* Convergence Results for Elliptic Variational-Hemivariational Inequalities

Abstract: We consider an elliptic variational-hemivariational inequality P in a re exive Banach space, governed by a set of constraints K, a nonlinear operator A, and an element f . We associate to this inequality a sequence {Pn}of variational-hemivariational inequalities such that, for each n ∈ N, inequalityPn is obtained by perturbing the data K and A and, moreover, it contains an additional term governed by a small parameter εn. The unique solvability ofP and, for each n ∈ N, the solvability of its perturbed versionPn, are guaranteed by an existence and uniqueness result obtained in literature. Denote by u the solution of Problem P and, for each n ∈ N, let un be a solution of Problem Pn. The main result of this paper states the strong convergence of un → u in X, as n → ∞. We show that the main result extends a number of results previously obtained in the study of Problem P. Finally, we illustrate the use of our abstract results in the study of a mathematical model which describes the contact of an elastic body with a rigid-deformable foundation and provide the corresponding mechanical interpretations.


Introduction
Variational-hemivariational inequalities represent a special class of inequalities which arise in the study of nonsmooth boundary value problems. They are governed by both convex functions and locally Lipschitz functions, which could be nonconvex. For this reason, their study requires prerequisites on both convex and nonsmooth analysis. Variational-hemivariational inequalities have been introduced by Panagiotopoulos [24] in the context of applications in engineering problems. Later, they have been studied in a large number of papers, including the books [21,23]. The mathematical literature concerning variational-hemivariational inequalities grew up rapidly in the last decade, motivated by important applications in Physics, Mechanics and Engineering Sciences. A recent reference is the book [27] which provides the state of the art in the eld, together with relevant applications in Contact Mechanics.
Recently, a considerable e ort was done to the study of variational-hemivariational inequalities in the functional framework that we describe below and we assume everywhere in this paper. Consider a real re exive Banach space X and denote by ·, · the duality pairing between X and its dual X * . Let K ⊂ X, A : X → X * , P, as n → ∞. Our main result on this matter is Theorem 2 which states an existence and convergence result. Our second aim is to show that Theorem 2 can be used to recover various convergence results in the study of Problems Pε n , P λn , P Kn described above. To this end, we use the theorem with a particular choice of sets, operators and parameters. Finally, our third aim is to illustrate the use of our abstract result in the study of a frictional contact problem and to provide the corresponding mechanical interpretations. The novelty of our paper arises from the generality of our main result which uni es various convergence results in the study of Problem P and provides a new and nonstandard mathematical tool in the variational analysis of frictional contact problems with elastic materials.
The rest of the manuscript is structured as follows. In Section 2 we introduce some preliminary material, then we recall the existence and uniqueness result obtained in [19,27]. In Section 3 we state and prove our main result, Theorem 2. Its proof is based on arguments of compactness, monotonicity, pseudomonotonicity, lower semicontinuity, combined with the properties of the Clarke subdi erential. In Section 4 we deduce some consequences of Theorem 2 that we present in a form of relevant particular cases. Finally, in Section 5 we illustrate the use of our abstract results in the analysis of a mathematical model of contact.

Preliminaries
We start with some notation and preliminaries and send the reader to [2,3,21,22,32] for more details on the material presented below in this section. We use · X and · X * for the norm on the spaces X and X * , and X , X * for the zero element of X and X * , respectively. We also use the notation X * w for the space X * endowed with the weak* topology. All the limits, upper and lower limits below are considered as n → ∞, even if we do not mention it explicitly. The symbols " " and "→" denote the weak and the strong convergence in various spaces which will be speci ed.
For multivalued and singlevalued operators de ned on X we recall the following de nitions.

De nition 1.
A multivalued operator T : X → X * is said to be pseudomonotone if: (a) For every u ∈ X, the set Tu ⊂ X * is nonempty, closed and convex.
(b) T is upper semicontinuous (u.s.c.) from each nite dimensional subspace of X into X * w .
(c) For any sequences {un} ⊂ X and {u * n } ⊂ X * such that un → u weakly in X, u * n ∈ Tun for all n ∈ N and lim sup u * n , un − u ≤ , we have that for every v ∈ X there exists u * (v) ∈ Tu such that

De nition 2.
A multivalued operator T : X → X * is said to be generalized pseudomonotone if for any sequences {un} ⊂ X and {u * n } ⊂ X * such that un → u weakly in X, u * n ∈ Tun for all n ∈ N, u * n → u * in X * w and lim sup u * n , un − u ≤ , we have u * ∈ Tu and lim u * n , un = u * , u .
De nition 3. An singlevalued operator A : X → X * is said to be: (c) bounded, if A maps bounded sets of X into bounded sets of X * ; (d) pseudomonotone, if it is bounded and un → u weakly in X with (e) demicontinuous, if un → u in X implies Aun → Au weakly in X * .
It is well known that if T : X → X * is a pseudomonotone operator then T is generalized pseudomonotone. Moreover, it can be proved that if A : X → X * is a pseudomonotone operator in the sense of De nition 3(d) then its multivalued extension de ned as X u → {Au} ∈ X * is pseudomonotone in the sense of De nition 1. In addition, the following results hold.
The generalized gradient (Clarke subdi erential) of j at x is a subset of the dual space X * given by The function j is said to be regular (in the sense of Clarke) We shall use the following properties of the generalized directional derivative and the generalized gradient.

Proposition 2.
Assume that j : X → R is a locally Lipschitz function. Then the following hold: a) For every x ∈ X, the function X v → j (x; v) ∈ R is positively homogeneous and subadditive, i.e., j (x; λv) = λj (x; v) for all λ ≥ , v ∈ X and j ( c) For every x ∈ X, the gradient ∂j(x) is a nonempty, convex, and compact subset of X * w which is bounded by the Lipschitz constant Lx > of j near x.
We proceed with some miscellaneous de nitions and results.

De nition 5.
Let {Kn} be a sequence of nonempty subsets of V and K a nonempty subset of X. We say that the sequence {Kn} converges to K in the sense of Mosco if the following conditions hold.
(a) For every v ∈ K, there exists a sequence {vn} ⊂ X such that vn ∈ Kn for each n ∈ N and vn → v in X.
(b) For each sequence {vn} such that vn ∈ Kn for each n ∈ N and vn v in X, we have v ∈ K.
Below in this paper we shall use the notation Kn M − → K for the convergence in the sense of Mosco de ned above.
Then, there exists y * ∈ C * such that For the proof of Proposition 3 we refer to [5].
De nition 6. An operator P : X → X * is said to be a penalty operator of the set K ⊂ X if P is bounded, demicontinuous, monotone and K = {x ∈ X | Px = X * }.
Note that the penalty operator always exists. Indeed, we recall that any re exive Banach space X can be always considered as equivalently renormed strictly convex space and, therefore, the duality map J : X → X * , de ned by x ∈ X is a single-valued operator. Then, as proved in [4,36], the following result holds. Proposition 4. Let K be a nonempty closed and convex subset of X, J : X → X * the duality map, I : X → X the identity map on X, and P K : X → K the projection operator on K. Then P K = J(I − P K ) : X → X * is a penalty operator of K.
We end this section with an existence and uniqueness result concerning the variational-hemivariational inequality (1.1) and, to this end, we consider the following assumptions on the data.
K is nonempty, closed and convex subset of X. (2.1) A : X → X * is pseudomonotone and strongly monotone with constant m A > .
: X → R is convex and lower semicontinuous, for all η ∈ X.
It can be proved that for a locally Lipschitz function j : X → R, hypothesis ( . )(c) is equivalent to the so-called relaxed monotonicity condition see, e.g., [20]. Note also that if j : X → R is a convex function, then ( . )(c) holds with α j = , since it reduces to the monotonicity of the (convex) subdi erential. Examples of functions which satisfy condition (2.4)(c) have been provided in [10,19,20], for instance.
The unique solvability of the variational-hemivariational inequality (1.1) is given by the following result. For the Proof of Theorem 1 we refer the reader to Theorem 18 in [19] and Remark 13 in [27].

An existence and convergence result
In this section we state and prove our main existence and convergence result, Theorem 2. To this end, we consider a family of subsets {Kn} of X, a family of operators {Gn} de ned on X with values in X * and two sequences {λn}, {εn} ⊂ R. Then, for each n ∈ N, we consider the following problem.
Problem Pn. Find un ∈ Kn such that (3.1) In the study of Problem Pn we assume that for each n ∈ N, the following hold.
Kn is a nonempty closed convex subset of X.
Gn : X → X * is a bounded demicontinuous monotone operator.
Moreover, we assume that the following conditions are satis ed.
There exists a set K ⊂ X such that Kn There exists an operator G : X → X * and a sequence {cn} ⊂ R such that (e) One of the two conditions below holds. (3.10) λn → as n → ∞. (3.11) εn → as n → ∞. (3.12) Our main result in this section is the following. b) If, in addition, ( . ) and ( . )-( . ) hold, u is the solution of Problem P and {un} ⊂ X is a sequence such that un is a solution of Problem Pn, for each n ∈ N, then un → u in X.
Proof. a) Let n ∈ N. Assumptions (3.3), (3.4) and Proposition 1 a) imply that the operator λn Gn : X → X * is pseudomonotone. Therefore (2.2) and Proposition 1 b) show that the operator An : X → X * de ned by An = A + λn Gn is pseudomonotone, too. Moreover, since Gn is monotone and λn > , using assumption (2.2), again, we deduce that An is strongly monotone with constant m A . We conclude from above that the operator An satis es condition (2.2). On the other hand, recall that the set Kn satis es condition (3.2). It follows from above that we are in a position to use Theorem 1 with Kn and An instead of K and A, respectively. In this way we deduce the existence of a unique element un ∈ Kn such that This proves the unique solvability of Problem Pn in the case when εn = . Next, for εn > , it follows that the solution un of (3.13) satis es inequality (3.1), too. This proves the existence of at least one solution to Problem Pn.
b) Let n ∈ N. We start by considering the auxiliary problem of nding an element un ∈ Kn such that (3.14) Note that the variational-hemivariational inequality (3.14) is similar to the variational-hemivariational inequality (3.13), the di erence arising in the fact that in (3.14) the rst argument of φ is the solution u of Problem P. The existence of a unique solution to inequality (3.14) follows from Theorem 1, by using the same arguments as those used in the proof of unique solvability of inequality (3.13). Next, we divide the rest of the proof into four steps. i) We claim that there is an element u ∈ K and a subsequence of { un}, still denoted by { un}, such that un u in X, as n → ∞.
To prove the claim, we establish the boundedness of the sequence { un} in X. Let n ∈ N and let u be a given element in K. We use assumption (3.6) to deduce that Then, by the strong monotonicity of the operator A we obtain and assumption (3.10) yields On the other hand, we write then we use assumption (2.4)(b) and Proposition 2 b) to see that And, obviously, Next, we combine inequalities (3.15)- (3.19) to nd that We now use condition (2.5) and the above inequality to deduce that { un} is a bounded sequence in X. Therefore, by the re exivity of X, there exists an element u ∈ X and a subsequence of { un}, still denoted by { un}, such that un u in X. Recall that un ∈ Kn for each n ∈ N. Then, assumption (3.8) and De nition 5 imply that u ∈ K.
ii) Next, we claim that u is the solution to Problem P, i.e., u = u.
To prove this claim we use assumption (3.8) and consider an element v ∈ K together with a sequence {vn} ⊂ X such that vn ∈ Kn for every n ∈ N and vn → v in X as n → ∞. We now use inequality (3.14) with v = vn and assumptions (2.2), (3.10), (2.4)(b) to see that Then, due to the convergence vn → v in X, the boundedness of sequence { un} and the boundedness of the operator A, we deduce that there exists a constant D > which does not depend on n such that Gn un , un − vn ≤ λn D.
Passing to the upper limit in above inequality and using assumption (3.11) we have On the other hand, we write and, using asumption (3.9)(a) we deduce that We now use hypotheses (3.9)(b), (c), the boundedness of sequence { un} and the convergence vn → v in X to see that Next, we pass to upper limit in inequality (3.21) and use (3.20), (3.22) and (3.23) to nd that Taking now v = u ∈ K in (3.24) we deduce that lim sup G un , un − u ≤ . Recall assumption (3.9)(c) which guarantees that the operator G : X → X * is pseudomonotone. Hence, using the pseudomonotonicity of G we deduce that We now combine inequalities (3.24) and (3.25) to see that Recall that this inequality is valid for any v ∈ K. Assume that condition (3.9)(e)(i) is satis ed. Then, inequality (3.26) implies that G u, u − v ≤ for all v ∈ X, which yields G u = X * and, therefore, u ∈ K. Assume now that condition (3.9)(e)(ii) is satis ed. Then, by assumptions (3.6) and (3.8) it is easy to see that K ⊂ K and, therefore, using (3.26) we obtain that On the other hand, from the assumption (3.9)(d) we have The last two inequalities imply that G u, v − u = for all v ∈ K and, using (3.9) (e)(ii), we infer that u ∈ K. We conclude from above that, either (3.9)(e)(i) or (3.9)(e)(ii) holds, we have Let n ∈ N. Then, using (3.6) and inequality (3.14), we nd that Next, using Proposition 2 b) we deduce that for each v ∈ K there exists an element , v − un , and, therefore, inequality (3.28) yields Recall that Proposition 2 c) guarantees the set is nonempty closed convex and bounded in X * w . Then, assumption (2.3)(a) allows us to use Proposition 3 with C = K and C * de ned by (3.30), x = v and y = un. In this way we nd that there exists an element ωn( un) ∈ ∂j( un) which does not depend on v such that Therefore, assumptions (3.4) and (3.7) yield We now use the regularity (3.27) to take v = u in (3.31). Then we pass to the upper limit in the resulting inequality, use the convergence un u in X and the lower semicontinuity of φ with respect to its second argument to infer that lim sup A un + ωn( un), un − u ≤ .
Then, by passing to upper limit in (3.31) and using assumption (2.3)(a) we have On the other hand, the de nition of the Clarke subdi erential yields Then, combining (3.38) and (3.39) we deduce that Finally, we use (3.27) and (3.40) to see that u is a solution to Problem P and, by the uniqueness of the solution we have that u = u, as claimed.
iii) We now prove the convergence of the whole sequence { un} to u.
A careful analysis of the proof in step ii) reveals that every subsequence of { un} which converges weakly in X has the same weak limit u. Moreover, we recall that the sequence { un} is bounded in X. Therefore, using a standard argument we deduce that the whole sequence { un} converges weakly in X to u, as n → ∞. This shows that all the statements in step ii) are valid for the whole sequence { un}. In particular, (3.36) combined with equality u = u shows that A un + ωn( un), un − u → . (3.41) Let n ∈ N and let ω(u) ∈ ∂j(u). We have We now use assumption (2.4)(c) to see that On the other hand, assumption (2.2) yields We now add the inequalities (3.42) and (3.43) to deduce that Next, we use the convergences (3.41), un u in X as well as the smallness assumption (2.5) to nd that which show that un → u in X as n → ∞, as claimed.
iv) In the nal step of the proof we prove that un → u in X, as n → ∞.
Let n ∈ N. We test with v = un in (3.1) and v = un in (3.14), then we add the resulting inequalities to see that +j (un; un − un) + j ( un; un − un) + εn un − un X .

Next, using assumptions (3.3), (2.3)(b), (2.4)(c), we deduce that
Aun − A un , un − un ≤ αφ un − u X un − un X +α j un − un X + εn un − un X and, therefore, the strong monotonicity of the opeator A yields We now write αφ un − u X ≤ αφ un − un X + αφ un − u X and substitute this inequality in (3.45) to deduce that Then, using the smallness assumption (2.5) we obtain that This inequality, the convergence (3.44) and assumption (3.12) imply that Finally, writing un − u X ≤ un − un X + un − u X and using the convergences (3.44), (3.46) we deduce that un → u in X which concludes the proof.

Relevant particular cases
In this section we present some relevant particular cases in which Theorem 2 can be applied. In particular, we show that using a convenient choice of sets, operators and parameters, Problem Pn reduces successively to Problems Pε n , P λn P Kn described in the Introduction. Then, we use Theorem 2 to recover various convergence results previously obtained in the study of these problems. Everywhere in this section we assume that ( . )-( . ) hold and we denote by u the solution of Problem P obtained in Theorem 1. We start by considering the following assumptions.
K is a nonempty closed convex subset of X. (4.1) Kn ⊂ K for each n ∈ N. G : X → X * is a penalty operator for K. fn ∈ X * for each n ∈ N. (4.7) fn → f in X * . (4.8) We are now in a position to introduce some relevant consequences of Theorem 2.

a) A rst penalty method.
Our rst particular case is when Kn = X and Gn = G for each n ∈ N, G being a penalty operator of K. In this case Theorem 2 leads to the following result.
Moreover, the solution is unique if εn = .
b) If {un} ⊂ X is a sequence such that un is a solution of ( . ), for each n ∈ N, then un → u in X.
Note that in the case when εn = inequality (4.9) reduces to inequality (1.3), used in the classical penalty method for variational-hemivariational inequalities. Therefore, Corollary 1 provides the unique solvability of Problem P λn , for each n ∈ N, and the convergence of the sequence of solutions to the solution of Problem P. This result was obtained in [19], in the particular case when φ(u, v) = φ(u) and extented in [25] in the case when φ depends on both u and v. b) A second penalty method. Our second particular case is when Kn = K where K is a given set which satis es condition (4.1) and Gn = G for each n ∈ N, G being a penalty operator of K. In this case Theorem 2 leads to the following result.
Moreover, the solution is unique if εn = .
b) If {un} ⊂ X is a sequence such that un is a solution of ( . ), for each n ∈ N, then un → u in X.
Proof. Since Note that in the case when εn = inequality (4.10) reduces to inequality A brief comparison between inequalities (1.1) and (4.11) shows that (4.11) is obtained from (1.1) by replacing the set K with the set K and the operator A with the operator A + λn G, in which λn is a penalty parameter. For this reason we refer to (4.11) as a penalty problem of (1.1). Corollary 2 establishes the link between the solutions of these problems and, at the best of our knowledge, it represents a new result. Roughly speaking, it shows that, in the limit when n → ∞, a partial relaxation of the set of constraints can be compensated by a perturbation of the nonlinear operator which governs Problem P.

c) A continuous dependence result. Our third particular case is when
Moreover, un → u in X.
Proof. The existence of a unique solution to inequality (4.12) is a direct consequence of Theorem 1. Let n ∈ N. Then, using (4.12) it is easy to see that and, denoting εn = f − fn X * , it follows that On the other hand, since (4.4) holds it follows that condition (3.8) is satis ed with K = K. Moreover, since Gn vanishes, it follows that conditions (3.3), (3.7), (3.9) hold, with Gv = X * for all v ∈ X and cn = . In addition, assumption (4.8) implies that (3.12) holds, too. We are now in a position to use Theorem 2 b) with λn = n , for instance, to deduce the convergence un → u in X, which concludes the proof.
Note that Corollary 3 represents a continuous dependence result of the solution to Problem P with respect to the set K and the element f . Similar convergence results have been obtained in [34,37], under di erent assumptions on functions and operators.

b) If {un} ⊂ X is a sequence such that un is a solution of Problem
Pn, for each n ∈ N, then un → u in X.
The proof of Corollary 4 is based on arguments similar to those presented above and, therefore, we skip it. We restrict ourselves to note that an elementary proof can be used to obtain the convergence result in Corollary 4, without assumption ( . ). The details can be found in [31]. Finally, using the de nitions in [29,31] we remark that Theorem 1 combined with Corollary 4 provides the well-posedness of Problem P in the sense of Tykhonov.
e) An existence, uniqueness and convergence result. We end this section with an existence, uniqueness and convergence result which completes our analysis of Problem P and has some interest in its own. To this end we assume in what follows that ( . ), ( . ) and ( . ) hold. Let J : X → X * be the duality map, I : X → X the identity map on X, P K : X → K the projection operator on K, P K : X → K the projection operator on K and, for each n ∈ N let P Kn : X → Kn be the projection operator on Kn. Consider the operators P, P, Pn, G, Gn, both de ned on X with values in X * , given by equalities P = J(I − P K ), P = J(I − P K ), Pn = J(I − P Kn ), (4.14) G = P + P, Gn = P + Pn . (4.15) We use these notation to state and prove the following result.
Moreover, un → u in X.
Proof. Recall that Proposition 4 guarantees that P, P and Pn are penalty operators of K, K and Kn, respectively. This implies that these operators are bounded demicontinuous and monotone. Therefore, so are the operators G and Gn de ned by (4.15). This shows that conditions (3.3) and (3.9)(c) are satis ed. The existence of a unique solution of inequality (4.16) results from Theorem 2 with εn = .
Assume now that u ∈ Kn, v ∈ K and recall assumption (3.6) which states that K ⊂ Kn. This implies that Pv = X * , Pn v = X * and, therefore, (4.15) yields We now use the monotonicity of the operators P and Pn to see that Gn u, v − u ≤ which implies that condition (3.7) is satis ed. A similar argument based on assumption (4.2), guaranteed by (3.6) and (4.3), shows that condition (3.9)(d) holds, too.
Let n ∈ N and u ∈ Kn. Then it follows that P Kn u = u and, since (4.3) guarantees that Kn ⊂ K, we deduce that P K u = u, too. We now use (4.15) and (4.14) to see that It follows from here that condition (3.9)(a) holds with cn = . This implies that condition (3.9)(b) is satis ed, too.
Assume now that u ∈ K is such that Then, since K ⊂ Kn ⊂ K we have that Pv = Pv = X * for all v ∈ K and, therefore, (4.15) yields We now combine (4.17) and (4.18) to deduce that On the other hand, using the monotonicity of the operators P and P we have (4.20) We now use (4.19), (4.20) and the elementary implication to deduce that which implies that We now take v = P K u in the previous equality and use the de nition (4.14) of the operator P and the properties of the duality mapping J to see that This implies that u ∈ K and, therefore, condition (3.9)(e) holds. We conclude from above that conditions (3.7), (3.9) are satis ed, the later with cn = . Moreover, we recall assumption (4.5), which implies (3.8). Therefore, we are in a position to apply Theorem 2 with εn = in order to conclude the proof.

A frictional contact problem
In this section we apply our abstract results in Section 3 in the study of a frictional contact problem with normal compliance and unilateral constraint. To this end we consider a bounded domain Ω ⊂ R d (d = , ) with smooth boundary Γ composed of three sets Γ , Γ and Γ with the mutually disjoint relatively open sets Γ , Γ and Γ , such that meas (Γ ) > . We use boldface letters for vectors and tensors, such as the outward unit normal on Γ, denoted by ν. A typical point in R d is denoted by x = (x i ). The indices i, j run between and d and the index that follows a comma indicates a partial derivative with respect to the corresponding component of the spatial variable x. Moreover, the indices ν and τ represent normal components and tangential parts of vectors and tensors. We denote by S d the space of second order symmetric tensors on R d . The zero element, the canonical inner product and the Euclidean norm on R d and S d will be denoted by , " · " and · , respectively. Then, the classical formulation of the contact problem we consider in this section is the following.
Problem Q. Find a displacement eld u : Ω → R d , a stress eld σ : Ω → S d and an interface function ξν : Γ → R such that in Ω, Problem Q describes the equilibrium of an elastic body acted upon by body forces and surface tractions, in frictional contact with a foundation made of a rigid body covered by a layer made of elastic material, say asperities. It was already considered in [27] and, therefore we skip the mechanical assumptions which lead to this model. We restrict ourselves to the following short description of the equations and boundary conditions. First, equation (5.1) represents the elastic constitutive law in which F is the elasticity operator, assumed to be nonlinear, and ε(u) represents the linearized strain tensor. Equation ( In the study of Problem Q we consider the following assumptions on the data. r) is measurable on Γ for all r ∈ R and there existsē ∈ L (Γ ) such that jν(·,ē(·)) ∈ L (Γ ), (c) |∂jν(x, r)| ≤c +c |r| for a.e. x ∈ Γ , for all r ∈ R withc ,c ≥ , (e) either jν(x, ·) or − jν(x, ·) is regular on R for a.e. x ∈ Γ .
a.e. x ∈ Γ . (5.9) Next, we use the space V de ned by which is real Hilbert space with the canonical inner product and the associated norm · V . Here and below, for every v ∈ V we use the notation We also use V * for the dual of V, ·, · for the duality pairing between V and V * and γ for the norm of the trace operator γ : V → L (Γ ) d . We denote by K the set of admissible displacement elds de ned by and, nally, we introduce the following notation.
for all u, v ∈ V. It can be proved that the function j is locally Lipschitz. Therefore, as usual, we shall use the notation j (u; v) for the generalized directional derivative of j at u in the direction v.
The variational formulation of Problem Q, obtained by a standard procedure, is as follows.
Next, for each n ∈ N we consider the following contact problem.
The di erence between Problems Qn and Q is twofold. First, in Problem Qn the densities of body forces f and surface tractions f as well as the bound k have been replaced by their perturbation f n , f n and kn, respectively. Second, the boundary contact condition (5.5) has been replaced by the contact boundary condition (5.24) in which λn > is a deformability coe cient, pν is a normal compliance function and gn is a given gap. This condition still models the contact with a rigid foundation covered by a layer of deformable material. Nevertheless, the thickness of this material changed (since k was replaced by kn) as well as its elastic response (since the additional term λn pν(unν − gn) was introduced in this condition).
In the study of Problem Qn we consider the following assumptions.
The variational formulation of Problem Qn is as follows.
Problem Q V n . Find a displacement eld un ∈ Kn such that Aun , v − un + λn Gn un , v − un) + φ(un , v) − φ(un , un) (5.34) Our main resut in this section is the following existence, uniqueness and convergence result. b) The solution un of Problem Q V n converges to the solution u of Problem Q V , i.e., un → u in V, as n → ∞.
Proof. a) The unique solvability of Problem Q V corresponds to Theorem 109 in [27] and, for this reason, we do not provide its proof. We restrict ourselves to mention that it represents a direct consequence of Theorem 1. The unique solvability of Problem Q V n follows from Theorem 2 a). Indeed, Problem Q V n is a special case of Problem Pn in which εn = .
b) Let n ∈ N. We use inequality (5.34) to see that  Therefore, since the integrand is negative, we deduce that pν(uν − k)uν = a.e. on Γ .
This equality combined with assumption (5.26)(d) implies that uν ≤ k a.e. on Γ . Thus, u ∈ K and, therefore, condition (3.9)(e) is satis ed. Finally, let u, v , v ∈ V. We use de nition (5.16) and assumption (5.9) to see that which shows that condition (3.10) holds with cφ(u) = L F b γ u V .
It follows from above that we are in a position to use Theorem 2. In this way we obtain that if { un} is a sequence of elements of V such that un is a solution of Problem Q V n , for each n ∈ N, then un → u in V. Recall now that for each n ∈ N the solution un of Problem Q V n is a solution of Problem Q V n . It follows from here that un → u in V which concludes the proof.
In addition to the mathematical interest in the convergence result in Theorem 3 b), it is important from the mechanical point of view, since it establishes the link between the solutions of two di erent contact models. It also shows that the weak solution of the elastic frictional contact problem Q depends continuously on the densities of body forces and surface tractions and the thickness of the deformable layer.