Patrick Maillé
email: patrick.maille@imt.fr

Bruno Tuffin
email: bruno.tuffin@inria.fr

Joshua Peignier

Estelle Varloot

Pricing of Coexisting Cellular and Community Networks

Community networks have emerged as an alternative to licensed-band systems (WiMAX, 4G, etc.), providing an access to the Internet with Wi-Fi technology while covering large areas. A community network is easy and cheap to deploy, as the network is using members' access points in order to cover the area. We study the competition between a community operator and a traditional operator (using a licensed-band system) through a game-theoretic model, while considering the mobility of each user in the area.

• If Φ D (0) ≥ 0 when D = {0} ∪ D 0

I. INTRODUCTION

Wireless technologies are becoming ubiquitous in Internet usage. Operators try to provide a whole wireless coverage on urban areas, in order to offer an Internet access to everyone, with a guaranteed quality. However, this system requires huge investment costs in terms of infrastructure and spectrum licenses. This has repercussions on the subscription fees, which can be large enough for users to prefer other options. Because of this, community networks have been imagined as an alternative. The principle is simple: when a user subscribes to a community network, he sets an access point where he lives (and is responsible for its maintenance), which can be used by all members of the community network. As a counterpart, he gains access to the Internet through every access point belonging to the community network. This approach presents the advantage that the infrastructure is cheaper and easier to maintain, from a provider perspective. However, the quality of service cannot be guaranteed, since it depends on the size of the community. Currently, the largest community operator is FON 1 .

From the user point of view, a community network has the particularity of having both positive and negative externalities, i.e., having more subscribers is both beneficial (larger coverage when roaming) and a nuisance (more traffic to serve from one's access point). An analysis of those effects and on the impact of prices, with users being heterogeneous in terms of their propension to roam, is carried out in [START_REF] Afrasiabi | Exploring user-provided connectivity[END_REF]. In the present paper, we add another dimension, that is, how-i.e., whereusers roam. Also, we consider that users can choose between two competing providers, a "classical" one and one operating a community network, that compete over prices.

Community networks have already been studied under a game-theoretic framework, with operators as players. In [START_REF] Manshaei | On wireless social community networks[END_REF], the authors first study how a community network evolves, depending on its initial price and coverage, and then investigate using a game-theoretic framework [START_REF] Osborne | A course in game theory[END_REF] the repartition of users having the choice between a community network and an operator on a licensed band. The competition is first studied when each player decides its price once and the size of the community network changes over time. Then a discrete-time dynamic model is studied, where operators can change their price at each time step, taking into account the preferences of the users concerning price and coverage. The authors show the existence of one or several Nash equilibria under specific conditions. An extension in [START_REF] Manshaei | Evolution and market share of wireless community networks[END_REF] investigates whether it is profitable for a licensed-band operator to complement the service it provides with a community network service. It is shown that this is generally not the case, as users will more likely choose the (less profitable) community network. In [START_REF] Mazloumian | Optimal pricing strategy for wireless social community networks[END_REF], the same authors study an optimal pricing strategy for a community network operator alone in both static and semidynamic models, while considering a mobility factor for each user (e.g., each user makes requests, but not all in the same spot). They also allow the operator to set different prices for each user. In the following, we will refer to the traditional operator as the classical ISP (Internet Service Provider).

In this article, we study a model similar to both [START_REF] Manshaei | On wireless social community networks[END_REF] and [START_REF] Mazloumian | Optimal pricing strategy for wireless social community networks[END_REF]. In [START_REF] Manshaei | On wireless social community networks[END_REF] the users all present the same characteristics while in [START_REF] Mazloumian | Optimal pricing strategy for wireless social community networks[END_REF] there is a mobility factor but the paper considers a community network alone. We consider here a more general and realistic framework: users are considered located in places heterogeneous in terms of attractiveness for connections (an urban area is more likely to see connections than the countryside). Moreover, their mobility behavior is also heterogeneous: they do not all plan to access the Internet from the same places. Instead of a mobility parameter, we rather consider a density function, which represents the probability that a user makes a request while being near the access point of another user. But in our paper, all users will have the same sensitivity toward quality; indeed, our goal is rather to focus on the impact of geographical locations of spots and connections on users' subscription and on the competition between the operator and the community network. The model is analyzed using noncooperative game theory [START_REF] Osborne | A course in game theory[END_REF], [START_REF] Maillé | Telecommunication Network Economics: From Theory to Applications[END_REF]. The decisions are taken at different time scales: first the networks fix their price, and then users choose which network to subscribe to. We illustrate on different scenarios that for fixed subscription prices to the ISP and the community network, several equilibria on the repartition of users can exist; the one we can expect depends on the initial mass of the subscribers to the community network. The pricing competition between operators is played anticipating the choice of users.

The paper is organized as follows. Section II presents the model; the basic notions are taken from the literature, but we extend it with the modeling of mobility via a continuous distribution. In Section III, we describe how, for fixed subscription prices, the repartition of users is determined. In Section IV, we introduce the pricing game between the operator and the community network, as well as our method to compute a Nash equilibrium. Section V presents two scenarios as examples of application of our method.

II. MODEL DEFINITION

We present here the basic elements of the model taken from the literature, mainly [START_REF] Manshaei | On wireless social community networks[END_REF], which we complement with more heterogeneity among users related to location and mobility, as well as the possible nuisance from providing service to other community members.

A. Actors and strategies

To study the competition between a community operator and a classical ISP, we need to define a model for the profit of each operator, but also for users, in order to explain what will make them choose a service rather than the other one. The decisions of these actors are taken at different time scales, defining a multilevel game:

1) First, the classical ISP and the community network play non-cooperatively on the subscription prices, in order to maximize their own revenue (expressed as the product of price and mass of users). 2) Given the prices and qualities of service, users choose their network based on price (we assume a flat-rate pricing is applied by each operator) and quality of service. We will describe how, depending on an initial repartition between operators, users can switch operators up to a situation when nobody has an interest to move. The results of the game for the operators are given once all users have settled on an operator (if any). Even though the operators play first (subscription is impossible until the price of subscription is set), they are assumed to make their decision strategically, anticipating the subsequent decisions of users. Hence the game is analyzed by backward induction [START_REF] Osborne | A course in game theory[END_REF]: we determine the user choices for any fixed prices, and consider that operators are able to compute those choices when selecting their prices.

B. Modeling of users, quality, and mobility

We consider a continuum of users characterized by their type u; this type typically represents a home location. In the following, we will not distinguish between a location u and the user u living there. Let Ω be the space of users and f their density over space Ω (with Ω f (v) dv = 1). We also assume that Ω is the support of f , i.e., f (u) > 0 for all u ∈ Ω.

Let D be the subset of Ω of users subscribing to the community network; we call it the domain, since it also represents the domain of coverage of the community network.

Each user makes requests when using the Internet. Let m(v) be the average number of communication requests of user v per time unit.

Depending on their location, users may also present different mobility patterns. To express this heterogeneity, define for each user u, the density function g(v|u) that a request from u occurs at v. Note that users may move to uninhabited regions: we aggregate those regions into one item, denoted by ⊥, and we define the set of mobility locations as Ω := Ω ∪ ⊥. Then, over a location area A ⊂ Ω, the probability that a type-u user's request is in A (rather than Ω \ A) is A g(v|u) dv.

If we define n(u) as the density (number per space unit) of requests at u from users of the community network, it can be computed as

n(u) = D g(u|v)m(v)f (v) dv.
The quality of a given service is defined as the probability that a request is fulfilled. For the ISP assumed to have a full coverage, it is therefore 1, in line with the literature. For a user u, the quality of the community network will depend on whether the requests by u are generated when in the coverage domain, hence it can be computed as

q u = D g(v|u)dv. (1)

C. User preferences

How will User u decide whether to subscribe to the classical ISP, to the community network, or to none of them? Following [START_REF] Manshaei | On wireless social community networks[END_REF], [START_REF] Manshaei | Evolution and market share of wireless community networks[END_REF], define U I (u), U C (u) and U ∅ (u) as the respective utility functions for choosing the classical ISP, the community network, or none. These functions depend on the price the user has to pay, the quality of the service he is provided, and his sensitivity toward quality. As in [START_REF] Manshaei | On wireless social community networks[END_REF], [START_REF] Manshaei | Evolution and market share of wireless community networks[END_REF], we consider a simple quasi-linear form for utilities: a user u, whose sensitivity toward quality is denoted by a and who benefits from service quality q (assumed in the interval [0, 1]) at price p perceives a utility a u q -p. Note that in [START_REF] Manshaei | On wireless social community networks[END_REF], [START_REF] Manshaei | Evolution and market share of wireless community networks[END_REF] the sensitivity parameter a depends on the user type, u, but we limit ourselves to a constant value a for all users since our goal is to focus on the geographical heterogeneity of users.

In addition, as in [START_REF] Afrasiabi | Exploring user-provided connectivity[END_REF] we consider a disturbance factor for the community network: satisfying requests for other members can indeed become an annoyance, which we model through a negative term -cn(u) in the utility function, with c a unit cost per request at u. Here we assume that the nuisance is due to WiFi spectrum usage, hence it depends on the total density of requests in u and is independent of the density of users at u.

Let p I and p C be the respective flat-rate subscription fees to the ISP and to the community network, respectively.

We assume users are rational: a type-u user will choose the network providing the largest utility (or no network), where the utilities at the community network, the ISP or for not subscribing to any are respectively (recall those functions depend on the set D of users in the community network)

U C (u) = aq u -p C -cn(u) U I (u) = a -p I U ∅ (u) = 0,
where U ∅ (u) is used to say that users with negative utilities at the operators do not subscribe to any of them; we also assume that users with null utilities at the operators do not subscribe.

In the following, we will therefore always assume that p I < a, because otherwise, the classical operator would get no subscriber (as for all u, we would have U I (u) ≤ 0 and all users prefer the no-subscription option over the classical operator). With the same argument, we assume that p C < a. We now have, for all u, U I (u) > 0, which implies that each user will necessarily subscribe to one operator, since they strictly prefer the classical operator over the no-subscription option.

However, the repartition between the classical ISP and the community network is not trivial, since the utilities expressed above, which determine user choices, also depend on user choices through the set D. Hence the notion of equilibrium (or fixed-point), which we define and analyze in Section III.

D. Operators' model

The utilities for the classical ISP and the community operator are simply defined as their profits. For each operator, the profit depends on the price it chose, and on the number of users subscribing to its service, which depends on both prices.

Let d I and d C be the number (or mass) of users subscribing respectively to the classical ISP and to the community operator. For a set D ⊂ Ω of users subscribing to the community network (which depends on prices as we see later on), those masses can be written as

d I = Ω\D f (v)dv d C = D f (v)dv.
The utilities are then expressed by

V C = d C p C V I = d I p I -χ I ,
where χ I is the infrastructure cost for the ISP. Each operator chooses (plays with) its price to maximize its revenue, but that revenue also depends on the decision of the competing operator which can attract some customers, hence the use of non-cooperative game theory to solve the problem.

III. USER EQUILIBRIUM

With the characterization of user behavior above, we aim in this section at determining if, for fixed subscription prices, there is an equilibrium user repartition among operators, and also if it is unique. We first define what such an equilibrium is. We consider here that prices p I and p C have already been decided.

A. Definition and characterization Definition 1.

A user equilibrium domain is a domain D ⊂ Ω such that no user, in D or in Ω\D has an interest to change his choice of network. Mathematically, this means that

∀u ∈ D U C (u) ≥ U I (u) ∀u ∈ Ω\D U C (u) ≤ U I (u).
Consider a User u. For a given domain D, he will prefer the community network if

U C (u) ≥ U I (u), that is, if a(q u - 1) + (p I -p C ) -cn(u) ≥ 0.
Let us define the domain-dependent function

Φ D : Ω → R as the difference U C (u) -U I (u), that is, ΦD(u) := a D g(v|u)dv -1 +(pI -pC )-c D g(u|v)m(v)f (v) dv. ( 2
)
Then D is a user equilibrium domain if and only if

Φ D (u) ≥ 0 ∀u ∈ D Φ D (u) ≤ 0 ∀u ∈ Ω \ D.
Example 1. Consider the case of users with homogeneous mobility behavior, that is, where g(v|u) does not depend on u so we only denote it by g(v). From (1), we also get that the quality q u does not depend on u: all the community network users experience the same quality, which we denote by q and equals D g(v)dv).

Moreover n(u) = g(u) D m(v)f (v)dv = M g(u) with M := D m(v)f (v)
dv the total request mass from community network users. At a user equilibrium D, User u prefers the community network if and only if

a(q -1) + (p I -p C ) -cM g(u) ≥ 0.
The domain D is then made of all users u with attractiveness g(u) below a threshold.

B. Existence and uniqueness

Proposition 1. A user equilibrium is not unique in general.

Proof. An example of non uniqueness is shown in Section V-A when users present a homogeneous mobility pattern g(v|u) = g(v) ∀u.

Proposition 2. D = ∅, that is no user subscribes to the community network, is a user equilibrium (but not necessarily the only one) if and only if p I ≤ p C + a. In words, if the difference of price between the community network and the ISP is not large enough (it has to be larger than a), no user subscribing to the community network is a user equilibrium, even if not necessarily the unique possibility.

Proof. D = ∅ is a user equilibrium if and only if, when there are no community network users,

U C (u) ≤ U I (u) ∀u ∈ Ω; that is, -a + (p I -p C ) ≤ 0, i.e., p I ≤ p C + a.
We can also consider the other case of "degenerate" equilibrium, that is, when all users in Ω subscribe to the community network.

Proposition 3. D = Ω, is a user equilibrium if and only if Φ Ω (u) ≥ 0 for all u ∈ Ω.

Corollary 1. Under our assumption p I < a, there always exists at least one user equilibrium.

Proof. Since we have assumed p I < a, Proposition 2 holds.

C. A dynamic view

Given that several user equilibria might exist, which one would be observed in practice? This may depend on a dynamic evolution of subscriptions: We can study how users make their choice, and how the repartition evolves, depending on an initial situation.

If a user u is associated to the ISP (resp. community network) but U C (u) > U I (u) (resp. U C (u) < U I (u)) then it will switch to the other operator. Without loss of generality, we can first partition users by assuming that those with the largest U C (u) -U I (u) subscribe to the community network and the others to the ISP (a natural move to that situation will occur otherwise). We can relate this to the function Φ D (x) defined in [START_REF] Manshaei | On wireless social community networks[END_REF]. For a given D, users u ∈ D (resp. u ∈ D) with the largest value Φ D (u) > 0 (resp. lowest value Φ D (u) < 0 will have an incentive to switch operator and join (resp. leave) D. Hence, D will change up to a moment when no user has an interest to move, that, up to reaching a user equilibrium as defined above.

All this will be made more specific and clearer in Section V on the analysis of two scenarios.

Depending on the initial situation (that is, the initial mass of users subscribing to the community network), we may end up in different user equilibria. We can assume that the community network will offer free subscriptions, or make offers to users so that an initial point will allow to lead to different equilibria.

D. Stability

Among user equilibrium domains, some are more likely to be observed. They are the so-called stable user equilibrium domains, which can basically be defined as domains that are stable to small perturbations in the following sense. Definition 2. A user equilibrium domain D is said to be stable if there exists ε > 0 such that ∀D with (D∪D )\(D∩D ) f (v) dv ≤ ε (that is, any D with "measure" close enough to D), then starting from D the user repartition will converge to D.

The following straightforward result establishes that there always exists at least one stable equilibrium. Proposition 4. If for all u ∈ Ω, the ratio of the densities g(•|u)/f (•) is upper-bounded on Ω, then for any price profile (p I , p C ) with p I < a, the situation D = ∅ is a stable equilibrium. Similarly, the other degenerate equilibrium D = Ω is stable if Φ Ω (u) > 0 for all u ∈ Ω.

Proof. From Corollary 1, D = ∅ is a user equilibrium domain. Since p I < a, (2) yields Φ D (u) = p I -a -p C < 0 ∀u ∈ Ω. We also have from (2) that for any domain D and any u,

Φ D (u) ≤ p I -a -p C + a D g(v|u) dv.
But when the ratio g(•|u)/f (•) is upper-bounded by some value L, then the integral a D g(v|u) dv is smaller than aL D f (v) dv. Therefore, with ε < p C +a-P I aL , for a domain D such that D f (v) dv ≤ ε all users in D would be better off switching back to the ISP, hence D = ∅ is a stable equilibrium domain.

Further characterizations are provided in Section V for specific scenarios.

IV. PRICING GAME

For any price pair (p I , p C ), being able to characterize all stable user equilibria, we can reasonably assume that the community network will set up things (again, by initial offers/bargains) such that the largest (defined in terms of demand d C ) stable user equilibrium domain is reached in the end. Therefore, for given prices, we will be able to compute the corresponding values of the utility functions V C and V I of each operator. Hence providers can non-cooperatively play the pricing game where the community network chooses p C and the ISP chooses p I , each operator trying to maximize its utility function. The solution concept is the classical Nash equilibrium [START_REF] Osborne | A course in game theory[END_REF], a pair (p * I , p * C ) from which no provider can improve its revenue from a unilateral price change.

V. ANALYSIS AND DISCUSSION OF TWO SCENARIOS

A. Users with a homogeneous mobility pattern

We first consider the simplest situation where the mobility pattern is the same for all users, which means that g(v|u) does not depend on u, that is g(v|u) = g(v) ∀u as treated in Example 1. From this assumption, q u = q does not depend on u (but still depends on D). We also get a much simpler expression for Φ D , which is now: ∀u ∈ Ω,

Φ D (u) = a(q -1) + (p I -p C ) -cM g(u), (3)
which depends on u only through the term g(u), with q = D g(v) dv and M = D m(v)f (v) dv. From such an expression, we can show that with an homogeneous mobility pattern, user equilibria have a specific form. 1) Characterization of user equilibria:

Proposition 5. Assume that location attractiveness values are distributed regularly over Ω: i.e. mathematically, that for all y ∈ R + , the mass of users with the specific value g(u) = y is null. Then a non-degenerate user equilibrium domain has the form

D x := {u ∈ Ω | g(u) ≤ x} for a given x ≥ 0, with x solution of a Dx g(v) dv -1 + p I -p C -cx Dx m(v)f (v) dv :=Ψ(x) = 0.
In the above characterization, x is a threshold such that all users u with mobility attractiveness density g(u) below x subscribe to the community network .Remark that Ψ(x) corresponds to Φ Dx (u) for a user u such that g(u) = x; and since D x is continuous in x under our assumption, Ψ is also a continuous function of x.

Proof. See Appendix A.

We can characterize, among all domains D x , which ones will actually be user equilibrium domains, and the corresponding dynamics.

Assume that the set of subscriber to the community network is of the form D x = {u ∈ Ω : g(u) ≤ x} for some x ∈ R + .

• If Ψ(x) > 0 and D x = Ω, it means that users u with g(u)

just above x are associated with the ISP and are those with the largest utility difference and incentive to switch to the community network (indeed, from (3) that utility difference Φ Dx (u) is continuous and strictly decreasing in g(u)); hence they switch such that x and D x increase; • If Ψ(x) < 0 and D x = ∅, it is the opposite situation: users u with value g(u) just below x are with the community network but have the largest incentive to switch to the ISP; hence x and D x decrease; • If Ψ(x) = 0, all users u ∈ Ω are such that Φ Dx (u) ≥ 0, hence have no interest to switch; we are then in an equilibrium situation. We thus end up with the following characterization of user equilibrium domains. Let y := sup u∈Ω {g(u)} (possibly ∞), such that D y = Ω.

• If Ψ(y) ≥ 0 then D = Ω (all users subscribe to the community network) is an equilibrium; • Since Ψ(0) = -a + p I -p C ≤ 0 by assumption, ∅ is always a user equilibrium domain (no users associated with the community network); • If Ψ(x) = 0, D x is a user equilibrium domain.

2) Stable equilibria: Among all user equilibrium domains, we can characterize the stable ones. Proposition 6. As suggested by the dynamics described in Subsection III-C, we consider that the community network subscriber set D is always of the form D y for some y. Then if Ψ(x) = 0 and Ψ (x) < 0, D x is a stable equilibrium.

Proof. Assume a small variation, from

x to x = x ± ε, in D (hence, from D x to D x ). If Ψ (x) < 0, for ε small enough, Ψ(x ) > 0 (resp. < 0) if x < x (resp. x > x)
; hence users u with g(u) between x and x are incentivized to switch back to their initial choice, driving back to the (then stable) equilibrium domain D x .

3) Nash equilibria for the pricing game between operators: For any pair (p C , p I ), we consider that the largest equilibrium domain is selected. Operators then play a non-cooperative game to determine their optimal strategy [START_REF] Osborne | A course in game theory[END_REF], [START_REF] Maillé | Telecommunication Network Economics: From Theory to Applications[END_REF]. The output concept is that of a Nash equilibrium, a point (p * C , p * I ) such that no operator has an interest to unilaterally move from, because it would decrease its utility (revenue). Because of analytical intractability, we are going to study the existence, and characterize, Nash equilibria numerically, for specific parameters values; the procedure can be repeated for any other set of parameters.

4) Examples:

We first show situations where there are several user equilibria, and even several stable user equilibria. We then discuss the solution of the pricing game between operators.

Example 2. Consider Ω = Ω = R + , i.e., users are placed over the positive line (negative values could be described as the sea). We assume:

• m(u) = 1, i.e., all users generate the same amount of requests; • f (u) = α/(1 + u) 1+α with α > 0. In other words, users are located according to a Pareto distribution, potentially with infinite expected value. The closer to the 0 value (which can be thought of as the town center), the more users you can find. • g(u) = λe -λu , meaning that connections are exponentially distributed with rate λ, with more connections close to 0; even far-away users are more likely to require connections there. With these functions, noting that g is strictly decreasing, the set D x is simply the interval [ln(λ/x)/λ, +∞) when x ∈ (0, λ], D x = Ω when x > λ, and D x = ∅ when x = 0. It gives

Ψ(x) = -a(1 -x/λ) + p I -p C -cx 1 1 + ln(λ/x)/λ α for x ∈ [0, λ], Ψ(x) = p I -p C -cx for x > λ,
and Ψ(0) = p I -a -p C . Note that the assumption made in Proposition 5 holds here, hence Ψ is continuous over R + .

Three outcomes are illustrated in the next three cases for λ when α = 1.2, a = 1, c = 1, p I = 0.95 and p C = 0.1. In Figure 1, there are two solutions for Ψ(x) = 0, with only the second one leading to a stable equilibrium domain (in addition to D 0 = ∅, which is stable too from Proposition 4, for which the assumptions also hold). In Figure 2, Ψ(x) = 0 has only one (unstable) solution but D 0 = ∅ and D 0.5 = Ω (because Ψ(0.5) > 0) are stable equilibrium domains. In Figure 3 there is no solution to Ψ(x) = 0, and ∅ is the only equilibrium domain. Remark that Ω is a (stable) equilibrium domain if

Ψ(λ) = p I -p C -cλ ≥ 0. 2
Assuming now that the community network plays such that the largest equilibrium domain is selected (thanks to discounts for example), we can can draw the best responses of the pricing game between operators. It is for example displayed in Figure 4 for specific parameter values, here when the infrastructure for the ISP is χ I = 0. With these parameters, the community network is able to get a positive demand only if p I ≥ 0.76. d C is then jumping from 0 to 0.574 and is readily and slightly increasing to 0.678 when p I = 0.99. We actually have here a price war where each operator has an interest to give a price just below that of the opponent, and we end up with a Nash equilibrium (p I = 0.23985, p C = 0) where one operator stops with the zero price. Due to price war, only one operator survives. With an infrastructure cost χ I = 0, it is the ISP, but we note that the threat of the development of a community network significantly decreases the price set by the ISP (whose monopoly price was 1 as can be seen on Figure 4 when p C is prohibitively high). Also, there is a threshold on χ I over which it is the community network that survives. Indeed, remark that the value of χ I does not change the best-response values since it appears as a constant in the expression of V I , but the game stops as soon as one of the two providers gets a zero revenue. The revenue of the ISP will go to zero before that of the community network if χ I > 0.23.

B. Several populations

We slightly modify the model such that Ω = R + with a mass of users in 0, seen as a town (probability/mass π 0 ) while others with u > 0 are regularly distributed over the "countryside" with conditional density f . In terms of mobility, we assume that users at u = 0 do not move, while those at u > 0 have a probability π 1 to call from 0 and a conditional density (when connecting from another place) g(v) to make a connection from v > 0. With those assumptions, remark that q 0 = 1 (resp. q 0 = 0) if the community network is chosen (resp., not chosen) at 0, and for u > 0,

q u = π 1 1l {0∈D} +(1-π 1 ) D 1 g(v)dv where D 1 = D∩(0, ∞) is D excluding 0 and 1l {•} is 1 if the condition is satisfied and 0 otherwise.
The number of connections to a member of the community network at 0 (assuming

0 ∈ D) is then n(0) = π 0 m(0) + (1 -π 0 )π 1 D 1 m(v)f (v)dv and n(u) = g(u)(1 - π 0 ) D 1 m(v)f (v)dv for u > 0.
The level of annoyance (interferences, etc.) is again assumed linear in n(•), leading to

Φ D (0) = (p I -p C ) -c π 0 m(0) + (1 -π 0 )π 1 D 1 m(v)f (v)dv Φ D (u) = a π 1 1l {0∈D} + (1 -π 1 ) D 1 g(v)dv -1 +(p I -p C ) -c(g(u)(1 -π 0 ) D 1 m(v)f (v)dv).
Assuming that 0 ∈ D or not, the above last equation tells us as in the previous subsection that D1 is of the form D 1

x = {u : g(u) ≤ x} for some value x. Exactly as in the previous homogeneous case, there might be several solutions, and we will assume that the selected one will be a stable one leading to the largest market share (revenue) for the community network.

The more subscribers of the community network, the less likely users in 0 will subscribe because since they do not move, they experience only losses from an increased number of subscribers. But there might be a risk of oscillations on the user equilibrium. Indeed, if x is small enough, then users in 0 subscribe. This increases the interest for others to subscribe, leading to a larger value of x, which might deter users in 0, and so on.

Example 3. We again consider m(u) = 1, f (u) = α/(1 + u) 1+α with α > 0, and g(u) = λe -λu . Then users in 0 join iff for D 1 = D 1

x = [ln(λ/x)/λ, +∞), they prefer the community network over the ISP, knowing that there is coverage in 0, i.e., iff 1 , there are oscillations in the user equilibrium domain, which does not exist with the chosen strategy from the community network. In that case, we are going to consider that when computing best responses in the pricing game, operators will make use of the "worst" scenario for them in the pricing game, in terms of market share: D = D ∅ 1 for the community network, and D = {0} ∪ D 0 1 for the ISP. Using the above user equilibrium domain, we can can draw the best responses of the pricing game between operators. It is for example displayed in Figure 5 for specific parameter values when the infrastructure cost for the ISP is χ I = 0. With these numerical values, the best response of the ISP to the community network price p C is such that d I = 1 for p C ≤ 0.63, and d I = 0.7 (just users in 0) when p C is above that value. We again have a price war (each provider setting its price just below its opponent) in the high-price region. Predicting the outcome of the competition is not trivial, since following a best-response dynamics leads to a cycle (appearing in the top-right corner in the figure): prices slide downwards until reaching (p i , p C ) ≈ (0.71, 0.63), at which point the ISP sets p I = 1, reinitiating the cycle.

Fig

  Fig. Ψ(x) when λ = 1.0.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Ψ(x) when λ = 0.5.

CFig. 4 .

 4 Fig. 4. Best responses in the pricing game when λ = 0.25, α = 1.2, a = 1, c = 1, and χ I = 0

.CFig. 5 .

 5 Fig. 5. Best responses in the pricing game with heterogeneous users when λ = 0.25, α = 1.2, a = 1, c = 1, π 0 = 0.7, π 1 = 0.1, and χ I = 0

https://fon.com/

Note that with other functions, depending on the variations of f and g, an arbitrary number of solutions of Ψ(x) = 0 can be obtained.

, it means that users 0 are satisfied with the gain in price to be in the community network, and D = {0} ∪ D 0 1 is the considered user equilibrium domain;

APPENDIX

A. All domains on equilibrium situations have the form D x for homogeneous and regular mobility patterns Proof. At an equilibrium situation, the community domain D must be such that

Let p := p I -p C . Under the assumption that g(v|u) does not depend on u and with the expressions of q and n we had before, we now have ∀u ∈ Ω,

So an equilibrium domain D should be made of u such that

plus possibly some of the users for which there is equality above, but which are of measure 0 under our assumption.

Hence the general form of the solution

Using that form D x for candidate domains, one can write Φ Dx (u) as a function of x and g(u), and an equilibrium is reached when the users wanting to subscribe to the community network (currently made of D x ) is exactly D x , i.e., when x is a root of the function Ψ(x) := a Dx g(v)dv -1 + pcx Dx m(v)f (v) dv.


Nicolas ; Belhomme
email: nicolas.belhomme@chu-rennes.fr.

Marine Le Noir De Carlan

Alain Lescoat

Thomas Le Gallou

Florence Rouget

Philippe Loget

Patrick Jego

Tomas Le 3 Gallou

Investigating in utero fetal death outcome of internal medicine consultation

Keywords: Clinical aspects < Anti-phospholipid antibody syndrome, Drug treatment < Anti-phospholipid antibody syndrome, Epidemiology < Anti-phospholipid antibody syndrome in utero fetal death, stillbirth, internal medicine, placental pathology, 50 antiphospholipid antibodies syndrome, placental vascular disorders 51 Tweetable

Results: From January 2007 to December 2014, 53 women who presented an IUFD at 14 32 weeks or more of gestational age were included. The main cause for each IUFD was 33 determined by expert agreement. Primary outcome was the prevalence of IUFD related to 34 placental disorders. Secondary outcomes included the frequency of antiphospholipid 35 antibodies syndrome (APS) among patients with IUFD of placental origin and the 36 pathological and clinical features associated to APS. IUFD resulted from placental disorders 37 in 36/53 (68%) patients, and remained unexplained in 11 cases (20.8%). Among the 36 38 patients with placental disorders, APS was diagnosed in 5 (13.9%) cases, and 4(11.1%) 39 patients were considered as having "non-criteria" APS. History of thrombosis (p=0.001) and 40 placental infarcts (p=0.047) were significantly associated to APS. 41 Conclusion: Placental disorders were the major cause for IUFD in patients who were 42 referred to internal medicine specialists. Importantly, APS was seldom found in patients with 43 placental disorders. Venous thromboembolism history and placental infarcts were both 44 significantly associated to APS. Further studies are needed in order to deepen our 45 understanding of the physiopathology of placental disorders and its underlying causes 46 among non-APS women, and to determine the best treatment regimen for future 47 There is currently no international consensus on the gestational age threshold which defines 56

In Utero Fetal Death (IUFD). The French National College of Gynecologists and 57 Obstetricians (CNGOF) defines IUFD as a spontaneous cessation of fetal cardiac activity 58 occurring before or during labor, after 14 weeks of amenorrhea 1 , whereas the threshold of 59 either 22 or 28 weeks or more is commonly used in many countries. 2

60

Bukowski et al [START_REF]Stillbirth Collaborative Research Network Writing Group ([END_REF] showed that obstetric complications, placental disease, genetic abnormality 61 and infections, were involved in respectively 29%, 26%, 13% and 13% of IUFD, then 62 followed by cord abnormalities (10.4%), hypertensive disorders (9.2%), and maternal 63 complications (7.8%). IUFD were considered idiopathic in 24% of cases. 64

The objectives of IUFD investigations are to identify the cause, to prepare for future 65 pregnancies, and to detect a maternal pathology requiring specific care, such as 66 antiphospholipid antibodies syndrome (APS). 1 67 APS diagnosis requires the presence of clinical and biological criteria meeting the Sydney 68 classification criteria. [START_REF] Miyakis | International consensus statement on an 260 update of the classification criteria for definite antiphospholipid syndrome (APS)[END_REF] However, many patients, while fulfilling the clinical criteria, cannot be 69 classified as having APS either because of low titers or non-persistent positivity of 70 conventional antibodies, or because of the detection of other antibodies that have currently 71 not been validated (eg anti-prothrombin ): in those cases, the appellation of "non-criteria" 72 APS is sometimes used. [START_REF] Rodríguez-García | 263 Examining the prevalence of non-criteria anti-phospholipid antibodies in patients with 264 anti-phospholipid syndrome: a systematic review[END_REF]

Study objectives 75

The objectives of our study were to bring to light the internist's role in the diagnosis of IUFD 76 by 1) analyzing the final etiologies diagnosed, on which depend the treatments that will be 77 proposed for future pregnancies 2) evaluating the prevalence of IUFD consecutive to 78 placental vascular disorders 3) assessing the prevalence of APS among these patients and 79 comparing their clinical and pathological features to non-APS patients, in order to find out 80 relevant factors that support APS testing. 81

METHODS

83

All women who consulted in the internal medicine department of Rennes University hospital, 84 a French tertiary care center, after IUFD (defined as death at 14 weeks of gestation or later 85 according to the CNGOF) were retrospectively included between January, 1 st 2007 and 86 December, 31 st 2014. All cases of IUFD were reviewed by four specialists (a pediatrician, a 87 pathologist specialised in foetopathology, an obstetrician and an internist) and the main 88 cause of death was determined through consensual agreement. 89

According to the Amsterdam Placental Workshop Group Consensus Statement 8 , the 90 following histological features were recorded from the pathological records: placental infarcts, 91 retroplacental hematoma, hypotrophia, decidual arteriopathy, fetal vascular malperfusion, 92 thrombosis, villitis, intervillitis, and cord abnormalities. IUFD were classified as placenta-93 related if the lesions observed were deemed sufficient to lead to fetal death. 94 APS testing results were recorded for all patients, and obstetrical APS was diagnosed 95 according to Sydney criteria. Clinical and pathological characteristics of patients with APS, 96 "non-criteria" APS and no APS were compared. Patients with APS were classified in the 97 "maternal cause" category according to the CODAC classification[9], and they were also 98 considered as placenta-related IUFD if their placentas exhibited significant lesions. 99 53 patients consulted in internal medicine for IUFD, all of whom were referred by their 109 obstetrician. 22 patients (42%) were primigravida, 5 patients (9%) had a history of IUFD, 9 110 patients (17%) had a history of miscarriage, 2 patients had an antecedent of preeclampsia, 2 111 patients an antecedent of HELLP syndrome. A history of severe fetal growth restriction (<5 112 percentile) was mentioned in two cases. One patient suffered from chronic hypertension, no 113 patients had diabetes mellitus and none had renal disease. 9 patients (13%) were overweight 114 or obese. Three patients were known to have APS, revealed in all cases by venous thrombo-115 embolism (VTE). APS was considered as primary in all cases as none of our patient had 116 associated connective tissue disease. The patients' mean age at the time of IUFD was of 30 117 +/-4.5 years (extremes: 19-42). The mean term was 29.2 weeks of amenorrhea +/-7.9 118 (extremes 15-40). There were no multiple pregnancies. Distribution of IUFD according to 119 gestational age is shown in Figure 1. 120

Results of the consultation 150

The main causes of IUFD are reported in Table 1. Etiological categories were extracted from 151 the CODAC classification 9 : 6 cases issued from maternal causes, including the 5 definite 152 APS patients plus one patient whose IUFD was secondary to severe pre-eclampsia. IUFD of 153 "non-criteria" (n=4) and non-APS patients (n=27) were ruled as having a placental cause. 154 IUFD resulted from placental disorders in 36/53 (68%) patients: placental insufficiency in 33 155 (62%) cases (including one case of pre-eclampsia), and placental inflammatory disorders in 3 156 (6%) cases (Table 1). None of our patients had isolated gestational hypertension, renal 157 disease or connective tissue disease. 158 IUFD was caused by fetal anemia in one case which was classified as a fetal cause, by 159 Parvovirus B19 infection in one case, and by funicular pathology in 4 cases. 160 IUFD remained unexplained in 11/53 (21%) cases, including all the patients whose placenta 161 was not examined. 162 163

Comparison between APS, "non-criteria" APS and non-APS patients 164

The 36 patients who had significant placental disorders were allocated to one of the following 165 groups according to their APS status: definite APS, "non-criteria" APS and non-APS (in which 166 the patient with pre-eclampsia was included as her placenta exhibited hypotrophia along with 167 massive infarct and retroplacental hematoma). 168

Only the three patients with known APS had VTE history. Comparisons between the three 169 groups revealed that antecedent of VTE (p=0.001) and placental infarcts (p=0.047) were 170 significantly associated with the definite APS group, whereas fetal vascular malperfusion or 171 inflammatory disorders (villitis/ intervillitis) were not. Patients with "non-criteria" APS were not 172 different from non-APS patients regarding VTE history or pathological findings. 173 There were no differences in mothers' age, time of pregnancy termination, presence of 174 livedo, history of fetal loss, or placental hypotrophia, between the 3 groups. 175 176 DISCUSSION 177 53 cases of IUFD were referred to an internist over the study period. In the meantime, 440 178 cases were registered in our center, meaning that internists were consulted in 12% of cases. 179 Placental disorders were over represented, as they accounted for 31 (58%) of cases, 180 whereas Bukowski et al. showed that they are usually involved in only 23.6% of IUFD. [START_REF]Stillbirth Collaborative Research Network Writing Group ([END_REF] 181 Moreover, no genetic causes were observed while infectious and fetal causes were rarely 182 found (one case of each), although Bukowski et al demonstrated that such etiologies are 183 involved in respectively 29.3, 13.7 and 12.9% of IUFD. This reveals the selection bias which 184 applied while referring the patients to our department: a thorough etiological assessment had 185 previously been conducted by the obstetrician, thus ruling out the most common causes such 186 as genetics, infections or fetal pathologies. Overall, 36/53 (68%) patients were referred to 187 investigate IUFD due to inflammatory or vascular placental disorders, which demonstrates 188 the obstetricians' expectations concerning internists in this field. 189 IUFD remains unexplained in 21% of cases which is comparable to the literature 4 10 11 . These 190 include the 4 patients whose placenta was not examined. This emphasizes the crucial 191 importance of performing placental pathological examinations. 192

In our study, as previously demonstrated, placental examination was highly contributive 12 13 , 193 since significant histological lesions were noted in 40/49 (81%) cases (including cord 194 abnormalities in 4 cases). 195

As for patients with placental abnormalities, only 5/36 (14%) were found to have APS 196 meeting Sydney criteria. [START_REF] Miyakis | International consensus statement on an 260 update of the classification criteria for definite antiphospholipid syndrome (APS)[END_REF] Unsurprisingly, history of thrombosis was significantly associated 197 with the diagnosis of definite APS (p=0.001). [START_REF] Reynaud | Risk of venous and arterial thrombosis 283 according to type of antiphospholipid antibodies in adults without systemic lupus 284 erythematosus: a systematic review and meta-analysis[END_REF] Among histological patterns, placental infarcts 198 were associated with APS (p=0.047), whereas other lesions such as fetal thrombosis or 199 villitis were not: this is consistent with previous studies which showed that infarcts is the most 200 common pathological feature encountered in patients with obstetrical APS. 15 16 These results 201 emphasize the need for APL testing in patients presenting a VTE history or placental infarcts. 202 Diagnosing obstetrical APS is essential since such patients may benefit from the association 203 of aspirin and heparin, which increases the live birth rate for future pregnancies up to 70%. [START_REF] Rai | Randomised controlled trial of aspirin and 291 aspirin plus heparin in pregnant women with recurrent miscarriage associated with 292 phospholipid antibodies (or antiphospholipid antibodies)[END_REF] others. In our study, "non-criteria" APS patients were not different from non-APS patients 208 regarding clinical or pathological features. Notably, they showed neither a larger history of 209 thrombosis, nor more placental infarcts. This finding does not support the relevance of an 210 entity such as "non-criteria" APS. The best treatment regimen whose would benefit to 211 patients with "non-criteria" APS for future pregnancies is still debated 19 20 , although Mekinian 212 et al recently showed that they may benefit from the same treatment as patients with definite 213 definite APS. [START_REF] Mekinian | Non-conventional antiphospholipid 302 antibodies in patients with clinical obstetrical APS: Prevalence and treatment efficacy in 303 pregnancies[END_REF] 214 Above all, 27/36 (75%) patients did not have APS although they exhibited significant 215 placental lesions. Placental inflammatory disorders, including villitis and intervillitis were 216 found in 3 of our patients (8%). Their physiopathology remains unclear in most of cases [START_REF] Derricott | 305 Characterizing Villitis of Unknown Etiology and Inflammation in Stillbirth[END_REF] ; 217 nevertheless previous studies have shown that such lesions might not be associated to 218 APS. 15 23-26 219

The final question is to find out how to manage the future pregnancies of the 24/36 (67%) 220

  Abbreviations: IUFD= In Utero Fetal Death. APS= Antiphospholipid Antibodies Syndrome. 49

  were analyzed by conducting a Chi square or Fisher exact 101 test. Quantitative data was analyzed by conducting student or Mann and Whitney U test. We 102 performed all tests with a significance level of P<0.05. Statistical analyzis was performed 103 using SPSS 20.0 software. 104This study was approved by the ethics committee of Rennes University Hospital.

  At the time of IUFD, 7/53 (13%) patients were receiving anticoagulant therapy or antiplatelet 121 agents: 4 patients were treated with low dose aspirin (LDA), because of previous fetal loss in 122 3 cases, and of severe fetal growth restriction in one case. The three patients with known 123 APS were receiving LDA combined with low-molecular-weight heparin (LMWH). 124 LA, aCL and β2-GPI) were tested in 49/53 patients (90%), and were 126 positive in 9 (17%) cases. 127 Among the 9 APL positive patients, 8 underwent repeat testing 12 weeks later. Among them, 128 5 still tested positive: they were classified as having definite APS. The four other patients 129 presented "non-criteria" APS: one of them was tested positive for LA without confirmation12 130 weeks apart and presented low titers aCL, the 3 others were negative for conventional APL, 131 but tested positive for anti-prothrombin antibodies. 132Pathological findings 133A placental histological analyzis was performed in 49/53 cases (92%). Anomalies were found 134 in 40/49 (81%) of the placentas. 135 Vascular disorders were noted in 33/49 (73%) placentas. Retroplacental hematoma was 136 found in 10 cases, infarcts in 19 cases. Thrombosis were observed in 2 cases, decidual 137 arteriopathy was found in 3 cases. Fetal vascular malperfusion was observed in 7 patients, 138 only one of whom had gestational diabetes. Inflammatory disorders were found in 3 cases: 2 139 cases of villitis and one case of intervillitis, in all of which TORCH (Toxoplasmosis, Rubella, 140 CMV and Herpes virus) screening was negative. IUFD was related to cord anomaly in 4 141 cases: cord thrombosis in 3 cases, and tight loops in one case. 142All the placentas of the 9 APL positive patients (definite and "non-criteria" APS) were 143 examined: all presented vascular disorders, which associated hypotrophy (constant) with 144 decidual arteriopathy in one case, infarcts in 8/9 cases, retroplacental hematoma in 2 cases, 145 and thrombi in two cases. In one patient was noted the co-existence of fetal vascular

  criteria" APS was discussed in 4/36 (11%) patients, because of intermittent LA 206 positivity and low-titers aCL in one of them, and isolated anti-prothrombin positivity in the 3 207

Figure 1 .

 1 Figure 1. Distribution of cases according to gestationnal age.

Table 1 .

 1 Main cause of In Utero Fetal Death, determined through expert agreement. Abbreviations: IUFD: In Utero Fetal Doss; PVD: Placental Vascular Disorders; PID: Placental inflammatory Disorders; APS: Antiphospholipid Syndrome; PE: Pre-Eclampsia.

	Main cause of IUFD	N	%
		Total: 53
	Placental	30	56.6
	(27 PVD, 3 PID)
	Maternal disease	6	11.3
	(5 proven APS, 1 PE)
	Cord conditions	4	7.5
	Infection	1	1.9
	Fetal	1	1.9
	Intrapartum or	0	0
	obstetric complication
	Congenital/ genetic	0	0
	Unknown	11	20.8
	Total	53	100

(APS) among women referred to the internal medicine department.

Huchon C, Deffieux X, Beucher G et al (2016) Pregnancy loss: French clinical practice

guidelines. Eur J Obstet Gynecol Reprod Biol 201:18-26

Alexander S, Zeitlin J (2016) Stillbirths and fetal deaths-Better definitions to monitor

(Codac): a utilitarian approach to the classification of perinatal deaths. BMC Pregnancy

Childbirth 9:22

Gardosi J, Kady SM, McGeown P, Francis A, Tonks A (2005) Classification of stillbirth by

136:9-15

Kaandorp SP, Goddijn M, van der Post JAM et al (2010) Aspirin plus heparin or aspirin

alone in women with recurrent miscarriage. N Engl J Med 362:1586-1596


Marius Cocou
email: cocou@lma.cnrs-mrs.fr

A variational inequality and applications to quasistatic problems with Coulomb friction

Keywords: Evolution inequalities, existence results, quasistatic problems, nonlinear materials, Coulomb friction MSC(2010): 35Q74, 49J40, 74D10, 74H20

The aim of this paper is to study an evolution variational inequality that generalizes some contact problems with Coulomb friction in small deformation elasticity. Using an incremental procedure, appropriate estimates and convergence properties of the discrete solutions, the existence of a continuous solution is proved. This abstract result is applied to quasistatic contact problems with a local Coulomb friction law for nonlinear Hencky and also for linearly elastic materials.

Introduction

This paper concerns the analysis of an evolution variational inequality that represents a generalization of some quasistatic elastic problems with pointwise Coulomb friction and relaxed unilateral contact.

Existence and approximation of solutions to the quasistatic elastic problems have been studied for various contact conditions. Based on the variational formulation proposed in [START_REF] Telega | Quasistatic Signorini's problem with friction and duality[END_REF], the quasistatic unilateral contact problems with local Coulomb friction have been studied in [START_REF] Andersson | Existence results for quasistatic contact problems with Coulomb friction[END_REF][START_REF] Rocca | Existence and approximation of a solution to quasistatic Signorini problem with local friction[END_REF][START_REF] Rocca | Numerical analysis of quasistatic unilateral contact problems with local friction[END_REF] and the normal compliance models have been investigated by several authors, see, e.g. [START_REF] Kikuchi | Contact problems in elasticity : a study of variational inequalities and finite element methods[END_REF][START_REF] Andersson | A quasistatic frictional problem with a normal compliance penalization term[END_REF][START_REF] Han | Quasistatic contact problems in viscoelasticity and viscoplasticity[END_REF] and references therein. Dynamic frictional contact problems with normal compliance laws for some viscoelastic bodies have been studied in [START_REF] Martins | Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws[END_REF][START_REF] Kikuchi | Contact problems in elasticity : a study of variational inequalities and finite element methods[END_REF][START_REF] Kuttler | Dynamic friction contact problems for general normal and friction laws[END_REF][START_REF] Chau | A dynamic frictional contact problem with normal damped response[END_REF][START_REF] Migórski | Nonlinear inclusions and hemivariational inequalities[END_REF].

1

A comprehensive presentation of contact models for the quasistatic processes can be found in [START_REF] Shillor | Models and analysis of quasistatic contact[END_REF]. An unified approach, which can be applied to various quasistatic problems, including unilateral and bilateral contact with nonlocal friction, or normal compliance conditions, has been considered in [START_REF] Badea | Internal and subspace correction approximations of implicit variational inequalities[END_REF], and different (quasi)static contact problems with nonlocal friction are analyzed in [START_REF] Capatina | Variational inequalities and frictional contact problems[END_REF].

A static contact problem with relaxed unilateral conditions and pointwise Coulomb friction was studied in [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF], based on abstract formulations and Ky Fan's fixed point theorem. Recently, an extension of these results to dynamic contact problems in viscoelasticity was treated in [START_REF] Cocou | A class of dynamic contact problems with Coulomb friction in viscoelasticity[END_REF][START_REF] Cocou | A variational analysis of a class of dynamic problems with slip dependent friction[END_REF].

This paper extends the results presented in [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF] to a new evolution variational inequality involving a nonlinear operator and with applications to two-field formulations of some nonsmooth elastic quasistatic contact problems with friction.

The paper is organized as follows. In Section 2, a general evolution variational inequality is analyzed by an incremental method. Using the Ky Fan's theorem, the existence of incremental solutions is proved. Then several estimates and compactness arguments enable to pass to limits in order to establish the existence of a continuous solution. In Section 3, applications to quasistatic contact problems with local Coulomb friction, for nonlinear Hencky and linearly elastic bodies, are presented.

An implicit variational inequality

For simplicity and also in view of applications to contact mechanics, we shall confine attention to the case when Ω is an open, bounded, connected set Ω ⊂ R d , d = 2, 3, with the boundary Γ ∈ C 1,1 and with Ξ an open part of Γ.

We denote Ξ T := Ξ × (0, T ), where 0 < T < +∞, and define the closed convex cones L 2 -(Ξ), L 2 -(Ξ T ) in the Hilbert spaces L 2 (Ξ), L 2 (Ξ T ), respectively, as follows:

L 2 -(Ξ) := {δ ∈ L 2 (Ξ); δ ≤ 0 a.e. in Ξ}, L 2 -(Ξ T ) := {δ ∈ L 2 (0, T ; L 2 (Ξ)); δ ≤ 0 a.e. in Ξ T }.
Let κ, κ : R → R be two mappings with κ lower semicontinuous and κ upper semicontinuous, satisfying the following conditions:

κ(s) ≤ κ(s) ≤ 0 ∀ s ∈ R, (1)
∃ r 0 ≥ 0 such that |κ(s)| ≤ r 0 ∀ s ∈ R. (2)
For every ζ ∈ L 2 (Ξ), define the following subset of L 2 -(Ξ):

Λ(ζ) = {η ∈ L 2 -(Ξ); κ • ζ ≤ η ≤ κ • ζ a.e. in Ξ }, (3)
which is clearly nonempty, because the bounding functions belong to the respective set, closed and convex. Since meas(Ξ) < ∞ and κ, κ satisfy (2), it is also readily seen that for all

ζ ∈ L 2 (Ξ) the set Λ(ζ) is bounded in norm in L 2 (Ξ) by R 0 = r 0 (meas(Ξ)) 1/2 and in L ∞ (Ξ) by r 0 .
The following compactness theorem proved in [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF] will be used in this paper.

Theorem 2.1. Let X, Û and Ŷ be three Banach spaces such that X ⊂ Û ⊂ Ŷ with compact embedding from X into Û .

(i) Let G be bounded in L p (0, T ; X), where 1 ≤ p < ∞, and ∂G/∂t := { ḟ ; f ∈ G} be bounded in L 1 (0, T ; Ŷ ). Then G is relatively compact in L p (0, T ; Û ).

(ii) Let G be bounded in L ∞ (0, T ; X) and ∂G/∂t be bounded in L r (0, T ; Ŷ ), where r > 1.

Then G is relatively compact in C([0, T ]; Û ).
If H is a Hilbert space, unless otherwise stated we shall denote by . , . H its inner product and by . H the corresponding norm.

Let (V, . , . , . ) and (U, . U ) be two Hilbert spaces such that V ⊂ U with continuous and compact embedding.

Consider a functional F : V → R differentiable on V and assume that its derivative F : V → V is strongly monotone and Lipschitz continuous, that is there exist two constants α, β > 0 for which

α v -u 2 ≤ F (v) -F (u), v -u (4)
and

F (v) -F (u) ≤ β v -u (5)
for all u, v ∈ V . Using the relations

F (v) -F (u) = 1 0 F (u + r(v -u)), v -u dr = F (u), v -u + 1 0 F (u + r(v -u)) -F (u), v -u dr
and (4), [START_REF] Kikuchi | Contact problems in elasticity : a study of variational inequalities and finite element methods[END_REF], it is easily seen that for all u, v ∈ V it results

F (u), v -u + α 2 v -u 2 ≤ F (v) -F (u) ≤ F (u), v -u + β 2 v -u 2 . ( 6
)
We remark that since F satisfies [START_REF] Andersson | A quasistatic frictional problem with a normal compliance penalization term[END_REF], it follows that F is strictly convex and sequentially weakly lower semicontinuous on V . Let (X, . X ) be a Hilbert space such that X ⊂ L 2 (Γ) with continuous and compact embedding, and l 0 : V → X, l : V → L 2 (Ξ), φ : L 2 -(Ξ) × V → R be three mappings satisfying the following conditions: l 0 is linear and continuous,

∃ k 1 > 0 such that ∀ v 1 , v 2 ∈ V, l(v 1 ) -l(v 2 ) L 2 (Ξ) ≤ k 1 v 1 -v 2 U , (7)
∀ γ, δ ∈ L 2 -(Ξ), ∀ v, w ∈ V verifying γ ∈ Λ(l(v)) and δ ∈ Λ(l(w)), γ -δ, l 0 (v -w) L 2 (Ξ) ≤ 0. (8)
∀ γ ∈ L 2 -(Ξ), ∀ θ ≥ 0, ∀ v 1 , v 2 , v ∈ V, φ(γ, v 1 + v 2 ) ≤ φ(γ, v 1 ) + φ(γ, v 2 ), (9)
φ(γ, θv) = θ φ(γ, v), (10)
∀ v ∈ V, φ(0, v) = 0, (11)
∃ k 2 , k 3 > 0 such that ∀ γ, δ ∈ L 2 -(Ξ), ∀ v ∈ V, |φ(γ, v) -φ(δ, v)| ≤ k 2 γ -δ L 2 (Ξ) v U , (12)
|φ(γ, v) -φ(δ, v)| ≤ k 3 γ -δ X v , (13)
∃ k 4 > 0 such that γ 1 -γ 2 X ≤ k 4 ( u 1 -u 2 + f 1 -f 2 ), ( (14)
) for all γ 1,2 ∈ L 2 -(Ξ), u 1,2 , f 1,2 , d 1,2 ∈ V verifying (Q 1 ) F (u 1 ), v -u 1 -γ 1 , l 0 (v -u 1 ) L 2 (Ξ) +φ(γ 1 , v -d 1 ) -φ(γ 1 , u 1 -d 1 ) ≥ f 1 , v -u 1 ∀ v ∈ V, (Q 2 ) F (u 2 ), v -u 2 -γ 2 , l 0 (v -u 2 ) L 2 (Ξ) +φ(γ 2 , v -d 2 ) -φ(γ 2 , u 2 -d 2 ) ≥ f 2 , v -u 2 ∀ v ∈ V, 15
and we assume that k 3 k 4 < α.

(

) 16
Let f ∈ W 1,2 (0, T ; V ), u 0 ∈ V , λ 0 ∈ Λ(l(u 0 )) be given and satisfy the following compatibility condition:

F (u 0 ), v -u 0 -λ 0 , l 0 (v -u 0 ) L 2 (Ξ) (17) +φ(λ 0 , v) -φ(λ 0 , u 0 ) ≥ f (0), v -u 0 ∀ v ∈ V.
Consider the following problem.

Problem Q : Find u ∈ W 1,2 (0, T ; V ), λ ∈ W 1,2 (0, T ; X ) such that u(0) = u 0 , λ(0) = λ 0 , λ(t) ∈ Λ(l(u(t)))
for almost all t ∈ (0, T ), and

F (u), v -u -λ, l 0 (v -u) L 2 (Ξ) + φ(λ, v) (18)
-φ(λ, u) ≥ f, v -u ∀ v ∈ V a.e. on (0, T ).

Incremental formulations

For n ∈ N * , we set ∆t := T /n, t i := i ∆t, i = 0, 1, ..., n. If θ is a continuous function of t ∈ [0, T ] valued in some vector space, we use the notations θ i := θ(t i ) unless θ = u, and if i , ∀ i ∈ {0, 1, ..., n}, are elements of some vector space, then we set

∂ i := i+1 -i ∆t , ∆ i := i+1 -i ∀ i ∈ {0, 1, ..., n -1}.
We approximate the problem Q using the following sequence of incremental problems

(Q i,n ) i=0,1,...,n-1 . Problem Q i,n : Find u i+1 ∈ V , λ i+1 ∈ Λ(l(u i+1 )) such that F (u i+1 ), v -∂u i -λ i+1 , l 0 (v -∂u i ) L 2 (Ξ) + φ(λ i+1 , v) (19)
-φ(λ i+1 , ∂u i ) ≥ f i+1 , v -∂u i ∀ v ∈ V.
It is easily seen that for all i ∈ {0, 1, ..., n -1} the problem Q i,n is equivalent to the following implicit variational inequality. Problem Qi,n : Find

u i+1 ∈ V , λ i+1 ∈ Λ(l(u i+1 )) such that F (u i+1 ), v -u i+1 -λ i+1 , l 0 (v -u i+1 ) L 2 (Ξ) + φ(λ i+1 , v -u i ) ( 20
) -φ(λ i+1 , u i+1 -u i ) ≥ f i+1 , v -u i+1 ∀ v ∈ V.
Let us define the following functions:

u n (0) = ûn (0) = u 0 , λ n (0) = λ 0 , f n (0) = f 0 and ∀ i ∈ {0, 1, ..., n -1}, ∀ t ∈ (t i , t i+1 ], u n (t) = u i+1 , λ n (t) = λ i+1 , ûn (t) = u i + (t -t i )∂u i , λn (t) = λ i + (t -t i )∂λ i , f n (t) = f i+1 .
Then for all n ∈ N * each of the sequences of inequalities

(Q i,n ) i=0,1,...,n-1 , ( Qi,n ) i=0,1,...,n-1 is equivalent to the following incremental formulation. Problem Q n : Find u n ∈ L 2 (0, T ; V ), λ n ∈ L 2 (Ξ T ) such that λ n (t) ∈ Λ(l(u n (t))) ∀ t ∈ (0, T ) and
F (u n (t)), v - d dt ûn (t) -λ n (t), l 0 (v - d dt ûn (t)) L 2 (Ξ) +φ(λ n (t), v) -φ(λ n (t), d dt ûn (t)) (21)
≥ f n (t), v - d dt ûn (t) ∀ v ∈ V, a.e. on (0, T ).
First, we prove the existence of a solution to the incremental problem Qi,n by a fixed point method.

Let Φ i : L 2 -(Ξ) → 2 L 2 -(Ξ) \ {∅} be the set-valued mapping defined by for all γ ∈ L 2 -(Ξ) Φ i (γ) = Λ(l(u γ )), (22)
where u γ is the solution of the following variational inequality of the second kind: find u γ ∈ V such that

F (u γ ), v -u γ -γ, l 0 (v -u γ ) L 2 (Ξ) + φ(γ, v -u i ) (23) -φ(γ, u γ -u i ) ≥ f i+1 , v -u γ ∀ v ∈ V. It is easily seen that λ is a fixed point of Φ i , i.e. λ ∈ Φ i (λ), iff (u i+1 , λ i+1 ) = (u λ , λ) is a solution of the problem Qi,n .
We shall prove the existence of a fixed point of the multifunction Φ i by using a corollary of the Ky Fan's fixed point theorem [START_REF] Fan | Fixed points and minimax theorems in locally convex topological linear spaces[END_REF], proved in [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF] in the particular case of a reflexive Banach space. Note that since Y is a reflexive Banach space and D is convex, closed and bounded, there is no assumption that Y is separable, see [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF][START_REF] Browder | Nonlinear operators and nonlinear equations of eyolution in Banach spaces[END_REF].

Theorem 2.2. Assume that (1 -5), (7 -14) hold. Then there exists λ ∈

L 2 -(Ξ) such that λ ∈ Φ i (λ) and (u i+1 , λ i+1 )=(u λ , λ) is a solution of the prob- lem Qi,n . Proof. We apply Proposition 2.1 to Φ = Φ i , Y = L 2 (Ξ) and D = L 2 -(Ξ) ∩ ζ ∈ L 2 (Ξ); ζ L 2 (Ξ) ≤ R 0 .
The set D ⊂ L 2 (Ξ) is clearly convex, closed and bounded. By ( 4), ( 5), ( 7), ( 8), (10 -14), for every γ ∈ D the classical variational inequality [START_REF] Brezis | Problèmes unilatéraux[END_REF] has a unique solution u γ .

Since for each ζ ∈ L 2 (Ξ) the set Λ(ζ) is nonempty, convex, closed, and bounded by R 0 , it follows that Φ i (γ) = Λ(l(u γ )) is a nonempty, convex and closed subset of D for every γ ∈ D.

In order to prove that the multifunction Φ i is sequentially weakly upper semicontinuous, let

γ p γ in L 2 (Ξ), γ p ∈ D, η p ∈ Φ i (γ p ) ∀ p ∈ N, η p η in L 2 (Ξ) and let us verify that η ∈ Φ i (γ).
Let u γp ∈ V be the solution of the variational inequality

F (u γp ), v -u γp -γ p , l 0 (v -u γp ) L 2 (Ξ) + φ(γ p , v -u i ) (24) -φ(γ p , u γp -u i ) ≥ f i+1 , v -u γp ∀ v ∈ V.
Taking v = 0 in (24) and using [START_REF] Chau | A dynamic frictional contact problem with normal damped response[END_REF], we obtain

F (u γp ) -F (0), u γp ≤ γ p , l 0 (u γp ) L 2 (Ξ) + φ(γ p , -u γp ) + f i+1 -F (0), u γp ,
so that, by ( 4), ( 7), ( 12), [START_REF] Capatina | Variational inequalities and frictional contact problems[END_REF],

α u γp 2 ≤ ( γ p X l 0 + k 3 γ p X + f i+1 + F (0) ) u γp ,
which implies

u γp ≤ α -1 ( l 0 + k 3 ) γ p X + f i+1 + F (0) ).
Thus

u γp ≤ C 1 ( γ p X + f i+1 + F (0) ) ∀ p ∈ N, (25)
where

C 1 = α -1 max( l 0 + k 3 , 1).
As the sequence (γ p ) p is bounded in L 2 (Ξ), it follows that (u γp ) p is bounded in V which implies that there exists a subsequence, still denoted by (u γp ) p , and an element u ∈ V such that

u γp u in V. (26)
By ( 6) and ( 10), the inequality [START_REF] Nečas | Mathematical theory of elastic and elasto-plastic bodies. An introduction[END_REF] implies

F (v) -F (u γp ) -γ p , l 0 (v -u γp ) L 2 (Ξ) + φ(γ p , v -u γp ) (27) ≥ f i+1 , v -u γp + α 2 v -u γp 2 ∀ v ∈ V,
and taking v = u, we obtain

F (u) -F (u γp ) -γ p , l 0 (u -u γp ) L 2 (Ξ) + φ(γ p , u -u γp ) ≥ f i+1 , u -u γp + α 2 u -u γp 2 .
As F is sequentially weakly lower semicontinuous, using the previous relation, [START_REF] Han | Quasistatic contact problems in viscoelasticity and viscoplasticity[END_REF], [START_REF] Badea | Internal and subspace correction approximations of implicit variational inequalities[END_REF] and the compact embeddings X ⊂ L 2 (Ξ), V ⊂ U , we have lim sup

p→∞ α 2 u -u γp 2 ≤ F (u) + lim sup p→∞ (-F (u γp )) + lim p→∞ | γ p , l 0 (u -u γp ) L 2 (Ξ) | + lim p→∞ φ(γ p , u -u γp ) -lim p→∞ f i+1 , u -u γp ≤ F (u) -lim inf p→∞ F (u γp ) + lim p→∞ γ p L 2 (Ξ) l 0 (u -u γp ) L 2 (Ξ) + lim p→∞ k 2 γ p L 2 (Ξ) u -u γp U -lim p→∞ f i+1 , u -u γp = F (u) -lim inf p→∞ F (u γp ) ≤ 0, which proves that u γp → u in V. ( 28
)
Passing to the limit in [START_REF] Nečas | Mathematical theory of elastic and elasto-plastic bodies. An introduction[END_REF], it follows that u is a solution of ( 23) and since its solution is unique we obtain that u γ = u = lim p→∞ u γp . Now, the relation η p ∈ Φ i (γ p ) implies

η p ∈ Λ(l(u γp )) that is κ • l p ≤ η p ≤ κ • l p a.e. in Ξ, (29)
for all p ∈ N, where l p := l(u γp ). The relations (29) are equivalent to

ω κ • l p ≤ ω η p ≤ ω κ • l p ,
for every measurable subset ω ⊂ Ξ and for all p ∈ N. Using (28), ( 8), the semi-continuity of κ and κ, the relations (1), ( 2), the convergence property ω η p → ω η, and passing to limits according to Fatou's lemma, we obtain

ω κ • l(u γ ) ≤ ω η ≤ ω κ • l(u γ ), (30)
for every measurable subset ω ⊂ Ξ, which implies η ∈ Φ i (γ). By Proposition 2.1 there exists a fixed point λ of Φ i and (u λ , λ) is clearly a solution to the problem Qi,n .

Remark 2.1. This existence result insures also that there exists (u 0 , λ 0 ) satisfying the compatibility condition (17).

Existence of a solution to the continuous problem

We now establish some useful estimates independent of n for the solutions of the incremental formulations Qi,n and Q n . Lemma 2.1. Under the above hypotheses, for all n ∈ N * and all i ∈ {0, 1, ..., n -1} the following estimates hold:

u i+1 ≤ C 1 ( λ i+1 X + f i+1 + F (0) ), (31)
∆u i ≤ k 3 α ∆λ i X + 1 α ∆f i ), ( 32
) ∆λ i X ≤ k 4 ( ∆u i + ∆f i ), ( 33
)
∆u i ≤ C 2 ∆f i , ( 34
) ∆λ i X ≤ C 3 ∆f i , ( 35
)
where

C 2 = k 3 k 4 + 1 α -k 3 k 4 , C 3 = (α + 1)k 4 α -k 3 k 4 .
Proof. By similar arguments to those that enabled to prove [START_REF] Zeidler | Nonlinear functional analysis and its applications[END_REF], using [START_REF] Browder | Nonlinear operators and nonlinear equations of eyolution in Banach spaces[END_REF] the estimate (31) follows.

If we take v = u i in [START_REF] Browder | Nonlinear operators and nonlinear equations of eyolution in Banach spaces[END_REF] then

F (u i+1 ), u i -u i+1 -λ i+1 , l 0 (u i -u i+1 ) L 2 (Ξ) (36) -φ(λ i+1 , u i+1 -u i ) ≥ f i+1 , u i -u i+1 ,
and taking v = u i+1 in (20), corresponding to i -

1 if i ≥ 1, or in (17) if i = 0, we have F (u i ), u i+1 -u i -λ i , l 0 (u i+1 -u i ) L 2 (Ξ) + φ(λ i , u i+1 -u i-1 ) (37) -φ(λ i , u i -u i-1 ) ≥ f i , u i+1 -u i .
By [START_REF] Chau | A dynamic frictional contact problem with normal damped response[END_REF], the inequalities (36) and (37) imply

F (u i+1 ) -F (u i ), u i+1 -u i ≤ λ i+1 -λ i , l 0 (u i+1 -u i ) L 2 (Ξ) (38) +φ(λ i , u i+1 -u i ) -φ(λ i+1 , u i+1 -u i ) + f i+1 -f i , u i+1 -u i .
As (u i+1 , λ i+1 ) and (u i , λ i ) are solutions of Qi+1,n and Qi,n , respectively, we have λ i+1 ∈ Λ(l(u i+1 )), λ i ∈ Λ(l(u i )), so that by ( 9)

λ i+1 -λ i , l 0 (u i+1 -u i ) L 2 (Ξ) ≤ 0.
Using this relation, ( 4) and ( 14) in (38), we have

α u i+1 -u i 2 ≤ k 3 λ i+1 -λ i X u i+1 -u i + f i+1 -f i u i+1 -u i ,
from which (32) follows.

From [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF], we obtain (33) and by (32), (33) the estimates (34), (35) can be easily verified.

Based on the previous lemma and the fact that f ∈ W 1,2 (0, T ; V ) is absolutely continuous, after possibly being redefined on a set of measure zero, the following estimates can be established by a straightforward computation, see, e.g. [START_REF] Cocou | Formulation and approximation of quasistatic frictional contact[END_REF], [START_REF] Capatina | A class of implicit variational inequalities and applications to frictional contact[END_REF].

Lemma 2.2. For all n ∈ N * u n (t) ≤ C 1 ( λ n (t) X + f n (t) + F (0) ) ∀ t ∈ [0, T ], (39)
u n (t) -ûn (t) ≤ T n d dt ûn (t) ≤ C 2 f n (t) -f n (t - T n ) (40) ≤ C 2 min{t+ T n ,T } t-T n ḟ (τ ) dτ ∀ t ∈ [0, T ], λ n (t) -λn (t) X ≤ T n d dt λn (t) X (41) ≤ C 3 f n (t) -f n (t - T n ) ∀ t ∈ [0, T ], u n -ûn L 2 (0,T ;V ) = T n √ 3 d dt ûn L 2 (0,T ;V ) (42) ≤ C 2 T n √ 3 ḟ L 2 (0,T ;V ) , λ n -λn L 2 (0,T ;X ) = T n √ 3 d dt λn L 2 (0,T ;X ) (43)
≤ C 3 T n √ 3 ḟ L 2 (0,T ;V ) .
Using Lemma 2.2, it follows that (û n ) n is bounded in W 1,2 (0, T ; V ), ( λn ) n is bounded in W 1,2 (0, T ; X ) ∩ L ∞ (Ξ T ), and since all these functions are absolutely continuous, after possibly being redefined on a set of measure zero, we have the following convergence results. Lemma 2.3. There exist subsequences of (u n , ûn ) n and (λ n , λn ) n , denoted by (u np , ûnp ) p and (λ np , λnp ) p , and two elements u

∈ W 1,2 (0, T ; V ), λ ∈ W 1,2 (0, T ; X ) ∩ L 2 (Ξ T ) such that u np (t) u(t) in V ∀ t ∈ [0, T ], ( 44
) ûnp u in W 1,2 (0, T ; V ), ( 45
)
λ np (t) λ(t) in X ∀ t ∈ [0, T ], ( 46
)
λ np , λnp λ in L 2 (0, T ; L 2 (Ξ)), ( 47
) λnp λ in W 1,2 (0, T ; X ). ( 48
)
Lemma 2.4. For the subsequences (û np ) p , (λ np ) p , the following relation holds:

lim inf p→∞ T 0 φ(λ np (t), d dt ûnp (t)) dt ≥ T 0 φ(λ(t), d dt û(t)) dt. ( 49
)
Proof. According to Theorem 2.1 with G = ( λnp ) p , X = L 2 (Ξ), Û = H ι-1/2 (Ξ), Ŷ = X , p = 2, 0 < ι < 1 2 , and to (47), (48), we obtain that

λ np , λnp → λ in L 2 (0, T ; X ). ( 50
)
By ( 14) it follows that

T 0 (φ(λ np (t), d dt ûnp (t)) -φ(λ(t), d dt ûnp (t))) dt ≤ k 3 T 0 λ np (t) -λ(t) X d dt ûnp (t) dt ≤ k 3 λ np -λ L 2 (0,T ;X ) d dt ûnp L 2 (0,T ;V ) , which implies lim p→∞ T 0 (φ(λ np (t), d dt ûnp (t)) -φ(λ(t), d dt ûnp (t))) dt = 0. ( 51
)
Since by ( 10), ( 11), [START_REF] Capatina | Variational inequalities and frictional contact problems[END_REF], φ(λ(t), •) is convex lower semicontinuous on V for a.e. t ∈ [0, T ], the mapping T 0 φ(λ(t), •) dt is convex lower semicontinuous on L 2 (0, T ; V ) (see, e.g. [START_REF] Brezis | Problèmes unilatéraux[END_REF]), so that

lim inf p→∞ T 0 φ(λ(t), d dt ûnp (t)) dt ≥ T 0 φ(λ(t), d dt û(t)) dt. (52)
From ( 51) and ( 52), (49) follows. Now, we prove the main strong convergence and existence result.

Theorem 2.3. Under the assumptions (1 -5), (7 -16), every convergent subsequence of Lemma 2.3, (u np , ûnp ) p , (λ np , λnp ) p , and their limits

u ∈ W 1,2 (0, T ; V ), λ ∈ W 1,2 (0, T ; X ) ∩ L 2 (Ξ T ) have the following strong convergence properties u np (t) → u(t) in V ∀ t ∈ [0, T ], (53)
λ np (t) → λ(t) in X ∀ t ∈ [0, T ], (54)
and (u, λ) is a solution to the problem Q.

Proof. In order to prove (53), we use the same method as the one that enabled to obtain (28). By [START_REF] Chau | A dynamic frictional contact problem with normal damped response[END_REF] the sequence ( Qi,n ) i=0,1,...,n-1 implies the following inequality: for every t ∈ [0, T ]

F (u n (t)), v -u n (t) -λ n (t), l 0 (v -u n (t)) L 2 (Ξ) (55) +φ(λ n (t), v -u n (t)) ≥ f n (t), v -u n (t) ∀ v ∈ V
and taking v = u, by ( 6) we derive

F (u(t)) -F (u np (t)) -λ np (t), l 0 (u(t) -u np (t)) L 2 (Ξ) (56) +φ(λ np (t), u(t) -u np (t)) ≥ f np (t), u(t) -u np (t) + α 2 u(t) -u np (t) 2 ∀ p ∈ N.
Using that F is sequentially weakly lower semicontinuous, ( 7), ( 13), the compact embeddings X ⊂ L 2 (Ξ), V ⊂ U and that for all t ∈ [0, T ] (λ np (t)) p is bounded in L 2 (Ξ) by R 0 , the previous relation implies

lim sup p→∞ α 2 u(t) -u np (t) 2 ≤ F (u(t)) + lim sup p→∞ (-F (u np (t))) + lim p→∞ | λ np (t), l 0 (u(t) -u np (t)) L 2 (Ξ) | + lim p→∞ φ(λ np (t), u(t) -u np (t)) -lim p→∞ f np (t), u(t) -u np (t) ≤ F (u(t)) -lim inf p→∞ F (u np (t)) + lim p→∞ λ np (t) L 2 (Ξ) l 0 (u(t) -u np (t)) L 2 (Ξ) + lim p→∞ k 2 λ np (t) L 2 (Ξ) u(t) -u np (t) U -lim p→∞ f np (t), u(t) -u np (t) = F (u(t)) -lim inf p→∞ F (u np (t)) ≤ 0, which proves (53). By Theorem 2.1 with G = ( λnp ) p , X = L 2 (Ξ), Û = H ι-1/2 (Ξ), Ŷ = X , r = 2, 0 < ι < 1 2 , it follows that λnp → λ in C([0, T ]; X ), (57)
so that by (41) we obtain (54). It remains to prove that (u, λ) is a solution of the problem Q. First, since λ np (t) ∈ Λ(l(u np (t))) for all t ∈ (0, T ), we have

ω κ • l(u np ) ≤ ω λ np ≤ ω κ • l(u np ), (58)
for every measurable subset ω ⊂ Ξ T and for all p ∈ N.

Using (53), ( 8), the semi-continuity of κ and κ, the relations (1), ( 2), (47), which implies the convergence property ω λ np → ω λ, and passing to limits according to Fatou's lemma, we obtain

ω κ • l(u) ≤ ω λ ≤ ω κ • l(u), (59)
for every measurable subset ω ⊂ Ξ T , which implies λ(t) ∈ Λ(l(u(t))) for almost all t ∈ (0, T ). Second, integrating both sides in ( 21) over [0, T ] and passing to the limit, by the relations (53), ( 54), ( 45), (49), it follows that for all v ∈ L 2 (0, T ; V )

T 0 F (u(t)), v(t) -u(t) dt - T 0 λ(t), l 0 (v(t) -u(t)) L 2 (Ξ) dt + T 0 φ(λ(t), v(t))dt - T 0 φ(λ(t), u(t))dt ≥ T 0 f (t), v(t) -u(t) dt.
By Lebesgue's theorem, it follows that (u, λ) is a solution of the variational inequality [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]. Assume the small deformation hypothesis and that the inertial effects are negligible.

We denote by u = u(x, t) the displacement field, by ε the infinitesimal strain tensor and by σ the stress tensor, with the components u = (u i ), ε = (ε ij ) and σ = (σ ij ), respectively. We use the classical decompositions

u = u N n + u T , u N = u • n, σn = σ N n + σ T , σ N = (σn) • n,
where n is the outward normal unit vector to Γ with the components n = (n i ). The usual summation convention will be used for i, j, k, l = 1, . . . , d.

Consider the Hilbert space V and the closed convex sets L 2 -(Γ 3 ), Λ 1 (ζ) as follows:

V = {v ∈ H 1 (Ω; R d ); v = 0 a.e. on Γ 1 }, L 2 -(Γ 3 ) := {δ ∈ L 2 (Γ 3 ); δ ≤ 0 a.e. in Γ 3 }, Λ 1 (ζ) = {η ∈ L 2 -(Γ 3 ); κ • ζ ≤ η ≤ κ • ζ a.e. in Γ 3 } ∀ζ ∈ L 2 (Γ 3 ).
Assume that in Ω a body force ϕ 1 ∈ W 1,2 (0, T ; L 2 (Ω; R d )) is prescribed, on Γ 1 the displacement vector equals zero and on Γ 2 a traction ϕ 2 ∈ W 1,2 (0, T ; L 2 (Γ 2 ; R d )) is applied. On Γ 3 , the contact between the body and a support is possible with the initial gap denoted by g 0 and the gap corresponding to the solution u denoted by [u N ] := u N -g 0 . We assume that there exists g ∈ V such that g N = g 0 on Γ 3 . Since the displacements, their derivatives and the gap are assumed small, we obtain the following unilateral contact condition at time t : [u N ] ≤ 0 on Γ 3 .

On the potential contact surface Γ 3 , the displacements and the stress vector will satisfy some contact conditions having the following form:

κ([u N ]) ≤ σ N ≤ κ([u N ]). Assume that, for all γ, δ ∈ L 2 -(Γ 3 ) and all v, w ∈ V such that γ ∈ Λ 1 ([v N ]), δ ∈ Λ 1 ([w N ]), γ -δ, v N -w N L 2 (Γ 3 ) ≤ 0. (60)
Different choices for κ, κ will give various contact and friction conditions as can be seen in the following examples. Example 1. (Friction conditions with controlled normal stress) Let M ≥ 0 be a constant and define

κ(s) = κ M (s) = 0 if s < 0, -M if s ≥ 0, κ(s) = κ M (s) = 0 if s ≤ 0, -M if s > 0.
The classical Signorini's conditions correspond formally to M = +∞.

Example 2. (Normal compliance conditions)

Various normal compliance conditions and friction laws can be obtained if one considers κ = κ = κ, where κ : R → R is some negative, decreasing, and bounded Lipschitz continuous function, so that σ N is given by the relation

σ N = κ([u N ]).
It is easily seen that these two examples verify the condition (60).

Let F ≥ 0 be the coefficient of friction, assumed to be a Lipschitz continuous function on Γ, which ensures to belong to the set of the multipliers on H 1/2 (Γ) denoted by M, see, e.g. [START_REF] Andersson | Existence results for quasistatic contact problems with Coulomb friction[END_REF], [START_REF] Rocca | Numerical analysis of quasistatic unilateral contact problems with local friction[END_REF]. Therefore the mapping

H 1/2 (Γ) v → Fv ∈ H 1/2 (Γ) is bounded with norm F M .
In order to describe the frictional contact conditions on Γ 3 , we define

∀ l ∈ V , S l := {v ∈ V ; Ω σ(v) • ε(ψ)dx = l, ψ V ∀ ψ ∈ V such that ψ = 0 a.e. on Γ 3 }, L ∈ V , L, w V = ϕ 1 , w L 2 (Ω; R d ) + ϕ 2 , w L 2 (Γ 2 ; R d ) ∀ w ∈ V , ∀ v ∈ S L , σ N (v), w Γ = Ω σ(v) • ε( w)dx -L, w V ∀ w ∈ H 1/2 (Γ),
where • , • Γ denotes the duality pairing on H -1/2 (Γ) × H 1/2 (Γ), w ∈ V satisfies wT = 0 a.e. on Γ 3 , wN = w a.e. on Γ 3 . It is easy to verify that for all v ∈ S L σ N (v) depends only on the values of w on Γ 3 and not on the choices of w having the above properties.

A contact problem with Coulomb friction for a nonlinear Hencky material

Assume that the elastic body satisfies the following nonlinear Hencky-Mises constitutive equation (see [START_REF] Nečas | Mathematical theory of elastic and elasto-plastic bodies. An introduction[END_REF], [START_REF] Zeidler | Nonlinear functional analysis and its applications[END_REF]):

σ(u) = σ(u) = (k - 2 3 µ(γ(u)))(tr ε(u)) I + 2 µ(γ(u)) ε(u),
where k is the constant bulk modulus, µ is a continuously differentiable function in [ 0, +∞) satisfying Let F 1 : V → R be defined by

0 < µ 0 ≤ µ(r) ≤ 3 2 k, 0 < µ 1 ≤ µ(r) + 2 ∂µ(r) ∂r r ≤ µ 2 , ∀ r ≥ 0, (61)
F 1 (v) = 1 2 k Ω ϑ 2 (v)dx + 1 2 Ω γ(v) 0 µ(r)dr dx ∀ v ∈ V , (67)
and J : L 2 -(Γ 3 ) × V → R be defined by

J(γ, v) = - Γ 3 F γ |v T |ds ∀ γ ∈ L 2 -(Γ 3 ), ∀ v ∈ V . ( 68
)
One can verify, see, e.g. [START_REF] Nečas | Mathematical theory of elastic and elasto-plastic bodies. An introduction[END_REF], Ch. 8, that F 1 is differentiable on V and for all u, v ∈ V

F 1 (u), v V = Ω [(k - 2 3 µ(γ(u))) ϑ(u) ϑ(v) + 2 µ(γ(u)) ε(u) • ε(v)]dx. ( 69
)
Let u 0 ∈ V , λ 0 ∈ Λ 1 ([u 0N ]) satisfy the following compatibility condition:

F 1 (u 0 ), v -u 0 V -λ 0 , v N -u 0N L 2 (Γ 3 ) (70)
+J(λ 0 , v) -J(λ 0 , u 0 ) ≥ L(0), vu 0 V ∀ v ∈ V .

We have the following variational formulation of problem P c 1 . Problem P v 1 : Find u ∈ W 1,2 (0, T ; V ), λ ∈ W 1,2 (0, T ; H -1/2 (Γ)) such that u(0) = u 0 , λ(0) = λ 0 , λ(t) ∈ Λ 1 ([u N (t)]) for almost all t ∈ (0, T ), and

F 1 (u), v -u V -λ, v N -uN L 2 (Γ 3 ) + J(λ, v) (71)
-J(λ, u) ≥ L, vu V ∀ v ∈ V a.e. on (0, T ).

The formal equivalence between the variational problem P v 1 and the classical problem (62)-(66) can be easily proved by using Green's formula. The Lagrange multiplier λ ∈ L 2 (Γ 3 ) satisfies the relation σ N = λ in H -1/2 (Γ) that is σ N (u), w Γ = λ, w L 2 (Γ 3 ) ∀ w ∈ H 1/2 (Γ).

Taking Ξ = Γ 3 , Λ = Λ 1 , V = V , U = H ι (Ω; R d ), 1 > ι > 1 2 , X = H 1/2 (Γ), F = F 1 , φ = J, f = L, and l 0 (v) = v N , l(v) = [v N ] = v N -g 0 ∀v ∈ V , it results that the problem P v 1 is a particular case of problem Q. As it is straightforward to verify the assumptions (1 -5), (7 -15), and also [START_REF] Cocou | A class of dynamic contact problems with Coulomb friction in viscoelasticity[END_REF] if F M is sufficiently small, by Theorem 2.3 we obtain the following existence result. Proposition 3.1. Under the previous assumptions and if F M is sufficiently small there exists a solution to problem P v 1 .

Definition 2 . 1 .

 21 Let Y be a reflexive Banach space, D a weakly closed set in Y , and Φ : D → 2 Y \ {∅} be a multivalued function. Φ is called sequentially weakly upper semicontinuous if z p z, y p ∈ Φ(z p ) and y p y imply y ∈ Φ(z).

Proposition 2 . 1 .

 21 ([15]) Let Y be a reflexive Banach space, D a convex, closed and bounded set in Y , and Φ : D → 2 D \ {∅} a sequentially weakly upper semicontinuous multivalued function such that Φ(z) is convex for every z ∈ D. Then Φ has a fixed point.

3

  Applications to two quasistatic contact problemsConsider an elastic body occupying the set Ω ⊂ R d , d = 2, 3, with Γ = Γ 1 ∪ Γ 2 ∪ Γ 3 , where Γ 1 , Γ 2 , Γ 3 are open, disjoint parts of Γ and meas(Γ 1 ) > 0.

c 1 :

 1 and, for all u, v ∈ V , γ(u) := γ(u, u), γ(u, v) = -2 3 ϑ(u) ϑ(v)+2 ε(u)•ε(v), ϑ(u) := tr ε(u) = div u.Consider the following quasistatic contact problem with Coulomb friction. Problem P Find u such that u(0) = u 0 and, for all t ∈ (0, T ), divσ(u) = -ϕ 1 in Ω, (62) σ(u) = σ(u) in Ω, (63) u = 0 on Γ 1 , σn = ϕ 2 on Γ 2 , (64) κ([u N ]) ≤ σ N ≤ κ([u N ]) on Γ 3 ,(65) |σ T | ≤ F |σ N | and (66) uT = 0 ⇒ σ T = -F|σ N | uT | uT | on Γ 3 .

A contact problem with local friction for a linearly elastic body

Let A denote the elasticity tensor, with the components A = (A ijkl ) satisfying the following classical symmetry and ellipticity conditions:

Consider the following elastic contact problem with Coulomb friction. Problem P c 2 : Find u such that u(0) = u 0 , satisfying

and ( 62), (64 -66) for all t ∈ (0, T ).

Let us define the bilinear and symmetric mapping a :

The form a is continuous on V × V and, since meas(Γ 1 ) > 0, by Korn's inequality is also

) satisfy the following compatibility condition:

We have the following variational formulation of problem P c 2 .

) for almost all t ∈ (0, T ), and

The Lagrange multiplier λ ∈ L 2 (Γ 3 ) satisfies again the relation

Taking Ξ, Λ, V , U , X, φ, f , l 0 , l as in 3.1 and F (v) = 1 2 a(v, v) ∀v ∈ V , we see that the problem P v 2 is a particular case of problem Q so that by using again Theorem 2.3 one obtains the following existence result. Proposition 3.2. Under the previous assumptions and if F M is sufficiently small there exists a solution to problem P v 2 .

Finally, we remark that viscoelastic or viscoplastic bulk behaviors can also be studied by using similar methods to those presented here.