Second order evolutionary problems driven by mixed quasi-variational–hemivariational inequalities

Abstract

In this paper we introduce a differential quasi-variational inequality which consists of a second order partial differential equation involving history-dependent operators and a mixed quasi-variational–hemivariational inequality in Banach spaces. At first, by using the KKM theorem and monotonicity arguments, we show that the solution set of the mixed quasi-variational–hemivariational inequality is nonempty, bounded, closed and convex. Then, we establish the measurability and upper semicontinuity of the solution set with respect to the time variable and state variable. Finally, based on the theory of strongly continuous cosine operators and a fixed point theorem for condensing set-valued operators, we build the existence of mild solutions for the differential quasi-variational–hemivariational inequality.

Introduction

Let $X,V$ be separable reflexive Banach spaces, $H$ a separable reflexive Hilbert space. In this paper, we always assume that $(V,H,V^{\ast })$ is an evolution triple and $V$ is embedded continuously densely and compactly in $H$ with the compact embedding operator $\gamma :V\to H$. For any Banach space $Z$ with the norm ${\Vert \cdot \Vert }_Z$, let $Z^{\ast }$ denote its topological dual and ${〈\cdot ,\cdot 〉}_Z$ stand for the duality pairing between $Z^{\ast }$ and $Z$. For simplicity, we sometimes omit the subscript $Z$ when no confusion arises. The symbols $\rightharpoonup$ and $\to$ denote the weak and strong convergence in a given Banach space, respectively. $K$ is a nonempty, closed, and convex subset of $V$. $0<b<\infty$. We consider the following differential quasi-variational–hemivariational inequality:

Find $x:[0,b]\to X$ and $u:[0,b]\to K$ such that

$$\left\{\begin{array}{c}{x}^{\prime \prime }\left(t\right)=Ax\left(t\right)+B(t,x\left(t\right))u\left(t\right)+f(t,x\left(t\right),\left(\mathcal{R}x\right)\left(t\right))\text{ for a.e. }t\in [0,b],\hfill \\ u\left(t\right)\in SOL(K,G(t,x,\cdot ),\phi ,J,g)\text{ for a.e. }t\in [0,b],\hfill \\ x\left(0\right)=x_0,x^{\prime }\left(0\right)=y_0.\hfill \end{array}\right.$$

where $A:D\left(A\right)\subset X\to X$ is a closed, linear and densely defined operator which generates a strongly continuous cosine family $\left\{C\left(t\right)\right|t\in [0,b]\}$ on $X$, $B\in \mathcal{L}(V,X)$, where $\mathcal{L}(V,X)$ means the collection of all bounded linear operators from $V$ to $X$, $\mathcal{R}$ is a history-dependent operator, $f$ is a nonlinear operator. $G$ is a multi-valued mapping, $\phi$ is a convex function and $J^0$ is the generalized directional derivative of a locally Lipschitz function $J(t,x,\cdot ):H\to \mathbb{R}$, for each $(t,x)\in [0,b]\times X$. $g\in C([0,b],V^{\ast })$ and $x_0,y_0\in X$. The set $SOL(K,G(t,x,\cdot ),\phi ,J,g)$ stands for the solution set of the following mixed quasi-variational–hemivariational inequality ((MQVI), for short): fixed $x\in C([0,b];X)$, for each $t\in [0,b]$, find $u\left(t\right)\in K$ and $u^{\ast }\in G(t,x\left(t\right),u\left(t\right))$ such that

$$〈u^{\ast },v-u\left(t\right)〉+\phi \left(v\right)-\phi \left(u\left(t\right)\right)+J^0(t,x\left(t\right),\gamma u\left(t\right);\gamma (v-u\left(t\right)))\ge 〈g\left(t\right),v-u\left(t\right)〉\forall v\in K.$$

In recent years, many scientists have made rich achievements in the research of elliptic, parabolic and hyperbolic variational and hemivariational inequalities. Including the existence, uniqueness, stability and numerical solution, a complete theoretical system has been established, for more details on these topics, the readers may consult [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] and the references therein.

While, the differential variational inequalities (DVIs, for short), as the dynamical systems, which are formulated as a combination of ordinary differential equations and time-dependent variational inequalities, have gradually become a hot area of research and fruitful results have been obtained. Recall that the notion of (DVIs) in finite-dimensional spaces was introduced and systematically studied by Pang and Stewart in [13]. It can be used as a powerful and useful mathematical tool to model and solve various problems in engineering field, such as dynamic vehicle routing problems, ideal diode circuit, Coulomb frictional problems for bodies in contact, economical dynamics, dynamic traffic networks, etc. For more details on these topics in finite dimensional spaces the readers may consult [14], [15], [16] and the references therein.

Liu–Zeng–Motreanu [17] firstly studied and proved the existence of solutions for an abstract evolution equation driven by a variational inequality in infinite-dimensional spaces, using the theory of semigroups, the Filippov implicit function lemma and the theory of measure of noncompactness. Furthermore, Liu–Migórski–Zeng [18], by using the upper semicontinuity and measurability properties and the measure of noncompactness, proved the existence of solutions for a class of mixed DVIs in infinite-dimensional Banach spaces. It is worth mentioning that the set of constraints is not necessarily bounded in this paper. Migórski-Zeng [19], by invoking the pseudomonotonicity of multivalued operators and a generalization of the KKM theorem, proved the existence of solutions for a fractional partial differential variational inequality which consists of a mixed quasi-variational inequality and a fractional partial differential equation in infinite-dimensional Banach spaces. While Liu–Zeng–Motreanu [20], studied a new class of problems called partial differential hemivariational inequalities, which consists of abstract evolution equations and hemivariational inequalities, and proved the existence of mild solutions for partial differential hemivariational inequalities, the main tools used are measure theory and Hausdorff MNC. Zeng–Migórski–Liu [21], analyzed a dynamical system called a differential variational–hemivariational inequality (DVHVI) which couples an abstract variational–hemivariational inequality of elliptic type with a nonlinear evolution inclusion in infinite-dimensional Banach spaces. For more research on this topic, we recommend readers to refer to [22], [23], [24], [25], [26], [27], [28], [29]

The novelties of this paper are following: First, for the first time, we study an abstract system of a second-order evolution equation with history-dependent operator coupled with a mixed quasi-variational–hemivariational inequality. Until now, all aforementioned works have treated only a first-order evolution equation, and a variational–hemivariational inequality. As far as we know the problem (1.1) has not been studied before in the literature.

Second, we work under general assumptions on the data which, on the one hand, allow to introduce a nonlinear multi-valued mapping strong coupling between the unknowns, and on the other hand no additional strong monotonicity assumptions on the nonlinear multi-valued mapping $G$.

Third, we do not need the constraint set $K$ to be bounded or compact comparing with previous many papers. We analyze a generalized second-order differential mixed quasi-variational–hemivariational inequality involving history-dependent operators.

The rest of this paper is organized as follows. In Section 2, we will introduce some useful preliminaries for our main results, concerning set-valued analysis and fixed point theorems. In Section 3, first, by using the KKM theorem and monotonicity arguments, we show that the solution set of the mixed quasi-variational–hemivariational inequality involved in the system is nonempty, bounded, convex and closed, then we establish the upper semicontinuity and measurability of the solution set of ($MQVI$) with respect to the time variable $t$ and state variable $x$. In Section 4, we will show the existence of mild solutions for the problem (1.1), by using the theory of strongly continuous cosine operator and a fixed point theorem.