various spaces which will be specified. Definition 2.1. Let (X, · X ) be a reflexive Banach space with its dual X * and A : X → X * . We say that is a bounded operator and for every sequence {x n } ⊆ X converging weakly to x ∈ X such that lim sup Ax n , x n −x ≤ 0, we have Ax, x − y ≤ lim inf Ax n , x n − y for all y ∈ X. (iv) A is hemicontinuous, if for all u, v, w ∈ X, the function λ → A(u + λv), w is continuous on [0, 1].
It is obvious that A : X → X * is pseudomonotone if and only if A is bounded and x n x in X with lim sup Ax n , x n −x ≤ 0 imply lim Ax n , x n − x = 0 and Ax n Ax in X * . Furthermore, if A ∈ L(X, X * ) is nonnegative, then it is pseudomonotone.
Let X be a Banach space with its dual space X * . A function f : X → R := R ∪ {+∞} is called proper, convex, and lower semicontinuous, if it fulfills, respectively, the following conditions: In the meantime, we review the definitions and properties of semicontinuous multivalued mappings. Definition 2.2. Let X and Y be topological spaces, and F : X → P (Y ) be a multivalued mapping. We say that F is If this holds for every x ∈ X, then F is called upper semicontinuous. the graph of the multivalued mapping F defined by We say that F is closed (or F has a closed graph), if it is closed at every x 0 ∈ X.
The following theorem gives a criterium for upper semicontinuity.
, with X and Y topological spaces. The statements below are equivalent: Definition 2.4. Let E and V be Banach spaces and let I ⊂ R be an interval. We say that F : I × E → P (V ) is superpositionally measurable if, for every measurable multivalued mapping Q : I → P (E) with compact values, the superposition Φ : Indeed, it is quite difficult to examine if a multivalued mapping is superpositionally measurable using definition. Fortunately, the following theorem provides a necessary criterion to validate whether a multivalued mapping is superpositionally measurable. Theorem 2.5. Let F : I ×E → P (V ) be a multivalued mapping. If t → F (t, u) is measurable on I for all u ∈ E and u → F (t, u) is upper or lower semicontinuous for a.e. t ∈ I, then F is superpositionally measurable.
Furthermore, we recall the well-known result, KKM principle, see Ky Fan [5], which will be used in Sect. 3 to verify the existence of solutions to generalized variational-hemivariational inequality (2). Lemma 2.6. Let K be a nonempty subset of a Hausdorff topological vector space V , and let G : K → P (V ) be a multivalued mapping satisfying Then it holds ∩ v∈K G(v) = ∅.
A function J : X → R is called locally Lipschitz continuous at u ∈ X, if there exist a neighborhood N (u) of u and a constant L u > 0 such that The generalized gradient of J : X → R at u ∈ X is given by In fact, the generalized gradient and generalized directional derivative of a locally Lipschitz function enjoy many nice properties and rich calculus. Here we just collect below some basic and critical results, see cf. [ Additionally, we recall the notion of the penalty operators, see [31]. Definition 2.9. Let X be a Banach space and K be a nonempty subset of X. An operator P : X → X * is called a penalty operator of set K if P is bounded, demicontinuous, monotone and K = { u ∈ X | P u = 0 X * }.

A well-posedness result for differential variational-hemivariatinal inequalities
This section is devoted to prove a well-posedness result for differential variational-hemivariational inequality, problem (1), including the existence, uniqueness, and continuous dependence of solution with respect to initial data. We assume that the data of problem (1) read as follows.
(ii) the following inequality holds: with r(s) → +∞ as s → +∞ and an element u 0 ∈ K such that The nonlinear function f : [0, T ] × E × V → E satisfies the following conditions: To establish the existence result for problem (1), first, we will exploit the following generalized elliptic variational-hemivariational inequality: find u ∈ K such that where Q : K → V * is a given mapping. For simplicity, in what follows, denote by SOL(K; Q, J, ϕ) the solution set of problem (3).
Then the solution set of problem (3), SOL(K; Q, J, ϕ), is nonempty, bounded and weakly closed in V .
Proof. We first show that the solution set of problem (3), SOL(K; Q, J, ϕ), is nonempty. To do so, we shall consider two situations that K is bounded and K is unbounded.
Obviously, for each v ∈ K, the set G(v) is nonempty, owing to v ∈ G(v) for each v ∈ K. Besides it asserts that G has weakly closed values. Let {u n } ⊂ G(v) be a weakly convergent sequence, namely u n u as n → ∞ for some u ∈ V . Hence, for each n, one has Notice that K is closed and convex, so it has u ∈ K. On the other side, the compactness of γ, hypothesis H(ϕ)(ii) and the fact ( Passing to the upper limit as n → ∞ in inequality (4) and taking into account the above inequalities, it finds Vol. 22 (2020) Differential variational-hemivariational inequalities Page 9 of 30 83 Multifunction G is a KKM mapping. Indeed, arguing by contradiction, there exist a finite subset {v 1 This means that The monotonicity of Q ensures that , as well as the positive homogeneity and subadditivity of v → J 0 (u; v), we obtain This generates a contradiction, so, we conclude that G is a KKM mapping.
Keeping in mind that K is bounded, closed and convex, it follows from reflexivity of V that K is also weakly compact. This implies that G(v) is weakly compact too, for every v ∈ K, since G has weakly closed values. By invoking KKM principle, Lemma 2.6, with respect to the weak topology of Hence, we can find an element u * ∈ K such that where we have utilized the condition H(ϕ)(iii). Passing to the lower limit as t → 0 + in the above inequality and using condition (i), one obtains Since w ∈ K is arbitrary, this signifies that u * ∈ K is a solution to problem (3).
Furthermore, we consider the situation that K is unbounded. For each n ∈ N, define K n ⊂ K by where u 0 is given in the hypothesis (ii). So, for each n ∈ N, we are able to find a solution u n ∈ K n to the following problem: for all v ∈ K n . We affirm that there exists an integer N 0 ≥ 1 such that If it does not hold, then for each n ∈ N, the equality u n − u 0 V = n is true. This points out u n V → ∞ as n → ∞. Putting v = u 0 into (5), it turns out However, Proposition 2.8(ii) says that there is an element ξ n ∈ ∂J(γu n ) such that Remembering that u n V → ∞ as n → ∞ and condition (ii), we imply This leads a contradiction. Hence, the claim in (6) is valid.
Let w ∈ K be arbitrary. Assume that N 0 ∈ N and u N0 are such that (6) holds. It allows us to pick a sufficiently small t > 0 satisfying Next, we show that SOL(K; Q, J, ϕ) is weakly closed in V . Let {u n } ⊂ SOL(K; Q, J, ϕ) be a weakly convergent sequence, i.e., u n u in V for some u ∈ V . Then we have The monotonicity of Q suggests The upper semicontinuity of w → ϕ(v, w) and (w, v) → J 0 (w; v) combined with the compactness of γ and hypothesis Passing to the upper limit in inequality (10) and taking into account the above inequalities, we have Now, we use the Minty approach again to obtain u ∈ SOL(K; Q, J, ϕ). Therefore, SOL(K; Q, J, ϕ) is weakly closed in V .
It is enough to demonstrate that SOL(K; Q, J, ϕ) is bounded. Suppose that SOL(K; Q, J, ϕ) is unbounded. Therefore, we can find a sequence {u n } ⊂ SOL(K; Q, J, ϕ) such that u n V → +∞ as n → ∞. As before we did, a simple calculation gives This reaches a contradiction. Consequently, we conclude that SOL(K; Q, J, ϕ) is bounded.
Additionally, we have the following two corollaries.
is nonempty, bounded, and weakly closed.
Proof. It is easy to verify that the function ϕ : hypothesis H(ϕ). The conclusion of the corollary is a direct consequence of Theorem 3.1.
is nonempty, bounded, and weakly closed.
The following lemma delivers a convexity result for set SOL(K; Q, J, ϕ).
Then the set SOL(K; Q, J, ϕ) is convex, when it is nonempty.
Proof. Assume that SOL(K; Q, J, ϕ) is nonempty. Let u 1 , u 2 ∈ SOL (K; Q, J, ϕ) and t ∈ (0, 1) be arbitrary. Then for i = 1, 2 we have Using the property, Proposition 2.8(ii), we are able to find an element ξ i ∈ ∂J(γu i ) such that for i = 1, 2. Taking into account (11) and monotonicity of u → Qu + γ * ∂J(γu), it yields The above inequalities and the concavity of u → ϕ(v, u) point out for all ξ v ∈ ∂J(γv) and all v ∈ K. For any w ∈ K, inserting v λ = λw + (1 − λ)u t into (12) implies Passing to the lower limit as λ → 0 in the above inequality and using condition (i) of Theorem 3.1, it emerges here we have applied the upper semicontinuity of (u, v) → J 0 (u; v), see Proposition 2.8(iii), and the fact, lim inf f (x n ) + g(x n ) ≤ lim inf f (x n ) + lim sup g(x n ). Since w ∈ K is arbitrary, so, we conclude that u t ∈ SOL (K; Q, J, ϕ), namely the set SOL(K; Q, J, ϕ) is convex.

Vol. 22 (2020)
Differential variational-hemivariational inequalities Page 13 of 30 83 As a byproduct of the proof of Lemma 3.4, we also provide a Minty type equivalence result for problem (3).

Lemma 3.5. Under the assumptions of Lemma 3.4, u ∈ SOL(K; Q, J, ϕ) if and only if u solves the following Minty variational-hemivariational inequal-
Combining Theorem 3.1 with Lemma 3.4, we obtain the following theorem.

Consider a multivalued mapping
for all (t, x) ∈ [0, T ] × E. Moreover, the following theorem reveals that U is well defined, strongly-weakly upper semicontinuous, and superpositionally measurable. Proof. In fact, Theorem 3.6 implies that for each (t, x) ∈ [0, T ] × E the set U (t, x) is nonempty, bounded, closed, and convex. So, the mapping U : [0, T ]× E → P (K) is well defined. We now apply Proposition 2.3 to verify the assertion (U 1 ). It reminds us to demonstrate that for each weakly closed subset C ⊂ K, the set Thus, there exists a sequence {u n } ⊂ K with u n ∈ U (t n , x n ) ∩ C for each n ∈ N, especially, and all n ∈ N. We prove that the sequence {u n } is uniformly bounded. Arguing by contradiction, passing to a subsequence if necessary, we may say that u n V → ∞ as n → ∞. Taking v = u 0 into (14) and using hypothesis for some c ϕ ∈ R and η ϕ ∈ V * . Then we have Recall that ρ is a bounded function, {(t n , x n )} is bounded in [0, T ] × E, and r(s) → +∞ as s → +∞, it takes the lower limit as n → ∞ in the above inequality to get This generates a contradiction, hence, {u n } is uniformly bounded. Without loss of generality, we may assume that u n u in V as n → ∞, for some u ∈ K.
On the other hand, Lemma 3.5 and (14) guarantee and all n ∈ N. Remember that u → ϕ(v, u) is concave and upper semicontinuous. Passing to the upper limit as n → ∞ in (15), it yields is weakly upper semicontinuous (because it is concave and upper semicontinuous). Employing Lemma 3.5 again, we conclude that u ∈ U (t, x). The latter coupled with the weak closedness of C implies u ∈ U (t, x) ∩ C, i.e., (t, x) ∈ U − (C). Therefore, U is strongly-weakly upper semicontinuous.
Concerning the proof of (U 2 ), Proposition 6.2.4 of [30] points out that if, for all x ∈ E and v ∈ V , the function t → d(v, U (t, x)) is measurable, then U (·, x) is measurable as well. In fact, if for each λ ≥ 0 the set U (t, x)) ≤ λ} is measurable, then the function t → d(v, U (t, x)) is measurable too. Moreover, here, we will show that for each λ ≥ 0 the set U (t, x)) ≤ λ} is closed, so, it is measurable. Notice that U has closed and convex values, hence, for every n ∈ N, we are able to take a unique element U (t n , x)) ≤ λ. Vol. 22 (2020) Differential variational-hemivariational inequalities Page 15 of 30 83 As before we have done, it is not difficult to see that the sequence {u n } is bounded. This allows us to suppose that u n u, as n → ∞, for some u ∈ K. Whereas the strongly-weakly upper semicontinuity of U ensures u ∈ U (t, x). This infers This indicates that M λ is closed, so, U (·, x) is measurable. Consequently, from assertion (U 1 ) and Theorem 2.5, we conclude the desired result (U 2 ).
Invoking the same arguments with the proof of [18,Lemma 4.2 and Theorem 4.4], [13,Lemma 3.6] and Theorem 3.6, we are now in a position to conclude the following existence result to problem (1).

Theorem 3.8. Assume that H(A), H(g), H(f ), H(J), H(γ), H(ϕ) and H(K) hold. Then the solution set of problem (1) in the sense of Definition 1.1 is nonempty, and the set of all mild trajectories x of problem (1) is compact in C([0, T ]; E).
Moreover, we shall examine a well-posedness result for problem (1). To do so, we need the following assumptions: H(J) : J : X → R is a locally Lipschitz function and enjoys the following properties (i) there exist constants α J ≥ 0 and b J > 0 such that (ii) there exists a constant m J ≥ 0 such that for all w, v ∈ X and all ξ ∈ ∂J(w), η ∈ ∂J(v).
x, u) is hemicontinuous and is uniformly strongly monotone, i.e., there exists a constant m g > 0 such that the following inequality holds  Remark 3.9. Assumption H(J) (ii) is usually called relaxed monotone condition (see, e.g. [25]) for the locally Lipschitz function J. It is equivalent to the inequality Proof. (i) For any u 0 ∈ K fixed, it follows from hypotheses H(g) and H(J) that for all ξ ∈ ∂J(γu) and all ξ 0 ∈ ∂J(γ0 V ). Hence, H(0) indicates that where r : R + → R and ρ : [0, T ] × R + → R are, respectively, defined by Let x ∈ C([0, T ]; E) be fixed. We now consider the following timedependent variational-hemivariational inequality: find u : [0, T ] → K such that and all t ∈ [0, T ]. We now claim that for t ∈ [0, T ] fixed inequality (17) (16), we can verify all conditions of Theorem 3.6. This permits us to use Theorem 3.6 to find an element u(t) ∈ K such that inequality (17) holds. Let u(t) and u(t) be two solutions to problem (17). Then one has Inserting v = u(t) into the above first inequality and v = u(t) into the second one, we sum the resulting inequalities to get The latter coupled with hypotheses H(g) (i), H(J) (ii), and the fact ϕ(w, v) However and i = 1, 2. Taking v = u(t 2 ) and v = u(t 1 ) into the above inequalities for i = 1 and i = 2, respectively, a simple calculation finds Then we have which implies that be the unique solution of problem (17), namely, Putting v = u 2 (t) and v = u 1 (t) into the above inequalities for i = 1, 2, accordingly, a easy verification gives for all t ∈ [0, T ].
For u ∈ C([0, T ]; K) fixed, we introduce the following Cauchy problem: This means that S is a history-dependent operator. Therefore, we are now in a position to invoke the fixed point principle, Lemma 2.11, that S has a unique fixed point x ∈ C([0, T ]; E). So, differential variational-hemivariational inequality (1) admits a unique solution ( (ii) Let x 1 0 and x 2 0 be two initial points in E. Assertion (i) allows us to find two unique solutions (x 1 , u 1 ) and (x 2 , u 2 ) to problem (1) associated with initial points x 1 0 and x 2 0 , respectively. Hence, it has for all t ∈ [0, T ]. Subtracting the above equalities, it emerges for all t ∈ [0, T ]. This coupled with inequality (18) finds Moreover, we put the above estimate to (18) to obtain To conclude, we can know that the map which completes the proof. (v) the function p : [0, T ] × Ω × R → R is a continuous function such that |p(t, y, s 1 ) − p(t, y, s 2 )| ≤ L p |s 1 − s 2 | for all t ∈ [0, T ], s 1 , s 2 ∈ R and a.e. y ∈ Ω with some L p > 0, then the function g : [0, T ]×L 2 (Ω)×H 1 (Ω) → H 1 (Ω) * defined by g (t, x, u), v := Ω (∇u(y), ∇v(y)) R n dy + Ω p(t, y, x(y))v(y) dy for all t ∈ [0, T ], x ∈ L 2 (Ω) and u, v ∈ H 1 (Ω), satisfies the condition H(g) .

Penalty method for differential variational-hemivariational inequalities
Penalty method as a useful tool has been widely used to the study of various optimization problems with constraints, such as Nash equilibrium problems, optimal control problems with state and input constraints of nonlinear systems, and convection-diffusion problems with characteristic layers.
Recently, penalty methods for variational inequalities and hemivariational inequalities have been investigated by many authors, for numerical purposes and for proofs of solution existence, see, e.g. [7,26]. However, until now, there are no results concerning penalty methods for generalized differential variational-hemivariational inequality (1). To fill this gap, therefore, this section is devoted to provide a theoretical analysis of penalty methods for differential variational-hemivariational inequality (1), see Theorem 4.1. More precisely, we introduce a penalized problem corresponding to problem (1), and prove that the penalized problem has a unique solution (x ρ , u ρ ) ∈ C([0, T ]; E) × C([0, T ]; V ). Then a convergence result, the solution of original differential variational-hemivariational inequality (1) can be approximated by the penalized problem (20), see below, as the penalty parameter ρ tends to zero, is established. Let ρ > 0 and P : V → V * be a penalty operator of constraint set K, see Definition 2.9. The penalized problem associated with differential variational-hemivariational inequality (1) is to find functions x ρ : [0, T ] → E and u ρ : x ρ (t), u ρ (t)) for a.e. t ∈ [0, T ], u ρ (t) ∈ V satisfying g(t, x ρ (t), u ρ (t)), v − u ρ (t) + 1 ρ P u ρ (t), v − u ρ (t) + J 0 (γu ρ (t); γ(v − u ρ (t))) +ϕ(v, u ρ (t)) ≥ 0 for all v ∈ V and for all t ∈ [0, T ], The main results of the section on existence, uniqueness and convergence for problem (20) is the following.