On convergence of solutions to variational–hemivariational inequalities

In this paper we investigate the convergence behavior of the solutions to the time-dependent variational–hemivariational inequalities with respect to the data. First, we give an existence and uniqueness result for the problem, and then, deliver a continuous dependence result when all the data are subjected to perturbations. A semipermeability problem is given to illustrate our main results.


Introduction
Variational-hemivariational inequalities represent a special class of inequalities which involve both convex and nonconvex functions. Elliptic hemivariational and variational-hemivariational inequalities were introduced by Panagiotopoulos in the 1980s and studied in many contributions, see [15,17] and the references therein. Various classes of such inequalities have been recently investigated, for instance, in [7,9,10,12,20,22]. They play an important role in describing many mechanical problems arising in solid and fluid mechanics.
In this paper we study the following time-dependent variational-hemivariational inequality: find u : R + = [0, +∞) → X such that, for all t ∈ R + , u(t) ∈ K and where K is a nonempty, closed and convex subset of a reflexive Banach space X, A : X → X * and ϕ : K × K → R are given maps to be specified later, j : X → R is a locally Lipschitz function, and f : R + → X * is fixed. The notation j 0 (u; v) stands for the generalized directional derivative of j at point u ∈ X in the direction v ∈ X. The goal of the paper is to study the convergence of solution of the variational-hemivariational inequality (1) when the data A, f , ϕ, j and K are subjected to perturbations. The dependence of solutions to elliptic variational-hemivariational inequalities on the data has been studied only recently. For such inequalities the dependence with respect to functions ϕ and j was investigated in [13], where A and K were not subjected to perturbations. A result on the dependence of solutions to elliptic variational inequalities with respect to perturbations of the set K of a special form was studied

Preliminaries
In this section we recall notation, basic definitions and a result on unique solvability of a variationalhemivariational inequality.
Let (X, · X ) be a Banach space. We denote by X * its dual space and by ·, · X the duality pairing between X * and X. The strong and weak convergences in X are denoted by " → and " , respectively.
Let C(R + ; X) be the space of continuous functions defined on interval R + = [0, +∞) with values in X. For a subset K ⊂ X the symbol C(R + ; K) denotes the set of continuous functions on R + with values in K. We also recall that the convergence of a sequence {x n } n≥1 to the element x, in the space C(R + ; X), can be described as follows ⎧ ⎨ ⎩ x n → x in C(R + ; X), as n → ∞ if and only if max t∈ [0,k] x n (t) − x(t) X → 0, as n → ∞, for all k ∈ N.
We recall the definitions of the convex subdifferential, the (Clarke) generalized gradient and the pseudomonotone single-valued operators, see [3,4]. Definition 1. A function f : X → R is said to be lower semicontinuous (l.s.c.) at u, if for any sequence {u n } n≥1 ⊂ X with u n → u, we have f (u) ≤ lim inf f (u n ). A function f is said to be l.s.c. on X, if f is l.s.c. at every u ∈ X. ZAMP On convergence of solutions to variational-hemivariational inequalities Page 3 of 20 87 Definition 2. Let ϕ : X → R∪{+∞} be a proper, convex and l.s.c. function. The mapping ∂ϕ c : for u ∈ X, is called the subdifferential of ϕ. An element u * ∈ ∂ c ϕ(u) is called a subgradient of ϕ in u.
Definition 3. Given a locally Lipschitz function ϕ : X → R, we denote by ϕ 0 (u; v) the (Clarke) generalized directional derivative of ϕ at the point u ∈ X in the direction v ∈ X defined by The generalized gradient of ϕ at u ∈ X, denoted by ∂ϕ(u), is a subset of X * given by Furthermore, a locally Lipschitz function ϕ : X → R is said to be regular (in the sense of Clarke) at u ∈ X, if for all v ∈ X the directional derivative ϕ (u; v) exists, and for all v ∈ X, we have ϕ (u; v) = ϕ 0 (u; v). The function is regular (in the sense of Clarke) on X if it is regular at every point in X.

Definition 4.
A single-valued operator F : X → X * is said to be pseudomonotone, if it is bounded (sends bounded sets into bounded sets) and satisfies the inequality The following result provides a useful characterization of a pseudomonotone operator.

Lemma 5.
(see [12,Proposition 1.3.66]) Let X be a reflexive Banach space and F : X → X * be a singlevalued operator. The operator F is pseudomonotone if and only if F is bounded and satisfies the following condition: if u n u in X and lim sup F u n , u n −u X ≤ 0, then F u n F u in X * and lim F u n , u n −u X = 0.
The following notion of the Mosco convergence of sets will be useful in the next sections. For the definitions, properties and other modes of set convergence, we refer to [4,Chapter 4.7] and [14]. Definition 6. Let (X, · ) be a normed space and {K ρ } ρ>0 ⊂ 2 X \{∅}. We say that K ρ converge to K in the Mosco sense, ρ → 0, denoted by K ρ M −→ K if and only if the two conditions hold (m1) for each x ∈ K, there exists {x ρ } ρ>0 such that x ρ ∈ K ρ and x ρ → x in X, For the following properties of the Mosco convergence, we refer to [14, p. 520].
Then, K = ∅ implies K ρ = ∅ and the opposite is not true. Also, if K ρ is a closed and convex set for all ρ > 0, then K is also closed and convex.
Finally, we recall a result on existence and uniqueness of solution to the following variationalhemivariational inequality.
Problem 8 was studied in [13] where results on its unique solvability, continuous dependence on the data and a penalty method were provided. We need the following hypotheses on the data of Problem 8. K is nonempty, closed and convex subset of X.
(c) there exists α j ≥ 0 such that The following existence and uniqueness result was established in Theorem 18 of [13].
Theorem 9. Assume that (4)-(8) hold and the following smallness conditions are satisfied Then Problem 8 has a unique solution u ∈ K.

Convergence of solutions
In this section we study the dependence of the solution to Problem 8 with respect to the operator A, functions f , ϕ and j, and the constraint set K.
Continuous dependence for Problem 8 has been investigated earlier in some particular cases. For example, it was studied in Theorem 23 in [13], where A and K are independent of ρ > 0 and the hypotheses on the behavior of ϕ ρ and j ρ are different than ours. Furthermore, the dependence of solution to an elliptic variational inequality with respect to perturbations of the set K ρ was studied in [19] under the hypotheses j ≡ 0, A, ϕ and f are independent of ρ, ϕ satisfies additional assumptions, and the constraint sets K ρ satisfy the following hypothesis K is a nonempty, closed and convex subset of X.
(b) 0 X ∈ K ρ and θ is a given element of X.
We make the following observation.
Since K is closed and convex, it is weakly closed. Hence, x ∈ K which implies that the condition (m2) in Definition 6 is satisfied.
Consider the following perturbed version of Problem 8.
We formulate the hypotheses needed for the continuous dependence result. Let ρ > 0. (5).
(c) there exists a nondecreasing function c ϕ : The following result ensures the existence, uniqueness and convergence of Problem 11. Proof. (i) The existence and uniqueness result for Problem 11 follows from Theorem 9.
(ii) Let ρ > 0 and u ρ ∈ K ρ be the unique solution to Problem 11. First, we will show that there exists a constant c > 0 independent of ρ such that for all ρ > 0 sufficiently small From conditions (8) and (16)(a), we have Taking v ρ = u 0ρ ∈ K ρ in inequality (11), we obtain Therefore, we have Hence, by hypothesis (17), we can find a constant c > 0 independent of ρ such that, for all ρ > 0 sufficiently small, condition (18) holds.
Exploiting (18) and the reflexivity of X, by passing to a subsequence if necessary, we may suppose that the sequence {u ρ }, u ρ ∈ K ρ for each ρ > 0, converges weakly to some u ∈ X, as ρ → 0. By the condition (m2) of Definition 6, we deduce that u ∈ K.
We now consider the following time-dependent variational-hemivariational inequality.
Problem 13. Find a function u : R + → X such that, for all t ∈ R + , u(t) ∈ K and We have the following existence and uniqueness result.

Theorem 14.
Assume that the hypotheses of Theorem 9 hold and f ∈ C(R + ; X * ). Then, Problem 13 has a unique solution u ∈ C(R + ; K).
Proof. We apply Theorem 9 for any t ∈ R + . We deduce that Problem 13 has a unique solution u(t) ∈ K. The fact u ∈ C(R + ; K) can be proved from the proof of Theorem 5 in [21]. The perturbed problem corresponding to Problem 13 reads as follows.
Problem 15. Find a function u ρ : R + → X such that, for all t ∈ R + , u ρ (t) ∈ K ρ and The following result concerns the pointwise convergence of solutions to Problem 15.

Theorem 16.
Assume that the hypotheses of Theorem 12 are satisfied. Suppose that for all ρ > 0, f ρ ∈ C(R + ; X * ) and f ρ (t) → f (t) in X * for all t ∈ R + , as ρ → 0. Then, (i) for each ρ > 0, Problem 15 has the unique solution u ρ ∈ C(R + ; K ρ ); is the unique solution to Problem 13.
Proof. By applying Theorems 12 and 14, we know that Problem 15 has a unique solution u ρ ∈ C(R + ; K ρ ) for all ρ > 0. Moreover, for each t ∈ R + , there is a subsequence {u ρ } such that u ρ (t) → u(t) in X, as ρ → 0, where u ∈ C(R + ; K) is the unique solution to Problem 13.
Finally, we conjecture that under additional hypotheses the convergence result of Theorem 16(ii) can be strengthen to the uniform convergence of u ρ → u in C(R + ; X), as ρ → 0, which will be studied in the future. We note that a convergence result for Problem 15 with j ≡ 0, A, f and ϕ independent of ρ, and K ρ of the form (10) was provided in [1] under assumption that A is Lipschitz continuous and ϕ depends on a history-dependent operator.

Semipermeability problem
In this section we consider a semipermeability problem to which our main results of Sect. 3 can be applied. First, we state the classical formulation of the problem, then we provide its variational formulation, and finally we obtain results on its weak solvability and convergence of solutions.
The motivation comes from semipermeability problems studied in [5, Chapter I] for monotone relations, and in [15, Chapter 5.5.3] and [16] for nonmonotone relations which lead to variational and hemivariational inequalities, respectively. We consider the stationary heat conduction problem with constraints and both the interior and the boundary semipermeability relations. Nevertheless, similar problems can be formulated in electrostatics and in flow problems through porous media, where the semipermeability relations are realized by natural and artificial membranes of various types, see [5,11,[15][16][17]. We will analyze a very general situation which leads to a variational-hemivariational inequality problem and provide examples which satisfy our hypotheses.
Let Ω be a bounded domain of R d with Lipschitz continuous boundary ∂Ω = Γ which consists of two disjoint measurable parts Γ 1 and Γ 2 such that m(Γ 1 ) > 0. The classical model for the heat conduction problem is described by the following boundary value problem.  (23) is the stationary heat equation related to the nonlinear operator in divergence form, with the time-dependent heat sourcef =f (t, u) where time plays a role of a parameter. The functionf in (24) admits an additive decomposition on f 1 = f 1 (t) which is prescribed and independent of the temperature u, and f 2 = f 2 (u) which is a multivalued function of u in the Clarke subgradient term. Here h = h(x, r) is a function which is assumed to be locally Lipschitz in the second argument. Condition (25) introduces an additional constraint for the temperature (or the pressure of the fluid). The temperature u is constrained to belong to a convex, closed set U . For example, the set U can represent a bilateral obstacle which means that we look for the temperature within prescribed bounds in the domain Ω, see Example 24. The homogeneous (for simplicity) Dirichlet boundary condition is supposed in (26). In the boundary condition (27) the expression ∂u ∂νa = a(x, ∇u) · ν represents the heat flux through the part Γ 2 , where ν denotes the outward unit normal on Γ. Here, g = g(x, r) is a prescribed function, convex in its second argument, ∂ c g stands for its convex subdifferential, and a given function k is positive. Note that in (27) we deal with the nonlinearity which is determined by a law of the form k∂ c g. In such a case we cannot deal with a variational inequality since there is not, in general, a function g 1 with ∂ c g 1 = k∂ c g.
We introduce the following spaces we denote the trace operator which is known to be linear, bounded and compact. Moreover, by γv we denote the trace of an element v ∈ H 1 (Ω). In order to study the variational formulation of Problem 17, we need the following hypotheses. (·, ξ) is measurable on Ω for all ξ ∈ R d , and a(x, 0) = 0 for a.e. x ∈ Ω. r) is measurable on Ω for all r ∈ R and there exists e ∈ L 2 (Ω) such that h(·, e(·)) ∈ L 1 (Ω).
Example 19. Consider the following example. Let h : R → R be defined by Then, its subdifferential is given by It can be proved that the function h satisfies condition (30) with c 0 = 2, c 1 = 1 and α h = 3. For more examples of functions which satisfy this condition, we refer to Examples 16 and 17 in [13].
We turn to the variational formulation of Problem 17. Let v ∈ U and t ∈ R + . We multiply (23) by v − u, use Green's formula, decompose the surface integral on two parts on Γ 1 and Γ 2 and take into account that v − u = 0 on Γ 1 .
From (23), (24) and definitions of subgradients, we have for all r ∈ R. Using these inequalities in (36), we obtain the following variational-hemivariational inequality.

ZAMP
On convergence of solutions to variational-hemivariational inequalities Page 13 of 20 87 Problem 20. Find u : The following result concerns the well posedness of Problem 20.

Theorem 21.
Assume that (29)-(33) hold and the following smallness condition is satisfied Then, Problem 20 has a unique solution u ∈ C(R + ; U ).
Proof. We apply Theorem 14 in the following functional framework: With this notation, we can see that Problem 20 is equivalent to Problem 13. We now check the hypotheses of Theorem 14.
First, since V is a closed linear subspace of the Sobolev space H 1 (Ω), containing H 1 0 (Ω), it is straightforward to prove that under hypotheses (29), the operator A is bounded and pseudomonotone, for details see, e.g., [18,Theorem 4.6] or [24,Proposition 26.12]. It is clear that condition (29)(d) implies that operator A is strongly monotone with constant m A = α a . Using the strong monotonicity condition, for u 0 ∈ K and u ∈ V , we have
We now turn to the dependence of solution to Problem 20 on the perturbation of the mapping a, functions k, g, h and f 1 , and the set U . We consider the following perturbation of Problem 20.
For the data of Problem 22, we introduce the following hypotheses.
U, U ρ are closed convex sets in V, and U ρ M −→ U, as ρ → 0.
We conclude this section with the following examples.
Example 24. Hypothesis (41) is satisfied for the following constraint sets for a bilateral obstacle problem.