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; +1). 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 ∗, ': 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 2 I; (1) x(0) = x0; and for all t 2 I, u(t) 2 K satisfies the inequality hG(t; x(t); u(t)); v − u(t)i + '(x(t); (Ru)(t); u(t); v) − '(x(t); (Ru)(t); u(t); u(t)) +j0(x(t); u(t); v − u(t)) ≥ 0 for all v 2 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 2 X in the direction v 2 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) = f ζ 2 X j h (u; v) ≥ hζ; viX∗×X for all v 2 X g for all u 2 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 2 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 fxng ⊂ X and ∗ ∗ fξng ⊂ X are such that ξn 2 @h(xn) and xn ! x in X, ξn ! ξ weakly in X∗, then it holds ξ 2 @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 2 J, and all x1; x2 2 C(I; X). Then, S has a unique fixed point in C(I; X), i.e., there exists a unique function u 2 C(I; X) such that u = Su. Given a function u 2 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 2 X and u 2 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 2 X, u; v 2 V and t 2 J. Then, for each function u 2 C(I; V ) and x0 2 X, there exists a unique solution x 2 C(I; X) to solve Cauchy problem (1). Moreover, given u1; u2 2 C(I; V ), if x1; x2 2 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 2 J for some LJ > 0.