Akhtar A Khan

Dumitru Motreanu
email: motreanu@univ-perp.fr

B Akhtar

Inverse problems for quasi-variational inequalities

Keywords: Inverse problems, Regularization, Output least-squares, Quasi-variational inequalities Mathematics Subject Classification 35R30, 49N45, 65J20, 65J22, 65M30

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.

Introduction

Applied models frequently lead to variational problems involving parameters characterizing physical features of the model. The direct problem in this setting is to solve the variational problem. By contrast, an inverse problem seeks the identification of the parameters from a measurement of a solution of the variational problem. In recent years, the field of inverse problems emerged as one of the most vibrant and expanding branches of applied mathematics.

Probably the main reason behind this is the increasing number of real-world situations that are being modeled and studied in a unified framework of inverse problems. However, the theoretical aspects of inverse problems are also quite challenging and require a delicate blending of various branches of mathematics.

In this paper, our aim is to study, for the first time, the inverse problem of parameter identification in quasi-variational inequalities. In our approach, the parameter identification in the considered quasi-variational inequality, which provides the solvability of the stated inverse problem, is achieved through the global minimization problem (see [START_REF] Gockenbach | An abstract framework for elliptic inverse problems. I. An output leastsquares approach[END_REF]) constructed with the regularization map. The global minimizers are found in Theorem 3.1.

The theory of quasi-variational inequalities has now established itself as one of the most promising areas of applied mathematics, offering a powerful mathematical apparatus for investigating a broad range of problems arising in diverse disciplines. Applications of quasivariational inequalities can be found in material science [START_REF] Kano | Existence theorems for elliptic quasi-variational inequalities in Banach spaces[END_REF], equilibrium models [START_REF] Barbagallo | A general quasi-variational problem of Cournot-Nash type and its inverse formulation[END_REF][START_REF] Scrimali | Evolutionary quasi-variational inequalities and the dynamic multiclass network equilibrium problem[END_REF], financial models [START_REF] Ciarciá | New existence theorems for quasi-variational inequalities and applications to financial models[END_REF], frictional elastostatic contact [START_REF] Motreanu | Quasivariational inequalities and applications in frictional contact problems with normal compliance[END_REF], image processing [START_REF] Lenzen | A class of quasi-variational inequalities for adaptive image denoising and decomposition[END_REF], sand-piles formation [START_REF] Barrett | A quasi-variational inequality problem arising in the modeling of growing sandpiles[END_REF], and numerous others.

We denote the parameter space by B which is a Banach space. We denote the set of admissible parameters by A which is a nonempty, closed, and convex subset of B. We will pose quasi-variational inequality in a Hilbert space V which we identify with its dual V * . We take the measured data in a Hilbert space Z such that V continuously embeds in Z . We specify the strong convergence by → and the weak convergence by ⇀. Assume that C is a nonempty, closed, and convex subset of V , K : C ⇒ C is a set-valued map such that for every u ∈ C, the set K (u) is a nonempty, closed, and convex subset of C, and m ∈ V * is fixed. Assume that there is a bounded set

C 0 ⊂ V such that K (v) ∩ C 0 =∅, for every v ∈ C.
We define a trilinear form T : B × V × V → R, with T (a, u,v)symmetric in u and v, and assume that there are constants α>0andβ>0 such that

T (a, u,v) ≤ β a B u V v V , for all u,v ∈ V, a ∈ B, (1)
T (a, u, u) ≥ α u 2 V , for all u ∈ V, a ∈ A. ( 2
)
We formulate the quasi-variational inequality: Given a ∈ A, find u ∈ K (u) such that

T (a, u,v-u) ≥ m,v-u , for every v ∈ K (u). (3)
Given a ∈ A, the quasi-variational inequality of finding u = u(a) constitutes the direct problem. In contrast, the inverse problem seeks to identify the coefficient a from a measurement z of a solution u of the quasi-variational inequality. Quasi-variational inequality (3), introduced in connection with an impulse control problem (see Bensoussan and Lions [4], Mosco [START_REF] Mosco | Implicit variational problems and quasi variational inequalities[END_REF]), is convenient for many applications such as implicit obstacle problem, dam problems, and others. See [START_REF] Giannessi | Regularization of non-coercive quasi variational inequalities[END_REF][START_REF] Khan | Existence theorems for elliptic and evolutionary variational and quasivariational inequalities[END_REF][START_REF] Khan | Regularization of quasi-variational inequalities[END_REF] for more details.

If

K (u) = C for all u ∈ C, then (3) becomes the variational inequality: find u ∈ C such that T (a, u,v-u) ≥ m,v-u , for every v ∈ C. ( 4
)
Variational inequality (4) has been extensively studied in the literature and has found numerous applications. Identification problems in variational inequalities have also been studied (see [START_REF] Hintermüller | Inverse coefficient problems for variational inequalities: optimality conditions and numerical realization[END_REF]).

Solvability of quasi-variational inequalities

The existence theory for quasi-variational inequalities is challenging. Although some existence results exploit the ordering structure of the underlying spaces, one of the most commonly adopted techniques for solving quasi-variational inequalities is by finding fixed points of the associated variational selection.

To define the variational selection, we fix an arbitrary element w ∈ C and consider the following variational inequality: Given a ∈ A, find u ∈ K (w) such that T (a, u,v-u) ≥ m,v-u , for every v ∈ K (w).

(

) 5
The variational selection is a (set-valued, in general) map S : C ⇒ C such that for any w ∈ C, the image S(w) is the set of all solutions of (5). Evidently, if u is a fixed point of the map S, that is, u ∈ S(u), then u solves quasi-variational inequality [START_REF] Barrett | A quasi-variational inequality problem arising in the modeling of growing sandpiles[END_REF].

We shall use the fixed point theorem of Kluge [START_REF] Kluge | On some parameter determination problems and quasi-variational inequalities[END_REF] to find a fixed point of the variational selection: Theorem 2.1 [START_REF] Kluge | On some parameter determination problems and quasi-variational inequalities[END_REF] Let Z be a reflexive Banach space and let C ⊂ Z be nonempty, convex, and closed. Assume that : C ⇒ C is a set-valued map such that for every u ∈ C, the set (u) is nonempty, closed, and convex, and the graph of is sequentially weakly closed. Assume that either the set C is bounded or the set (C) is bounded. Then the map has at least one fixed point in C.

We have the following existence result for quasi-variational inequality (3). Theorem 2.2 Besides the general setting described above, assume that the map K : C ⇒ C is M-continuous, that is, it satisfies the following conditions: (M1) For any sequence {x n }⊂C with x n ⇀ x, and for each y ∈ K (x), there exists a sequence {y n } such that y n ∈ K (x n ) and y n → y. (M2) For y n ∈ K (x n ) with x n ⇀ x and y n ⇀ y, we have y ∈ K (x).

Then, for every a ∈ A, the quasi-variational inequality (3) has a nonempty solution set. Proof We will divide the proof into several parts. Our aim is to show that the variational selection S : C ⇒ C satisfies the assumptions imposed on the map in Theorem 2.1.

For a given a ∈ A and a fixed w ∈ C, the parametric variational inequality ( 5) is uniquely solvable. Therefore, for the given a ∈ A, and each w ∈ C, the solution set S(w) is nonempty, and (trivially) closed and convex.

We shall now prove that the graph of S is sequentially weakly closed. Let {(v n , y n )}⊂ C × C be such that y n = S(v n ) with y n ⇀ y and v n ⇀v.We will show that y = S(v). The set C being convex and closed is also weakly closed, and hence v ∈ C. Since

y n = S(v n ), we infer that y n ∈ K (v n ) and T (a, y n , z -y n ) ≥ m, z -y n , for every z ∈ K (v n ). (6)
The containment y n ∈ K (v n ), by using the M-continuity of the map K and taking into account the convergence y n ⇀ y and v n ⇀v ,implies that y ∈ K (v) (see (M2)). Let z ∈ K (v) be arbitrary. Using the M-continuity once again (precisely, condition (M1)) ensure that there is a sequence {z n } converging strongly to z and satisfying 6) and rearrange the resulting inequality as follows

z n ∈ K (v n ). We insert z = z n ∈ K (v n ) in (
T (a, z n , z n -y n ) ≥ T (a, y n -z n , y n -z n ) + m, z n -y n ≥ m, z n -y n ,
which when passed to the limit n →∞implies that for every z ∈ K (v), we have

T (a, z, z -y) ≥ m, z -y .
We now insert y + t (zy) with t ∈ (0, 1) in place of z in the above inequality to get

T (a, y, z -y) + tT(a, z -y, z -y) ≥ m, z -y .
We pass in the above inequality to limit t → 0 obtaining

T (a, y, z -y) ≥ m, z -y , for every z ∈ K (v),
which ensues that y ∈ S(v). Finally, the boundedness of S(C) follows from the ellipticity of T . Theorem 2.1 then confirms that there is a fixed point for S, which solves (3). The proof is complete.

⊓ ⊔ Remark 2.1
The linearity of T (a, u,v)with respect to a does not play any role in the above result.

Solvability of the inverse problem

We will now introduce an optimization framework to study the inverse problem of parameter identification in quasi-variational inequalities. To counter the adverse effects of the ill-posedness of the inverse problem, we develop an abstract regularization framework by assuming:

(A1) The Banach space B is continuously embedded in a Banach space L . There is another Banach space B that is compactly embedded in L.ThesetA consists of real-valued functions and is a subset of B ∩ B, closed and bounded in B and also closed in L. (A2) For any sequence {b k }⊂B with b k → 0inL, any bounded sequence {u k }⊂V, and fixed v ∈ V, we have

T (b k , u k ,v) → 0. (7)
(A3) R : B → R is convex, and lower-semicontinuous with respect to

• L such that R(a) ≥ τ 1 a B -τ 2 , for every a ∈ A, for some τ 1 > 0,τ 2 > 0. ( 8
)
Using the above framework, which will be exemplified shortly, we consider the following regularized output-least-squares (OLS) optimization problem: Find a ∈ A by solving

min a∈A J κ (a) := 1 2 u(a) -z 2 Z + κ R(a), ( 9
)
where κ>0 is a regularization parameter, R is the regularization map introduced in (A3), u(a) is the solution of (3), and z ∈ Z is the measured data. For recent developments in inverse problems, see [START_REF] Boiger | An online parameter identification method for time dependent partial differential equations[END_REF][START_REF] Clason | L ∞ fitting for inverse problems with uniform noise[END_REF][START_REF] Gockenbach | An abstract framework for elliptic inverse problems. I. An output leastsquares approach[END_REF][START_REF] Jadamba | A new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location[END_REF][START_REF] Liu | Regularization of nonlinear ill-posed variational inequalities and convergence rates[END_REF][START_REF] Resmerita | Regularization of ill-posed problems in Banach spaces: convergence rates[END_REF] and the cited references therein.

We have the following existence result: Theorem 3.1 Fo r κ>0, regularized output-least-squares problem (9) has a nonempty solution set.

Proof The functional J κ is bounded from below and hence there exists a minimizing sequence. Let {a n }⊂A be a minimizing sequence such that lim n→∞ J κ (a

n ) = inf{J κ (b)| b ∈ A}. Let u n = u(a n
) be the solution of quasi-variational inequality (3)forthe parameter a n . Due to the definition of {a n } and the inequality κ R(a n ) ≤ J κ (a n ), the sequence {a n } is bounded in B. Due to the compact embedding of B into L , {a n } has a subsequence which converges strongly in • L . Keeping the same notation for subsequences as well, let {a n } be the subsequence converging in • L to some ā ∈ A.

On the other hand, by the definition of u n (see ( 3)), we have u n ∈ K (u n ) and

T (a n , u n ,v-u n ) ≥ m,v-u n , for every v ∈ K (u n ). (10)
Let {u n } be a sequence of solutions of (10) that correspond to the subsequence of parameters {a n } converging to ā ∈ A. We claim that {u n } is bounded. By assumption made before (1) there exists a bounded sequences {s n } with s n ∈ K (u n ) ∩ C 0 for each n.Bytakingv = s n in [START_REF] Hintermüller | Inverse coefficient problems for variational inequalities: optimality conditions and numerical realization[END_REF], we obtain

T (a n , u n , s n -u n ≥ m, s n -u n ,
which by using the ellipticity of T in (2)a sw e l la s( 1) ensures that the sequence {u n } is bounded.

Since {u n } is bounded, there exists a weakly convergent subsequence in V . Keeping the same notation for the subsequence, let {u n } be the subsequence which converges weakly to some ū ∈ V. We shall show that ū = u( ā). Since u n ∈ K (u n ) and since u n converges weakly to ū, by using condition (M2) of the M-continuity of the map K , we obtain ū ∈ K ( ū). Let z ∈ K ( ū) be arbitrary. Using condition (M1) of the M-continuity, we ensure that there is a sequence {z n } converging strongly to z in V and satisfies [START_REF] Hintermüller | Inverse coefficient problems for variational inequalities: optimality conditions and numerical realization[END_REF] and using the identity

z n ∈ K (u n ). Setting v = z n ∈ K (u n ) in
T (a n , u n , z n -u n ) = T (a n , u n -z n , z n -u n ) + T (a n , z n -z, z n -u n ) + T (a n -ā, z, z n -u n ) + T ( ā, z, z n -u n )
and the ellipticity of T in (2), we obtain

T (a n , z n -z, z n -u n ) + T (a n -ā, z, z n -u n ) + T ( ā, z, z n -u n ) ≥ m, z n -u n ,
which, by [START_REF] Clason | L ∞ fitting for inverse problems with uniform noise[END_REF] and the symmetry of T (a, u,v)in u and v,whenpassedtothelimitn →∞, yields

T ( ā, z, z -ū) ≥ m, z -ū , for every z ∈ K ( ū).
We set ū + t (z -ū), t ∈ (0, 1), in place of z in the above inequality to get

T ( ā, ū + t (z -ū), z -ū) ≥ m, z -ū .
We pass in the above inequality to limit t → 0toget T ( ā, ū, z -ū) ≥ m, z -ū , which, due to the fact that z ∈ K ( ū) was chosen arbitrarily, ensures that ū solves (3)for ā.

The weak lower-semicontinuity of the norm and lower-semicontinuity of R in

• L yield J κ ( ā) = 1 2 u( ā) -z 2 Z + κ R( ā) ≤ lim inf n→∞ 1 2 u n -z 2 Z + lim inf n→∞ κ R(a n ) ≤ lim inf n→∞ 1 2 u n -z 2 Z + κ R(a n ) = lim n→∞ J κ (a n ) = inf {J κ (b)| b ∈ A} ,
which confirms that ā is a solution of ( 9). The proof is complete.

⊓ ⊔

The above framework simplifies nicely (and does not need B and L), if the regularization space B and the parameter space are linked. The following result sheds some light on this aspect: Theorem 3.2 Assume that a Hilbert space H is compactly embedded into the Banach space B, A ⊂ H is nonempty, closed, and convex, the map R : H → R is convex, lowersemicontinuous, and there exist α 1 > 0 and

α 2 > 0 such that R(a) ≥ α 1 a 2 H -α 2 ,f o r every a ∈ A. Then (9) is solvable.
Proof The proof follows by the arguments used above but relying on (1) (instead of ( 7)). ⊓ ⊔

Finite-dimensional approximation

We shall discretize the identification problem for quasi-variational inequalities. Recall that we have been dealing with the quasi-variational inequality: Given a ∈ A,

find u = u(a) ∈ K (u) such that T (a, u,v-u) ≥ m,v-u , for every v ∈ K (u). ( 11
)
We considered the following regularized optimization problem: Find a ∈ A by solving

min a∈A J κ (a) = 1 2 u(a) -z 2 Z + κ R(a), ( 12
)
where κ>0 is a regularization parameter, R is the regularization map, u(a) is the unique solution of ( 11), and z ∈ Z is the measured data. We now proceed to describe the discretization framework. Assume that we have a family {V n } of finite-dimensional subspaces of V, and for each n, P n : V → V n is a projection operator such that v -P n v V → 0 for each v ∈ V. We assume that {B n } is a family of finite-dimensional subspaces of B. We set A n = B n ∩ A and assume that ∩ n A n =∅ . For each n ∈ N, let C n ⊂ V n be a nonempty, closed, and convex set. Let K n : C n ⇒ C n be a set-valued map such that for every v ∈ C n , the set K n (v) is nonempty, closed, and convex. We assume that the maps K n approximate K in the following sense: (M3) For any sequence {x n } with x n ∈ C n and x n ⇀ x in V , and for each y ∈ K (x), there exists a sequence {y n } such that y n ∈ K n (x n ) and y n → y. (M4)F o ry n ∈ K n (x n ) with x n ⇀ x and y n ⇀ y in V ,wehavey ∈ K (x).

We denote by R n : A n → R the discrete analogue of R and assume the following conditions:

1. For any a ∈ A, there exists a sequence {a n } with a n ∈ A n such that a n → a in L and

lim sup n→∞ R n (a n ) ≤ R(a). ( 13
)
2. For every sequence {a n } with a n ∈ A n and a n → a in L , we have

R(a) ≤ lim inf n→∞ R n (a n ). ( 14
)
The discrete analogue of ( 11) then reads: Given

a n ∈ A n ,findu n ∈ K n (u n ) such that T (a n , u n ,v-u n ) ≥ m,v n -u n , for every v ∈ K n (u n ). (15)
Consider the following finite-dimensional minimization problem: Find ān ∈ A n by solving min

a n ∈A n J n κ (a n ) = 1 2 u n (a n ) -z 2 Z + κ R n (a n ), (16)
where u n (a n ) is the solution of [START_REF] Kluge | On some parameter determination problems and quasi-variational inequalities[END_REF]. Evidently, the data z could be replaced by discrete data z n such that z n → z as n → 0.

The following is a convergence result for the finite-dimensional optimization problem (16): Theorem 4.1 Fo r e a ch n , discrete problem (16) has a minimizer ān ∈ A n . Moreover, there is a subsequence of {ā n } that converges in • L to a solution of [START_REF] Kano | Existence theorems for elliptic quasi-variational inequalities in Banach spaces[END_REF].

Proof The solvability follows from the arguments used before. Due to the assumption ∩ n A n =∅ , there is a constant c such that J n κ ( ān ) ≤ c, for every n. Therefore, {ā n } is bounded in • B . Due to the compact embedding of B into L, there is a subsequence that converges strongly, in • L , to an element of A. Keeping the same notation for the subsequences, we assume that {ā n } converges, in • L , to some ā ∈ A. Let ū be the solution of quasi-variational inequality [START_REF] Jadamba | A new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location[END_REF] for the parameter ā and let ūn be the solution of quasivariational inequality [START_REF] Kluge | On some parameter determination problems and quasi-variational inequalities[END_REF] for the parameter ān . It can be shown as in Theorem 3.1 that the sequence {ū n } is bounded in V . Therefore, there exists a subsequence, still denoted by { ūn } , such that ūn converges weakly to some ū ∈ V as n →∞ . By the definition of the solution of (15), we have ūn ∈ K n ( ūn ) which implies that ū ∈ K ( ū) (see assumption (M4)). We will show that ū = u( ā).

By the definition of ūn , we have ūn ∈ K n ( ūn ) and

T ( ān , ūn ,v-ū n ) ≥ m,v-ū n , for every v ∈ K n ( ūn ).
Let z ∈ K ( ū) be arbitrary and let {z n } be a sequence with z n ∈ K n ( ūn ) such that z n → z in V (see assumption (M3)). We set v = z n in the above inequality, and after a rearrangement, obtain,

T ( ān , ūn -z, z n -ū n ) + T ( ān -ā, z, z n -ū n ) + T ( ā, z, z n -ū n ) ≥ m,v-ū n
which, in view of ( 2)and( 7), when passed to the limit n →∞, implies that

T ( ā, z, z -ū) ≥ m, z -ū , for every z ∈ K ( ū)
ensuring that ū = u( ā).

This observation, in view of ( 14), yields

J κ ( ā) = 1 2 ū -z 2 Z + κ R( ā) ≤ lim n→∞ 1 2 ūn -z 2 Z + lim inf h→0 κ R n ( ān ) ≤ lim inf n→∞ 1 2 ūn -z 2 Z + κ R n ( ān ) .
Let a ∈ A be arbitrary. Then there exists a sequence {a n } such that a n ∈ A n , a n → a in L and R n (a n ) → R(a) as n →∞(see [START_REF] Khan | Existence theorems for elliptic and evolutionary variational and quasivariational inequalities[END_REF]a n d( 14)). We have shown above that the corresponding solutions {u n } with u n = u n (a n ) converges weakly to u(a). We need that in fact the convergence is strong. We note that u n ∈ K n (u n ) and

T (a n , u n , z -u n ) ≥ m, z -u n , for every z ∈ K n (u n ).
Due to the containment u n ∈ K n (u n ) and the fact that u n converges weakly to some u, we have u ∈ K (u) (see assumption (M4)). Moreover, there exists w n ∈ V n such that w n ∈ K n (u n ) and w n → u strongly (see assumption (M3)). By setting z = w n in the above inequality, we obtain

T (a n , u n ,w n -u n ) ≥ m,w n -u n ,
which can be rearranged as follows

T (a n , u n -u,w n -u) + T (a n , u n -u, u -u n ) + T (a n , u,w n -u n ) + m, u n -w n ≥ 0.
By [START_REF] Barbagallo | A general quasi-variational problem of Cournot-Nash type and its inverse formulation[END_REF], this implies that

T (a n , u n -u,w n -u) + T (a n , u,w n -u n ) + m, u n -w n ≥ α u n -u 2 ,
or equivalently

T (a n , u n -u,w n -u) + T (a n -a, u,w n -u n ) + T (a, u,w n -u n ) + m, u n -w n ≥α u n -u 2
which by (7) when passed to the limit n →∞implies that {u n } converges strongly to u.

In view of the above observations, we have

J κ ( ā) ≤ lim inf n→∞ 1 2 u( ān ) -z 2 Z + κ R n ( ān ) ≤ lim inf h→0 1 2 u n (a n ) -z 2 Z + κ R n (a n ) = 1 2 u(a) -z 2 Z + κ R(a) = J κ (a),
and hence ā is a solution of [START_REF] Kano | Existence theorems for elliptic quasi-variational inequalities in Banach spaces[END_REF]. This completes the proof. ⊓ ⊔

An application

We now justify our framework by identifying a parameter in a gradient obstacle problem. Let be a bounded domain in R N , where 1 ≤ N < ∞, with a sufficiently smooth boundary. We set V = H 1 0 ( ) and B = L ∞ ( ). Let κ * be a constant, and let κ c be a Lipschitz continuous real function on R such that 0 <κ c (r ) ≤ κ * ,fore v eryr ∈ R. Define a convex set C and

K : C ⇒ C by C ={w ∈ V ||∇w|≤κ * , a.e. on }, K (v) ={w ∈ V ||∇w|≤κ c (v), a.e. on },
which satisfy conditions (M1) and (M2) of Theorem 2.2 (see [START_REF] Kano | Existence theorems for elliptic quasi-variational inequalities in Banach spaces[END_REF]). We focus on identifying a in the quasi-variational inequality: Find u ∈ K (u) such that

T (a, u,v-u) ≥ m,v-u , for every v ∈ K (u),
where T (a, u,v) = a∇u∇v. To introduce the regularization space, we recall that the total variation of f ∈ L 1 ( ) is given by

TV( f ) = sup f (∇•g) : g ∈ C 1 ( ) N , |g(x)|≤1forallx ∈ where |•|represents the Euclidean norm. Clearly, if f ∈ W 1,1 ( ),thenTV( f ) = |∇ f |. If f ∈ L 1 ( ) satisfies TV( f )<∞,t h e
n f is said to have bounded variation, and the Banach space BV( ) is defined by BV( ) = f ∈ L 1 ( ) : TV( f )<∞ being endowed with the norm f BV( ) = f L 1 ( ) + TV( f ). The functional TV(•) is a seminorm on BV( ) and is often called the BV-seminorm.

We set L = L 1 ( ), B = BV( ), and R(a) = TV(a), and define two sets

A 1 ={a ∈ L ∞ ( )| c 1 ≤ a(x) ≤ c 2 , a.e. in , }, A 2 ={a ∈ L ∞ ( )| c 1 ≤ a(x) ≤ c 2 , a.e. in , TV(a) ≤ c 3 },
where c 1 , c 2 and c 3 are positive constants. Clearly, both sets are compact in L, whereas A 2 is bounded in • B . It is known that L ∞ ( ) is continuously embedded in L 1 ( ),BV( ) is compactly embedded in L 1 ( ),andTV(•) is convex and lower-semicontinuous in L 1 ( )norm, see [START_REF] Acar | Analysis of bounded variation penalty methods for ill-posed problems[END_REF][START_REF] Nashed | Least squares and bounded variation regularization with nondifferentiable functionals[END_REF]. Let u,v ∈ H 1 0 ( ) be fixed and let {a n } be a bounded sequence in L ∞ ( ).Ifa n → 0in the L 1 ( ) norm, then, by the Lebesgue dominated convergence theorem, a k ∇u •∇v → 0. Furthermore, for any sequence {v n }, bounded in V, and for any v ∈ V ,wehave

a n ∇v n •∇v ≤ |a n ∇v n •∇v| = |a n | 1/2 ∇v • |a n | 1/2 ∇v n ≤ |a n |∇v n •∇v n 1/2 |a n |∇v •∇v 1/2
, which confirms [START_REF] Clason | L ∞ fitting for inverse problems with uniform noise[END_REF]. Here we use the fact that a n → 0i nt h eL 1 ( ) implies |a n |∇v •∇v 1/2 → 0, whereas |a n |∇v n •∇v n 1/2 remains bounded due to the boundedness of v n in V and the boundedness of a n in L ∞ . We note that the boundedness of a n in L ∞ ( ) has been obtained by optimally choosing the feasible set (see the sets A 1 and A 2 given above). Conditions (1)and( 2) are also verified by standard arguments. Summarizing, the developed framework is applicable and ensures the identification of a discontinuous parameter in the considered gradient obstacle problem.

Concluding remarks

As a first step towards developing a theory for identifying variable parameters in quasivariational inequalities, we provided new existence, convergence, and discretization results. It remains an open question to establish a penalization approach for quasi-variational inequality so that an optimization problem with an equality constraint can be considered. Of course, derivation of implementable necessary optimality conditions remains a priority as well. Note the parameter-to-solution map does not exhibit any smoothness and hence some smoothing needs to be done. Numerical results are of utmost importance, and some derivative-free methods can be used as the initial step.

Acknowledgements

The authors are grateful to the reviewers for the valuable remarks. Akhtar Khan is supported by National Science Foundation Grant 005613-002 and RIT's COS FEAD Grant for 2016-2017.


Mircea Sofonea
email: sofonea@univ-perp.fr

Andaluzia Matei

History-dependent mixed variational problems in contact mechanics

Keywords: History-dependent operator, Mixed variational problem, Lagrange multiplier, Viscoplastic material, Frictionless contact, Normal compliance, History-dependent stiffness coefficient, Unilateral constraint, Variational formulation, Weak solution

We consider a new class of mixed variational problems arising in Contact Mechanics. The problems are formulated on the unbounded interval of time [0, +∞) and involve history-dependent operators. For such problems we prove existence, uniqueness and continuous dependence results. The proofs are based on results on generalized saddle point problems and various estimates, combined with a fixed point argument. Then, we apply the abstract results in the study of a mathematical model which describes the frictionless contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic and the contact is modelled with normal compliance and unilateral constraint, in such a way that the stiffness coefficient depends on the history of the penetration. We prove the unique weak solvability of the contact problem, as well as the continuous dependence of its weak solution with respect to the viscoplastic constitutive function, the applied forces, the contact conditions and the initial data.

Introduction

Mixed variational problems involving Lagrange multipliers are used both in analysis and mechanics, in the study of minimization problems. They provide a useful framework in Problem 1 Find the functions u : R + → X and λ : R + → such that

(Au(t), v) X + (S(u, λ)(t), v) X + b(v, λ(t)) = ( f (t), v) X ∀ v ∈ X, (1.1) b(u(t), µ -λ(t)) ≤ b(h(t), µ -λ(t)) ∀ µ ∈ , (1.2)
for all t ∈ R + .

Our aim in this paper is threefold. The first one is to study the unique solvability of Problem 1. To this end, we use a result related to a generalized saddle point problem proved in [START_REF] Matei | Contact problems for nonlinearly elastic materials: weak solvability involving dual Lagrange multipliers[END_REF], combined with a fixed point result obtained in [START_REF] Sofonea | A Fixed point result with applications in the study of viscoplastic frictionless contact problems[END_REF] and show that, under appropriate conditions, Problem 1 has a unique solution (u,λ)such that u ∈ C(R + ; X ) and λ ∈ C(R + ; Y ).The second aim is to study the behavior of the solution of Problem 1 with respect to a perturbation of the data. To this end we use monotonicity properties and arguments of convergence in the spaces C(R + ; X ) and C(R + ; Y ) which allow us to prove a convergence result. Finally, our third aim is to show how our abstract results can be used in the analysis of mathematical models in Contact Mechanics. To this end we consider a quasistatic process of contact between a viscoplastic body and a deformable foundation. The contact is with normal compliance and finite penetration, in such a way that the stiffness coefficient depends on the history of the penetration. Considering such kind of model leads to a new and interesting mathematical problem, governed by two history-dependent operators. We provide the analysis of this problem, which includes its unique weak solvability and the continuous dependence of the solution with respect to the data. The proofs are based on the abstract results obtained in the study of Problem 1. In this way we fully exemplify the cross fertilization between the models and applications, in one hand, and the nonlinear functional analysis, on the other hand.

The rest of the paper is structured as follows. In Sect. 2 we introduce the mixed variational problem, list the assumptions on the data, then we state and prove our main abstract existence and uniqueness result, Theorems 2.1. In Sect. [START_REF] Barboteu | Analysis of quasistatic viscoplastic vontact problems with normal compliance[END_REF] we introduce a perturbation of the problem, then we state and prove a convergence result, Theorem 3.1. Then, in Sect. [START_REF] Barboteu | On the behavior of the solution of a viscoplastic contact problem[END_REF], we describe our mathematical model of contact, list the assumptions on the data and derive its variational formulation. In Sect. 5 we prove the unique weak solvability of the model. The proof is based on the abstract result provided by Theorem 2.1. Finally, in Sect. 6 we prove the continuous dependence of the weak solution of the contact problem with respect to the data. The proof is based on the abstract result provided by Theorem 3.1.

We end this introduction with some notation and preliminaries. First, everywhere in this paper we denote by N the set of positive integers. Given a normed space U and a subset K ⊂ U we use the symbol C(R + ; K ) for the set of continuous functions defined on R + with values in K . It is well known that, if U is a Banach space, then C(R + ; U ) can be organized in a canonical way as a Fréchet space, i.e. a complete metric space in which the corresponding topology is induced by a countable family of seminorms. Details can be found in [START_REF] Corduneanu | Problèmes globaux dans la théorie des équations intégrales de Volterra[END_REF]a n d [START_REF] Massera | Linear Differential Equations and Function Spaces[END_REF], for instance. Here we restrict ourselves to recall that the convergence of a sequence (u k ) k to the element u, in the space C(R + ; U ), can be described as follows:

u k → u in C(R + ; U ) as k →∞ if and only if max r ∈[0,n] u k (r ) -u(r ) U → 0a sk →∞, for all n ∈ N.
(1.3)

We also recall the following fixed point result.

Theorem 1.1 Let (X, • X ) be a real Banach space and let L : C(R + ; X ) → C(R + ; X ) be a nonlinear operator. Assume that there exists m ∈ N with the following property: for each n ∈ N there exist two constants c n ≥ 0 and k n ∈[0, 1) such that

Lu(t) -Lv(t) m X ≤ c n t 0 u(s) -v(s) m X ds + k n u(t) -v(t) m X (1.4)
for all u,v∈ C(R + ; X ) and for any t ∈[0, n]. Then the operator L has a unique fixed point η * ∈ C(R + ; X ).

Note that in (1.4) and below the notation Lη(t) represents the value of the function Lη at the point t, i.e. Lη(t) = (Lη)(t). The proof of Theorem 1.1 can be found in [START_REF] Sofonea | A Fixed point result with applications in the study of viscoplastic frictionless contact problems[END_REF]. We shall use this fixed point result twice, in Sects. 2 and 5 of the paper.

An abstract existence and uniqueness result

In this section we prove the unique solvability of Problem 1. To this end we assume that the data satisfy the following condition.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (a) There exists m A > 0 such that (Au -Av, u -v) X ≥ m A u -v 2 X ∀ u,v ∈ X. (b) There exists L A > 0 such that Au -Av X ≤ L A u -v X ∀ u,v ∈ X.
(2.1)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ For each n ∈ N there exist d n ≥ 0andr n ≥ 0 such that S(u 1 ,λ 1 )(t) -S(u 2 ,λ 2 )(t) X ≤ d n ( u 1 (t) -u 2 (t) X + λ 1 (s) -λ 2 (s) Y ) + r n t 0 ( u 1 (s) -u 2 (s) X + λ 1 (s) -λ 2 (s) Y ) ds ∀ u 1 , u 2 ∈ C(R + ; X ), ∀ λ 1 ,λ 2 ∈ C(R + ; Y ), ∀ t ∈[0, n].
(2.2)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ b : X × Y → R is a bilinear form such that (a) There exists M b > 0 such that |b(v, µ)|≤M b v X µ Y ∀ v ∈ X,µ∈ Y. (b) There exists α>0 such that inf µ∈Y,µ =0 Y sup v∈X,v =0 X b(v, µ) v X µ Y ≥ α.
(2.3)

f ∈ C(R + ; X ), h ∈ C(R + ; X ). (2.4) is a closed convex unbounded subset of Y that contains 0 Y . (2.5)
Our main result in this section is the following. The proof of Theorem 2.1 will be carried out in several steps. To this end, we assume in what follows that (2.1)-(2.5) hold. The first step is given by the following result.

Lemma 2.2 Given g, k ∈ X, there exists a unique pair (u,λ) ∈ X × such that (Au,v) X + b(v, λ) = (g,v) X ∀ v ∈ X, (2.6) b(u,µ-λ) ≤ b(k,µ-λ) ∀ µ ∈ . (2.7)
In addition, if (u 1 ,λ 1 ) and (u 2 ,λ 2 ) are the solutions of the problem (2.6)-(2.7) corresponding to the data g 1 , k 1 ∈ X and g 2 , k 2 ∈ X, respectively, then there exists c 0 which depends only on A and b such that

u 1 -u 2 X + λ 1 -λ 2 Y ≤ c 0 ( g 1 -g 2 X + k 1 -k 2 X ). (2.8)
Proof The existence and uniqueness part of the lemma corresponds to Theorem 5.2 in [START_REF] Matei | Contact problems for nonlinearly elastic materials: weak solvability involving dual Lagrange multipliers[END_REF] and, for this reason, we skip its proof. The estimate (2.8) shows the Lischitz continuous dependence of the solution with respect to the data and corresponds to Theorem 5.8 in [START_REF] Matei | Contact problems for nonlinearly elastic materials: weak solvability involving dual Lagrange multipliers[END_REF]. Nevertheless, since the size of the constant c 0 in this estimate will play an important role in what follows, for the convenience of the reader, we present the proof of (2.8). Thus, consider g i , k i ∈ X and denote by (u i ,λ i ), the solution of the problem (2.6)-(2.7), corresponding to the data g i , k i ∈ X , for each i = 1, 2. Then, using (2.6) it follows that

(Au 1 -Au 2 ,v) X + b(v, λ 1 -λ 2 ) = (g 1 -g 2 ,v) X ∀ v ∈ X (2.9)
and, using (2.1)(b) we find that

b(v, λ 1 -λ 2 ) ≤ g 1 -g 2 X v X + L A u 1 -u 2 X v X ∀ v ∈ X.
We now use (2.3)(b) and the previous inequality to obtain that

α λ 1 -λ 2 Y ≤ g 1 -g 2 X + L A u 1 -u 2 X . (2.10)
On the other hand, (2.7) yields

b(u 1 -u 2 ,λ 2 -λ 1 ) ≤ b(k 1 -k 2 ,λ 2 -λ 1 )
and, therefore, using condition (2.3)(a) we find that

b(u 1 -u 2 ,λ 2 -λ 1 ) ≤ M b k 1 -k 2 X λ 1 -λ 2 Y . (2.11)
We now take v = u 1u 2 in (2.9) and use (2.11) in the resulting inequality to deduce that

(Au 1 -Au 2 , u 1 -u 2 ) X ≤ (g 1 -g 2 , u 1 -u 2 ) X + M b k 1 -k 2 X λ 1 -λ 2 Y .
Therefore, using the assumption (2.1)(a) it follows that

m A u 1 -u 2 2 X ≤ g 1 -g 2 X u 1 -u 2 X + M b k 1 -k 2 X λ 1 -λ 2 Y .
(2.12)

Inequalities (2.12)and(2.10) imply that

m A u 1 -u 2 2 X ≤ g 1 -g 2 2 X 2c 1 + c 1 u 1 -u 2 2 X 2 (2.13) + M 2 b k 1 -k 2 2 X 2c 2 + c 2 λ 1 -λ 2 2 Y 2 , λ 1 -λ 2 2 Y ≤ 2 α 2 g 1 -g 2 2 X + L 2 A u 1 -u 2 2 X (2.14)
respectively, where c 1 , c 2 are arbitrary positive constants. We now choose c 1 and c 2 such that

m A - c 1 2 - c 2 L 2 A α 2 > 0.
Therefore, from (2.13)and(2.14) we deduce that there exists c 3 > 0, which depends only on A and b, such that

u 1 -u 2 2 X ≤ c 3 g 1 -g 2 2 X + k 1 -k 2 2 X . (2.15)
Finally, combining (2.14)a n d( 2.15) we obtain (2.8) with c 0 depending only on A and b, which concludes the proof.

⊓ ⊔

The next step is given by the following result.

Lemma 2.3 Given η ∈ C(R + ; X ), there exists a unique couple of functions

(u η ,λ η ) ∈ C(R + ; X ) × C(R + ; ) such that (Au η (t), v) X + (η(t), v) X + b(v, λ η (t)) = ( f (t), v) X ∀ v ∈ X, (2.16) b(u η (t), µ -λ η (t)) ≤ b(h(t), µ -λ η (t)) ∀ µ ∈ , (2.17)
for all t ∈ R + . In addition, given η 1 ,η 2 ∈ C(R + ; X ) and denoting by u η 1 , u η 2 the corresponding couples of functions which verify (2.16)-(2.17)

at each t ∈ R + , then u η 1 (t) -u η 2 (t) X + λ η 1 (t) -λ η 2 (t) Y ≤ c 0 η 1 (t) -η 2 (t
i ) = η i , u(t i ) = u i ,λ(t i ) = λ i , f (t i ) = f i , h(t i ) = h i and g i = f i -η i .
Then, using inequality (2.8), it follows that

u 1 -u 2 X + λ 1 -λ 2 Y ≤ c 0 ( f 1 -f 2 X + η 1 -η 2 X + h 1 -h 2 X ).
Therefore, since f,η ,h ∈ C(R + ; X ) we conclude that u η ∈ C(R + ; X ) and λ η ∈ C(R + ; ).

Lη 1 (t) -Lη 2 (t) X ≤ d n ( u η 1 (t) -u η 2 (t) X + λ η 1 (t) -λ η 2 (t) Y ) + r n t 0 ( u η 1 (s) -u η 2 (s) X + λ η 1 (s) -λ η 2 (s) Y ) ds.
Therefore, inequality (2.18) yields

Lη 1 (t) -Lη 2 (t) X ≤ c 0 d n η 1 (t) -η 2 (t) X + c 0 r n t 0 η 1 (s) -η 2 (s) X ds. (2.

20)

Let d 0 = 1 c 0 and note that Lemma 2.2 shows that d 0 depends only on A and b. Assume now that d n < d 0 for all n ∈ N. Then inequality (2.20)andTheorem1.1 show that L has a unique fixed point, which concludes the proof.

⊓ ⊔

We now have all the ingredients to prove Theorem 2.1.

Proof Let d 0 > 0 be defined as above and recall that d 0 depends only on A and b. Assume that d n < d 0 for all positive integers n, and denote by η * the unique fixed point of the operator L provided in Lemma 2.4. Then, using (2.16), (2.17) and definition (2.19) of the operator L it is easy to see that (u η * ,λ η * ) is a solution of Problem 1 and, moreover, it has the regularity

(u η * ,λ η * ) ∈ C(R + ; X ) × C(R + ;
). This concludes the existence part of the theorem. The uniqueness part follows from the uniqueness of the fixed point of the operator L, guaranteed by Lemma 2.4. ⊓ ⊔

A convergence result

We now turn to the dependence of the solution with respect to the data. To this end, everywhere in this section we assume that (2.1)-(2.5) hold and we denote by (u,λ) the solution of Problem 1 provided by Theorem 2.1. Moreover, we assume that for each ρ>0 the operator

S ρ : C(R + ; X ) × C(R + , Y ) → C(R + ; X )
, and the functions f ρ , h ρ : R + → X are given, and represent perturbations of the data S, f and h, respectively. With these data, for each ρ>0, we consider the following problem.

Problem 2 Find the functions u ρ : R + → X and λ ρ : R + → such that

(Au ρ (t), v) X + (S ρ (u ρ ,λ ρ )(t), v) X + b(v, λ ρ (t)) = ( f ρ (t), v) X ∀ v ∈ X, (3.1) b(u ρ (t), µ -λ ρ (t)) ≤ b(h ρ (t), µ -λ ρ (t)) ∀ µ ∈ , (3.2)
for all t ∈ R + .

We assume that, for each ρ>0, the following conditions hold.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ For each n ∈ N there exist d ρn ≥ 0andr ρn ≥ 0 such that S ρ (u 1 ,λ 1 )(t) -S ρ (u 2 ,λ 2 )(t) X ≤ d ρn ( u 1 (t) -u 2 (t) X + λ 1 (t) -λ 2 (t) Y ) + r ρn t 0 ( u 1 (s) -u 2 (s) X + λ 1 (s) -λ 2 (s) Y ) ds ∀ u 1 , u 2 ∈ C(R + ; X ), ∀ λ 1 ,λ 2 ∈ C(R + ; Y ), ∀ t ∈[0, n].
(3.3)

f ρ ∈ C(R + , X ), h ρ ∈ C(R + , X ). (3.4)
Under these assumptions, if d ρn < d 0 for all n ∈ N, Theorem 2.1 guarantees the existence of a unique solution (u ρ ,λ ρ ) to Problem 2 such that u ρ ∈ C(R + ; X ) and λ ρ ∈ C(R + ; ). Our interest lies in the behavior of the solution of the perturbed problem as ρ tends to zero. To this end we consider the following additional assumptions, in which d 0 represents the constant in Theorem 2.1.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ For each n ∈ N there exist H n : R + → R + , J n : C(R + ; X ) × C(R + ; Y ) → R + and R n ≥ 0 such that (a) S ρ (u, λ)(t) -S(u, λ)(t)) X ≤ H n (ρ)J n (u,λ) ∀ (u,λ) ∈ C(R + ; X × ), ∀ t ∈[0, n], ∀ ρ>0. (b) r ρn ≤ R n ∀ ρ>0. (c) lim ρ→0 H n (ρ) = 0. (3.5) There exists d 0 such that d ρn ≤ d 0 < d 0 ∀ n ∈ N, ∀ ρ>0. (3.6
)

f ρ → f, h ρ → h in C(R + ; X ) as ρ → 0. (3.7)
We have the following convergence result.

u ρ → u in C(R + ; X ), λ ρ → λ in C(R + ; Y ) as ρ → 0. (3.8) Proof Let ρ>0, n ∈ N and let t ∈[0, n].
We note that the system (3.1)-(3.2)representsa system of the form (2.6)-(2.7)inwhich

g = f ρ (t) -S ρ (u ρ ,λ ρ )(t) and h = h ρ (t). Also, the system (1.1)-(1.2) is a system of the form (2.6)-(2.7)inwhich g = f (t) -S(u, λ)(t) and h = h(t).
Therefore, using the estimate (2.8) yields

u ρ (t) -u(t) X + λ ρ (t) -λ(t) Y (3.9) ≤ c 0 ( f ρ (t) -f (t) X + S ρ (u ρ ,λ ρ )(t) -S(u, λ)(t) X + h ρ (t) -h(t) X ).
Next, we remark that

f ρ (t) -f (t) X ≤ max s∈[0,n] f ρ (s) -f (s) X := δ ρn , (3.10
)

h ρ (t) -h(t) X ≤ max s∈[0,n] h ρ (s) -h(s) X := ω ρn (3.11)
and, in order to simplify the writing we denote

ϕ ρ (t) := u ρ (t) -u(t) X + λ ρ (t) -λ(t) Y . (3.12)
Then, inequalities (3.9)-(3.11) imply that

ϕ ρ (t) ≤ c 0 (δ ρn + ω ρn + S ρ (u ρ ,λ ρ )(t) -S(u, λ)(t) X ). (3.13)
On the other hand

S ρ (u ρ ,λ ρ )(t) -S(u, λ)(t) X ≤ S ρ (u ρ ,λ ρ )(t) -S ρ (u, λ)(t) X + S ρ (u, λ)(t) -S(u, λ)(t) X ,
and, therefore, assumptions (3.3)and(3.5)(a) combined with definition (3.12) show that

S ρ (u ρ ,λ ρ )(t) -S(u, λ)(t) X ≤ d ρn ϕ ρ (t) + r ρn t 0 ϕ ρ (s) ds + H n (ρ)J n (u, λ). (3.14)
We now use (3.13)and(3.14) to see that

ϕ ρ (t) ≤ c 0 (δ ρn + ω ρn ) + c 0 d ρn ϕ ρ (t) + c 0 r ρn t 0 ϕ ρ (s) ds + c 0 H n (ρ)J n (u, λ).
Therefore, the hypothesis (3.6) allows us to write

(1 -c 0 d 0 )ϕ ρ (t) ≤ c 0 (δ ρn + ω ρn ) + c 0 r ρn t 0 ϕ ρ (s) ds + c 0 H n (ρ)J n (u,λ).
Recall now that d 0 = 1 c 0 , as shown in the proof of Lemma 2.4. Thus, assumption (3.5)(b)

combined with inequality d 0 < d 0 in (3.6) imply that ϕ ρ (t) ≤ c (δ ρn + ω ρn ) + cR n t 0 ϕ ρ (s) ds + cH n (ρ)J n (u,λ)
where c is a positive constant independent of ρ and n. Using now a Gronwall argument we obtain

ϕ ρ (t) ≤ c (δ ρn + ω ρn + H n (ρ)J n (u, λ))e cR n t
and, therefore, max

t∈[0,n] ϕ ρ (t) ≤ c(δ ρn + ω ρn + H n (ρ)J n (u, λ))e cn R n . ( 3
u ρ (t) -u(t) X → 0, max t∈[0,n] λ ρ (t) -λ(t) Y → 0a sρ → 0. (3.17)
Finally, we use (3.17)and(1.3) to see that the convergences (3.8) hold, which concludes the proof. ⊓ ⊔

A history-dependent contact problem

In this section we introduce a model of frictionless contact which can be studied by using the abstract results presented in Section 2. The physical setting is as follows. A viscoplastic body occupies the bounded domain ⊂ R d (d = 2, 3), with the boundary ∂ = Ŵ partitioned into three disjoint measurable parts Ŵ 1 , Ŵ 2 and Ŵ 3 , such that meas Ŵ 1 > 0. We assume that the boundary Ŵ is Lipschitz continuous and we denote by v its unit outward normal, defined almost everywhere. The body is clamped on Ŵ 1 and, therefore, the displacement field vanishes there. A volume force of density f 0 acts in , surface tractions of density f 2 act on Ŵ 2 and, finally, we assume that the body is in contact with a deformable foundation on Ŵ 3 . The contact is frictionless and we model it with a normal compliance condition with unilateral constraint, in which the stiffness coefficient depends on the history of the penetration. The process is quasistatic and we study it in the unbounded interval of time [0, +∞). We denote by S d the space of second order symmetric tensors on R d and, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable. Then, the classical formulation of the contact problem is the following.

Problem 3 Find a displacement field u: ×[ 0, +∞) → R d and a stress field σ : ×[0, +∞) → S d such that

σ (t) = Eε( u(t)) + G(σ , ε(u(t))) in , (4.1)
Div

σ (t) + f 0 (t) = 0 in , (4.2
)

u(t) = 0 on Ŵ 1 , (4.3)
σ (t)v = f 2 (t) on Ŵ 2 , (4.4
)

u ν (t) ≤ g(t), σ ν (t) + k(ζ u(t)) p(u ν (t)) ≤ 0, (u ν (t) -g(t))(σ ν (t) + k(ζ u(t)) p(u ν (t))) = 0, ζ u(t) = t 0 u + ν (s) ds ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ on Ŵ 3 , (4.5)
σ τ (t) = 0 on Ŵ 3 , (4.6)
for all t ∈ R + and, moreover,

u(0) = u 0 , σ (0) = σ 0 in . ( 4.7)
We now provide a brief description of the equations and conditions in Problem 3 and refer the reader to [START_REF] Han | Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity[END_REF][START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF][START_REF] Sofonea | Mathematical Models in Contact Mechanics[END_REF] for details and additional comments on the classical formulation of the contact problems. First, Eq. (4.1) represents the viscoplastic constitutive law of the material, in which ε(u) denotes the linearized stress tensor, E is the elasticity tensor and G is a given constitutive function. Moreover, the dot above represents the derivative with respect to the time variable t. Quasistatic frictionless and frictional contact problems for such kind of materials were studied in various works, see for instance [START_REF] Barboteu | Analysis of quasistatic viscoplastic vontact problems with normal compliance[END_REF][START_REF] Barboteu | On the behavior of the solution of a viscoplastic contact problem[END_REF][START_REF] Han | Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity[END_REF][START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF] and the references therein.

Equation (4.2) is the equilibrium equation in which Div represents the divergence operator for tensor-valued functions; we use it here since the process is assumed to be quasistatic. Conditions (4.3)a n d( 4.4) are the displacement and traction boundary conditions, respectively, and condition (4.5) represents a new version of the normal compliance condition with unilateral constraint; here u ν and σ ν represent the normal component of the displacement and the stress field, respectively; g ≥ 0 is a time-dependent bound for the penetration, p represents a given normal compliance function, k is a positive function and ζ u(t) represents the accumulated contact penetration depth at time t, u + ν being the positive part of u ν .W e interpret k = k(ζ u) as a stiffness coefficient which, clearly, depends on the history of penetration. Note that such kind of dependence models the surface hardening or softening which appears in various applications, when cycles of contact and no contact arise. Details can be found in [START_REF] Piotrowski | Smoothing dry friction damping by dither generated in rolling contact of wheel and rail and its influence on ride dynamics of freight wagons[END_REF]. Condition (4.5) was introduced for the first time in [START_REF] Jarušek | On the solvability of dynamic elastic-visco-plastic contact problems[END_REF] in the case when g is a constant and k ≡ 1. In this particular form it was recently used in [START_REF] Barboteu | Analysis of quasistatic viscoplastic vontact problems with normal compliance[END_REF][START_REF] Barboteu | On the behavior of the solution of a viscoplastic contact problem[END_REF], in the study of a quasistatic viscoplastic problem. Condition (4.6) shows that the tangential stress on the contact surface, denoted σ τ , vanishes. We use it here since we assume that the contact process is frictionless. Finally, (4.7) represents the initial conditions in which u 0 and σ 0 denote the initial displacement and the initial stress field, respectively.

We turn now to the variational formulation of Problem 3. To this end, we need further notation and preliminaries. First, we use the notation x = (x i ) for a typical point in ∪ Ŵ and we denote by ν i the components of v, i.e. v = (ν i ). Here and below the indices i, j, k, l runbetween1andd and, unless stated otherwise, the summation convention over repeated indices is used. An index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e.g. u i, j = ∂u i /∂ x j . Recall that the inner product and norm on R d and S d are defined by

u • v = u i v i , v =(v • v) 1 2 ∀ u, v ∈ R d , σ • τ = σ ij τ ij , τ =(τ • τ ) 1 2 ∀ σ , τ ∈ S d .
We use standard notation for the Lebesgue and Sobolev spaces associated to and Ŵ and, moreover, we consider the spaces

V ={v = (v i ) ∈ H 1 ( ) d : v = 0 on Ŵ 1 }, Q ={τ = (τ ij ) ∈ L 2 ( ) d×d : τ ij = τ ji }.
These are real Hilbert spaces endowed with the inner products

(u, v) V = ε(u) • ε(v) dx,( σ , τ ) Q = σ • τ dx,
and the associated norms • V and • Q , respectively. Here ε represents the deformation operator given by

ε(v) = (ε ij (v)), ε ij (v) = 1 2 (v i, j + v j,i ) ∀ v ∈ H 1 ( ) d .
Completeness of the space (V, • V ) follows from the assumption meas Ŵ 1 > 0, which allows the use of Korn's inequality. For an element v ∈ V we still write v for the trace of v on the boundary and we denote by v ν and v τ the normal and tangential components of v on Ŵ,

givenbyv ν = v • v, v τ = v -v ν v.
Let Ŵ 3 be a measurable part of Ŵ. Then, by the Sobolev trace theorem, there exists a positive constant c tr which depends only on , Ŵ 1 and Ŵ 3 such that

v L 2 (Ŵ 3 ) d ≤ c tr v V for all v ∈ V. (4.8)
Inequality (4.8) represents a consequence of the Sobolev trace theorem. We also consider the space

S ={w = v |Ŵ 3 : v ∈ V },
where v |Ŵ 3 denotes the restriction of the trace of the element v ∈ V to Ŵ 3 . Thus, S ⊂

H 1/2 (Ŵ 3 ; R d ) where H 1/2 (Ŵ 3 ; R d
) is the space of the restrictions on Ŵ 3 of traces on Ŵ of functions of H 1 ( ) d . It is known that S can be organized as a Hilbert space, in a canonical way, see for instance [START_REF] Adams | Sobolev Spaces[END_REF][START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF][START_REF] Lions | Problèmes aux limites non homogènes[END_REF]. The dual of the space S will be denoted by D and the duality paring between D and S will be denoted by •, • Ŵ 3 . Nevertheless, for simplicity, we write

µ, v Ŵ 3 instead of µ, v |Ŵ 3 Ŵ 3 ,whenµ ∈ D and v ∈ V .
For a regular function σ ∈ Q we use the notation σ ν and σ τ for the normal and the tangential traces, i.e. σ ν = (σ v) • v and σ τ = σ vσ ν v. Moreover, we recall that the divergence operator is defined by the equality Div σ = (σ ij, j ) and, in addition, the following Green's formula holds:

σ • ε(v) dx + Div σ • v dx = Ŵ σ v • v da ∀ v ∈ V. ( 4.9)
Finally, we denote by Q ∞ the space of fourth order tensor fields given by

Q ∞ = E = (E ijkl ) : E ijkl = E jikl = E kli j ∈ L ∞ ( ), 1 ≤ i, j, k, l ≤ d ,
and we recall that Q ∞ is a real Banach space with the norm

E Q ∞ = max 1≤i, j,k,l≤d E ijkl L ∞ ( ) .
Moreover, a simple calculation shows that

Eτ Q ≤ d E Q ∞ τ Q ∀ E ∈ Q ∞ , τ ∈ Q. (4.10)
In the study of the mechanical problem (4.1)-(4.7) we assume that the elasticity tensor E, the nonlinear constitutive function G, the normal compliance function and the stiffness function k satisfy the following conditions.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (a) E = (E ijkl ) : × S d → S d . (b) E ijkl = E kli j = E jikl ∈ L ∞ ( ), 1 ≤ i, j, k, l ≤ d. (c) There exists m E > 0 such that Eτ • τ ≥ m E τ 2 ∀ τ ∈ S d , a.e. in . (4.11) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (a) G : × S d × S d → S d . (b) There exists L G > 0 such that G(x, σ 1 , ε 1 ) -G(x, σ 2 , ε 2 ) ≤ L G ( σ 1 -σ 2 + ε 1 -ε 2 ) ∀ σ 1 , σ 2 , ε 1 , ε 2 ∈ S d , a.e. x ∈ . (c) The mapping x → G(x, σ , ε) is measurable in , for any σ , ε ∈ S d . (d)
The mapping x → G(x, 0, 0) belongs to Q. We also assume that the densities of the body forces and surface tractions have the regularity

f 0 ∈ C R + ; L 2 ( ) d , f 2 ∈ C R + ; L 2 (Ŵ 2 ) d , (4.15)
the penetration bound satisfies

g ∈ C(R + ; R + ), (4.16)
and the initial data are such that

u 0 ∈ V, σ 0 ∈ Q. (4.17)
Finally, we assume that there exists θ ∈ V such that θ ν = 1a .e. on Ŵ 3 (4.18) where, recall,

θ ν = θ • v.
Next, we define the sets K ⊂ V and ⊂ D, the bilinear form b : V × D → R and the function f : R + → V by equalities

K ={v ∈ V : v ν ≤ 0 a.e. on Ŵ 3 }, (4.19) ={µ ∈ D : µ, v Ŵ 3 ≤ 0 ∀ v ∈ K }, (4.20) b(v, µ) = µ, v Ŵ 3 ∀ v ∈ V, µ ∈ D, (4.21) (f (t), v) V = f 0 (t) • v dx + Ŵ 2 f 2 (t) • v da ∀ v ∈ V, t ∈ R + . (4.22)
Assume now that u and σ are regular functions which verify (4.1)-(4.7), t ∈ R + , v ∈ V and µ ∈ . Then, we integrate (4.1) with the initial condition (4.7)tofindthat

σ (t) = E ε(u(t)) + t 0 G(σ (s), ε(u(s))) ds + σ 0 -Eε(u 0 ). (4.23)
Next, using Green's formula (4.9) and the equation of equilibrium (4.2)wehave

(σ (t), ε(v)) Q = (f 0 (t), v) L 2 ( ) d + Ŵ σ (t)v • v da. (4.24)
Then, since v = 0 on Ŵ 1 ,using(4.4), (4.6)and(4.22) we obtain that

(σ (t), ε(v)) Q = (f (t), v) V + Ŵ 3 σ ν (t)v ν da. (4.25)
Let λ(t) ∈ D be the Lagrange multiplier defined by

λ(t), w Ŵ 3 =- Ŵ 3 (σ ν (t) + k(ζ u(t)) p(u ν (t))w ν da ∀ w ∈ S. (4.26)
Then, taking into account (4.21) we can write

Ŵ 3 σ ν (t)v ν da =-b(v, λ(t)) - Ŵ 3 k(ζ u(t)) p(u ν (t))v ν da ∀ v ∈ V (4.27)
and, combining this equality with (4.25) we obtain that Problem 4 Find a displacement field u:R + → V, a stress field σ :R + → Q and a Lagrange multiplier λ:R + → such that

(σ (t), ε(v)) Q + b(v, λ(t)) + Ŵ 3 k(ζ u(t)) p(u ν (t))v ν da = (f (t), v) V . ( 4
(u(t), µ -λ(t)) = b(u(t) -g(t) θ , µ -λ(t)) + b(g(t) θ , µ -λ(t)) = µ -λ(t), u(t) -g(t) θ Ŵ 3 + b(g(t) θ , µ -λ(t)) and, therefore, b(u(t), µ -λ(t)) = µ, u(t) -g(t) θ Ŵ 3 -λ(t), u(t) -g(t) θ Ŵ 3 + b(g(t) θ , µ -λ(t)).
u(t) -g(t) θ ∈ K , λ(t), u Ŵ 3 = λ(t), g(t) θ Ŵ 3 , which show that µ, u(t) -g(t) θ Ŵ 3 ≤ 0, λ(t), u -g(t) θ Ŵ 3 = 0. ( 4
σ (t) = E ε(u(t)) + t 0 G(σ (s), ε(u(s))) ds + σ 0 -Eε(u 0 ), (4.32) (σ (t), ε(v)) Q + b(v, λ(t)) + Ŵ 3 k(ζ u(t)) p(u ν (t))v ν da = (f (t), v) V ∀ v ∈ V, (4.33) b(u(t), µ -λ(t)) ≤ b(g(t) θ , µ -λ(t)) ∀ µ ∈ , (4.34)
for all t ∈ R + .

Note that Problem 4 represents a mixed variational formulation which couples a nonlinear implicit integral equation for the stress field, a history-dependent variational equation for the displacement field, and a first-order time-dependent variational inequality for the Lagrange multiplier. This formulation is quite different to that in Problem 1. Nevertheless, we shall see in the next section that we can associate to Problem 4 a mixed variational formulation of the form (1.1)-(1.2) and, therefore, the analysis of Problem (4.32)-(4.34) can be carried out by using the abstract results we obtained in the previous two sections of this paper.

Weak solvability

In the study of Problem 4 we have the following existence and uniqueness result.

Theorem 5.1 Assume (4.11)-(4.18).

There exists e 0 > 0 which depends only on E, , Ŵ 1 and Ŵ 3 such that, if k 0 L p < e 0 , then Problem 4 has a unique solution (u, σ , λ). Moreover, the solution satisfies

u ∈ C(R + ; X ), σ ∈ C(R + ; Q), λ ∈ C(R + ; ).
(

The proof of Theorem 5.1 will be carried out in several steps. To this end, we assume in what follows that (4.11)-(4.18) hold. The first step is given by the following result.

Lemma 5.2 For each function u

∈ C(R + ; V ) there exists a unique function σ I (u) ∈ C(R + ; Q) such that σ I (u)(t) = t 0 G(σ I (u)(s) + Eε(u(s)), ε(u(s))) ds + σ 0 -Eε(u 0 ) ∀ t ∈ R + .
(5.2)

Moreover, the operator σ I : C(R + ; V ) → C(R + ; Q) satisfies the following property: for every n ∈ N there exists r n > 0 such that

σ I (u 1 )(t) -σ I (u 2 )(t) Q ≤ r n t 0 u 1 (s) -u 2 (s) V ds (5.3) ∀ u 1 , u 2 ∈ C(R + ; V ), ∀ t ∈[0, n]. Proof Let u ∈ C(R + ; V ) and consider the operator L : C(R + ; Q) → C(R + ; Q) defined as follows Lτ (t) = t 0 G(τ (s) + Eε(u(s)), ε(u(s)))ds + σ 0 -Eε(u 0 ) (5.4) ∀ τ ∈ C(R + ; Q), t ∈ R + .
The operator L depends on u but, for simplicity, we do not indicate explicitly this dependence. Let τ 1 , τ 2 ∈ C(R + ; Q) and let t ∈ R + . Then, using (5.4)and(4.12)wehave

Lτ 1 (t) -Lτ 2 (t) Q ≤ t 0 G(τ 1 (s) + Eε(u(s)), ε(u(s))) -G(τ 2 (s) + Eε(u(s)), ε(u(s))) Q ds ≤ L G t 0 τ 1 (s) -τ 2 (s) Q ds.
Next, we use Theorem 1.1 to see that L has a unique fixed point in C(R + ; Q), denoted σ I (u).

And, finally, we combine (5.4) with equality Lσ I (u) = σ I (u) to see that (5.2) holds.

To proceed, let u 1 , u 2 ∈ C(R + ; V ), n ∈ N and let t ∈[0, n]. Then, using (5.2) and taking into account (4.10)-(4.12) we write

σ I (u 1 )(t) -σ I (u 2 )(t) Q ≤ L G t 0 d E Q ∞ u 1 (s) -u 2 (s) V ds + t 0 σ I (u 1 )(s) -σ I (u 2 )(s) Q ds = ω t 0 u 1 (s) -u 2 (s) V ds + t 0 σ I (u 1 )(s) -σ I (u 2 )(s) Q ds , where ω = L G (d E Q ∞ + 1)
. Using now a Gronwall argument we deduce that

σ I (u 1 )(t) -σ I (u 2 )(t) Q ≤ ω e n ω t 0 u 1 (s) -u 2 (s) V ds.
This inequality shows that (5.3) holds with r n = ω e n ω .

⊓ ⊔

We now use the Riesz's representation theorem and Lemma 5.2 to define the operators

A : V → V and R : C(R + ; V ) → C(R + ; V ) by equalities (Av, w) V = (Eε(v), ε(w)) Q ∀ v, w ∈ V, ( 5.5)
(Ru(t), v) V = (σ I (u)(t), ε(v)) Q (5.6) + Ŵ 3 k(ζ u(t)) p(u ν (t))v ν da ∀ v ∈ V, t ∈ R + .
Then, we have the following equivalence result.

Lemma 5.3 Let (u, σ , λ) be a triple of functions with regularity (5.1).Then(u, σ , λ) is a solution of Problem 4 if and only if

σ (t) = Eε(u(t)) + σ I (u)(t), (5.7)
(Au(t), v) V + (Ru(t), v) V + b(v, λ(t)) = (f(t), v) V ∀ v ∈ V, (5.8) b(u(t), µ -λ(t)) ≤ b(g(t) θ , µ -λ(t)) ∀ µ ∈ , (5.9)
for all t ∈ R + .

Using now estimate (5.3) in Lemma 5.2 and the definition of the function ζ we can write

|(Ru 1 (t) -Ru 2 (t), v) V |≤ r n t 0 u 1 (s) -u 2 (s) ds v V + c 2 tr k 0 L p u 1 -u 2 V + p 0 L k t 0 u 1 (s) -u 2 (s) V ds v V
and, therefore,

Ru 1 (t) -Ru 2 (t) V ≤ c 2 tr k 0 L p u 1 -u 2 V
(5.11)

+ ( r n + c 2 tr p 0 L k ) t 0 u 1 (s) -u 2 (s) V ds.
Inequality (5.11) combined with definition (5.10) shows that the operator S satisfies condition (2.2) with d n = c 2 tr k 0 L p and r n = r n + c 2 tr p 0 L k . We are now in position to apply Theorem 2.1. According to this theorem there exists d 0 > 0 which depends only on A and b such that if d n < d 0 for all positive integers n,then there exists a unique couple of functions (u, λ) which satisfies (5.8)-(5.9)f o ra l lt ∈ R + . We now take e 0 = d 0 c -2 tr (5.12) which, clearly, depends only on E, , Ŵ 1 and Ŵ 3 . We note that d n < d 0 iff k 0 L p ≤ e 0 which concludes the proof.

⊓ ⊔

We now have all the ingredients to prove Theorem 5.1.

Proof Let e 0 be defined by (5.12) and assume that k 0 L p < e 0 . Under this condition, Lemma 5.4 implies that there exists a unique couple of functions (u, λ), such that (5.8)-(5.9) hold, for all t ∈ R + .Defineσ = Eε(u) + σ I (u) and note that, obviously, σ ∈ C(R + ; Q).Then the triple (u, σ , λ) represents a solution to problem (5.7)-(5.9) with regularity (5.1). The existence part of the theorem follows now from Lemma 5.3. The uniqueness part follows from Lemma 5.3 combined with the uniqueness of the solution of the system (5.8)-(5.9) guaranteed by Lemma 5.4.

⊓ ⊔

We end this section with the remark that the inequality k 0 L p < e 0 , which guarantees uniqueness solvability of Problem 4, represents a smallness condition on the normal compliance function p and the stiffness function k. It is satisfied if, for instance, either the Lipschitz constant L p or the bound k 0 is small enough.

Continuous dependence with respect to the data

In this section we study the behavior of the solution of Problem 4 with respect to a perturbation of the data. To this end we assume in what follows that (4.11)-(4.18) hold and k 0 L p < e 0 , where e 0 is defined in Theorem 5.1. Also, we denote by (u, σ , λ) the solution of Problem 4 obtained in Theorem 5.1. In addition, for each ρ>0 we denote by G ρ , p ρ , k ρ , f 0ρ , f 2ρ , g ρ , u 0ρ , σ 0ρ a perturbation of G, p, k, f 0 , f 2 , g, u 0 and σ 0 , respectively, which satisfies the following conditions.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (a) G ρ : × S d × S d → S d . (b) There exists L ρ G > 0 such that G ρ (x, σ 1 , ε 1 ) -G ρ (x, σ 2 , ε 2 ) ≤ L ρ G ( σ 1 -σ 2 + ε 1 -ε 2 ) ∀ σ 1 , σ 2 , ε 1 , ε 2 ∈ S d , a.e. x ∈ . (c) The mapping x → G ρ (x, σ , ε) is measurable in , for any σ , ε ∈ S d . (d)
The mapping x → G ρ (x, 0, 0) belongs to Q.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (a) k ρ : R + → R + . (b) There exists L ρ k > 0 such that |k ρ (r 1 ) -k ρ (r 2 )|≤L ρ k |r 1 -r 2 |∀ r 1 , r 2 ∈ R. (c) There exists k ρ 0 > 0 such that |k ρ (r )|≤k ρ 0 ∀ r ∈ R. (6.3) f 0ρ ∈ C(R + ; L 2 ( ) d ), f 2ρ ∈ C(R + ; L 2 (Ŵ 2 ) d ), (6.4
)

g ρ ∈ C(R + ; R + ) (6.5) u 0ρ ∈ V, σ 0ρ ∈ Q. ( 6.6)
With these data we define the function f ρ : R + → V by equality

(f ρ (t), v) V = f 0ρ (t) • v dx + Ŵ 2 f 2ρ (t) • v da ∀ v ∈ V, t ∈ R + (6.7)
and we consider the following problem.

Problem 5 Find a displacement field u ρ : R + → V, a stress field σ ρ : R + → Q and a Lagrange multiplier λ ρ : R + → such that

σ ρ (t) = E ε(u ρ (t)) + t 0 G ρ (σ ρ (s), ε(u ρ (s))) ds + σ 0ρ -Eε(u 0ρ ), (6.8)
(σ ρ (t), ε(v)) Q + b(v, λ ρ (t)) + Ŵ 3 k ρ (ζ u ρ (t)) p ρ (u ρν (t))v ν da (6.9) = (f ρ (t), v) V ∀ v ∈ V, b(u ρ (t), µ -λ ρ (t)) ≤ b(g ρ (t) θ , µ -λ ρ (t)) ∀ µ ∈ , ( 6
u ρ ∈ C(R + ; V ), σ ρ ∈ C(R + ; Q), λ ρ ∈ C(R + ; ). (6.11)
Our interest in what follows lies in the behavior of the solution as ρ tends to zero. To this end we consider the following additional assumptions. (6.12)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ There exist L 0 > 0, p 0 > 0, k 0 > 0, e 0 > 0andK 0 > 0 such that (
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ There exist M, N , P : R + → R + such that (a) G ρ (x, σ , ε) -G(x, σ , ε) ≤M(ρ)( σ + ε +1) for all σ , ε ∈ S d , a.e. x ∈ . (b) | p ρ (r ) -p(r )|≤N (ρ)(|r |+1) for all r ∈ R. (c) |k ρ (r ) -k(r )|≤P(ρ)(|r |+1) for all r ∈ R. (d) lim ρ→0 M(ρ) = 0, lim ρ→0 N (ρ) = 0, lim ρ→0 P(ρ) = 0. (6.13) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (a) f 0ρ → f 0 in C(R + ; L 2 ( ) d ) as ρ → 0. (b) f 2ρ → f 2 in C(R + ; L 2 (Ŵ 2 ) d ) as ρ → 0. (c) g ρ → g in C(R + ; R) as ρ → 0. (d) u 0ρ → u 0 in V as ρ → 0. (e) σ 0ρ → σ 0 in Q as ρ → 0. (6.14)
Our main result in this section is the following. Proof We use the operators A, σ I , R and S defined by (5.5), (5.2), (5.6)and(5.10), respectively. Moreover, for each ρ>0 we define the operators σ I ρ :

C(R + , V ) → C(R + , Q), R ρ : C(R + ; V ) → C(R + ; V ) and S ρ : C(R + ; V ) × C(R + ; D) → C(R + ; V ) by equalities σ I ρ (u)(t) = t 0 G ρ (σ I ρ (u)(s) + Eε(u(s)), ε(u(s))) ds + σ 0ρ -Eε(u 0ρ ), (6.16)
(R ρ u(t), v) V = (σ I ρ (u)(t), ε(v)) Q + Ŵ 3 k ρ (ζ u(t)) p ρ (u ν (t))v ν da, (6.17) S ρ (u, λ) = R ρ (u) (6.18) for all u ∈ C(R + ; V ), t ∈ R + , v ∈ V and λ ∈ C(R + ; D).
Then, Lemma 5.3 states that (5.7)-(5.9) hold for all t ∈ R + and, moreover,

σ ρ (t) = Eε(u ρ (t)) + σ I ρ (u ρ )(t), (6.19)
(Au ρ (t), v) V + (R ρ u ρ (t), v) V + b(v, λ ρ (t)) = (f ρ (t), v) V ∀ v ∈ V, (6.20) b(u ρ (t), µ -λ ρ (t)) ≤ b(g ρ (t) θ , µ -λ ρ (t)) ∀ µ ∈ , (6.21)
for all t ∈ R + . We note that the system (5.8)-(5.9) is of the form (1.

N, t ∈[ 0, n], u ∈ C(R + ; V ), λ ∈ C(R + ; D) and v ∈ V .W e write |(R ρ u(t) -Ru(t), v) V |≤ σ I ρ (u)(t) -σ I (u)(t) Q v V + Ŵ 3 k ρ (ζ u(t)) p ρ (u ν (t)) -k(ζ u(t)) p(u ν (t)) v ν da. ( 6.22)
We now use (6.16), (5.2) to see that

σ I ρ (u)(t) -σ I (u)(t) Q ≤ t 0 G ρ (σ I ρ (u)(s) + Eε(u(s)), ε(u(s))) -G(σ I (u)(s) + Eε(u(s)), ε(u(s))) Q ds + σ 0ρ -σ 0 Q + Eε(u 0ρ ) -Eε(u 0 ) Q
and, therefore, (4.10) yields

σ I ρ (u)(t) -σ I (u)(t) Q ≤ t 0 G ρ (σ I ρ (u)(s) + Eε(u(s)), ε(u(s))) -G ρ (σ I (u)(s) + Eε(u(s)), ε(u(s))) Q ds + t 0
G ρ (σ I (u)(s) + Eε(u(s)), ε(u(s))) -G(σ I (u)(s) + Eε(u(s)), ε(u(s))) Q ds

+ σ 0ρ -σ 0 Q + d E Q ∞ u 0ρ -u 0 V .
Using now (6.1), (6.12)(a), (6.13)(a) and (4.10) we obtain

σ I ρ (u)(t) -σ I (u)(t) Q ≤ L 0 t 0 σ I ρ (u)(s) -σ I (u)(s) Q ds + cM(ρ) t 0 ( σ I (u)(s) Q + ε(u(s)) Q + 1) ds + c σ 0ρ -σ 0 Q + u 0ρ -u 0 V
where, here and below, c is a positive constant which does not depend on ρ and n,a n d whose value will change from line to line. Applying now a Gronwall argument and using the inequality t ≤ n we have

σ I ρ (u)(t) -σ I (u)(t) Q ≤ c σ 0ρ -σ 0 Q + u 0ρ -u 0 V + M(ρ) n 0 ( σ I (u)(s) Q + u(s) V + 1
) ds e L 0 n . (6.23)

We turn now to the second term into the right side of the inequality (6.22). We write Then, using (6.13)(b),(c) and the trace inequality (4.8), after some algebra we obtain that Then, it follows from the previous inequality that

R ρ u(t) -Ru(t) V ≤ c σ 0ρ -σ 0 Q + u 0ρ -u 0 V + M(ρ) + N (ρ) + P(ρ) J n (u).
Therefore, denoting

H (ρ) = c σ 0ρ -σ 0 Q + u 0ρ -u 0 V + M(ρ) + N (ρ) + P(ρ) , (6.25)
we deduce that R ρ u(t) -Ru(t) V ≤ H (ρ) J n (u). (

We now combine equalities (6.18), (5.10) and inequality (6.26) to conclude that condition (3.5)(a) holds with H n (ρ) = H (ρ) and J n (u, λ) = J n (u).

Also, the proofs of Lemmas 5. Therefore, using assumption (6.12)(a),(b) and (e) it follows that r ρn ≤ L 0 (d E Q ∞ + 1)e nL 0 (d E Q∞ +1) + c 2 tr p 0 K 0 . We conclude from above that condition (3.5)(b) holds with R n = L 0 (d E Q ∞ + 1)e nL 0 (d E Q∞ +1) + c 2 tr p 0 K 0 . In addition, using assumptions (6.13)(d), (6.14)(d), (e), we deduce that the function H n (ρ) = H (ρ) defined by (6.25) is such that lim ρ→0 H n (ρ) = 0. Therefore, condition (3.5)(c) is satisfied, too.

Moreover, using assumption (6.12)(d) and equality (5.12) we deduce that

Theorem 3 . 1

 31 Assume (3.5)-(3.7). Then the solution (u ρ ,λ ρ ) of Problem 2 converges to the solution (u,λ)of Problem 1 i.e.

  p : R → R + . (b) There exists L p > 0 such that | p(r 1 )p(r 2 )|≤L p |r 1r 2 |∀ r 1 , r 2 ∈ R. (c) There exists p 0 > 0 such that | p(r )|≤ p 0 ∀ r ∈ R.

  k : R + → R + . (b) There exists L k > 0 such that |k(r 1 )k(r 2 )|≤L k |r 1r 2 |∀ r 1 , r 2 ∈ R. (c) There exists k 0 > 0 such that |k(r )|≤k 0 ∀ r ∈ R.

( 4 .

 4 29)In addition, (4.5)and(4.18) imply that

  p ρ : R → R + . (b) There exists L ρ p > 0 such that | p ρ (r 1 )p ρ (r 2 )|≤L ρ p |r 1r 2 |∀ r 1 , r 2 ∈ R. (c) There exists p ρ 0 > 0 such that | p ρ (r )|≤ p ρ 0 ∀ r ∈ R.

  a) L ρ G < L 0 for all ρ>0. (b) p ρ 0 ≤ p 0 for all ρ>0. (c) k ρ 0 ≤ k 0 for all ρ>0. (d) k ρ 0 L ρ p ≤ e 0 < e 0 for all ρ>0. (e) L ρ k ≤ K 0 for all ρ>0.

Theorem 6 . 1

 61 Assume (6.12)-(6.14). Then the solution (u ρ , σ ρ , λ ρ ) of Problem 5 converges to the solution(u, σ , λ) of Problem 4,i.e. ⎧ ⎨ ⎩ (a) u ρ → u in C(R + ; V ) as ρ → 0. (b) σ ρ → σ in C(R + ; Q) as ρ → 0. (c) λ ρ → λ in C(R + ; D) as ρ → 0.(6.15)

Ŵ 3 k 0 ( 0 (

 300 ρ (ζ u(t)) p ρ (u ν (t))k(ζ u(t)) p(u ν (t)) v ν da (6.24) ≤ c N (ρ) + P(ρ) + N (ρ) max r ∈[0,n] u(r ) V + P(ρ) n 0 u(s) V ds v V .We now combine the inequalities (6.22)-(6.24) to deduce that|(R ρ u(t) -Ru(t), v) V |≤c σ 0ρσ 0 Q + u 0ρ -u 0 V + M(ρ) n σ I (u)(s) Q + u(s) V + 1) ds e L 0 n v V + c N (ρ) + P(ρ) + N (ρ) max r ∈[0,n] u(r ) V + P(ρ) n 0 u(s) V ds v V .Therefore, since v is an arbitrary element in V ,wefindthatR ρ u(t) -Ru(t) V ≤ c σ 0ρσ 0 Q + u 0ρ -u 0 V + N (ρ) + P(ρ) + cM(ρ) n 0 ( σ I (u)(s) Q + u(s) V + 1) ds e L 0 n + c N (ρ) max r ∈[0,n] u(r ) V + P(ρ) n 0 u(s) V ds .DenoteJ n (u) = 1 + e L 0 n n σ I (u)(s) Q + u(s) V + 1) ds

G

  4 and 5.2 show that the operator S ρ satisfies condition (3(d E Q ∞ + 1)e nL ρ G (d E Q∞ +1).

c 2 tr k ρ 0 L ρ p ≤ c 2

 2 tr e 0 < d 0 ∀ n ∈ N, ∀ ρ>0 and, using the first equality in (6.27) we obtain thatd ρn ≤ c 2 tr e 0 < d 0 ∀ n ∈ N, ∀ρ>0. We conclude from here that (3.6) holds with d 0 = c 2 tr e 0 .

  Theorem 2.1 Assume (2.1)-(2.5). There exists d 0 > 0 which depends only on A and b such that, if d n < d 0 for all positive integers n, then Problem 1 has a unique solution (u,λ).

Moreover, the solution satisfies u ∈ C(R + ; X ) and λ ∈ C(R + ; ).

  .28) On the other hand, using (4.5), (4.19)a n d( 4.20) we deduce that λ(t) ∈ . Moreover, using assumption (4.18) and the definition (4.21) of the bilinear form b it it is easy to see that b

  .10) for all t ∈ R + .Note that, here and below, u ρν (t) represents the normal component of the function u ρ (t), i.e. u ρν (t) = u

ρ (t) • v,forallt ∈ R + . Under the assumptions above, if k ρ 0 L ρ p < e 0 , Theorem 5.1 guarantees the existence of a unique solution (u ρ , σ ρ , λ ρ ) to Problem 5 such that

  , in order to apply Theorem 3.1, we check in what follows the validity of the conditions (3.5)-(3.7).To start, we fix ρ>0, n ∈

	Therefore

1)-(1.2) with S given by (5.10) while the system (6.20)-(6.21) is of the form (3.1)-(3.2) with S ρ givenby(6.18).

Acknowledgments

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project Number PN-II-ID-PCE-2011-3-0257.

Proof Assume that (u, σ , λ) is a solution of Problem 4 and let t ∈ R + . Then, using (4.32) we have σ (t) -Eε(u(t)) = t 0 G(σ (s) -Eε(u(s)) + Eε(u(s)), ε(u(s))) ds + σ 0 -Eε(u 0 ) and, using the definition (5.2) of the operator σ I ,weobtain (5.7). Equality (5.8) follows from (4.33) combined with (5.7) and the definition of the operators A and R and, finally, (5.9) coincides with (4.34).

Conversely, assume that (u, σ , λ) satisfies (5.7)-(5.9). Then by (5.7) and the definition (5.2)ofσ I (u) we obtain (4.32). Moreover, using (5.7), (5.8) and the definition of the operators A and S we obtain (4.33), which concludes the proof.

⊓ ⊔

We now proceed with the following existence and uniqueness result.

Lemma 5. [START_REF] Barboteu | On the behavior of the solution of a viscoplastic contact problem[END_REF] There exists e 0 > 0 which depends only on E, , Ŵ 1 and Ŵ 3 such that, if k 0 L p < e 0 , then there exists a unique couple of functions (u, λ), which satisfies (5.8)-(5.9)

To this end, we use assumption (4.11) to see that the operator A defined by (5.5) verifies condition (2.1). Moreover, the bilinear form b(•, •) is continuous and satisfies the "inf-sup" condition, i.e. there exists α>0 which depends only on , Ŵ 1 and

see [START_REF] Matei | Contact problems for nonlinearly elastic materials: weak solvability involving dual Lagrange multipliers[END_REF], for instance. We conclude from here that condition (2.

Then, by a standard calculus based on the trace inequality (4.8) and the properties of the functions p and k we deduce that

Finally, we note that assumption (6.14)(a)-(c) imply that

as ρ → 0 and, therefore, (3.7) holds. The convergence (6.15)(a) and (c) represent now a direct consequence of Theorem 3.1, applied for the systems (6.20)-(6.21)and(5.8)-(5.9).

Next, to provide the convergence (6.15)(b) we use equalities (6.16), (5.2), assumptions (6.1)(b), (6.12)(a) and arguments similar to those used in the proof of (6.23)tofindthat

) ds e L 0 n .

Then, using (6.19), (5.7)and(4.10) we deduce that

We now combine inequality (6.28) with assumptions (6.14)(d), (e), (6.13)(d) and the convergence (6.15)(a). As a result we obtain that max

Finally, we use (6.29)a n d( 1.3) to deduce that (6.15)(b) holds, which concludes the proof.

⊓ ⊔

In addition to the mathematical interest in the convergence result (6.15) it is of importance from mechanical point of view, since it states that the weak solution of the problem (4.1)-(4.7) depends continuously on the viscoplastic function, the normal compliance function, the stiffness coefficient, the penetration bound, the densities of body forces and surface tractions, and the initial data, as well.