ON A CLASS OF SADDLE POINT PROBLEMS AND CONVERGENCE RESULTS

We consider an abstract mixed variational problem consisting of two inequalities. The ﬁrst one is governed 
 by a functional φ, possibly non-diﬀerentiable. The second inequality is governed by a nonlinear term depending on a non negative 
 parameter ǫ. We study the existence and the uniqueness of the solution by means of the saddle point theory. In addition to existence 
 and uniqueness results, we deliver convergence results for ǫ → 0. Finally, we illustrate the abstract results by means of two examples 
 arising from contact mechanics.


Introduction
In the present paper we study the following variational problem.
Problem 1. Given f, h ∈ X and ≥ 0, find u ∈ K ⊆ X and λ ∈ Λ ⊆ Y such that (h 2 ) The form a : X × X → R is symmetric, bilinear, continuous (of rank M a > 0) and X−elliptic (of rank m a > 0); (h 3 ) The form b : X × Y → R is bilinear, continuous (of rank M b > 0); (h 4 ) The functional φ : X → R + is Lipschitz continuous (of rank L φ > 0) and convex; (h 5 ) Λ is a closed, convex subset of Y that contains 0 Y ; (h 6 ) K is a closed, convex subset of X that contains 0 X . We can associate to Problem 1 the following functional: Assuming that Problem 1 has a solution (u , λ ) ∈ K × Λ, then this solution is a saddle point of the functional L . Conversely, assuming that the functional L has a saddle point (u , λ ) ∈ K × Λ, then this saddle point verifies Problem 1. Thus, Problem 1 can be called saddle point problem.
The present study is motivated by mechanical and numerical reasons. If > 0, Problem 1 is a saddle point problem with penalty term. Saddle point problems with penalty term can arise in elasticity theory, see, e.g., [3], pages 137-138. If = 0, Problem 1 can be seen as a generalization of Problem (S) in [3], page 129. Also, if = 0, Problem 1 can be seen as a mixed variational formulation with Lagrange multipliers for a class of contact problems; see, e.g., [11]. Due to the interest on the mixed variational formulations via Lagrange multipliers in contact mechanics, a lot of work has been done in the last decade. For recent related papers we refer to, e.g., [2,7,12,13,14,15,18].
Notice that L is strictly convex in the first argument and concave in the second one. If > 0, L is also strictly concave in the second argument; algorithms of type multi-level can be envisaged in this case in order to approximate the solution (u , λ ).
The present work focuses on existence and uniqueness results as well as on the convergence of the sequence (u , λ ) when → 0. Then, two examples are delivered. Both examples are related to the weak solvability via mixed variational formulations with Lagrange multipliers of a contact model with two-contact zones, by considering a deformable body in unilateral frictionless contact on a part of the boundary and in bilateral frictional contact on another one. The "differentiable case" is related to the description of the friction by means of a regularized friction law. To simplify the presentation, the examples we deliver in the present work are only "inspired" from realistic models. By using some specific function spaces in 3D contact mechanics, other examples, more realistic models from the physical point of view, can be delivered.
The reader can consult [9,10] for helpful techniques in the saddle point theory. However, for the convenience of the reader, we recall here two main tools: the definition of the saddle point and an existence theorem. Definition 1. Let A and B be two non-empty sets. A pair (u, λ) ∈ A × B is said to be a saddle point of a functional L : Theorem 1. Let (X, (·, ·) X , · X ), (Y, (·, ·) Y , · Y ) be two Hilbert spaces and let A ⊆ X, B ⊆ Y be non-empty, closed, convex subsets. Assume that a functional L : A × B → R satisfies the following conditions: v → L(v, µ) is convex and lower semi-continuous for all µ ∈ B, µ → L(v, µ) is concave and upper semi-continuous for all v ∈ A.

Moreover,
Then, the functional L has at least one saddle point.
The proof of Theorem 1 can be found in [9]. The rest of the paper has the following structure. In Section 2 we prove the existence of at least one solution (u , λ ) ∈ K × Λ assuming that the functional φ is non-differentiable. The uniqueness in the first argument is also obtained. Furthermore, we give some convergence results for → 0. In Section 3 the study is devoted to the case when φ is a Gâteaux differentiable functional. This additional hypothesis allows us to obtain uniqueness as well as strong convergence in the second component of the pair solution. In Section 4 we illustrate the abstract results through two examples related to contact models involving multi-contact zones.

The non-differentiable case
In this section we deliver existence, uniqueness and convergence results under the hypotheses (h 1 )-(h 6 ), assuming that the functional φ is non-differentiable. Let us start with an auxiliary result. Lemma 1. If (u , λ ) ∈ K × Λ is a solution of Problem 1, then this pair is a saddle point of the functional L : K × Λ → R, Conversely, assuming that the functional L has a saddle point (u , λ ) ∈ K × Λ, then this pair is a solution of Problem 1.
Proof. Let (u , λ ) ∈ K × Λ be a solution of Problem 1. By summing (1.2) On the other hand, Therefore, (u , λ ) is a saddle point of L . Let us assume now that (u , λ ) ∈ K × Λ is a saddle point of the functional L . Since L (u , µ) ≤ L (u , λ ) for all µ ∈ Λ, by the definition of L we immediately get (1.2). In addition, because then, Dividing by t > 0 and then passing to the limit as t → 0, we obtain (1.1).
Proof. In order to obtain the existence part, we consider two cases.
1. K ⊆ X is a bounded subset. By Theorem 1, we immediately deduce that the functional L has at least one saddle point (u , λ ) ∈ K × Λ. So, Problem 1 has at least one solution.
2. K ⊆ X is an unbounded subset.
We have to verify that Thus, Passing to the limit as v X → ∞, we get (2.1). Using Theorem 1 and Lemma 1 we obtain the existence of a solution (u , λ ) ∈ K × Λ of Problem 1 in this case too.
then Problem 1 has a solution, unique in its first argument. Moreover, Proof. Let us consider the following two cases. 1. Λ ⊆ Y is a bounded subset. As in Theorem 2 we obtain the existence of a solution (u , λ ) ∈ K × Λ of Problem 1 and the uniqueness of the first argument u ∈ K.
Let µ ∈ Λ and let u µ ∈ K be the unique solution of the variational inequality of the second kind, where f µ ∈ X is defined by Riesz's representation theorem as follows, Since u µ minimizes the functional Taking v = 0 X in (2.5) and then summing with 1 2 a(u µ , u µ ), we are lead to Due to the inf-sup property of the form b, we get where c > 0 depends on the positive constant α from (h 7 ). Consequently, we can write, wherec(α, m a , M a ) > 0. We pass to the limit as µ Y → ∞ in this last relation to obtain (2.4). By Theorem 1 we get the existence of a saddle point (u , λ ) ∈ K × Λ of the functional L and then, by Lemma 1, we deduce that this saddle point verifies Problem 1. Furthermore, as in the previous theorem we obtain the uniqueness of the first component of the pair solution of Problem 1, u ∈ K.
Let us prove now (2.2) and (2.3). To this end in view, we take v = 0 X in (1.1) and µ = 0 Y in (1.2). Hence, Combining these last relations, we get Consequently, where By the inf-sup property of the form b, we get Combining (2.7) with (2.9) and taking Therefore, Let us draw the attention to the case = 0. Problem 1 leads us to the following particular problem.
Problem 2. For given f, h ∈ X, find u 0 ∈ K ⊆ X and λ 0 ∈ Λ ⊆ Y such that, for all v ∈ K and µ ∈ Λ we have, If K = X, then this problem was already studied, see, e.g., [5]. Let us introduce a new hypothesis as follows.
(h 8 ) If (u n ) n ⊂ X such that u n → u in X as n → ∞ and (λ n ) n ⊂ Y such that λ n λ in Y as n → ∞, then b(u n , λ n ) → b(u, λ) as n → ∞.
Proof. Let > 0, let (u , λ ) ∈ K × Λ be a solution of Problem 1 and let (u 0 , λ 0 ) ∈ K×Λ be a solution of Problem 2, unique in their first arguments, u ∈ K and u 0 ∈ K, respectively. Due to (2.2) and (2.3), passing to a subsequence, we deduce that there exists u ∈ K such that u u and there exists λ ∈ Λ such that λ λ, as → 0. Let > 0. We take v = u in (2.11) and v = u 0 in (1.1) to obtain, (2.13) Recall that (u 0 , λ 0 ) verifies (2.11)-(2.12), because, in this proof, as mentioned in the beginning, (u 0 , λ 0 ) denotes a solution of Problem 2. Setting now µ = λ in (2.12) and µ = λ 0 in (1.2), we get (2.14) Combining (2.13)-(2.14) and taking into account the X-ellipticity of the form a, we have By passing to the limit as → 0 in the relation above, we obtain that u → u 0 . Due to the uniqueness of the limit, we conclude that u = u 0 .
Passing now to the limit as → 0 in Problem 1 and keeping in mind (h 8 ), we deduce that (u 0 , λ) ∈ K × Λ verifies Problem 2. We conclude the proof of the theorem by considering λ 0 = λ.

The differentiable case
In this section we pay attention to the case when φ is a differentiable functional. Precisely, in addition to (h 1 )-(h 8 ), we admit the following new hypothesis.
Problem 3. Given f, h ∈ X and ≥ 0, find u ∈ K ⊆ X and λ ∈ Λ ⊆ Y such that, for all v ∈ K and µ ∈ Λ, we have: . Theorem 5. Assume that the hypotheses (h 1 )-(h 7 ) and (h 9 ) hold true. If, in addition, Λ ⊆ Y is a bounded subset, then Problem 1 has an unique solution. Moreover, where Proof. Theorem 2 ensures the existence of a solution (u , λ ) ∈ K × Λ of Problem 1 as well as the uniqueness of its first component, u ∈ K. We proceed by proving the uniqueness in the second component, λ ∈ Λ. Let (u 1 , λ 1 ), (u 2 , λ 2 ) ∈ K × Λ be two solutions of Problem 1 and hence of Problem 3. For all v ∈ K and i ∈ {1, 2}, we can write According to the inf-sup property of the form b, we obtain Since u 1 = u 2 , it results that λ 1 = λ 2 . Let us prove (3.2). Firstly, we observe that with p 4 , p 5 , p 6 > 0. Next, by (3.1) and (h 7 ), we get Thus, Setting p 4 = p 6 = ma 2 and p 5 =
Proof. The existence of at least one solution is ensured by Theorem 3. For the uniqueness part we use similar arguments with those used in the proof of Theorems 2 and 5, respectively. Furthermore, as in Theorem 5, (3.2) take place.
If = 0, Problem 3 leads to the following simplified problem.
Next, we focus on the first convergence result of this section.
With similar arguments we can prove the following convergence result.

Examples
In this section we illustrate the previous abstract results by two examples. Let us consider X = {v ∈ H 1 (Ω), γ v = 0 a.e. on Γ I }, where Ω ⊂ R 2 is a bounded domain with smooth boundary Γ partitioned in three parts Γ I , Γ II , Γ III with meas(Γ i ) > 0, i ∈ {I, II, III}. Herein γ : H 1 (Ω) → L 2 (Γ ) is the Sobolev's trace operator. We introduce also the Hilbert space, Let Y = S be the dual of the Hilbert space S. The space Y is a Hilbert space too. Thus, (h 1 ) is fulfilled.
For Lebesgue and Sobolev spaces we use standard notation; the reader can consult, e.g., [1,4,16]. We also sent the reader to, e.g., [4,6,8] for details on Hilbert spaces. Let us introduce the bilinear forms By · we denote the inner product on R 2 and by ·, · we denote the duality product between Y and S. The form a in (4.1) verifies (h 2 ) with M a = ξ L ∞ (Ω) and m a = ξ * . By using the trace theorem and an inequality of Poincaré type we deduce that (h 3 ) holds true. The hypothesis (h 7 ) holds true due to the properties of the trace operator and its right invers. Moreover, keeping in mind (4.2) we immediately verify (h 8 ). Let K = X and let Λ be the nonempty, closed, convex set of Lagrange multipliers, where K 1 = {v ∈ X, γv ≤ 0 a.e. on Γ II }. Clearly, (h 5 ) and (h 6 ) are fulfilled.
To end, it is worth to mention that the previous examples were "inspired" from contact models. Due to the interest into the study of the interaction between bodies, recently, several papers were devoted to the mathematical analysis of the contact models; see, for instance, [19,20] and the references therein.