1 Introduction

In this paper, we study various convergence results at three different levels. The first one concerns generic abstract problems in metric spaces. The second one concerns hemivariational inequalities in abstract reflexive Banach space. Finally, the third level concerns boundary value problems, which model the contact of an elastic body with an obstacle, the so-called foundation. For this reason, we use notation and arguments arising from the abstract well-posedness theory in the sense of Tykhonov, the theory of hemivariational inequalities and the mathematical theory of contact mechanics. A brief description of these three topics is the following.

The concept of well-posedness in the sense of Tykhonov was introduced in [29] for a minimization problem. It is based on two main ingredients: the existence and uniqueness of the solution and the convergence to the unique solution of any approximating sequence. This concept was generalized to variational inequalities in [15, 16] and to hemivariational inequalities in [8]. References in the field include [1, 13, 27, 28, 31]. An extension of this concept in the study of generic problems in metric spaces was considered in our recent paper [26]. There some abstract results have been proved and then used in the study of nonlinear equations, history-dependent equations, variational and hemivariational inequalities. In this way, a number of convergence results have been obtained.

Hemivariational inequalities represent a special class of inequalities, which arise in the study of nonsmooth boundary value problems. In contrast with the variational inequalities (which are governed by convex functions), hemivariational inequalities are governed by locally Lipschitz functions, which could be nonconvex. For this reason, their study requires prerequisites on nonsmooth analysis. Hemivariational inequalities have been introduced by Panagiotopoulos in early eighties in the context of applications in engineering problems. Later, they have been studied in a large number of papers, including the books [17,19,22]. The mathematical literature on hemivariational inequalities concerns existence, uniqueness, regularity and convergence results, among others. It grew up rapidly in the last decade, motivated by important applications in physics, mechanics and engineering sciences. A recent reference is the book [25], which provides the state of the art in the field together with relevant applications in contact mechanics.

The mathematical theory of contact mechanics deals with the study of systems of partial differential equations, which describe processes of contact with different constitutive laws, different geometries and different interface laws. Such kind of processes arise in industry and daily life, and therefore, a large effort has been put into their modeling, analysis and numerical simulations. The literature on this field is extensive. It deals with the analysis of various models of contact, which are expressed in terms of strongly elliptic, time-dependent or evolutionary nonlinear boundary value problems. References in the field include [4, 5, 10, 14, 20, 22] and, more recently, [2, 3, 17, 23,24,25]. There various existence and uniqueness results have been proved by using arguments of variational and hemivariational inequalities. Once existence and uniqueness of solutions have been established, related important questions arise, such as convergence results, which provide the link between the solutions of different models.

The aim of this paper is to introduce a functional framework in which various convergence results can be studied. More precisely, if u represents the solution of a given problem, we provide a framework in which two convergence results u′n → u and u′n′ → u can be compared.

To this end, we use arguments and ingredients on Tykhonov well-posedness, including a new result in the study of Tykhonov triples, Theorem 2. The theorem is nonstandard and represents the first trait of novelty of this current paper. Then we illustrate the use of Theorem 2 in the study of hemivariational inequalities of the form

Here X is a real reflexive Banach space,( , ) denotes the duality pairing between X and its dual X∗, K Ϲ X, A : X → X∗, j : X → R and f ϵ X∗. We assume that j is a locally Lipschitz function, and we denote by j0(u; v) the Clarke directional derivative of j at the point u in the direction v. Using Theorem 2 in the study of inequality problems of the form (1) allows us to recover, to compare and to complete our convergence results in [18, 25,26,27] obtained by using different functional methods. Presenting these results in a unified way consists the second trait of novelty of this paper. Finally, we provide a new and nonstandard model of contact for which our results hold, which consists the third novelty of this work.

The rest of the paper is structured as follows. In Section 2, we provide several elementary convergence results for inequalities of the form (1) in the particular case when j vanishes. Moreover, we establish the link between these convergence results. In Section 3, we introduce the concept of Tykhonov triple, and based on the examples in Section 2, we prove our main abstract result, Theorem 2. It provides sufficient conditions which allow us to construct a majorant for a finite family of Tykhonov triples which can be used in the proof of convergence results. In Sections 4 and 5 we apply Theorem 2 in the study of hemivariational inequalities of the form (1) and establish a general convergence result, respectively. Finally, in Section 6 we consider a mathematical model which describes the equilibrium of an elastic body in contact with a foundation. In a variational formulation, the model leads to a hemivariational inequality for the displacement field. We illustrate the use of the abstract results in Section 4 in the study of this inequality and provide the corresponding mechanical interpretations.

We end this section with the remark that in Sections 4 and 5 of this paper, X is a real reflexive Banach space, unless stated otherwise. We use ‖.‖ X and ‖.‖ X∗ for the norm on X and its dual X∗ and 0X , 0X∗ for the zero element of X and X∗, respectively. We denote by Kn → K the convergence in the sense of Mosco recalled in (M1), (M2), and ∂j will represent the Clarke subdifferential of the function j assumed to be locally Lipschitz. All the limits, upper and lower limits will be considered as n → ∞ even if we do not mention it explicitly. Moreover, the symbols “→” and “→” denote the weak and the strong convergence in various spaces, which will be specified. In addition, for a nonempty set F , we use the notation (F ) for the set of sequences whose elements belongs to F and 2F for the set of nonempty parts of F .

2 Problem statement

We start with an existence and uniqueness result for a class of hemivariational inequalities, which extends (1) and which is needed in the rest of the paper. Thus, besides the data K, A, j and f , we consider a function ϕ : X × X → R. Then we formulate the following inequality problem: find an element u ∈ K such that

Note that the function ϕ is supposed to be convex with respect to the second argument, and for this reason, following the terminology in [21], we refer to inequality (2) as a variational-hemivariational inequality. In the case when j vanishes, it represents a pure variational inequality, and in the case when ϕ vanishes, it is a pure hemivariational inequality.

In the study of (2), we consider the following assumptions:

(A1) K is nonempty, closed and convex subset of X;

(A2) A : X → X∗ is pseudomonotone and strongly monotone, i.e.,

(i) if A is bounded and un → u in X with lim sup⟨Aun, un − u⟩ ≤ 0, then

lim inf⟨Aun, un − v⟩ ≥ ⟨Au, u − v⟩ for all v ∈ X,

(ii) there exists mA > 0 such that for all u, v ∈ X,

(A3) j : X → R is such that

(i) j is locally Lipschitz,

(ii) ǁξǁX∗ ≤ c0 + c1ǁvǁX for all v ∈ X, ξ ∈ ∂j(v) with c0, c1 ≥ 0,

(iii) there exists αj ≥ 0 such that for all v1, v2 ∈ X,

(A4) f ∈ X∗;

(A5) ϕ : X × X → R is such that

(i) ϕ(η, ·) : X → R is convex lower semicontinuous for all η ∈ X,

(ii) for all η1, η2, v1, v2 ∈ X, there exists αϕ ≥ 0 such that

(A6) αϕ + αj < mA.

Concerning condition (A2), we follow [25] and recall that an operator A : X X∗ is pseudomonotone, i.e., it satisfies condition (A2)(i) if and only if it is bounded and un u weakly in X together with lim sup Aun, un u ≤ 0 implies lim Aun, un u = 0 and Aun Au weakly in X∗.

The unique solvability of inequality (2) is given by the following result [18, 25].

Theorem 1. Assume (A1)–(A6). Then inequality (2) has a unique solution u ∈ K.

We now consider the following versions of inequality (2):

where

(A7) Kn ⊂ X, fn ∈ X∗, θn ≥ 0, ϕn is a seminorm on X for each n ∈ N.

Then, assuming (A1), (A2) and (A4), we deduce the unique solvability of inequality (3). Denote in what follows by u the unique solution to this inequality and consider the following assumptions:

(B1) un is a solution to inequality (4), and θn → 0;

(B2) un is a solution to inequality (5), for each n ∈ N, there exists εn > 0 such that ϕ n(v) ≤ εnǁvǁX for all v ∈ X and, moreover, εn → 0;

(B3) un is a solution to inequality (6) and fn → f in X∗;

(B4) un is a solution to inequality (7) and Kn → K.

Recall that notation Kn → K represents a short-hand notation for the convergence in

the sense of Mosco, i.e., by definition it means that the following properties hold:

(M1) For every v ∈ K, there exists a sequence {vn} ⊂ X such that vn ∈ Kn for each n ∈ N and vn → v in X;

(M2) For each sequence {vn} such that vn ∈ Kn for each n ∈ N and vn v in X, we have v ∈ K.

Then we have the following result.

Proposition 1. Under conditions (A1), (A2), (A4) and (A7), each of assumptions (B1)– (B4) above implies the convergence

The proof of the convergence (8) under assumptions (B1)–(B3) is standard, and therefore, we skip it. We restrict ourselves to mention that it follows by using estimates based on the strong monotonicity of the operator A, (A2)(ii). The proof of the convergence (8) under assumptions (B4) is more delicate since it is based on arguments of compactness and pseudomonotonicity. Nevertheless, it follows from the proof of a more general convergence result in the study of variational-hemivariational inequality (2) obtained in [30, 32], and therefore, we skip it, too.

We now complete Proposition 1 with the following remark.

Remark 1. Assume (A1), (A2), (A4) and (A7) and let n ∈ N. Then:

(a) If ϕn is a seminorm on X, then assumption (B2) implies (B1) with θn = εn;

(b) If fn ∈ X∗, then assumption (B3) implies (B1) with θn = ǁfn − f ǁX∗ ;

(c) If fn ∈ X∗, then assumption (B3) implies (B2) with ϕn(v) = ⟨f − fn, v⟩ for all

(d) In general, if (B4) holds, we cannot find θn ≥ 0 such that (B1) holds. Similarly, we cannot find a seminorm ϕn such that (B2) is satisfied nor an element fn X∗ such that (B3) holds.

The proof of statements (a)–(c) is elementary. For instance, assume that ϕn is a seminorm and (B2) holds. We have ϕn(v) ϕn(un) ≤ ϕn(v un) ≤ εn ‖v - un‖ X for all v ϵ K, and therefore, inequality (5) implies (4) with θn = εn, which proves (a). In addition, the proof of (b) and (c) can be easily obtained, so we omit it. The proof of (d) is a consequence of the following counter-example. Let X = R, K = [−1, 1], Kn = [−1 − 1/n, 1 + 1/n], Au = u and f > 2. Then it is easy to see that Kn → K. Moreover, the solution of inequality (7) is the projection of f in the set Kn, i.e., un = 1 + 1/n. This shows that un cannot satisfy inequalities (4)–(6) since un Ɇ K.

It follows from Remark 1 that the convergence (8) under assumptions (B2) or (B3) is a consequence of the convergence (8) under assumption (B1). Also, the convergence (8) under assumption (B3) can be viewed as a consequence of the convergence (8) under assumption (B2). Moreover, the convergence (8) under assumption (B4) cannot be deduced from fact that it holds under assumptions (B1), (B2) or (B3). In other words, it follows from above that some of the convergences imply other convergences and some do not.

Assume now that (X, d) is a metric space and consider Problem , which has a unique solution denoted by u. An example of Problem is given by the problem of finding a solution to the variational inequality (3) in a reflexive space X. It follows from the results above in this section that, in general, there exist several sequences {un} , which converge to u in X. Moreover, for each n ϵ N, un is a solution of Problem n, which represents a perturbation of Problem . Examples are provided by the variational inequalities (4)–(7). Nevertheless, these convergences do not have the same status since, as shown above, some of the convergences could imply others and some of the convergences cannot be implied by others. Therefore, there is a need to compare two convergences, i.e., to establish an order relation between them. And on this matter, we formulate the following question:

(Q) Is it possible to compare two convergence results u′n → u and u′n′ → u, both in X, in the study of Problem P?

A possible answer to this question will be provided in the next section. There we also construct a relevant example of convergence un → u such that the two convergences in the statement of the question Q represent particular cases of this third convergence.

3 Tykhonov triples for abstract problems

Everywhere in this section, we assume that (X, d) is a metric space and P is an abstract mathematical object called generic “problem”. We also assume that Problem P has a unique “solution” u ϵ X, and we use the notation SP = u . Next, following our previous paper [31] and using the notation S(F ) and 2F in introduction, we recall the following definition.

Definition 1.

(a) A Tykhonov triple is a mathematical object of the form T = (I, Ω, Ϲ ), where I is a given nonempty set, Ω : I → 2X and C is a nonempty subset of the set S(I).

(b) Given a Tykhonov triple T = (I, Ω, C ), a sequence {un} ϵ S (X) is called a T -approximating sequence if there exists a sequence {θn} such that un ϵ Ω(θn) for each n ϵ N.

(c) Given a Tykhonov triple T = (I, Ω, C ), Problem P is said to be well-posed with or, equivalently, -well-posed if it has a unique solution and every T -approximating sequence converges in X to this solution.

Below in this paper, we refer to I as the set of parameters. A typical element of I will be denoted by θ. We refer to the family of sets {Ω(θ)}θ∈I as the family of approximating sets, and moreover, we say that C define the criterion of convergence. Note that T -ap-proximating sequences always exist since, by assumption, C ≠ ∅ and, moreover, for any sequence {θn} ϵϹ and any n ϵ N, the set Ω(θn) is not empty. We also remark that in most of the references in introduction, including [26], the well-posed of a problem was studied with respect to a canonical triple, and for this reason, in that papers, the terminology “approximating sequence” and ”well-posedness” was used without any reference to that Tykhonov triple. In contrast, in the current paper, we consider several Tykhonov triples in the study of the same problem, which implicitly requires the use of terminology “T -approximating sequence” and “T -well-posedness”.

Let T = (I, Ω, C) be a Tykhonov triple. We denote SP the set of sequences of X,

Next, we use Definition 1(c) and equalities (9), (10) to see that

(C) Problem P is T -well-posed if and only if ST ⊂ SP .

Moreover, the set ST of T -approximating sequences suggests us to introduce the following definition.

Definition 2. Given two Tykhonov triples 1 = (I1, Ω1, 1) and 2 = (I2, Ω2, 2), we say that:

(a) T1 and T2 are equivalent if their sets of approximating sequences are the same, i.e., ST1 = ST2 . In this case, we use the notation T1 ≈ T2.

(b) T1 is smaller than T2 if ST1 ⊂ ST2 . In this case, we use the notation T1 ≈ T2.

It is easy to see that “.” represents an equivalence relation on the set of Tykhonov triples, while “≤” defines a relation of order on the same set. Moreover, using (C), we deduce that the following statements hold:

(C1) If T1 ≈ T2, then Problem P is T1-well-posed if and only if it is T2-well posed.

(C2) If T1 ≤ T2 and Problem P is T2-well-posed, then Problem P is T1-well-posed, too.

Next, we remark that any Tykhonov triple such that Problem P is T -well-posed gives rise to a convergence result to the solution of Problem P since, following Definition 1(c), we have

(C3) If {un} ∈ ST and Problem P is T -well-posed, then un → u in X. Assume in what follows that the converse of this implication is also true, i.e.,

(C4) If un → u in X, then there exists a Tykhonov triple T such that Problem P is T -well-posed and {un} ∈ S˜T .

Note that condition (C4) is valid for the examples presented below in this paper as well as for most of the convergence results in the literatures, see, for instance [26, 27] and the references therein. Moreover, in all these examples, T can be chosen in a canonical way. Then, using implications (C3) and (C4), we can associate to each convergence result un → u in X a (canonical) Tykhonov triple T such that Problem P is T -well-posed and vice versa. This allows us to replace the study of the convergence results by the study of the Tykhonov triples. Moreover, since the implications above are governed by the set ST , using Definition 2(a), we deduce that a convergence result to the solution of Problem P is characterized by an equivalence class of Tykhonov triples with whom Problem P is well posed. In addition, since Definition 2(b) allows us to compare two (equivalence classes of) Tykhonov triples, we deduce that it allows us to compare two convergence results, too. This provides a possible answer to the question formulated at the end of Section 2.

We turn now to the construction of a relevant example of Tykhonov triple, which will be used in the next sections. Let p ϵ N, T i = (Ii, Ωi, C i) be Tykhonov triples with i = 1, . . . , p and consider the following assumptions:

(D1) For each i = 1, . . . , p, there exists ci ∈ Ii such that the sequence θi = {θni } defined by θni = ci for each n ∈ N belongs to Ci;

(D2) There exists a multifunction Ω : I1 × I2 × · · · × Ip → 2X such that

Our main result in this section is the following.

Theorem 2. Assume (D1) and (D2). Then there exists a Tykhonov triple T = (I, Ω, C) such that Ti ≤ T for all i = 1, . . . , p.

Proof. Define I = I1 × I2 × …· · · × Ip, C = C1 × C2 × …· · · × Cp, and let T = (I, Ω, C), where Ω : I → 2X is the multifunction provided by assumption (D2). We shall prove that Ti ≤ T for all i = 1, . . . , p.

Let {u1 } be a T1-approximating sequence for Problem P. Then it follows from

Definition 1(b) that there exists a sequence {θn1 } ∈ C1 such that un1 ∈ Ω1(θn1 ) for each n ∈ N. We now use the first inclusion in assumption (D2) to see that

On the other hand, assumption (D1) guarantees that the sequence θ = {θn} with θn = (θ1 , c2, c3, . . . , cp) belongs to C. Combining this result with (11), we deduce that

We now use (12) and Definition 1(b) to see that the sequence {un1 } is a T –approximating sequence for Problem P . It follows from here that ST1, CST and using Definition 2(b), we find that T1 ≤ T . A similar argument leads to the inequalities T2 ≤ T , . . . , T p ≤ T , which concludes the proof.

We now use implication (C2) and end this section with the following remark: under the assumption of Theorem 2, if Problem P is T -well-posed, then it is Ti-well-posed for each i = 1, 2, . . . , p. In terms of convergence, this remark can be stated as follows: the convergence generated by T implies each of the convergences generated by Ti with i = 1, 2, . . . , p. The importance of this abstract result arises in the fact that it can be used in order to obtain various convergence results in an unified way as we exemplify in Sections 4–6 below.

4 The case of hemivariational inequalities

In this section, we assume that X is a reflexive Banach space, and we apply Theorem 2 in the study of the hemivariational inequality (1). Therefore, Problem P under consideration is as follows.

Problem P. Find u ∈ X such that (1) holds.

Everywhere in this section, we assume that (A1)–(A4) hold, and moreover, (A6′) αj < mA.

Then, using Theorem 1 with ϕ≡ 0, we deduce that Problem P has a unique solution denoted in what follows by u. Next, we consider a function h : [0, + ) X → R such that

(E1) h(ε, u) ≥ 0 and h(0, u) = 0 for all u ∈ X, ε ≥ 0,

(E2) h(εn, un) → 0 whenever 0 ≤ εn → 0, and {un} ⊂ X is bounded, (E3) there exists Lh : [0, +∞) → R such that

(iii) |h(ε, u) − h(ε, v)| ≤ Lh(ε)ǁu − vǁX for all u, v ∈ X, ε > 0,

(iv) Lh(εn) → 0 whenever 0 ≤ εn → 0.

Moreover, we assume the following additional condition on the function j:

(E4) For all sequences {un}, {vn} ⊂ X such that un u in X, vn → v in X, we have lim sup j0(un; vn − un) ≤ j0(u; v − u).

Note that this condition can be avoided in the proof of Theorem 3 below. Nevertheless, we keep it for two reasons: first, it allows us to simplify the proof of this theorem; second, it is satisfied in the example we present in Section 6 below. Moreover, we mention that examples of functions j, which satisfy this conditions, are given in [25].

Next, we consider the Tykhonov triples T1 = (I1, Ω1, C1), T2 = (I2, Ω2, C2), T3 = (I3, Ω3, C3) defined as follows: I1 = [0, +∞), I2 = {g: g ∈ X∗}, I3 = {K: K is a nonempty closed convex subset of X},

Our main result in this section is the following.

Theorem 3. Assume (A1)–(A4), (A6′), (E1)–(E4), and let T1, T2, T3 be defined by

(13) –(15), respectively. Then there exists a Tykhonov triple T = (I, Ω, C) such that:

(a) T1 ≤ T , T2 ≤ T and T3 ≤ T ;

(b) Problem P is T -well-posed.

Proof. (a) Consider the sequences θ1 = {θn1 }, θ2 = {θn2 }, θ3 = {θn3 } defined by θn1 = 0,

θ2 = f , θ3 = K for all n ∈ N. Then, using definitions (13)–(15) and assumptions (A1), (A4), we see that 0 ϵ I1, f ϵ I2, K ϵ I3 and, moreover, θ1 ϵ Ϲ1, θ2 ϵ Ϲ2, θ3 ϵ Ϲ3.

We conclude from here that condition (D1) is satisfied with p = 3, c1 = 0, c2 = f and c3 = K.

On the other hand, define the multifunction Ω : I1 × I2 × I3 → 2X by equality

Then, using assumptions (E1)–(E3) and (13)–(16), it is easy to see that

which shows that condition (D2) is satisfied, too.

It follows from above that we are in a position to apply Theorem 2. Therefore, we conclude that the Tykhonov triple T = (I, Ω, C), defined by

satisfies the inequalities T 1 ≤ , T 2 ≤ T and T3 ≤ T, which concludes the proof of the first part of the theorem.

(b) For the second part of the theorem, we consider a T -approximating sequence of Problem P denoted by {un}. Then Definition 1(a) and (17) show that there exists a sequence θ = {θn} ∈ C with θn = (εn, gn, Kn) such that

Recall also that inclusion θ = {θn} ∈ C implies the following convergences:

We shall prove that un → u in X, and to this end, we divide the proof in three steps described below.

Step 1. The sequence {un} is bounded in X.

Let v ∈ K be a given element. Then the convergence Kn → K as n → ∞, guaranteed by (21), implies that there exists a sequence {vn} ⊂ X such that vn ∈ Kn for all n ∈ N and vn → v in K. Letting v = vn in inequality (18), we have

We now use assumption (A2)(ii) to see that

and therefore, inequality (22) yields

Moreover, note that condition (E3)(i) implies that

On the other hand, using assumption (A3) and the properties of the Clarke directional derivative, we have

We now combine inequalities (23)–(25) to see that

and using assumptions (E3)(ii) and (A6′), we find that there exists a positive constant C0, which does not depend on n such that

for n large enough. Therefore, since the sequences {vn} and {gn} are bounded in X and X∗, respectively, and A is a bounded operator, using the convergence (19) and assumption (E2), we deduce from inequality (26) that the sequence {un vn} is bounded in X. This implies that un is a bounded sequence in X, which concludes the proof of this step.

Step 2. The sequence {un} converges weakly to the solution u of Problem P .

Using the step 1 and the reflexivity of the space X, we deduce that passing to a subsequence if necessary, we have that

with some ũ ∈ X. Our aim in what follows is to prove that u ˜ is a solution to Problem P.

To this end, we remark that the convergences (21) and (27) imply that

Consider now an arbitrary element v ∈ K and a sequence {vn} ⊂ X such that vn ∈ Kn for all n ∈ N and vn → v in K. Then we use inequality (22) to see that

Passing to the upper limit in this inequality, we find that

We now use the convergences (19), (20), (27), vn v in X and assumptions (E2), (E4) to deduce that

We now combine relations (29)–(32) to find that

Next, we write

then use assumptions (A2)(i) and the convergence vn → v in X to see Aun, v - vn → 0, and therefore,

We now combine inequalities (33) and (34) to obtain that

Next, we take v = ũ in (35) and use the property j0(ũ; 0X) = 0 of the Clarke directional derivative to deduce that

Exploiting now the pseudomonotonicity of the operator A, from (27) and (36) we have

Next, from (28), (35) and (37) we obtain that u is a solution to Problem P as claimed.

Thus, by the uniqueness of the solution we find that ũ = u.

A careful analysis of the results presented above indicates that every subsequence of {un}, which converges weakly in X, has the same weak limit u. On the other hand, {un} is bounded in X. Therefore, we deduce that the whole sequence {un} converges weakly to u in X as n → ∞, which concludes the proof of this step.

Step 3. The sequence {un} converges strongly to the solution u of Problem P.

We take v = u ∈ K in both (35) and (37), then we use equality u˜ = u to obtain

which shows that lim (Aun, un u) = 0. Therefore, using the strong monotonicity of the operator A and the convergence un → u in X, we have

Taking limit at both sides of the above inequality yields un → u in X, which concludes the proof of this step.

So, we proved that any T-approximating sequence converges to the solution of Problem P

. Therefore, using Definition 1(c) it follows that Problem P is T -well-posed, which concludes the proof of the theorem.

We end this section with the following corollary, which represents a direct consequence of Theorem 3.

Corollary 1. Assume (A1)–(A4), (A6′), (E1)–(E4), and let T1, T2 and T3 be the Tykhonov triples defined by (13), (14) and (15), respectively. Then Problem P isT i-well-posed for each i = 1, 2, 3.

5 A convergence result

In this section, we provide a consequence of Theorem 3. To this end, besides the data K, A, j and f in Problem P , for each n ϵ N, we consider a set Kn, an operator An, a function ϕn and an element fn such that the following hold:

(F1) Kn is nonempty, closed and convex subset of X, and Kn → K as n → ∞;

(F2) An : X → X∗, and there exist T : X → X∗ and ωn ≥ 0 such that

(v) Anv = Av + ωnTv for all v ∈ X,

(vi) ǁTu − TvǁX∗ ≤ LT ǁu − vǁX for all u, v ∈ X with LT > 0,

(vii) ⟨Tu − Tv, u − v⟩ ≥ 0 for all u, v ∈ X,

(viii) ωn → 0 as n → ∞;

(F3) ϕn : X x X → R satisfies condition (A5) with αn = αϕn ≥ 0, and there exists δn ≥ 0 such that

(i) ϕn(η, v1) − ϕn(η, v2) ≤ δnǁηǁX ǁv1 − v2ǁX for all η, v1, v2 ∈ X,

(ii) δn → 0 as n → ∞;

(F4) α j + αn < mA;

(F5) fn ∈ X∗ and fn → f in X∗ as n → ∞; (F6) αn → 0 as n → ∞.

With this data, we consider a perturbation of the variational-hemivariational inequality (2),

together with the following problem.

Problem Pn. Find un ∈ X such that (38) holds. Our main result in this section is the following.

Theorem 4. Assume (A1)–(A4), (E4) and (F1)–(F6). Then the following statements hold:

(a) There exists a unique solution u to Problem P, and for each n ∈ N, there exists a unique solution un to Problem Pn;

(b) The sequence {un} converges to the unique solution u in X.

Proof. (a) Note that since αn ≥ 0, assumption (F4) implies (A6′). Therefore, the existence of the unique solution to Problem P is a direct consequence of Theorem 1 with ϕ 0.

Let n ϵ N. It is well known that a monotone and Lipschitz continuous operator is pseudomonotone and the sum of two pseudomonotone operators is pseudomonotone. Therefore, assumptions (A2) and (F2) show that the operator An is pseudomonotone. Moreover, it is strongly monotone with the constant mA. It follows from here that operator An satisfies condition (A2), too. On the other hand, assumption (F3) implies that (A5) holds and (F4) implies (A6), both with αϕ = αn. Recall also that Kn is a nonempty, closed and convex subset of X, fn ∈ X∗ and j satisfies (A3). All these ingredients show

that we are in a position to use Theorem 1 with Kn, An, ϕn and fn instead of K, A, ϕ and f , respectively. In this way, we deduce the unique solvability of Problem n, which concludes the proof of the first part of the theorem.

(b) For this part of the proof, we use the Tykhonov triple T = (I, Ω, C) defined by (17). Let n ∈ N and v ∈ X. We use assumption (F2)(i) and (38) to see that

Then we write

and using assumption (F2)(ii), we find that

Next, assumption (F3)(i) implies that

We now denote

and then we combine (39)–(42) to see that un satisfies inequality (18) with gn = fn and

Note that the function h satisfies assumptions (E1)–(E3) with Lh(ε) = ε(LT + 1). On the other hand, (F2)(iv), (F3)(iii) and (42) show that εn → 0, and therefore, (19) holds. Moreover, (F5) implies (20) with gn = fn, and (F1) implies (21). It follows from here that the sequence {θn} with θn = (εn, fn, Kn) belongs to the set C defined in (17), and therefore, {un} is a T -approximating sequence for Problem P. The convergence un → u in X follows from the T -well-posedness of Problem P guaranteed by Theorem 3.

We end this section with the following comments. First, the convergence of the solution of inequality (38) to the solution of inequality (2) was obtained in [30, 32] under different assumptions on the data. The aim of these papers was to obtain results on the variational-hemivariational inequality (2). In contrast, in the current paper, we focus on the hemivariational inequality (1), and we consider its perturbation (38) in order to show that its solution approaches the solution of (1) as n →∞. Such kind of results are important in various applications since they could establish the link between the solutions of different mathematical models. An example arising in contact mechanics will be provided in Section 6 below.

Next, consider the following particular versions of inequalities (18) and (38):

Then, using Theorem 3 and Theorem 4(b), we can obtain the convergence of the solution of each of inequalities (43)–(46) to the solution of inequality (1) under appropriate assumptions. The convergence of the solution of (44) and (45) to the solution of inequality (1) stands for a continuous dependence result of the solution of Problem P with respect to the element f and the operator A, respectively. Moreover, the convergence of the solution of (46) to the solution of inequality (1) extends a result proved in [27] in the particular case when h(ε, u) = ε.

The convergence of the solution of inequality (43) to the solution of (1) shows the continuous dependence of the solution of Problem P with respect to the convex set K. This result is important in the numerical analysis of the hemivariational inequality (1).

There assumption Kn → K shows that Kn represents an approximation of the set K in the sense used in [6, 7]. The approximation is external if Kn C K and is internal if Kn C K. The internal approximation of hemivariational inequalities with the choice Kn = Xn ᴖ K, where Xn is a finite dimensional subspace of X, was used in [12], for instance. More details on abstract approximation of hemivariational inequalities can be found in [9, 11, 12]. There, besides the convergence of the solution of the discrete scheme (38) to the solution of Problem P, various error estimates have been obtained.

6 A contact problem

Hemivariational inequalities of the forms (1), (2) arise in the study of a number of contact problems with elastic materials. Details on the construction and variational analysis of these problems, including some technical results, can be found in our books [17, 23, 25], and therefore, we skip them. We restrict in this section to illustrate the convergence result provided by Theorem 4 in the study of two contact models.

Let d ϵ {2, 3} . We denote by Sd the space of second-order symmetric tensors on Rd and use the notation “ ”, ‖.‖ for the inner product and norm on Rd and Sd. The zero element of the spaces Rd and Sd will be denoted by 0. Also, consider a domain Ω Rd with smooth boundary Γ divided into three measurable disjoint parts Γ1, Γ2 and Γ3 such that meas(Γ1) > 0 and denote by ν the unit outward normal vector to Γ .

We use the standard notation for Sobolev and Lebesgue spaces associated to Ω and Γ , and for an element v ∈ H1(Ω)d, we still write v for the trace of v to Γ . In addition, we consider the space V = {v ∈ H1(Ω)d: v = 0 on Γ∫1}, which is real Hilbertassociated norm ǁ·ǁV . Here and below ε represents the symmetric part of the gradient of v. We denote by 0V the zero element of V and, for an element v ∈ V , vν and vτ will represent its normal and tangential components on Γ given by vν = v · ν and vτ = v − vνν, respectively. Finally, V ∗ represents the dual of V , ⟨·, ·⟩ denotes the duality pairing between V ∗ and V , and ǁγǁ is norm of the trace operator γ : V → L2(Γ )d. Recall that the following inequality holds:

The first contact model we consider in this section is constructed by using the data F, f0, f2, k and p assumed to satisfy the following conditions:

(G1) F : Sd → Sd and

(ix) there exists LF > 0 such that ǁF(ε1) − F(ε2)ǁ ≤ LF ǁε1 − ε2ǁ for all ε1, ε2 ∈ Sd,

(x) there exists mF > 0 such that (F(ε1)−F(ε2))·(ε1−ε2) ≥ mF ǁε1−ε2ǁ2 for all ε1, ε2 ∈ Sd,

(xi) F(0) = 0;

(G2) f0 L2(Ω)d, f2 L2(Γ2)d; (G3) k ≥ 0;

(G4) p : R → R is a continuous function such that

(i) |p(r)| ≤ c¯0 + c¯1|r| for all r ∈ R with c¯0, c¯1 ≥ 0,

(ii) (p(r1) − p(r2))(r2 − r1) ≤ αp|r1 − r2|2 for all r1, r2 ∈ R with αp ≥ 0.

Let q : R → R is the function defined by

and introduce the following notations:

for all u, v ∈ V . Then we consider the following problem.

Problem S. Find a displacement field u ∈ K such that

Following the arguments in [25], it can be shown that Problem represents the variational formulation of a mathematical model, which describes the equilibrium of an elastic body in frictionless contact with a foundation under the action of external forces. It is assumed that the foundation is made of a rigid material covered by a layer of deformable material. The function F is a constitutive function, while f0 and f2 denote the density of body forces and applied tractions, which act on the body and the surface Γ2, respectively. Finally, k represents the thickness of the deformable layer, and p is a normal compliance function, which describes its reaction towards the elastic body.

For the second contact model, we consider a set B such that

(H1) B is a closed convex subset of Sd and 0 ∈ B.

Moreover, for each n ∈ N, we assume that ωn, µn, f0n, f2n, kn are given and satisfy

(H2) ωn > 0, ωn → 0; (H3) µn > 0, µn → 0;

(H4) f0n ∈ L2(Ω)d, f2n ∈ L2(Γ2)d;

(H5) f0n → f0 in L2(Ω)d, f2n → f2 in L2(Γ2)d; (H6) kn ≥ 0, kn → k;

(H7) (αp + µn)ǁγǁ2 < mF .

Note that here we consider only the homogeneous case, for simplicity. Nevertheless, we remark that the results below can be easily extended to the case when the functions F , p, as well as ωn, µn, depend on the spatial variable x ϵ Ω U Γ .

We now introduce the following notations:

for all u, v ϵ V . Here and below PB : Sd → B denotes the projection operator on the set B, and r+ represents the positive part of r, i.e., r+ = max {r, 0} . Then, for each n ϵ N, we consider the following problem.

Problem Sn. Find a displacement field un such that

Note that, in contrast to Problem , Problem n represents the variational formulation of a mathematical model, which describes the equilibrium of an elastic body in frictional contact. The friction is described with the function ϕn in which µn represents the coefficient of friction. Moreover, in the statement of Problem n, the constitutive law is perturbed by using the elasticity coefficient ωn and the projection on the convex set B, the densities of body forces and surface tractions are replaced by their perturbation f0n and f2n, respectively, and the thickness of the deformable material is replaced by its perturbation kn.

Our result in the study of Problem S and Problem Sn is the following.

Theorem 5. Assume (G1)–(G4), (H1)–(H6). Then the following statements hold:

(a) There exists a unique solution u to Problem S, and for each n ∈ N, there exists a unique solution un to Problem Sn.

(a) The sequence {un} converges to the unique solution u in V .

Proof. We use Theorem 4 with X = V , and to this end, we have check the validity of conditions (A1)–(A4), (E4) and (F1)–(F6). First, we remark that condition (A1) is obviously satisfied. Moreover, the operator A defined by (50) satisfies condition (A2). Indeed, using assumption (G1)(i), we find that

This implies that

and shows that A is Lipschitz continuous. On the other hand, using assumption (G1)(ii) yields

This shows that condition (A2)(ii) is satisfied with mA = mF . Since A is Lipschitz continuous and monotone, it follows that A is pseudomonotone, and therefore, (A2)(i) holds.

On the other hand, it is obvious to see that the function q defined by (48) is a locally Lipschitz function. Moreover, using the properties of p and equality q0(r; s) = p(r)s, valid for all r, s ϵ R, it follows that the function q satisfies condition (A3) on X = R. Therefore, using the arguments in [25, p. 219], we deduce that the function j given by (51) satisfies (A3) with αj = αp ‖γ‖ 2. In addition, condition (A4) is guaranteed by assumption (G2). Next, we remark condition (F1) is a direct consequence of definitions (49), (52) and assumption (H5).

Consider now the operator T : V → V ∗ given by

Then using the nonexpansivity of the projection yields

Therefore, assumption 6 and notation (53) imply that condition (F2) is satisfied.

Next, we apply the trace inequality (47) to see that the function ϕn defined by (54) satisfies condition (F3) with αn = δn = µn ‖γ‖ 2, and therefore, assumption (H6) shows that condition (F4) holds. Moreover, we note that assumptions (H3) and (H4) imply condition (F5) and, since αn = µnǁγǁ2, (H2) shows that (F6) holds, too.

Finally, condition (E4) follows from the compactness of the trace operator γ. Indeed, since the function q is regular, Lemma 8 in [25] guarantees that

Therefore, if un u and vn → v in V , using (55), we deduce that

which shows that condition (E4) holds, as claimed.

It follows from above that we are in a position to use Theorem 4 in order to conclude the proof of Theorem 5.

We end this section with the following mechanical interpretations. First, Theorem 5 provides the unique weak solvability of two contact models: a frictionless model in which the stress tensor σ satisfies the elastic constitutive law σ = F ε(u) and a frictional model in which the stress tensor σ satisfies the elastic constitutive law F σ = ε(u) + ω(ε(u) PBε(u)). Second, it establishes the link between the weak solutions of these contact models constructed by using different mechanical assumptions. Third, it provides the continuous dependence of the weak solution of the first model with respect to the density of body forces, the surface tractions and the thickness of the deformable layer. All these ingredients show that, in addition of the mathematical interest in Theorem 5, it is important from mechanical point of view.

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