EVOLUTION EQUATIONS AND doi:10.3934/eect.2020044 CONTROL THEORY Volume 9, Number 4, December 2020 pp. 1073–1087

HISTORY-DEPENDENT DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH APPLICATIONS TO CONTACT MECHANICS

Zhenhai Liu

College of Sciences, Guangxi University for Nationalities Nanning 530006, Guangxi, China and Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China

Van Thien Nguyen

Departement of Mathematics, FPT University Education zone, Hoa Lac high tech park, Km29 Thang Long highway Thach That ward, Hanoi, Vietnam

Jen-Chih Yao

Center for General Education, China Medical University, Taichung, Taiwan

Shengda Zeng∗

Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China and Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science ul. Lojasiewicza 6, 30348 Krakow, Poland

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday.

Abstract. The primary objective of this paper is to explore a complicated dif- ferential variational-hemivariational inequality involving a history-dependent operator in Banach spaces. A well-posedness result for the inequality, includ- ing the existence, uniqueness, and continuous dependence on the initial data of the solution is established by using a fixed point principle for history-dependent operators. Moreover, to illustrate the applicability of the theoretical results, an elastic contact problem with wear and long time dependent effort is explored.

2000 Mathematics Subject Classification. Primary: 35L86, 35L87; Secondary: 74Hxx, 74M15. Key words and phrases. Differential variational-hemivariational inequality, well-posedness, history-dependent operator, compliance, wear. This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 – CON- MECH. It is also supported by the National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611, NNSF of China Grant No. 11671101, NSF of Guangxi Grant No. 2018GXNSFDA138002, and International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0. ∗ Corresponding author: Shengda Zeng.

1073 1074 ZHENHAI LIU, VAN THIEN NGUYEN, JEN-CHIH YAO AND SHENGDA ZENG

1. Introduction. Differential variational inequalities (DVIs, for short) as the pow- erful mathematical tools, which were systematically introduced and investigated by Pang-Stewart [20] in finite-dimensional spaces, have been applied to the study of various problems involving both dynamics and constraints in the form of inequali- ties, which arise in many applied problems in our real life, for instance, mechanical impact problems, electrical circuits with ideal diodes, the Coulomb friction problems for contacting bodies, economical dynamics, dynamic traffic networks, and so on. Over the past decade, the theory and applications of DVIs grew rapidly. Compre- hensive references in the area include: Liu et al. [8,9, 10, 11] employed the theory of semigroups, Filippov implicit function lemma and fixed point theorems for con- densing multivalued operators to establish the existence of mild solutions for a class of differential mixed variational inequalities in Banach spaces; Mig´orski-Zeng[16] introduced a temporally semi-discrete algorithm based on the backward Euler dif- ference scheme together with a feedback iterative method to explore a differential hemivariational inequality, which is formulated by a parabolic hemivariational in- equality and a nonlinear evolution equation in the framework of an evolution triple of spaces; by adopting the idea of DVIs, Chen-Wang [1] in 2014 discussed a dynamic Nash equilibrium problem of multiple players with shared constraints and dynamic decision processes; Zeng-Liu-Mig´orski[26] applied Rothe method combined with surjectivity of multivalued pseudomonotone operators to deliver an existence theo- rem to a class of fractional differential hemivariational inequalities in Banach spaces, and then utilized the theoretical results to study a frictional quasistatic contact problem for viscoelastic materials with adhesion; Gwinner [5] in 2013 established a stability result of a class of differential variational inequalities by using the mono- tonicity method and technique of the Mosco convergence. For more details on these topics the reader is welcome to consult [2,7, 12, 13, 14, 17, 18, 19, 25] and the references therein. Everywhere in this paper, symbol I will denote either a bounded interval of the form [0,T ] with T > 0, or the unbounded interval R+ := [0, +∞). Let X,V be reflexive Banach spaces, Y be a normed space, and K be a nonempty, closed and convex subset of V . Given functions F : I × X × V → X, G: I × X × K → V ∗, ϕ: X × Y × K × K → R, R: C(I; V ) × C(I; Y ) and j : X × Y → R, in the current research, we are interesting in the study of the following generalized differential variational-hemivariational inequality involving history-dependent operator Problem 1. Find functions x: I → X and u: I → V such that ( x0(t) = F (t, x(t), u(t)) for a.e. t ∈ I, (1) x(0) = x0, and for all t ∈ I, u(t) ∈ K satisfies the inequality hG(t, x(t), u(t)), v − u(t)i + ϕ(x(t), (Ru)(t), u(t), v) − ϕ(x(t), (Ru)(t), u(t), u(t))

+j0(x(t), u(t); v − u(t)) ≥ 0 for all v ∈ K. (2) The aim of the paper is twofold. The first goal is to explore a well-posedness result for Problem1, including the existence, uniqueness, and continuous dependence on the initial data of the solution. However, the second intention is to employ the abstract results to the study of a complicated elastic contact problem with wear in which a generalized elastic constitutive law with long memory effect is considered, and the wear is described by a nonlinear ordinary differential equation. DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES 1075

The outline of the paper is organized as follows. In Section2, we recall some preliminary materials, which will be needed in the sequel. Section3 is devoted to explore a well-posedness result to Problem1 by using the fixed point principle for history-dependent operators. In Section4, an application of the abstract results, established in Section3, to a complicated history-dependent elastic contact problem with wear is discussed.

2. Preliminaries. In the section, we recall some important notation, definitions and preliminary materials, which will be needed in the sequel. For more details, we refer to [3,4, 15]. Let X be a reflexive Banach space with its dual space X∗ and h: X → R be a locally Lipschitz function. The (Clarke) generalized directional derivative of h at u ∈ X in the direction v ∈ X is defined by h(y + λv) − h(y) h0(u; v) = lim sup . y→u, λ↓0 λ

∗ In the meantime, the Clarke subdifferential operator ∂h: X → 2X of h is given by ∗ 0 ∂h(u) = { ζ ∈ X | h (u; v) ≥ hζ, viX∗×X for all v ∈ X } for all u ∈ X. The generalized gradient and generalized directional derivative of a locally Lip- schitz function enjoy many nice properties and rich calculus. Here we just collect below some basic and crucial results, see for instance, [15, Proposition 3.23].

Proposition 1. Let h: X → R be a locally Lipschitz function, then the following statements are ture (i) for each x ∈ X, ∂h(x) is nonempty, convex and weakly compact in X∗. ∗ ∗ (ii) the graph of ∂h is closed in X × (w − X ) topology, i.e., if {xn} ⊂ X and ∗ ∗ {ξn} ⊂ X are such that ξn ∈ ∂h(xn) and xn → x in X, ξn → ξ weakly in X∗, then it holds ξ ∈ ∂h(x). (iii) the multivalued mapping X 3 x → ∂h(x) ⊆ X∗ is upper semicontinuous from X into w∗ − X∗. Next, we recall the following fixed point principle for history-dependent opera- tors, see cf. [23]. Lemma 2.1. Let X be a Banach space and S : C(I; X) → C(I; X) be a history- dependent operator, namely, for each compact set J ⊂ I, there exists MJ > 0 satisfying Z t kSx1(t) − Sx2(t)kX ≤ MJ kx1(s) − x2(s)kX ds 0 for all t ∈ J, and all x1, x2 ∈ C(I; X). Then, S has a unique fixed point in C(I; X), i.e., there exists a unique function u ∈ C(I; X) such that u = Su. Given a function u ∈ C(I; V ), we recall the following uniqueness and existence result for Cauchy problem (1). Lemma 2.2. Let X and V be Banach spaces and the following conditions hold: H(F ): F : I × X × V → X is such that (i) for all x ∈ X and u ∈ V , the function t 7→ F (t, x, u): I → X is continuous. 1076 ZHENHAI LIU, VAN THIEN NGUYEN, JEN-CHIH YAO AND SHENGDA ZENG

(ii) for each compact set J ⊂ I, there exists a constant LJ > 0 satisfying
kF (t, x, u) − F (t, y, v)kX ≤ LJ kx − ykX + ku − vkV for all x, y ∈ X, u, v ∈ V and t ∈ J.

Then, for each function u ∈ C(I; V ) and x0 ∈ X, there exists a unique solution x ∈ C(I; X) to solve Cauchy problem (1). Moreover, given u1, u2 ∈ C(I; V ), if x1, x2 ∈ C(I; X) are the unique solutions to Cauchy problem (1) corresponding to u1 and u2, accordingly, then for each compact set J ⊂ I, we have t ∗ Z ∗ LJ t kx1(t) − x2(t)kX ≤ LJ e ku1(s) − u2(s)kV ds (3) 0 ∗ for all t ∈ J for some LJ > 0. In the end of the section, we shall introduce the usual notation and symbols, which will be used in the study of the elastic contact problem in Section4. Let Ω be a bounded and connected domain in Rd, where (d = 2, 3), such that the boundary Γ = ∂Ω is Lipschitz continuous. The normal and tangential components of a vector field ξ ∈ Rd on the boundary are given by

ξν = ξ · ν and ξτ = ξ − ξν ν, d where ν = (νi) denotes the outward unit normal at the boundary. Also, S denotes the space of real symmetric d × d matrices. However, the notation σν and στ represents the normal and tangential components of the stress field σ ∈ Sd on the d d boundary, that is, σν = (σν) · ν and στ = σν − σν ν. On R and S we use the standard notation for inner products and norms defined by 1/2 d ξ · η = ξiηi, kξk = (ξ · ξ) for ξ = (ξi), η = (ηi) ∈ R ,

1/2 d σ : τ = σijτij, kσk = (σ : σ) for σ = (σij), τ = (τij) ∈ S . Here, i, j, k, l ∈ {1, . . . , d} and the summation convention over repeated indices is used. In addition, we use the notation u = (ui), σ = (σij), and 1 ε(u) = (ε (u)), ε (u) = u + u
, i, j = 1, . . . , d, ij ij 2 i,j j,i to denote the displacement vector, the stress tensor, and the linearized strain tensor, respectively.

3. Well-posedness result. This section is devoted to explore a well-posedness result for a comprehensive differential variational-hemivariational inequality, Prob- lem1, including the existence, uniqueness, and continuous dependence on the initial data of the solution. Under the framework mentioned in Section1, we impose the following assump- tions. H(K): K is a nonempty, closed and convex subset of V . H(G): G: I × X × K → V ∗ is such that (i) for every t ∈ I and x ∈ X, the function u 7→ G(t, x, u) is strongly monotone, i.e., there is a constant mG > 0 such that 2 hG(t, x, u) − G(t, x, v), u − vi ≥ mGku − vkV for all u, v ∈ K. DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES 1077

(ii) there exists a constant LG > 0 such that
kG(t1, x1, u) − G(t2, x2, u)kV ∗ ≤ LG |t1 − t2| + kx1 − x2kX

for all t1, t2 ∈ I, x1, x2 ∈ X and u ∈ K. H(R): R: C(I; V ) → C(I; Y ) is a history-dependent operator, namely, for each compact set J0 ⊂ I, there is a constant L0 > 0 such that Z t k(Ru)(t) − (Rv)(t)kY ≤ L0 ku(s) − v(s)kV ds 0 for all u, v ∈ C(I; V ) and all t ∈ J0. H(j): j : X × V → R is such that (i) for each x ∈ X, the function u 7→ j(x, u): V → R is locally Lipschitz. (ii) there is a constant cj ≥ 0 such that
k∂j(x, u)kV ∗ ≤ cj 1 + kxkX + kukV for all x ∈ X and u ∈ V . (iii) there exist αj ≥ 0 and βj ≥ 0 satisfying 0 0 j (x1, u1; u2 − u1) + j (x2, u2; u1 − u2)

2 ≤ αjku1 − u2kV + βjku1 − u2kV kx1 − x2kX

for all x1, x2 ∈ X and u1, u2 ∈ V .

H(ϕ): ϕ: X × Y × K × K → R is such that (i) for all x ∈ X, y ∈ Y and v ∈ K, the function u 7→ ϕ(x, y, v, u): K → R is convex and l.s.c. on K. (ii) there exist αϕ ≥ 0, βϕ ≥ 0, and γϕ ≥ 0 such that

ϕ(x1, y1, u1, v2) − ϕ(x1, y1, u1, v1) + ϕ(x2, y2, u2, v1) − ϕ(x2, y2, u2, v2)

≤ αϕku1 − u2kV kv1 − v2kV + βϕky1 − y2kY kv1 − v2kV

+γϕkx1 − x2kX kv1 − v2kV

for all x1, x2 ∈ X, y1, y2 ∈ Y and u1, u2, v1, v2 ∈ K.

H(0): αϕ + αj < mG. We now provide the main results of the section as follows. Theorem 3.1. Assume that H(K), H(F ), H(G), H(ϕ), H(j), H(R) and H(0) are satisfied. Then, for each x0 ∈ X, Problem1 has a unique solution (x, u) ∈ C(I; X) × C(I; V ). Moreover, for x0, y0 ∈ X fixed, for each compact set J ⊂ I, there exists a constant MJ > 0 such that

kx1(t) − x2(t)kX + ku1(t) − u2(t)kV ≤ MJ kx0 − y0kX (4) for all t ∈ J, where (x1, u1) and (x2, u2) are the unique solutions to Problem1 corresponding to x0 and y0, respectively. Proof. For any x ∈ C(I; X) fixed, it follows from [22, Theorem 93, p.160] that history-dependent variational-hemivariaitonal inequality (2) admits a unique solu- tion ux ∈ C(I; X). Consider the mapping P : C(I; X) → C(I; V ) by

P x = ux for all x ∈ C(I; X), (5) 1078 ZHENHAI LIU, VAN THIEN NGUYEN, JEN-CHIH YAO AND SHENGDA ZENG where ux ∈ C(I; V ) is the unique solution to problem (2) corresponding to x ∈ C(I; X). Let xi ∈ C(I; X) and ui = P xi for i = 1, 2, hence,

hG(t, x1(t), u1(t)), v − u1(t)i + ϕ(x1(t), (Ru1)(t), u1(t), v)

0 −ϕ(x1(t), (Ru1)(t), u1(t), u1(t)) + j (x1(t), u1(t); v − u1(t)) ≥ 0 (6) and

hG(t, x2(t), u2(t)), v − u2(t)i + ϕ(x2(t), (Ru2)(t), u2(t), v)

0 −ϕ(x2(t), (Ru2)(t), u2(t), u2(t)) + j (x2(t), u2(t); v − u2(t)) ≥ 0 (7) for all v ∈ K and all t ∈ I. Inserting v = u2(t) and v = u1(t) into (6) and (7), respectively, we sum the resulting inequalities to yield

hG(t, x1(t), u1(t)) − G(t, x1(t), u2(t)), u1(t) − u2(t)i

≤ hG(t, x2(t), u2(t)) − G(t, x1(t), u2(t)), u1(t) − u2(t)i

+ϕ(x1(t), (Ru1)(t), u1(t), u2(t)) − ϕ(x1(t), (Ru1)(t), u1(t), u1(t))

+ϕ(x2(t), (Ru2)(t), u2(t), u1(t)) − ϕ(x2(t), (Ru2)(t), u2(t), u2(t))

0 0 +j (x1(t), u1(t); u2(t) − u1(t)) + j (x2(t), u2(t); u1(t) − u2(t)) (8) for all t ∈ I. The strong monotonicity of u 7→ G(t, x, u) and Lipschitz continuity of x 7→ G(t, x, u) imply

2 mGku1(t) − u2(t)kV ≤ hG(t, x1(t), u1(t)) − G(t, x1(t), u2(t)), u1(t) − u2(t)i (9) and

hG(t, x2(t), u2(t)) − G(t, x1(t), u2(t)), u1(t) − u2(t)i

≤ LGkx1(t) − x2(t)kX ku1(t) − u2(t)kV . (10) Besides, conditions H(ϕ)(ii) and H(j)(iii) reveal

ϕ(x1(t), (Ru1)(t), u1(t), u2(t)) − ϕ(x1(t), (Ru1)(t), u1(t), u1(t))

+ϕ(x2(t), (Ru2)(t), u2(t), u1(t)) − ϕ(x2(t), (Ru2)(t), u2(t), u2(t))

2 ≤ αϕku1(t) − u2(t)kV + βϕk(Ru1)(t) − (Ru2)(t)kY ku1(t) − u2(t)kV

+γϕkx1(t) − x2(t)kX ku1(t) − u2(t)kV (11) and

0 0 j (x1(t), u1(t); u2(t) − u1(t)) + j (x2(t), u2(t); u1(t) − u2(t))

2 ≤ αjku1(t) − u2(t)kV + βjkx1(t) − x2(t)kX ku1(t) − u2(t)kV . (12) Recall that R is a history-dependent operator, for each compact set J ⊂ I, it finds Z t k(Ru1)(t) − (Ru2)(t)kY ≤ LJ ku1(s) − u2(s)kV ds for all t ∈ J, (13) 0 DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES 1079 with some LJ > 0. Taking account of (8)–(13), and the smallness condition H(0), we have

ku1(t) − u2(t)kV Z t (LG + βj + γϕ) βϕLJ ≤
kx1(t) − x2(t)kX +
ku1(s) − u2(s)kV ds mG − αj − αϕ mG − αj − αϕ 0 for all t ∈ I. Invoking Gronwall inequality deduces

kP x1(t) − P x2(t)kV = ku1(t) − u2(t)kV

β L t ϕ J
β L (L + β + γ )e mG−αj −αϕ Z t ≤ ϕ J G j ϕ kx (s) − x (s)k ds
2 1 2 X mG − αj − αϕ 0

(LG + βj + γϕ) +
kx1(t) − x2(t)kX (14) mG − αj − αϕ for all t ∈ I. On the other hand, it follows from Lemma 2.2 and hypotheses H(F ) that for each u ∈ C(I; V ) fixed, there exists a unique function x ∈ C(I; X) to solve the Cauchy problem (1) and the identity holds Z t x(t) = x0 + F (s, x(s), u(s)) ds for all t ∈ I. 0

Let u1, u2 ∈ C(I; V ) be arbitrary, and x1, x2 ∈ C(I; X) be the unique solutions of Cauchy problem (1) associated to u1 and u2, respectively. Employing Lemma 2.2 ∗ again, it concludes that for each compact set J ⊂ I, we are able to find LJ > 0 such that (3) holds for all t ∈ J. Further, consider the function Q: C(I; V ) → C(I; X) defined by

Qu = xu for all u ∈ C(I; V ), (15) where xu is the unique solution to Cauchy problem (1) corresponding to u ∈ C(I; V ). It is not difficult to see that x ∈ C(I; X) is a fixed point of Q ◦ P : C(I; X) → C(I; X) if and only if (x, P x) ∈ C(I; X)×C(I; V ) is a solution to Problem1. Taking advantage of the fact, next, we shall illustrate that Q ◦ P : C(I; X) → C(I; X) has a unique fixed point in C(I; X). Let x1, x2 ∈ C(I; X) be arbitrary. From (14) and (3), it yields t ∗ Z ∗ LJ t kQ(P x1)(t) − Q(P x2)(t)kX ≤ LJ e kP x1(s) − P x2(s)kV ds 0

∗ ∗ LJ t Z t (LG + βj + γϕ)LJ e ≤
kx1(s) − x2(s)kX ds mG − αj − αϕ 0

β L s ϕ J
Z t mG−αj −αϕ Z s ∗ L∗ t βϕLJ (LG + βj + γϕ)e +L e J kx (τ) − x (τ)k dτ ds J
2 1 2 X 0 mG − αj − αϕ 0   ∗ βϕLJ LJ + t t  ∗
 Z ∗ LJ t ∗ mG−αj −αϕ ≤ RJ LJ e + LJ e kx1(s) − x2(s)kX ds (16) 0 1080 ZHENHAI LIU, VAN THIEN NGUYEN, JEN-CHIH YAO AND SHENGDA ZENG for all t ∈ J, where RJ > 0 is defined by (L + β + γ ) β L (L + β + γ ) R := G j ϕ + ϕ J G j ϕ . J

2 mG − αj − αϕ mG − αj − αϕ Exploring Lemma 2.1 and (16), we conclude that Q ◦ P : C(I; X) → C(I; X) has a unique fixed point, say, x ∈ C(I; X). Therefore, Problem1 has a unique solution (x, P x) ∈ C(I; X) × C(I; V ). Let x0, y0 ∈ X be fixed. Assume that (x1, u1), (x2, u2) ∈ C(I; X) × C(I; V ) are the unique solutions to Problem1 corresponding to x0 and y0, respectively. Therefore, for each compact set J ⊂ I, it holds

kx1(t) − x2(t)kX Z t
≤ kx0 − y0kX + LJ kx1(s) − x2(s)kX + ku1(s) − u2(s)kV ds 0 for all t ∈ J with some LJ > 0, where we have used the condition H(F )(ii). The latter together with the Gronwall inequality deduces  Z t  LJ t kx1(t) − x2(t)kX ≤ kx0 − y0kX + LJ ku1(s) − u2(s)kV ds e (17) 0 for all t ∈ J. Combining (14) and (17) with Gronwall inequality, we can find MJ > 0 such that (4) holds.

4. A history-dependent elastic contact problem with wear. To illustrate the applicability of the theoretical results obtained in Section3, the current section is concerned with the study of an impressive application to Contact Mechanics. More precisely, an elastic contact problem with wear and long time dependent effort is explored. In what follows, the time interval of interest is I, which could be either a bounded interval of the form [0,T ] for some T > 0, or the unbounded interval R+ = [0, +∞). Additionally, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable x. The physical setting of the contact problem is given as follows. An elastic body occupies a bounded domain Ω in Rd for d = 2, 3 with a Lipschitz continuous bound- ary Γ, which is considered to be divided into four measurable parts Γ1,Γ2,Γ3 and Γ4 such that meas(Γ1) > 0, where Γ3 is particularly assumed to be a plane. Also let v∗ be the velocity of the foundation. Besides, we set Q = Ω × I, Σ = Γ × I, Σ1 = Γ1 × I,Σ2 = Γ2 × I,Σ3 = Γ3 × I and Σ4 = Γ4 × I. We now provide the classical formulation of the contact problem as follows.

Problem 2. Find a displacement field u: Q → Rd, a stress field σ : Q → Sd, and a wear function w :Σ3 → R such that Z t σ(t) = At, ε(u(t))
+ B(t − s, ε(u(s)) ds in Q, (18) 0

Div σ(t) + f 0(t) = 0 in Q, (19)

u(t) = 0 on Σ1, (20)

σ(t)ν = f 2(t) on Σ2, (21)

− σν (t) = pν (uν (t) − w(t)) on Σ3, (22) DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES 1081

∗ − στ (t) = η|σν (t)|n (t) on Σ3, (23)

0 w (t) = h(t, uν (t), w(t)) on Σ3, (24)

w(0) = w0 on Γ3, (25)  uν (t) ≤ g   σν (t) + ξν (t) ≤ 0 on Σ4, (26) (uν (t) − g)(σν (t) + ξν (t)) = 0   ξν (t) ∈ ∂jν (uν (t))

στ (t) = 0 on Σ4. (27) We now provide a brief description on the equations, conditions and relations appeared in Problem2. Equation (18) is a generalized elastic constitutive law with long memory effect (see e.g. [15, Section 6, p. 182]), where A: Q × Sd → Sd is a nonlinear elasticity operator, and B : Q × Sd → Sd is a relaxation operator in which both operators A and B depend on the location of the point t ∈ I. Additionally, we assume that the elasticity operator A and relaxation function B in the constitutive law (18) satisfy the following conditions.  A: Q × d → d is such that  S S   d (a) A(·, ·, ε) is measurable on Q, for all ε ∈ S .   (b) there exists LA > 0 such that   kA(x, t , ε ) − A(x, t , ε )k d ≤ L (|t − t | + kε − ε k d )  1 1 2 2 S A 1 2 1 2 S  d for a.e. x ∈ Ω, all t1, t2 ∈ I and all ε1, ε2 ∈ S . (28)  (c) ther exists αA > 0 such that  

2  A(x, t, ε1) − A(x, t, ε2) : ε1 − ε2 ≥ αAkε1 − ε2k d  S  d  for all ε1, ε2 ∈ S and a.e. (x, t) ∈ Q.   (d) A(x, t, 0 d ) = 0 d for a.e. (x, t) ∈ Q. S S

 d d B : Q × S → S is such that for each compact set J ⊂ I,   there exists a constant LJ > 0 such that (29) kB(x, t, ε ) − B(x, t, ε )k d ≤ L kε − ε k d  1 2 S J 1 2 S  d  for a.e. x ∈ Ω, all t ∈ J and all ε1, ε2 ∈ S . However, (19) is the equation of equilibrium, and we use it since we assume that the inertial term in the equation motion is neglected. Here, Div stands for the divergence operator, i.e.,
Divσ = σij,j . The boundary conditions (20) and (21) characterize the physical phenomena that the elastic body is clamped on Γ1 and it is subjected to the density f 2(t) of surface tractions on Γ2 at time t ∈ I. The boundary condition (22) represents a normal contact condition with wear effect on boundary Γ3 (see for example, [6, 21]) in which 1082 ZHENHAI LIU, VAN THIEN NGUYEN, JEN-CHIH YAO AND SHENGDA ZENG the normal compliance function p reads the following conditions.  p:Γ × → is such that  3 R R+  (a) for all r ∈ , the function x 7→ p(x, r) is measurable on Γ .  R 3  (b) there exists a constant L > 0 satisfying  p   |p(x, r1) − p(x, r2)| ≤ Lp|r1 − r2| (30) for a.e. x ∈ Γ3 and all r1, r2 ∈ R.   (c) for all r1, r2 ∈ and a.e. x ∈ Γ3, we have  R 

 p(x, r1) − p(x, r2) r1 − r2 ≥ 0.   (d) p(x, r) = 0 for all r ≤ 0 and a.e. x ∈ Γ3. For the velocity of the foundation v∗, we suppose that it is a non-vanishing time- ∗ dependent function (i.e., v (t) 6= 0 d for all t ∈ I) in the plane of Γ and reads the R 3 condition ∗ d v ∈ C(I; R ). (31) However, under the assumption that the velocity of the foundation v∗ is large in comparison with the tangential velocity of the elastic body, we have the equation (23), which stands for the sliding version of the classical Coulomb law of dry friction (see [24], for more details), where n∗ is the unitary vector given by ∗ ∗ v (t) n (t) = − ∗ for all t ∈ I. kv (t)k d R Besides, in the equation (23), the coefficient η is assumed to enjoy the following regularity ∞ η ∈ L (Γ3) with η(x) > 0 for a.e. x ∈ Γ3. (32) On the other side, the wear function w is considered to be governed by a nonlinear ordinary differential equation (29) depending on the normal displacement uν (t), and formulated on contact surface Γ3, where the function h:Σ3 × R × R → R satisfies the following conditions.

 h:Σ3 × R × R → R is such that   (a) for all r, s ∈ and t ∈ I, the function x 7→ h(x, t, r, s)  R   is measurable on Γ3.   (b) for a.e. x ∈ Γ3 and all r, s ∈ R, the function t 7→ h(x, t, r, s) (33) is continuous on I.   (c) for each compact set J ⊂ I, there exists a constant L > 0 satisfying  J 
 |h(x, t, r1, s1) − h(x, t, r2, s2)| ≤ LJ |r1 − r2| + |s1 − s2|   for a.e. (x, t) ∈ Σ3 and all r1, r2, s1, s2 ∈ R. Particularly, if h is specialized to be the formulation ∗ h(t, u (t), w(t)) = kkv (t)k d p(u (t) − w(t)) for all (x, t) ∈ Σ , ν R ν 3 then equation (24) reduces the classical version of the Archard’s law for wear func- tion w. However, condition (25) represents the initial condition for the wear func- tion. In fact, if we take w0 = 0, then we can see that at the initial moment the material is new. DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES 1083

Whereas, boundary condition (26) models the contact with a foundation made of a rigid body covered by a deformable layer of thickness g on Γ4, where g is assumed to be a positive constant. In the meanwhile, we assume that jν is a locally Lipschitz function (or superpotential) to satisfy the following conditions  jν :Γ4 × R → R is such that   (a) for all r ∈ R, the function x 7→ jν (x, r) is measurable on Γ4,   and there exists e ∈ L2(Γ ) such that j (·, e(·)) ∈ L1(Γ ).  4 ν 4   (b) for a.e. x ∈ Γ4, r 7→ jν (x, r) is locally Lipschitz on R.  (c) there exist c ≥ 0 and c > 0 such that 0 1 (34)  |∂j (x, r)| ≤ c + c |r|  ν 0 1   for a.e. x ∈ Γ4 and all r ∈ R.   (d) there is αjν ≥ 0 such that   j0(x, r ; r − r ) + j0(x, r ; r − r ) ≤ α |r − r |2  ν 1 2 1 ν 2 1 2 jν 1 2   for a.e. x ∈ Γ4 and all r1, r2 ∈ R.

Here, ∂jν denotes the Clarke subdifferential of jν with respect to the last variable. Finally, condition (27) represents the frictionless contact condition on Γ4. To obtain the variational formulation of Problem2, we consider the following function spaces:  1 d V := v ∈ H (Ω; R ) | v = 0 on Γ1 ,

 2 d Q := τ = (τij) ∈ L (Ω; S ) | τij = τji for all i, j = 1, . . . , d . It is not difficult to prove that V and Q are both Hilbert spaces endowed with the inner products Z Z
hu, viV := ε(u) · ε(v) dx, σ, τ Q := σ : τ dx Ω Ω for all u, v ∈ V , and all σ, τ ∈ Q, respectively. Also, the associated norms on spaces V and Q are denoted by k · kV and k · kQ, respectively. Moreover, we introduce the set of admissible displacement fields K by  K := v ∈ V | vν ≤ g on Γ4 . (35)

From the Sobolev trace theorem, we are able to find the smallest constant r0 > 0 (which only relies on Ω, Γ1 and Γ4) such that

2 kvkL (Γ4) ≤ r0kvkV for all v ∈ V. Recall that the Green’s formula Z Z Z σ : ε(v) dx + Divσ · v dx = σν · v dΓ for all v ∈ V, Ω Ω Γ it is not difficult to use this formulate to obtain the variational formulation of Problem2 as follows. 2 Problem 3. Find a wear function w : I → L (Γ3) and a displacement field u: I → K such that ( 0 w (t) = h(t, uν (t), w(t)) for a.e. t ∈ I (36) w(0) = w0, 1084 ZHENHAI LIU, VAN THIEN NGUYEN, JEN-CHIH YAO AND SHENGDA ZENG and Z t

u(t) ∈ K, A t, ε(u(t)) + B(t − s, ε(u(s)) ds, ε(v) − ε(u(t)) Q 0 Z Z
∗
+ p uν (t) − w(t) (vν − uν (t)) dΓ + ηp(uν (t) − w(t))n (t) vτ − uτ (t) dΓ Γ3 Γ3 Z 0 + jν (uν (t); vν − uν (t)) dΓ ≥ hf(t), v − u(t)iV (37) Γ4 for all v ∈ K and all t ∈ I, where for each t ∈ I, f(t) ∈ V ∗ is such that Z Z hf(t), viV = f 0(t) · v dx + f 2(t) · v dΓ for all v ∈ V. Ω Γ2 2 ∞ Theorem 4.1. Assume that (28)–(34) hold. If αA > LpkηkL (Γ3) + αjν r0, then, 2 Problem3 has a unique solution (u, w) ∈ C(I; K) × C(I; L (Γ3)). 2 ∗ ∗ Proof. Let X = L (Γ3), Y = V , and consider the functions G: I × X × V → V , R: C(I; V ) → C(I; V ∗) and ϕ: X × Y × V × V → R by Z
hG(t, w, u), viV = A(t, ε(u(t))), ε(v) Q + p(uν − w)vν dΓ − hf(t), viV(38) Γ3 Z t
h(Rz)(t), viV = B(t − s, ε(z(s))) ds, ε(v) Q (39) 0 Z ∗ ∗ ∗ ϕ(w, v , u, v) = hv , viV + ηp(uν − w)n (t) · vτ dΓ (40) Γ3 for all t ∈ I, u, v ∈ V , v∗ ∈ V ∗, w ∈ X, and all z ∈ C(I; V ). Also, we introduce functions F : I × X × V → X and j : V → R defined by

F (t, w, u) := h(t, uν , w), (41) Z j(u) := jν (x, uν (x)) dΓ. (42) Γ4 for all t ∈ I, w ∈ X, and all u ∈ V . Consider the following intermediate problem: find functions u: I → K and w : I → X such that ( w0(t) = F (t, w(t), u(t)) for a.e. t ∈ I, (43) w(0) = w0, and for all t ∈ I, u(t) ∈ K is a solution to the inequality 0 hG(t, w(t), u(t)), v − u(t)iV + j (u(t); v − u(t)) + ϕ(w(t), (Ru)(t), u(t), v) −ϕ(w(t), (Ru)(t), u(t), u(t)) ≥ 0 for all v ∈ K. (44) Invoking hypotheses (34) and [15, Theorem 3.47], we conclude that j : V → R is locally Lipschitz continuous (i.e., H(j)(i) is valid) and satisfies Z 0 0 j (u; v) ≤ jν (x, uν (x); vν (x)) dΓ, (45) Γ4 Z ∂j(u) ⊂ ∂jν (x, uν (x)) dΓ (46) Γ4 DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES 1085 for all u, v ∈ V . Hence, it is obvious that if (u, w) ∈ C(I; K)×C(I; X) is a solution to problem (43)–(44), then it is also a solution to Problem3. Based on this fact, we will utilize Theorem 3.1 to show that Problem3 is solvable. To end this, first, we assert that G, ϕ and R read the conditions H(G), H(ϕ) and H(R), respectively. For any t ∈ I, w ∈ X, conditions (28)(c) and (30)(c) deduce

hG(t, w, v) − G(t, w, u), v − uiV
= A(t, ε(v)) − A(t, ε(u)), ε(v) − ε(u) Q Z
+ p(vν − w) − p(uν − w) (vν − uν ) dΓ Γ3

2 ≥ αAkv − ukV for all u, v ∈ K, so, H(G)(i) holds with mG := αA. From hypotheses (28)(b) and (30)(b), we apply H¨olderinequality directly to obtain

kG(t1, w1, u) − G(t2, w2, u)kV ∗ = sup hG(t1, w1, u) − G(t2, w2, u), viV v∈V,kvkV =1

≤ LA|t1 − t2| + Lpm0kw1 − w2kX

2 for all u ∈ K, where m0 > 0 is such that kvν kL (Γ3) ≤ m0kvkV for all v ∈ V . Therefore, H(G)(ii) is valid for LG := max{LA,Lpm0}. Let J ⊂ I be any compact set, and z1, z2 ∈ C(I; V ). Using (29) and H¨older inequality, it is not difficult to reveal Z t 0 k(Rz1)(t) − (Rz2)(t)kV ∗ ≤ LJ kz1(s) − z2(s)kV ds 0 0 for all t ∈ J for some LJ > 0, namely, R is a history-dependent operator. From the formulation of ϕ, see (40), we can see that v 7→ ϕ(w, v∗, u, v) is linear and continuous. Additionally, a simple calculating gives ∗ ∗ ∗ ∗ ϕ(w1, v1, u1, v2) − ϕ(w1, v1, u1, v1) + ϕ(w2, v2, u2, v1) − ϕ(w2, v2, u2, v2) Z
∗
= η p(uν,1 − w1) − p(uν,2 − w2) n (t) · vτ,2 − vτ,1 dΓ Γ3 ∗ ∗ +hv1 − v2, v2 − v1iV

∗ ∗ ≤ αϕku1 − u2kV kv1 − v2kV + βϕkv1 − v2kV ∗ kv1 − v2kV

+γϕkw1 − w2kX kv1 − v2kV ∗ ∗ ∗ ∞ for all w1, w2 ∈ X, v1, v2, u1, u2 ∈ V and all v1, v2 ∈ V , where αϕ := LpkηkL (Γ3) and for some βϕ, γϕ > 0. Here, we have used hypotheses (30) and H¨olderinequality. Furthermore, it is obvious that the function F : I × X × V → X defined in (43) reads H(F ) (see assumptions (33)). For function j defined in (42), using inequality (45) and relaxed monotone condition (34)(d) implies j0(u; v − u) + j0(v; u − v) Z 0 0 ≤ jν (x, uν (x); vν (x) − uν (x)) + jν (x, vν (x); uν (x) − vν (x)) dx Γ4 1086 ZHENHAI LIU, VAN THIEN NGUYEN, JEN-CHIH YAO AND SHENGDA ZENG Z 2 ≤ αjν |uν (x) − vν (x)| dΓ Γ4

2 2 ≤ αjν r0ku − vkV 2 for all u, v ∈ V . So, H(j)(iii) holds for αj = αjν r0. Because of (34)(c) and (46), it is not difficult to verify that H(j)(ii) is available for some cj > 0. We have verified all of conditions in Theorem 3.1, so, invoking this theorem con- 2 cludes that problem (43)–(44) has a unique solution (u, w) ∈ C(I; K)×C(I; L (Γ3)). Finally, we will show that Problem3 is unique solvability. Let ( u1, w1), (u2, w2) ∈ 2 C(I; K) × C(I; L (Γ3)) be two solutions to Problem3. A simple calculating gives

ku1(t) − u2(t)kV Z t (LG + γϕ) βϕLJ ≤
kw1(t) − w2(t)kX +
ku1(s) − u2(s)kV ds mG − αj − αϕ mG − αj − αϕ 0 and Z t LJ t kw1(t) − w2(t)kX ≤ LJ e ku1(s) − u2(s)kV ds 0 for all t ∈ J with some LJ > 0, where J ⊂ I is compact. Combining the above inequalities and Gronwall’s inequality, we conclude that w1 = w2 and u1 = u2. 2 Therefore, Problem3 has a unique solution ( u, w) ∈ C(I; K) × C(I; L (Γ3)).

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