Penalty method for a class of differential nonlinear system arising in contact mechanics

The main goal of this paper is to study a class of diﬀerential nonlinear system involving parabolic variational and history-dependent hemivariational inequalities in Banach spaces by using the penalty method. We ﬁrst construct a penalized problem for such a nonlinear system and then derive the existence and uniqueness of its solution to obtain an approximating sequence for the nonlinear system. Moreover, we prove the strong convergence of the obtained approximating sequence to the solution of the original nonlinear system when the penalty parameter converges to zero. Finally, we apply the obtained convergence result to a long-memory elastic frictional contact problem with wear and damage in mechanics. the studies on variational inequalities (VIs) and hemivariational inequalities


w(t), u(t)),
(1.1) Moreover, they gave a unique solvability result for (1.1) by using Banach's fixed-point theorem and applied it to the long-memory elastic frictional contact problem with wear and damage in mechanics.
We would like to mention that (1.1) is an extended model that can be used to describe many real problems such as the long-memory elastic frictional contact problem with wear and damage in mechanics, engineering operation research, network equilibrium problems, and so on [2, 4-8, 23, 30]. Moreover, to choose suitable spaces and maps, many known differential variational inequalities (DVIs) and differential hemivariational inequalities (DHVIs) can be considered as special cases of (1.1) (see, for example, [12-15, 18, 25, 26, 29] and the references therein).
Among the studies on variational inequalities (VIs) and hemivariational inequalities (HVIs), constructing approximating sequences for their solutions and further discussing their convergence analysis are crucially important [10,11]. It is well known that the penalty method is a kind of efficient approximating method forvarious problems. It is also constantly used for the study of VIs and HVIs (see, for example, [3,21,28,31]). Due to the close relationship with VIs and HVIs, differential variational inequalities (DVIs) and differential hemivariational inequalities (DHVIs) are studied by employing the penalty method, such as Liu and Zeng [16,17] and Weng et al. [27]. As the generalization of DVIs and DHVIs, the differential variational-hemivariational inequalities (DVHVIs) have drawn the attention of researchers in operations research and contact mechanics. With the penalty method, Tang et al. [25], Liu et al. [16], and Lu et al. [19] recently studied different DVHVIs, obtained their convergence results, and gave the corresponding applications in contact mechanics. However, to the best of our knowledge, there are no results in the literature concerning the penalty method for (1.1). The motivation of the present work is to make an attempt in this direction.
The main goal of this paper is to obtain a convergence result for (1.1) by employing the penalty method. The main contributions of this paper are twofold. First, we construct a penalized problem for (1.1) and show a convergence result, i.e., the solution of (1.1) can be approached as the penalty parameter converges to zero. Secondly, we apply the obtained convergence result to the long-memory elastic frictional contact problem with wear and damage in mechanics.
The rest of the paper is structured as follows. In Sect. 2, we introduce some preliminary materials that will be used in the following sections. In Sect. 3, we construct approximating sequences of solutions to (1.1) by the penalty method and derive its convergence. Finally, in Sect. 4, we apply the obtained convergence result to a long-memory elastic frictional contact problem with wear and damage in mechanics.

Preliminaries
Let (X, · X ) be a real Banach space with its dual X * and ·, · X * ×X denote the duality pairing between X * and X. In this section, we recall some known definitions and lemmas that will be used subsequently (see [20,22] for more details). Moreover, the symbols "→" and " " represent the strong and weak convergence in various spaces, respectively. Definition 2.1 A functional j : X → R is lower semicontinuous if and only if for any convergence sequence {u n } ∞ n=1 ⊂ X satisfying u n → u ∈ X, one has lim inf n→∞ j(u n ) ≥ j(u).
there exists a point u ∈ X such that j(u) < +∞.

Definition 2.3
Let j : X → R ∪ {+∞} be a proper, convex and lower semicontinuous functional. Define the convex subdifferential of j at u by

Definition 2.4
Let j : X → R be a locally Lipschitz function. The Clarke directional derivative of j at x in the direction v ∈ X is given by The Clarke subdifferential of j at x is a subset of the dual space X * defined by For a set-valued operator A : X → 2 X * , the graph of A is denoted by G(A), i.e.,

Definition 2.5 A set-valued operator
Moreover, a monotone operator A is called maximal monotone if for any (u, u * ) ∈ X × X * satisfying For a proper, convex and lower semicontinuous functional j : X → R ∪ {∞}, it is well known that ∂ C j : X → 2 X * is maximal monotone. Definition 2.6 A single-valued operator A : X → X * is said to be (1) strongly monotone, if there exists m A > 0 such that (2) bounded, if A maps bounded sets of X into bounded sets of X * ; (3) pseudomonotone, if it is bounded and u n u in X with lim sup n→∞ Au n , u nu X * ×X ≤ 0, Definition 2.7 An operator P : X → X * is said to be a penalty operator of the set K ⊂ X if P is bounded, demicontinuous, monotone, and K = {x ∈ X | Px = 0 X * }, where 0 X * represents the zero element of X * .

Convergence result for (1.1)
In this section, we first use the penalty method to construct a penalized problem of (1.1) and show that the penalized problem has a unique solution by employing Theorem 3.1 of Chen at al. [4]. Then, we show a convergence result that the solution of (1.1) can be approximated by the penalized problem as the penalty parameter converges to 0. We assume that (V , H, V * ) and (Y , Y 1 , Y * ) are two Gelfand triplets of Banach spaces that have continuous, compact, and dense embeddings, M is the embedding operator of V → H, M * is the adjoint operator of M, and M and M * are the norms of M and M * , respectively. K V is a convex subset of V . Let P : V → V * be a penalty operator of K V . In order to develop the approximation procedure of (1.1), we need to construct the penalized problem of (1.1). For any given ρ > 0, the penalized problem of (1.1) can be constructed as follows.
Remark 3.1 We note that in Problem 3.1, we can consider the penalty operators for both K V and K Y . Since our main interest is to provide tools in analyzing Problem 3.1, we restrict ourselves to study penalty operators for K V . The case in which K Y is considered can be solved likewise.
In order to study Problem 3.1, we need the following assumptions on the data.
(b) For any given t ∈ I, A(t, ·) is hemicontinuous, pseudomonotone, and strongly monotone with m A > 0 on V , i.e., (c) There exists ∈ L 2 (I; R + ) such that (c) There exist α 0 > 0 and α 1 > 0 such that H(F): The operator F : The functional a : Y × Y → R is a continuous bilinear symmetric coercive functional and there exist a 1 ∈ R and a 2 > 0 such that

Remark 3.3 H(j)(c) is equivalent to the following condition
First, to solve the history-dependent hemivariational inequality in Problem 3.1, we consider the following auxiliary problem.

Problem 3.2 For any given
Since P is bounded, demicontinuous, monotone, and is continuous for any given v ∈ V and A ρ (t, ·) is hemicontinuous, pseudomonotone, and strong monotone with A ρ (t, 0 V ) = 0 V * for all t ∈ I. This shows that Problem 3.2 satisfies all the hypotheses of Lemma 3.2 in [4] and so Problem 3.2 has a unique solution u ρwζ ∈ C(I; V ).
(ii) For fixed η ∈ C(I; K V ), we consider the auxiliary problem for (3.5) as follows: find a map u ρwζ η : (0, T) → V such that, for any t ∈ I and any v ∈ V , where f η is defined by It follows from the strong monotonicity of A that for all t ∈ [0, T]. As P is monotone, Pv = 0 for all v ∈ K V and u 0 ∈ K V , one has for all t ∈ [0, T]. Thus, Remark 3.3 implies that From (3.8), (3.9), and H(j)(b), we have Moreover, A is pseudomonotone, so A is bounded, then there exists a constant N , such that and H(B)(c) implies that which implies that the sequence {u ρwζ (t)} ρ>0 is uniformly bounded. Therefore, for any Next, we show thatũ ∈ C(I; K V ). In fact, according to the monotonicity of A, we have Taking v =ũ(t) into (3.13), one has Combining H(j), the continuity of f η and the compactness of M, we have Because of Lemma 2.1, P is pseudomonotone, it follows from (3.13) and (3.14) that Since v ∈ V is arbitrary, we know that Pũ(t) = 0 and soũ(t) ∈ K V . Moreover, according to (3.7) and Pv = 0 for all v ∈ K V , one has Taking v =ũ(t) in (3.15) and passing to the upper limit as ρ → 0, we have Moreover, the pseudomonotonicity of A implies that Passing to the upper limit as ρ → 0 in (3.15), we obtain

Combining (3.16) and (3.17), we have
Since (3.6) has a unique solution, we know thatũ(t) = u wζ (t) and soũ ∈ C(I, K V ). Finally, we show the strong convergence of {u ρwζ (t)}. Indeed, because {u ρwζ (t)} is bounded and for any weakly convergent subsequence of {u ρwζ (t)} converges weakly to the same limit u wζ (t), by Theorem 1.20 in [24], we know that the whole sequence {u ρwζ (t)} converges weakly to u wζ (t) for any t ∈ I. On the other hand, using the monotonicity of P, one has Similar to the proof of (3.15), we have Taking v = u wζ (t) in (3.18) and (3.19), and then passing to the limit as ρ → 0, one has Using u ρwζ (t) u wζ (t) in V as ρ → 0, it follows from the strong monotonicity of A that for all t ∈ I. Consequently, we conclude for each t ∈ I, u ρwζ (t) → u wζ (t) in V as ρ → 0. Next, we consider the following auxiliary problem that includes a history-dependent hemivariational inequality and a differential equation in Problem 3.1.

Problem 3.3
For any given ρ > 0, ζ ∈ H 1 (I; Y 1 ) ∩ L 2 (I; Y ) and f ∈ C(I; V * ), consider the following problem: find u ρζ : I → V and w ρζ : I → W such that, for any t ∈ I,
(ii) Consider an operator : C(I; W ) → C 1 (I; K V ) defined as follows: Then, by the proof of Lemma 3.3 in [4], we know that has a unique fixed point w ζ (t). It follows from H(F) that Now, Gronwall's inequality yields Since for each s ∈ I, u ρwζ (s) → u wζ (s) in V as ρ → 0 and u ρwζ , u wζ ∈ C(I; V ), one has Letting u wζ (t) = u ζ (t) and u ρwζ (t) = u ρζ (t), we can conclude that Finally, we only need to solve the following parabolic variational inequality.

Problem 3.4
For any given ρ > 0, consider the following problem: find ζ ρ : I → K Y such that, for all t ∈ I, with ζ ρ (0) = ζ 0 . 1,9]) Suppose that condition H(a) holds. Then, for any given λ ∈ L 2 (I; Y 1 ), there exists a unique ζ ∈ H 1 (I; Y 1 ) ∩ L 2 (I; Y ) such that with d 1 > 0. (ii) ζ ρ converges strongly to ζ as ρ → 0, where ζ is the the unique solution of the following problem: find ζ : I → K Y such that, for all t ∈ I,

Theorem 3.1 Suppose that the assumptions H(A), H(B), H(j), H(F), H(φ), and H(a) hold and m A
Then, one has the following conclusions: (i) for any given ρ > 0, Problem 3.1 has a unique solution Proof Let (u ρζ , w ρζ ) be the same as in Lemma 3.2, ζ ρ be the same as in Lemma 3.4, and (ζ ρ , u ρζ , w ρζ ) = (ζ * , u * , w * ) be the same as in Remark 3.2. Then, it is easy to see that the conclusions (i) and (ii) are true. This finishes the proof.

An application
In this section, we use the abstract results obtained in Sect. 3 to study the long-memory elastic frictional contact problem with wear and damage. To this end, we first recall some notations. Let S d denote the second-order symmetric tensors on R d . For any given σ , τ ∈ S d , define σ ij τ ij and τ = √ τ · τ . We use notations u = (u i ), σ = (σ ij ) and ε(u) = (ε ij (u)) = ( 1 2 (u i,j + u j,i )), i, j = 1, 2, . . . , d to denote the displacement vector, the stress tensor and the linearized strain tensor, respectively, where u i,j := ∂u i ∂x j . Here and below, the spatial derivative is defined in the sense of distribution. Let be a bounded domain in R d (d = 2, 3) with Lipschitz continuous boundary := ∂ . Let ν denote the unit outward normal vector defined a.e. on . The normal and tangential components of stress field σ and displacement field u on are denoted by σ ν = (σ ν) · ν, u ν = u · ν, σ τ = σ νσ ν ν and u τ = uu ν ν, respectively.
Consider a viscoelastic body that occupies . The boundary can be divided into three disjoint measurable parts 1 , 2 , and 3 with meas( 1 ) > 0. We are interested in the evolution of the body on the time interval I := [0, T] with T > 0. We also use the following abbreviations to simplify the notations Q = × I, = × I, i = i × I, i = 1, 2, 3. The time partial derivative for a function f (x, t) is denoted byḟ (x, t). For the sake of simplicity, we do not mention the dependence of different functions on variable x.
Thus, the long-memory elastic frictional contact problem with wear and damage can be modeled as follows (see [4]).
Then, the variational formulation of Problem 4.1 can be described as follows (see [4]). ≥ f(t), vu(t) V * ×V , ∀v ∈ K V , (4.14) Now, we turn to introduce the following penalized problem concerning Problem 4.1.