A New Class of History–Dependent Evolutionary Variational–Hemivariational Inequalities with Unilateral Constraints

Abstract

In this paper we study a new abstract evolutionary variational–hemivariational inequality which involves unilateral constraints and history–dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected functions together with a fixed-point principle for history–dependent operators. Then, we apply the abstract result to show the unique weak solvability to a dynamic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity combined with the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

1 Introduction

Mathematical models based on variational and hemivariational inequalities are particulary useful to determine and characterize the qualitative properties of systems in mechanics, economics, engineering sciences, etc. The theory of hemivariational and variational–hemivariational inequalities has been initiated in early 1980s with the pioneering works of Panagiotopoulos, see [35,36,37]. Since then, because of new and efficient approaches which combine convex and nonsmooth analysis, the theory has undergone a remarkable development in pure and applied mathematics, see monographs [12, 16, 20, 21, 33, 34, 37].

Hemivariational inequalities are variational descriptions of physical phenomena that include nonconvex, nondifferentiable and locally Lipschitz functions. They play an important role in a description of diverse mechanical problems arising in solid and fluid mechanics. We refer to [2,3,4, 12, 19, 20, 26, 28, 34, 38, 39] and the references therein for the recent results on the mathematical theory of contact mechanics and related issues.

In this paper we study the Cauchy problem for a new class of evolutionary first order variational–hemivariational inequalities in infinite dimensional spaces which involve history–dependent operators and a set of constraints. Given a reflexive Banach space V, and a nonempty, closed and convex set of constraints $K \subset V$, the problem reads as follows: find $w :(0, T) \rightarrow V$ such that $w(t) \in K$ for a.e. $t \in (0, T)$ and

$$\begin{aligned}&\langle w'(t) + A(t, w(t)) + (R_1 w)(t) - f(t), v - w(t) \rangle _{V^* \times V} \nonumber \\&\quad + \, j^0 (t, (Sw)(t), M w(t); Mv - Mw(t))\nonumber \\&\quad + \, \varphi (t, (R_2w)(t), Mv) - \varphi (t, (R_2w)(t), Mw(t)) \ge 0 \end{aligned}$$

(1.1)

for all $v \in K$, a.e. $t \in (0,T)$. Here, the nonlinear operators A, $R_1$, $R_2$, S, and an affine operator M are prescribed, and (1.1) is supplemented by an initial condition. Since the inequality (1.1) contains two potentials, $\varphi$ which is a convex function and j which is a locally Lipschitz function, it is called a variational–hemivariational inequality. Operators $R_1$, $R_2$ and S are called history–dependent, for their properties and applications, see e.g. [18, 25, 29, 30, 32, 40].

The main novelty of the paper is to establish a new result on existence, uniqueness and regularity of solution to the inequality (1.1). We extend the main theorem of [18, Theorem 9], where a particular case $K = V$ (no constraints) has been studied. It is worth mentioning that in the study of (1.1), one can not proceed as in [18]. Specifically, since the set of constraints is nonempty, closed and convex, the indicator function $I_K$ of K is proper, convex and lower semicontinuous, so its subdifferential $\partial I_K$ is maximal monotone. After inserting the term $I_K(v) - I_K(w(t))$ in (1.1) makes an equivalent problem with v in the whole space. However, it is clear that $I_K$ is not a locally Lipschitz function, so the existence results of [18, Theorems 6 and 9] are not applicable in this case. Note also that virtually in every mechanical system there is a situation in which unilateral effects occur, see [16]. We study the problem (1.1) in the framework of evolution triple of spaces, exploit recent result on the mixed equilibrium inequality, see [6, Theorem 3.4] and [7, Theorem 2.4], and a fixed-point principle for history–dependent operators. The second novelty is to obtain the unique weak solvability to a dynamic frictional contact problem for a viscoelastic material with long memory and unilateral constraints in velocity. The model of this contact problem is new and, to the best of our knowledge, it has not be solved in the literature yet. The contact condition involves a unilateral Signorini-type condition for the normal velocity combined with the nonmonotone normal damped response condition. The friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip. Moreover, a similar unique weak solvability result can be proved for other unilateral boundary condition and a nonmonotone frictional law governed by a nonconvex potential. Results on existence of a weak solution to dynamic viscoelastic unilateral contact problems with constraints in displacement can be found in [1, 9, 13, 15, 23] and with constraints in velocity in [17, 31].

The outline of this paper is the following. In Sect. 2 we recall the notation and some preliminary materials. The main result of the paper on constrained history–dependent variational–hemivariational inequality is stated and proved in Sect. 3. Finally, in Sect. 4, we examine a dynamic nonsmooth contact problem of viscoelasticity with multivalued contact and friction conditions for which we prove existence and uniqueness of a weak solution.

2 Essential Tools

In this section we introduce the notation and recall some preliminary results.

Let $(X, \Vert \cdot \Vert _{X})$ be a Banach space, $X^*$ denote its dual space and $\langle \cdot ,\cdot \rangle _{X^* \times X}$ be the duality pairing between $X^*$ and X. By ${\mathscr {J}} :X \rightarrow 2^{X^*}$ we denote the duality mapping defined by

$$\begin{aligned} {\mathscr {J}}u = \left\{ \, u^* \in X^* \mid \langle u^*,u \rangle _{X^* \times X} = \Vert u \Vert _X^2 = \Vert u^* \Vert _{X^*}^2 \, \right\} \ \ \text{ for } \text{ all }\ \ u \in X. \end{aligned}$$

It is known, see e.g. [10, 41], that if X is a reflexive Banach space, then an equivalent norm can be introduced so that X is a strictly convex space Banach space and, therefore, the duality map is a single-valued and continuous operator. The symbols “$\rightarrow$” and “$\rightharpoonup$” denote the strong and the weak convergence, respectively. For a set $D \subset X$, $\mathrm{conv}(D)$ is the convex hull of D. The notation ${{\mathcal {L}}}(E, F)$ stands for the space of linear bounded operators from a Banach space E to a Banach space F, and it is well known that ${{\mathcal {L}}}(E, F)$ is a Banach space endowed with the usual norm $\Vert \cdot \Vert _{{{\mathcal {L}}}(E, F)}$. For a set $S \subset X$, we write $\Vert S \Vert _X = \sup \{ \Vert u \Vert _X \mid u \in S \}$.

We recall some concepts for single-valued operators and bifunctions which can be found in [2, 6, 7, 11, 28].

Definition 1

A single-valued operator $A :X \rightarrow X^*$ is said to be

• (i) demicontinuous, if $u_n\rightarrow u$ in X implies $A u_n\rightharpoonup A u$ in $X^*$,

• (ii) monotone, if $\langle Au-Av,u-v\rangle _{X^*\times X}\ge 0$ for all u, $v\in X$,

• (iii) maximal monotone, if it is monotone and the conditions $(u,u^*)\in X\times X^*$ and $\langle u^*-Av, u-v\rangle _{X^*\times X}\ge 0$ for all $v\in X$ imply $u^*= Au$,

• (iv) quasimonotone, if $\limsup \langle Au_n,u_n-u\rangle _{X^*\times X}\ge 0$ for any sequence $\{u_n\}\subset X$ with $u_n\rightharpoonup u$ in X,

• (v) pseudomonotone, if for any sequence $\{u_n\}\subset X$ such that $u_n\rightharpoonup u$ in X and $\limsup \langle Au_n,u_n-u\rangle _{X^*\times X}\le 0$, we have $\liminf \langle Au_n,u_n-v\rangle _{X^*\times X}\ge \langle Au, u-v\rangle _{X^*\times X}$ for all $v \in X$,

• (vi) bounded, if it maps bounded subsets of X into bounded subsets of $X^*$.

Definition 2

A function $f :X \rightarrow {\mathbb {R}}$ is said to be

• (i) (resp. weakly) upper semicontinuous (usc) at $x_{0}\in X$, if for any sequence $\{x_{n}\} \subset X$ with (resp. $x_{n}\rightharpoonup x_{0})$ $x_{n}\rightarrow x_{0}$, we have $\limsup f(x_{n})\le f(x_{0})$,

• (ii) (resp. weakly) lower semicontinuous (lsc) at $x_{0}\in X$, if for any sequence $\{x_{n}\} \subset X$ with (resp. $x_{n}\rightharpoonup x_{0})$ $x_{n}\rightarrow x_{0}$, we have $f(x_0) \le \liminf f(x_{n})$,

• (iii) f is said to be (resp. weakly) usc (lsc) on X, if f is (resp. weakly) usc (lsc) at x, for all $x\in X$.

Definition 3

Let K be a nonempty, closed and convex subset of X. A real-valued function $F :K\times K \rightarrow {\mathbb {R}}$, called later a bifunction, is said to be

• (i) monotone, if $F(u,v)+F(v,u)\le 0$ for all u, $v\in K$,

• (ii) quasimonotone, if for all $\{u_n\}\subset K$ with $u_n\rightharpoonup u$ in X, we have $\liminf F(u_n,u)\le 0$,

• (iii) pseudomonotone, if for all $\{u_n\}\subset K$ with $u_n\rightharpoonup u$ in X and $\liminf F(u_n,u) \ge 0$, we have $\limsup F(u_n,v)\le F(u,v)$ for all $v \in K$.

Definition 4

Let K be a nonempty, closed and convex subset of X. Let $F :K\times K \rightarrow {\mathbb {R}}$ be a real-valued bifunction with $F(u,u)=0$ for all $u\in K$. The bifunction F is said to be maximal monotone if for every $u\in K$ and for every convex function $\psi :K\rightarrow {\mathbb {R}}$ with $\psi (u)=0$, we have

$$\begin{aligned} \psi (v)\ge F(v,u)\ \ \text{ for } \text{ all } \ v \in K \ \ \Longrightarrow \ \ \psi (v)\ge -F(u,v)\ \ \text{ for } \text{ all } \ v \in K. \end{aligned}$$

The following properties collected from [6, 7] will be useful in next sections. Below, K denotes a nonempty, closed and convex subset of X.

Remark 5

• (i) If $A :X\rightarrow X^*$ is pseudomonotone (respectively, quasimonotone), then the bifunction $F:K\times K\rightarrow {\mathbb {R}}$ defined by $F(u,v)=\langle A u, v - u \rangle _{X^*\times X}$ for u, $v \in K$, is pseudomonotone (respectively, quasimonotone).

• (ii) If the bifunction F is usc with respect to the first argument for the weak topology, then it is pseudomonotone, and in particular, if the condition $F(u,u)\ge 0$ for all $u\in K$ is satisfied, then F is quasimonotone.

• (iii) If F, $G:K\times K\rightarrow {\mathbb {R}}$ are pseudomonotone bifunctions such that $F(u,u)\le 0$ and $G(u,u)\le 0$ for all $u \in K$, then $F + G$ is also pseudomonotone.

• (iv) If $A :X\rightarrow X^*$ is maximal monotone, then the bifunction $F :K\times K\rightarrow {\mathbb {R}}$ defined by $F(u,v)=\langle A u,v-u\rangle _{X^*\times X}$ for u, $v \in K$, is monotone and maximal monotone.

• (v) If the bifunction $F:K\times K\rightarrow {\mathbb {R}}$ is such that $F(u,v)+F(v,u) = 0$ for all u, $v \in K$, then F is monotone and maximal monotone.

We recall an abstract result on existence of solution to the mixed equilibrium problem involving bifunctions of the following form. Let U be a subset of a reflexive Banach space X.

Problem 6

Find $u \in U$ such that

$$\begin{aligned} F(u,v)+G(u,v)+H(u,v)\ge 0 \ \ \text{ for } \text{ all } \ \ v \in U. \end{aligned}$$

We need the following hypotheses on the data of Problem 6.

$\underline{H(U)}:$    U is a nonempty, closed and convex subset of X.

$\underline{H(F)}:$    $F :U\times U\rightarrow {\mathbb {R}}$ is such that

• (1) F is monotone and maximal monotone,

• (2) $F(u,\cdot )$ is convex and lsc for all $u\in U$,

• (3) $F(u,u)=0$ for all $u\in U$.

$\underline{H(G)}:$    $G :U\times U\rightarrow {\mathbb {R}}$ is such that

• (1) G is pseudomonotone,

• (2) for each finite subset D of U, $G(\cdot ,v)$ is usc on conv(D), for all $v \in U$,

• (3) $G(u,\cdot )$ is convex for all $u\in U$,

• (4) $G(u,u) = 0$ for all $u \in U$.

$\underline{H(H)}:$    $H :U\times U\rightarrow {\mathbb {R}}$ is such that

• (1) H is quasimonotone,

• (2) $H(\cdot ,v)$ is usc for all $v \in U$,

• (3) $H(u,\cdot )$ is convex for all $u\in U$,

• (4) $H(u,u) = 0$ for all $u \in U$.

$\underline{(H_{coer})}:$ There exists a nonempty weakly compact subset W such that for each $\lambda >0$ small enough, there exists a weakly compact and convex subset $B_{\lambda }$ of U satisfying the following condition:

$$\begin{aligned} \forall \, u\in U \setminus W,\ \exists \, v \in B_{\lambda } \ \ \text{ such } \text{ that } \ \ G(u,v)+H(u,v)+\lambda \, \langle {\mathscr {J}}u, v-u\rangle _{X^* \times X} < F(v,u). \end{aligned}$$

We have the following existence result whose proof can be found in [6, Theorem 3.4] or [7, Theorem 2.4].

Theorem 7

Assume that the hypotheses H(U), H(F), H(G), H(H) and $(H_{coer})$ hold. Then, Problem 6 has at least one solution $u\in U$.

Remark 8

• (i) If, additionally, U is a weakly compact set, then the condition $(H_{coer})$ can be omitted in Theorem 7.

• (ii) If $F(u,\cdot )$ is convex and lsc for all $u \in U$, and $F(u,u) = 0$ for all $u \in U$, then the condition $(H_{coer})$ is satisfied, if for some $v_0 \in U$, we have

  $$\begin{aligned} \frac{G(u,v_0)+H(u,v_0)+\lambda \, \langle {{\mathscr {J}}} u, v_0-u\rangle }{\Vert u-v_0\Vert _{X}} \rightarrow -\infty \ \mathrm{uniformly~in}\ \lambda , \ \mathrm{as}\ \Vert u-v_0\Vert _{X}\rightarrow +\infty . \end{aligned}$$

Next, we recall the generalized directional derivative and the generalized gradient for a locally Lipschitz function, see [8, 10, 28].

Definition 9

Let X be a Banach space and $j :X \rightarrow {\mathbb {R}}$ be a locally Lipschitz function, that is, for each $x \in X$, there are a neighborhood ${{{\mathcal {N}}}} = {{{\mathcal {N}}}}(x)$ and a constant $k_{{{\mathcal {N}}}} > 0$ such that $|j(w) - j(z)| \le k_{{{\mathcal {N}}}} \Vert w - z \Vert _X$ for all w, $z \in {{{\mathcal {N}}}}$. The generalized directional derivative of j at $x \in X$ in the direction $v \in X$, denoted by $j^{0}(x; v)$, is defined by

$$\begin{aligned} j^{0}(x; v) = \limsup _{y \rightarrow x, \ \lambda \downarrow 0} \frac{j(y + \lambda v) - j(y)}{\lambda }. \end{aligned}$$

The generalized gradient of j at x, denoted by $\partial j (x)$, is given by

$$\begin{aligned} \partial j (x) = \{\, x^* \in X^* \mid {\langle x^*, v \rangle } \le j^{0}(x; v) \ \ \text{ for } \text{ all } \ \ v \in X \, \}. \end{aligned}$$

Finally, we state the fixed-point result which is a consequence of the Banach contraction principle, see [22, Lemma 7] or [40, Corollary 27].

Lemma 10

Let ${{\mathbb {X}}}$ be a Banach space, $0< T < \infty$. Let $\Lambda :L^2(0,T; {{\mathbb {X}}}) \rightarrow L^2(0,T; {{\mathbb {X}}})$ be an operator such that

$$\begin{aligned} \Vert (\Lambda \eta _1)(t) - (\Lambda \eta _2)(t)\Vert ^2_{{\mathbb {X}}} \le c \int _0^t \Vert \eta _1(s) - \eta _2(s)\Vert ^2_{{\mathbb {X}}} \, ds \end{aligned}$$

for all $\eta _1$, $\eta _2 \in L^2(0,T;{{\mathbb {X}}})$, a.e. $t \in (0,T)$ with a constant $c > 0$. Then $\Lambda$ has a unique fixed point in $L^2(0, T; {{\mathbb {X}}})$, i.e., there exists a unique $\eta ^* \in L^2(0,T;{{\mathbb {X}}})$ such that $\Lambda \eta ^* = \eta ^*$.

3 Existence and Uniqueness Result

The goal of this paper is to study the evolutionary variational–hemivariational inequality with constraints and history–dependent operators.

Let $(V,H,V^*)$ be an evolution triple of spaces, that is, V is a separable, reflexive Banach space, H is a separable Hilbert space, the embedding $V\subset H$ is continuous and V is dense in H, see e.g. [28, Definition 1.52]. We also assume that the embedding $V\subset H$ is compact. Given $0< T < +\infty$, we introduce the following Bochner spaces

$$\begin{aligned} {\mathcal {V}}=L^2(0,T;V),\quad {\mathcal {V}}^*=L^2(0,T;V^*), \quad {\mathcal {W}}=\{\, v \in {\mathcal {V}} \mid v'\in {\mathcal {V}}^*\}, \end{aligned}$$

where $v'$ denotes the distributional derivative of v. From the reflexivity of V, it follows that both ${\mathcal {V}}$ and ${\mathcal {V}}^*$ are reflexive Banach spaces, see [41, p. 411]. Endowed with the graph norm, ${{\mathcal {W}}}$ is a separable, reflexive Banach space. It is well known that the embeddings ${\mathcal {V}}\subset L^2(0,T;H) \subset {\mathcal {V}}^*$ are continuous and ${\mathcal {W}}\subset L^2(0,T;H)$ is compact, see, e.g., [11, Theorem 3.4.13]. The duality pairing between ${{\mathcal {V}}^*}$ and ${{\mathcal {V}}}$ is denoted by

$$\begin{aligned} \langle w, v\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} = \int _0^T\langle w(t), v(t) \rangle _{V^* \times V} \,dt \ \ \text{ for } \ \ w\in \mathcal {V}^*, v \in {{\mathcal {V}}}, \end{aligned}$$

where $\langle \cdot , \cdot \rangle _{V^* \times V}$ is the duality pairing between $V^*$ and V. We recall that the operator $L:D(L)\subset {\mathcal {V}} \rightarrow {\mathcal {V}}^*$ defined by

$$\begin{aligned} Lv = v' \ \ \text{ for } \text{ all } \ \ v\in D(L), \end{aligned}$$

(3.1)

where $D(L) = \{v\in {\mathcal {W}} \mid v(0)=0 \}$ is linear and maximal monotone, see, e.g., [41, Proposition 32.10, p. 855].

Let X, Y and Z be Banach spaces. The evolutionary history–dependent variational–hemivariational inequality is of the form.

Problem 11

Find $w \in {{\mathcal {W}}}$ such that $w(t) \in K$ for a.e. $t \in (0, T)$ and

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \langle w'(t) + A(t, w(t)) + (R_1 w)(t) - f(t), v - w(t) \rangle _{V^* \times V} \\ \ \ \qquad + \, j^0 (t, (Sw)(t), M w(t); Mv - Mw(t))\\ \ \ \qquad + \, \varphi (t, (R_2w)(t), Mv) - \varphi (t, (R_2w)(t), Mw(t)) \ge 0 \\ \qquad \qquad \qquad \qquad \ \ \text{ for } \text{ all } \ \ v \in K, \ \text{ a.e. } \ \ t \in (0,T), \\ w(0) = w_0 . \end{array}\right. } \end{aligned}$$

The hypotheses for Problem 11 are following.

$\underline{H(A)}:$    $\displaystyle A :(0, T) \times V \rightarrow V^*$ is such that

• (1) $A(\cdot , v)$ is measurable on (0, T) for all $v \in V$,

• (2) $A(t,\cdot )$ is demicontinuous on V for a.e. $t \in (0, T)$,

• (3) $\Vert A(t, v) \Vert _{V^*} \le a_0(t) + a_1 \Vert v \Vert _V$ for all $v \in V$, a.e. $t \in (0,T)$ with a function $a_0 \in L^2(0,T)$ satisfying $a_0 \ge 0$ a.e. in (0, T), and a constant $a_1 \ge 0$,

• (4) $A(t, \cdot )$ is strongly monotone for a.e. $t \in (0,T)$, i.e., for a constant $m_A > 0$,

  $$\begin{aligned} \langle A (t, v_1) - A(t, v_2), v_1 - v_2 \rangle _{V^* \times V} \ge m_A \Vert v_1 - v_2 \Vert _V^2 \end{aligned}$$

  for all $v_1$, $v_2 \in V$, a.e. $t \in (0, T)$.

$\underline{H(j)}:$    $j :(0, T) \times Z \times X \rightarrow {\mathbb {R}}$ is such that

• (1) $j(\cdot , z, v)$ is measurable on (0, T) for all $z \in Z$, $v \in X$,

• (2) $j(t, \cdot , v)$ is continuous on Z for all $v \in X$, a.e. $t \in (0, T)$,

• (3) $j(t, z, \cdot )$ is locally Lipschitz on X for all $z \in Z$, a.e. $t \in (0, T)$,

• (4) $\Vert \partial j(t, z, v) \Vert _{V^*} \le c_{0j} (t) + c_{1j} \Vert z \Vert _Z + c_{2j} \Vert v \Vert _X$ for all $z \in Z$, $v \in X$, a.e. $t \in (0, T)$ with $c_{0j} \in L^2(0, T)$, $c_{0j}$, $c_{1j}$, $c_{2j} \ge 0$,

• (5) $j^0(t, z_1, v_1; v_2 - v_1) + j^0(t, z_2, v_2; v_1 - v_2) \le {\overline{m}}_{2j} \, \Vert z_1 - z_2 \Vert _Z \Vert v_1 - v_2 \Vert _X + m_{2j} \, \Vert v_1 - v_2 \Vert ^2_X$ for all $z_i \in Z$, $v_i \in X$, $i = 1$, 2, a.e. $t \in (0, T)$ with ${\overline{m}}_{2j}\ge 0$, $m_{2j} \ge 0$.

$\underline{H(\varphi )}:$    $\varphi :(0, T) \times Y \times X \rightarrow {\mathbb {R}}$ is such that

• (1) $\varphi (\cdot , y, v)$ is measurable on (0, T) for all $y \in Y$, $v \in X$,

• (2) $\varphi (t, \cdot , v)$ is continuous on Y for all $v \in X$, a.e. $t \in (0, T)$,

• (3) $\varphi (t, y, \cdot )$ is convex and lsc on X for all $y \in Y$, a.e. $t \in (0, T)$,

• (4) $\varphi (t, y_1, v_2) - \varphi (t, y_1, v_1) + \varphi (t, y_2, v_1) - \varphi (t, y_2, v_2) \le \beta _{\varphi } \, \Vert y_1 - y_2 \Vert _Y \Vert v_1 - v_2 \Vert _X$ for all $y_i \in Y$, $v_i \in X$, $i = 1$, 2, a.e. $t \in (0, T)$ with $\beta _{\varphi } \ge 0$,

• (5) $\Vert \partial \varphi (t, y, v)\Vert _{X^*} \le c_{0\varphi } + c_{1\varphi } \Vert y \Vert _Y + c_{2\varphi } \Vert v \Vert _V$ for all $y \in Y$, $v \in V$, a.e. $t \in (0, T)$ with $c_{0\varphi } \in L^2(0, T)$, $c_{0\varphi }$, $c_{1\varphi }$, $c_{2\varphi } \ge 0$.

$\underline{H(R,S)}:$    $R_1 :{{\mathcal {V}}} \rightarrow {{\mathcal {V}}^*}$, $R_2 :{{\mathcal {V}}} \rightarrow L^2(0, T; Y)$, and $S :{{\mathcal {V}}} \rightarrow L^2(0, T; Z)$ are such that

• (1) $\displaystyle \Vert (R_1 v_1)(t) - (R_1 v_2)(t) \Vert _{V^*} \le c_{R_1} \int _0^t \Vert v_1(s) - v_2(s) \Vert _V \, ds$

  for all $v_1$, $v_2 \in {{\mathcal {V}}}$, a.e. $t\in (0, T)$ with $c_{R_1} > 0$,

• (2) $\displaystyle \Vert ({R_2} v_1)(t) - ({R_2} v_2)(t) \Vert _Y \le c_{R_2} \int _0^t \Vert v_1(s) - v_2(s) \Vert _V \, ds$

  for all $v_1$, $v_2 \in {{\mathcal {V}}}$, a.e. $t\in (0, T)$ with $c_{R_2} > 0$,

• (3) $\displaystyle \Vert ({S} v_1)(t) - ({S} v_2)(t) \Vert _Z \le c_{S} \int _0^t \Vert v_1(s) - v_2(s) \Vert _V \, ds$

  for all $v_1$, $v_2 \in {{\mathcal {V}}}$, a.e. $t\in (0, T)$ with $c_{S} > 0$.

$\underline{H(K)}:$    K is a closed and convex subset of V with $0 \in K$.

$\underline{H(M)}:$    $M :V\rightarrow X$ is such that

• (1) M is an affine operator.

• (2) the Nemitsky operator ${{\mathcal {M}}} :{{\mathcal {V}}} \rightarrow L^2(0,T;X)$ corresponding to M is compact.

$\underline{(H_1)}:$    $f \in {{\mathcal {V}}}^*$, $w_0 \in V$.

$\underline{(H_2)}:$$m_A > m_{2j} \Vert A_M \Vert ^2$, where $A_M :V \rightarrow X$ is defined by $A_M v = M v - M0$ for $v \in V$.

Remark 12

Recall that in the hypothesis H(M), the operator M is an affine operator if and only if the operator $A_M$ is linear. The operator $A_M$ is called the linear part of M. In $(H_2)$ and in what follows, the constant $\Vert A_M \Vert$ denotes the norm in ${{{\mathcal {L}}}}(V, X)$ of the linear part $A_M$ of M. It is obvious that if $M \in {{{\mathcal {L}}}}(V, X)$, then $A_M = M$.

First, we consider Problem 11 in the case of $w_0 = 0$. We have the following existence and uniqueness result.

Theorem 13

Under hypotheses H(A), H(j), $H(\varphi )$, H(R, S), H(K), H(M), $(H_1)$ with $w_0 = 0$, and $(H_2)$, Problem 11 has a unique solution.

Proof

It is performed in several steps.

Step 1 Let $\xi \in {\mathcal {V}}^*$, $\eta \in L^2(0, T; Y)$ and $\zeta \in L^2(0, T; Z)$ be fixed and consider the following auxiliary problem.

Problem 14

Find $w = w_{\xi \eta \zeta } \in {{\mathcal {W}}}$ with $w(t) \in K$ for a.e. $t \in (0,T)$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \langle w'(t) + A(t, w(t)) - f(t) + \xi (t), v - w(t) \rangle _{V^* \times V}\\ \ \ \quad \qquad + \, j^0 (t, \zeta (t), Mw(t); Mv - Mw(t)) \\ \ \ \quad \qquad + \, \varphi (t, \eta (t), Mv) - \varphi (t, \eta (t), Mw(t)) \ge 0 \\ \qquad \qquad \qquad \qquad \ \ \text{ for } \text{ all } \ v \in K, \ \text{ a.e. } \ t \in (0,T), \\ w(0) = 0 . \end{array}\right. } \end{aligned}$$

(3.2)

We show that the solution to Problem 14 is unique. For simplicity, we skip the subscripts $\xi$, $\eta$ and $\zeta$ for this part of the proof. Let $w_i \in {{{\mathcal {W}}}}$, $i=1$, 2 be solutions to Problem 14, i.e., $w_i(t) \in K$ for a.e. $t \in (0, T)$, $w_i(0) = 0$ and

$$\begin{aligned}&\langle w_i'(t) + A(t, w_i(t)) - f(t) + \xi (t), v - w_i(t) \rangle _{V^* \times V} + \, j^0 (t, \zeta (t), Mw_i(t); Mv - Mw_i(t)) \\&\quad + \, \varphi (t, \eta (t), Mv) - \varphi (t, \eta (t), Mw_i(t)) \ge 0 \ \ \text{ for } \text{ all } \ v \in K, \ \text{ a.e. } \ t \in (0,T), \ i=1, 2. \end{aligned}$$

From the above inequalities, we obtain

$$\begin{aligned}&\langle w'_{1}(t)-w'_{2}(t), w_{1}(t)-w_{2}(t)\rangle _{V^*\times V}+\langle A(t,w_{1}(t))-A(t,w_{2}(t)), w_{1}(t)-w_{2}(t)\rangle _{V^*\times V} \\&\quad \le j^0 (t, \zeta (t), Mw_1(t); Mw_2(t) - Mw_1(t)) +j^0 (t, \zeta (t), Mw_2(t); Mw_1(t) - Mw_2(t)) . \end{aligned}$$

Integrating the above inequality on (0, t), using the integration by parts formula, and the hypothesis H(A)(4) on the left hand side, and applying the hypothesis H(j)(5) to the right hand side, we have

$$\begin{aligned}&\frac{1}{2} \Vert w_1(t) - w_2(t) \Vert ^2_H + m_A \int _0^t \Vert w_1(s) - w_2(s)\Vert _V^2 \, ds \\&\quad \le m_{2j} \int _0^t \Vert Mw_1(s) - Mw_2(s)\Vert ^2_X \, ds \end{aligned}$$

for all $t \in [0,T]$. Then, by H(M), we obtain

$$\begin{aligned} (m_{A}-m_{2j}\, \Vert A_M \Vert ^2) \Vert w_{1}-w_{2}\Vert _{L^2(0,t;V)}^{2}\le 0, \end{aligned}$$

for all $t \in [0,T]$. Hence, due to condition $(H_2)$, we have $w_{1}=w_{2}$. This completes the proof of uniqueness of the solution.

Step 2 We now prove that Problem 14 has a solution. To this end, we introduce

$$\begin{aligned} {\mathcal {K}}= L^2(0, T; K) := \{\, v \in {\mathcal {V}} \mid v (t)\in K \ \text{ for } \text{ a.e. } \ t\in (0,T)\, \}, \end{aligned}$$

(3.3)

and ${{\mathcal {K}}}_1=D(L)\cap {\mathcal {K}}$, where, recall, $D(L) = \left\{ w\in {\mathcal {W}} \mid w(0)=0\right\}$. Consider the following problem.

Problem 15

Find $w\in {{\mathcal {K}}}_1$ such that

$$\begin{aligned}&\int _0^T\langle w'(t)+A(t,w(t))-f(t) +\xi (t), z(t)-w(t)\rangle _{V^* \times V} \, dt \\&\quad +\int _0^T j^0(t, \zeta (t), Mw(t); Mz(t)-Mw(t))\, dt \\&\quad + \int _0^T (\varphi (t,\eta (t),Mz(t)) -\varphi (t, \eta (t),Mw(t)))\, dt \ge 0 \ \ \ \text{ for } \text{ all } \ \ z\in {{\mathcal {K}}}_1. \end{aligned}$$

Lemma 16

Problems 14 and 15 are equivalent.

Proof

Let $w\in {{\mathcal {W}}}$ be a solution to Problem 14. This means that $w \in {{\mathcal {K}}}$ and $w(0) = 0$ which entails $w\in {{\mathcal {K}}}_1$. Let $z \in {\mathcal {K}}_1$. Then, we have

$$\begin{aligned}&\langle w'(t) + A(t, w(t)) - f(t) + \xi (t), z(t) - w(t) \rangle _{V^* \times V} \\&\quad + \, j^0 (t, \zeta (t), Mw(t); Mz(t) - Mw(t)) \\&\quad + \, \varphi (t, \eta (t), Mz(t)) - \varphi (t, \eta (t), Mw(t)) \ge 0 \ \ \text{ for } \text{ a.e. } \ t\in (0, T). \end{aligned}$$

Integrating the last inequality on (0, T), since $z \in {\mathcal {K}}_1$ is arbitrary, we deduce that $w\in {{\mathcal {K}}}_1$ is a solution to Problem 15.

Vice versa, assume that $w\in {{\mathcal {K}}}_1$ solves Problem 15. Since $0 \in K$ and ${{\mathcal {K}}}$ is a convex set, by using [24, Theorem 9.1, p.270], we deduce that the condition (9.18) in [24, p.269] holds, which, in particular, implies that

$$\begin{aligned} D(L) \cap {{\mathcal {K}}} \ \ \text{ is } \text{ dense } \text{ in } \ \ {{\mathcal {K}}}. \end{aligned}$$

(3.4)

Exploiting the density (3.4), we obtain that $w \in {{\mathcal {K}}}_1$ is a solution to the following problem.

Problem 17

Find $w\in {{\mathcal {K}}}_1$ such that

$$\begin{aligned}&\int _0^T\langle w'(t)+A(t,w(t))-f(t) +\xi (t), {\overline{z}}(t)-w(t)\rangle _{V^* \times V} \, dt \\&\quad +\int _0^T j^0(t, \zeta (t), Mw(t); M{\overline{z}}(t)-Mw(t))\, dt \\&\quad + \int _0^T (\varphi (t,\eta (t),M{\overline{z}}(t)) -\varphi (t, \eta (t),Mw(t)))\, dt \ge 0 \ \ \ \text{ for } \text{ all } \ \ {\overline{z}}\in {{\mathcal {K}}}. \end{aligned}$$

Indeed, let ${\overline{z}} \in {{\mathcal {K}}}=L^2(0,T; K)$. From (3.4), there exists a sequence $z_n \in D(L) \cap {{\mathcal {K}}}$ such that $z_n \rightarrow {\overline{z}}$ in ${{\mathcal {V}}}$. By Problem 15, for all $n \in {\mathbb {N}}$, we have

$$\begin{aligned}&\int _0^T\langle w'(t)+A(t,w(t))-f(t) +\xi (t), z_n(t)-w(t)\rangle _{V^* \times V} \, dt \nonumber \\&\quad +\int _0^T j^0(t, \zeta (t), Mw(t); Mz_n(t)-Mw(t))\, dt \nonumber \\&\quad + \int _0^T (\varphi (t,\eta (t),Mz_n(t)) -\varphi (t, \eta (t),Mw(t)))\, dt. \end{aligned}$$

(3.5)

We will pass to the limit in (3.5), as $n\rightarrow \infty$. By passing to a subsequence if necessary, we deduce, see e.g. [28, Theorem 2.39], that $z_n(t) \rightarrow {\overline{z}}(t)$ in V, for a.e. $t \in (0, T)$ and there is $g \in L^2(0, T)$ such that $\Vert z_n(t) \Vert _V \le g(t)$ for a.e. $t \in (0, T)$. Hence, $Mz_n(t) \rightarrow M {\overline{z}}(t)$ in X for a.e. $t \in (0, T)$. The upper semicontinuity of $j^0$ with respect to its last variable entails

$$\begin{aligned} \limsup j^0(t, \zeta (t), Mw(t); Mz_n(t) - Mw(t)) \le j^0(t, \zeta (t), Mw(t); M{\overline{z}} - Mw(t)) \end{aligned}$$

for a.e. $t \in (0, T)$. The latter together with H(j)(4) and the following estimate

$$\begin{aligned}&|j^0(t, \zeta (t), Mw(t); Mz_n(t) - Mw(t))| \le \Vert \partial j(t, \zeta (t), Mw(t))\Vert _{X^*} \Vert Mz_n(t) - Mw(t)\Vert _X \\&\quad \le (c_{0j}(t) + c_{1j} \Vert \zeta \Vert _Z + c_{2j} \Vert Mw(t)\Vert _X) (\Vert M\Vert g(t) + \Vert M w(t)\Vert _X) =: h(t) \end{aligned}$$

for a.e. $t \in (0, T)$ with $h \in L^1(0, T)$, allows to apply the Fatou lemma to get

$$\begin{aligned}&\limsup \int _0^T j^0(t, \zeta (t), Mw(t); Mz_n(t)-Mw(t))\, dt \nonumber \\&\quad \le \int _0^T j^0(t, \zeta (t), Mw(t); M {\overline{z}}(t)-Mw(t))\, dt. \end{aligned}$$

(3.6)

Next, since the function $\varphi (t, \eta (t), \cdot )$ is continuous on X for a.e. $t \in (0, T)$, see e.g. [14, Corollary 2.5, p.13], we have

$$\begin{aligned} \varphi (t, \eta (t), Mz_n (t)) - \varphi (t, \eta (t), Mw (t)) \rightarrow \varphi (t, \eta (t), M {\overline{z}} (t)) - \varphi (t, \eta (t), Mw (t)) \end{aligned}$$

for a.e. $t \in (0, T)$. It follows from $H(\varphi )(5)$ that

$$\begin{aligned}&|\varphi (t, \eta (t), Mz_n (t)) - \varphi (t, \eta (t), Mw (t))| \le \Vert \partial \varphi (t, \eta (t), Mz_n (t)) \Vert _{X^*} \Vert Mz_n(t) - Mw(t)\Vert _X \\&\quad \le (c_{0\varphi }(t) + c_{1\varphi } \Vert \eta (t) \Vert _Y + c_{2\varphi } \Vert M\Vert g(t)) (\Vert M\Vert g(t) + \Vert M w(t)\Vert _X) =: {\widetilde{h}}(t) \end{aligned}$$

for a.e. $t \in (0, T)$ with ${\widetilde{h}} \in L^1(0, T)$. By the Lebesgue dominated convergence theorem, we deduce

$$\begin{aligned}&\lim \int _0^T (\varphi (t,\eta (t),Mz_n(t)) -\varphi (t, \eta (t),Mw(t)))\, dt \nonumber \\&\quad = \int _0^T (\varphi (t,\eta (t),M {\overline{z}}(t)) -\varphi (t, \eta (t),Mw(t)))\, dt. \end{aligned}$$

(3.7)

Using (3.6) and (3.7) and taking the upper limit in (3.5), we conclude that $w\in {{\mathcal {K}}}_1$ is a solution to Problem 17.

In the final part of the proof, we show that Problems 14 and 17 are equivalent. It is obvious that Problem 14 implies Problem 17. The converse implication follows from [5, Lemma 2.3] (applied with $\varphi :V \times V \rightarrow {\mathbb {R}}$, $\varphi (u, v) = \langle Au, v - u \rangle$ which under H(A) is monotone-convex with a suitable growth condition). We conclude that Problems 14 and 15 are equivalent, which completes the proof of the lemma. $\square$

It follows from Step 1 and Lemma 16 that Problems 14 and 15 have the same unique solution. Therefore, to show the existence of a solution to Problem 14, it is enough to prove that there exists a solution to Problem 15.

In what follows, we prove that Problem 15 has at least one solution. This can be done in several ways. For our purpose, it is sufficient to apply Theorem 7. We introduce the following additional notation which allow to rewrite Problem 15 in the form of the mixed equilibrium inequality as stated in Problem 6. Let the operators ${\mathcal {A}} :{\mathcal {V}}\rightarrow {\mathcal {V}}^*$, ${{\mathcal {M}}} :{{\mathcal {V}}} \rightarrow L^2(0,T;X)$, and the functions $\Phi :{\mathcal {V}} \rightarrow {\mathbb {R}}$, $J :{{\mathcal {V}}} \rightarrow {\mathbb {R}}$ be defined by

$$\begin{aligned}&({\mathcal {A}}w)(t)=A(t, w(t)) \ \ \text{ for } \text{ all } \ w \in {\mathcal {V}}, \ \text{ a.e. } \ t\in (0,T), \\&({{\mathcal {M}}} w)(t) = M w(t) \ \ \text{ for } \text{ all } \ \ w \in {{\mathcal {V}}}, \ \text{ a.e. } \ t\in (0,T), \\&\Phi (w)=\int _0^T\varphi (t, \eta (t),Mw(t))\, dt \ \ \text{ for } \ \ w \in {{\mathcal {V}}},\\&J(w)= \int _0^T j(t,\zeta (t),Mw(t))\,dt \ \ \text{ for } \ \ w \in {{\mathcal {V}}}. \end{aligned}$$

Next, we introduce the bifunctions F, G, $H :{{\mathcal {K}}}_1 \times {{\mathcal {K}}}_1 \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned}&F(w,z)= \langle Lw, z-w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} +\Phi (z)-\Phi (w), \end{aligned}$$

(3.8)

$$\begin{aligned}&G(w,z)=\langle {\mathcal {A}}w-f+\xi ,z-w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}, \end{aligned}$$

(3.9)

$$\begin{aligned}&H(w,z)=J^{0}(w; z-w) \end{aligned}$$

(3.10)

for w, $z \in {{\mathcal {K}}}_1$. Under these notation, Problem 15 can be equivalently formulated as follows.

Problem 18

Find $w\in {{\mathcal {K}}}_1$ such that

$$\begin{aligned} F(w,z)+G(w,z)+H(w,z)\ge 0 \ \ \text{ for } \text{ all } \ \ z \in {{\mathcal {K}}}_1. \end{aligned}$$

(3.11)

The existence of a solution to Problem 18 is demonstrated by using Theorem 7. We will verify the hypotheses of this theorem for $U = {{{\mathcal {K}}}}_1$.

Claim 1

The bifunction F defined by (3.8) satisfies the condition H(F).

First, we prove H(F)(1). Since $F(w, z) + F(z, w) = - \langle Lw - Lz, w-z \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} \le 0$ for all w, $z \in {{\mathcal {K}}}_1$, it is clear that F is monotone. We show that F is maximal monotone. Let $F_1(w,z)=\langle Lw,z-w\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}$ for w, $z\in {{\mathcal {K}}}_1$. Assume that for every $w\in {{\mathcal {K}}}_1$ and every convex function $\psi :{{\mathcal {K}}}_1 \rightarrow {\mathbb {R}}$ with $\psi (w)=0$, we have

$$\begin{aligned} \psi (z)\ge F(z,w)\ \ \text{ for } \text{ all } \ \ z\in {{\mathcal {K}}}_1, \end{aligned}$$

i.e., $\psi (v)\ge F_1(z,w)+\Phi (w)-\Phi (z)$ for all $z \in {{\mathcal {K}}}_1$. Hence

$$\begin{aligned} \psi (z)+\Phi (z)-\Phi (w)\ge F_1(z,w) \ \ \text{ for } \text{ all } \ \ z \in {{\mathcal {K}}}_1. \end{aligned}$$

(3.12)

By the maximal monotonicity of the operator $L:{{\mathcal {V}}} \rightarrow {{\mathcal {V}}^*}$, it follows that $F_1$ is maximal monotone. Thus, the condition (3.12) implies

$$\begin{aligned} \psi (z)+\Phi (z)-\Phi (w)\ge -F_1(w,z) \ \ \text{ for } \text{ all } \ \ z \in {{\mathcal {K}}}_1. \end{aligned}$$

Thus

$$\begin{aligned} \psi (z)\ge -F_1(w,z)-\Phi (w)+\Phi (z) = -F(w,z) \ \ \text{ for } \text{ all } \ z \in {{\mathcal {K}}}_1, \end{aligned}$$

which means that bifunction F is maximal monotone.

Condition H(F)(2) follows from hypothesis H(M) and $H(\varphi )$(3). We will show that $F(w,\cdot )$ is convex and lsc for all $w\in {{\mathcal {K}}}_1$. Let w, $z_1$, $z_2\in {{\mathcal {K}}}_1$ and $\lambda \in (0,1)$. By $H(\varphi )$(3), we have

$$\begin{aligned}&F(w,\lambda z_1+(1-\lambda )z_2)\\&\quad = \langle Lw, \lambda z_1+(1-\lambda )z_2-w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} +\Phi (\lambda z_1+(1-\lambda )z_2)-\Phi (w)\\&\quad \le \lambda \langle Lw, z_1-w\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}+(1-\lambda )\langle Lw, z_2-w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}\\&\qquad +\lambda (\Phi (z_1)-\Phi (w))+(1-\lambda )(\Phi (z_2)-\Phi (w))\\&\quad =\lambda F(w,z_1)+(1-\lambda )F(w,z_2), \end{aligned}$$

which implies that $F(w,\cdot )$ is convex for all $w\in {{\mathcal {K}}}_1$. Moreover, for $z_n$, $z\in {{\mathcal {K}}}_1$ with $z_n\rightarrow z$ in ${\mathcal {V}}$, as $n\rightarrow \infty$, from H(M), we obtain ${{\mathcal {M}}}z_n \rightarrow {{\mathcal {M}}}z$ in $L^2(0,T;X)$. Hence, by passing to a subsequence, if necessary, we may suppose that $({{\mathcal {M}}}z_n)(t) \rightarrow ({{\mathcal {M}}}z)(t)$ in X for a.e. $t\in (0,T)$, i.e., $Mz_n(t)\rightarrow Mz(t)$ in X for a.e. $t\in (0,T)$. Then, from Fatou’s lemma and $H(\varphi )$(3), it follows that

$$\begin{aligned}&\liminf \Phi (z_n)=\liminf \int _0^T\varphi (t, \eta (t),Mz_n(t))\, dt \\&\quad \ge \int _0^T\liminf \varphi (t, \eta (t),Mz_n(t))\, dt \ge \int _0^T\varphi (t, \eta (t),Mz(t))\, dt = \Phi (z). \end{aligned}$$

Thus for $w\in V$, we have

$$\begin{aligned}&\liminf F(w,z_n)\\&\quad =\liminf \, [\langle Lw, z_n-w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} +\Phi (z_n)-\Phi (w)]\\&\quad \ge \liminf \langle Lw, z_n-w\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}+\liminf \Phi (z_n)-\Phi (w)\\&\quad \ge \langle Lw, z-w\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} +\Phi (z)-\Phi (w) =F(w,z), \end{aligned}$$

which implies that $F(w,\cdot )$ is lsc for all $w\in {{\mathcal {K}}}_1$. The condition H(F)(3) is obvious. Hence, we deduce that Claim 1 holds.

Claim 2

The bifunction G defined by (3.9) satisfies the condition H(G).

We first show that, for each finite subset D of ${{\mathcal {K}}}_1$, $G(\cdot ,z)$ is usc on $\mathrm{conv}(D)$ for all $z\in {{\mathcal {K}}}_1$. Let $\{w_n\}\subset \mathrm{conv}(D)$ be such that $w_n\rightarrow w$ in ${\mathcal {V}}$. Since $\mathrm{conv}(D)$ is a closed and convex set, we deduce that $w\in \mathrm{conv}(D)$. From hypothesis H(A)(2) and [41, Proposition 27.7(b)], it follows that ${{\mathcal {A}}} :{{\mathcal {V}}} \rightarrow {{\mathcal {V}}^*}$ is demicontinuous. Thus, we have ${{\mathcal {A}}} w_n \rightharpoonup {{\mathcal {A}}}w$ in ${\mathcal {V}}^{*}$ and

$$\begin{aligned}&\limsup G(w_n,z) = \limsup (\langle {\mathcal {A}}w_n,z-w_n\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}+\langle \xi -f,z-w_n\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}})\\&\quad = \limsup \langle {\mathcal {A}}w_n,z-w_n\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} +\lim \langle \xi -f,z-w_n\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}\\&\quad =\langle {\mathcal {A}}w,z-w\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}+\langle \xi -f,z-w\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} = G(w,z). \end{aligned}$$

Hence $G(\cdot ,z)$ is usc on $\mathrm{conv}(D)$ for all $z \in {{\mathcal {K}}}_1$, i.e., the condition H(G)(2) holds. It follows from hypothesis H(A)(2),(4) that G is pseudomonotone on ${{\mathcal {K}}}_1$. The proof is analogous to the one in [7, Lemma 3.4] or in [6, Lemma 4.4], and, for this reason, it is omitted. Thus, H(G)(1) is satisfied. The conditions H(G)(3),(4) can be easily verified. In conclusion, Claim 2 holds.

Claim 3

The bifunction H defined by (3.10) satisfies the condition H(H).

For the proof of H(H)(1), let $\{w_n\}\subset {{\mathcal {K}}}_1$ with $w_n\rightharpoonup w$ in ${\mathcal {W}}$. From H(M), we obtain ${{\mathcal {M}}}w_n \rightarrow {{\mathcal {M}}}w$ in $L^2(0,T;X)$. From the converse Lebesgue dominated convergence theore, see, e.g., [28, Theorem 2.39], we can find $\eta \in L^2(0, T)$ such that passing to a subsequence, if necessary, we may suppose that $\Vert M w_n(t)\Vert _X \le \eta (t)$ for a.e. $t \in (0, T)$ and $Mw_n(t)\rightarrow Mw(t)$ in X for a.e. $t\in (0,T)$. Next, we consider the function $h_n :(0, T) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} h_n(t) = j^{0}(t,\zeta (t),Mw_n(t);Mw(t)-Mw_n(t)) \ \ \text{ for } \text{ a.e. } \ \ t \in (0, T). \end{aligned}$$

Using H(j)(4) and [28, Proposition 3.23(iii)], we obtain

$$\begin{aligned}&|h_n(t)| = |j^{0}(t,\zeta (t),Mw_n(t);Mw(t)-Mw_n(t))| \\&\quad \le \Vert \partial j(t, \zeta (t), Mw_n(t)) \Vert _{X^*} \Vert M w(t) - Mw_n(t)\Vert _X \\&\quad \le (c_{0j}(t) + c_{1j} \Vert \zeta (t) \Vert _Z + c_{2j} \Vert Mw_n (t) \Vert _X) (\Vert M w(t)\Vert _X + \Vert Mw_n(t)\Vert _X) \end{aligned}$$

for a.e. $t \in (0, T)$. Hence, we have $|h_n(t)| \le {\overline{h}}(t)$ a.e. $t \in (0, T)$ with ${\overline{h}} \in L^1(0, T)$, ${\overline{h}}(t) = (c_{0j}(t) + c_{1j} \Vert \zeta (t) \Vert _Z + c_{2j} \eta (t)) (\Vert M w(t)\Vert _X + \eta (t))$ for a.e. $t \in (0, T)$. Moreover, from [28, Proposition 3.23(ii)], we know that $j^0(t,z,\cdot ; \cdot )$ is usc on $X \times X$ for $t\in (0,T)$, $z\in Z$. So, by applying Fatou’s lemma, we obtain

$$\begin{aligned}&\liminf H(w_n,w)=\liminf J^{0}(w_n;w-w_n) \\&\quad \le \limsup J^{0}(w_n;w-w_n)\\&\quad \le \limsup \int _0^T j^{0}(t,\zeta (t),Mw_n(t);Mw(t)-Mw_n(t))\,dt\\&\quad \le \int _0^T \limsup j^{0}(t,\zeta (t),Mw_n(t);Mw(t)-Mw_n(t))\,dt\\&\quad \le \int _0^T j^{0}(t,\zeta (t),Mw(t);0)\,dt = 0 . \end{aligned}$$

Hence H(H)(1) holds, i.e., H is quasimonotone. From the upper semicontinuity of $J^0(\cdot ; z)$ for all $z \in {\mathcal {V}}$, see [28, Proposition 3.23(ii)], we infer the condition H(H)(2). Next, since $J^0(w; \cdot )$ is positively homogeneous and subadditive for all $w \in {\mathcal {V}}$, see e.g. [28, Proposition 3.23(i)], it is convex. This implies the condition H(H)(3). Condition H(H)(4) is obvious. We conclude that Claim 3 holds.

Claim 4

The condition $(H_{coer})$ holds.

From Remark 8(ii), the condition $(H_{coer})$ holds if

$$\begin{aligned} \frac{G_\lambda (w,v_0)}{\Vert w-v_0\Vert _{{\mathcal {V}}}}\rightarrow -\infty \ \ \mathrm{uniformly~in}\ \lambda , \ \ \mathrm{as}\ \ \Vert w-v_0\Vert _{{\mathcal {V}}}\rightarrow +\infty , \end{aligned}$$

(3.13)

for some $v_0 \in {{\mathcal {K}}}_1$, where $G_\lambda (w,v_0)=G(w,v_0)+H(w,v_0)+\lambda \langle {{\mathscr {J}}}w,v_0-w\rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}$ for $w \in {{{\mathcal {K}}}}_1$.

By hypotheses H(A)(3), (4) and H(j)(4), (5), we have

$$\begin{aligned}&G_\lambda (w,0) = \langle {{\mathcal {A}}} w - f, -w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} + J^0(w; -w) + \lambda \langle {{\mathscr {J}}} w, -w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} \\&\quad =\langle {{\mathcal {A}}} w -{{\mathcal {A}}}0, 0-w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}+\langle {{\mathcal {A}}}0, 0-w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} +\langle f, w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}}\\&\qquad + J^0(w; 0-w) +J^0(0; w-0)-J^0(0;w-0)+ \lambda \langle {{\mathscr {J}}} w, -w \rangle _{{\mathcal {V}}^*\times {{\mathcal {V}}}} \\&\quad \le -m_A\Vert w\Vert _{{\mathcal {V}}}^2+\Vert a_0\Vert _{L^2}\Vert w\Vert _{{\mathcal {V}}} +m_{2j}(\Vert A_M\Vert \Vert w\Vert _{{\mathcal {V}}}+\Vert M0\Vert _X)^2\\&\qquad +(c_{0j}+c_{1j}\Vert \zeta \Vert _{L^2(0,T;Z)}) (\Vert A_M\Vert \Vert w\Vert _{{\mathcal {V}}}+\Vert M0\Vert _X)+\Vert f\Vert _{{\mathcal {V}}^*}\Vert w\Vert _{{\mathcal {V}}}-\lambda \Vert w\Vert _{{\mathcal {V}}}^2\\&\quad \le -(m_{A}-m_{2j}\Vert A_M\Vert ^2) \Vert w\Vert _{{\mathcal {V}}}^2+(\Vert a_0\Vert _{L^2}+2m_{2j}\Vert A_M\Vert \Vert M0\Vert _X\\&\qquad +(c_{0j}+c_{1j}\Vert \zeta \Vert _{L^2(0,T;Z)}) \Vert A_M\Vert + \Vert f\Vert _{{\mathcal {V}}^*}) \Vert w\Vert _{{\mathcal {V}}}\\&\qquad +(c_{0j}+c_{1j}\Vert \zeta \Vert _{L^2(0,T;Z)}) \Vert M0\Vert _X + m_{2j} \Vert M0\Vert _X^2, \end{aligned}$$

and hence

$$\begin{aligned} \frac{G_\lambda (w,0)}{\Vert w\Vert _{{\mathcal {V}}}}\le & {} -(m_{A}-m_{2j}\Vert A_M\Vert ^2) \Vert w\Vert _{{\mathcal {V}}} +\Vert a_0\Vert _{L^2}+2m_{2j}\Vert A_M\Vert \Vert M0\Vert _X\\&+\,(c_{0j}+c_{1j}\Vert \zeta \Vert _{L^2(0,T;Z)}) \Vert A_M\Vert + \Vert f\Vert _{{\mathcal {V}}^*} \\&+\frac{(c_{0j}+c_{1j}\Vert \zeta \Vert _{L^2(0,T;Z)}) \Vert M0\Vert _X + m_{2j} \Vert M0\Vert _X^2}{\Vert w\Vert _{{\mathcal {V}}}}. \end{aligned}$$

We deduce that the condition (3.13) is satisfied with $v_0 = 0$.

Having verified all hypotheses of Theorem 7, we deduce that Problem 18 has a solution $w \in {{{\mathcal {K}}}}_1$. Hence, Problem 15 has a solution. By Step 1 and Lemma 16, we conclude that Problem 14 has a unique solution $w \in {{{\mathcal {W}}}}$. This completes the proof of Step 2.

Step 3 Let $(\xi _i, \eta _i, \zeta _i) \in L^2(0,T;V^* \times Y \times Z)$, $i=1$, 2 and $w_1 = w_{\xi _1 \eta _1 \zeta _1}$, $w_2 = w_{\xi _2 \eta _2 \zeta _2} \in {{{\mathcal {W}}}}$ with $w_1(t)$, $w_2(t) \in K$ for a.e. $t \in (0,T)$, be the unique solutions to Problem 14 corresponding to $(\xi _1, \eta _1, \zeta _1)$ and $(\xi _2, \eta _2, \zeta _2)$, respectively. We will show the following estimate

$$\begin{aligned} \Vert w_1 - w_2 \Vert _{L^2(0, t; V)} \le c \, \big ( \Vert \zeta _1 - \zeta _2 \Vert _{L^2(0, t; Z)} + \Vert \eta _1 - \eta _2 \Vert _{L^2(0, t; Y)} + \Vert \xi _1 - \xi _2 \Vert _{L^2(0, t; V^*)} \big )\nonumber \\ \end{aligned}$$

(3.14)

for all $t \in [0,T]$, where c is a positive constant.

By the definition of solution to Problem 14, it follows that

$$\begin{aligned}&\langle w_1'(t) + A(t, w_1(t)) - f(t) + \xi _1(t), w_2(t) - w_1(t) \rangle _{V^* \times V} \\&\quad + j^0(t, \zeta _1(t), Mw_1(t); Mw_2(t) - Mw_1(t)) \\&\quad + \varphi (t, \eta _1(t), Mw_2(t)) - \varphi (t, \eta _1(t), Mw_1 (t)) \ge 0 \end{aligned}$$

for a.e. $t \in (0, T)$ and

$$\begin{aligned}&\langle w_2'(t) + A(t, w_2(t)) - f(t) + \xi _2(t), w_1(t) - w_2(t) \rangle _{V^* \times V} \\&\quad + j^0(t, \zeta _2(t), Mw_2(t); Mw_1(t) - Mw_2(t)) \\&\quad + \varphi (t, \eta _2(t), Mw_1(t)) - \varphi (t, \eta _2(t), Mw_2 (t)) \ge 0 \end{aligned}$$

for a.e. $t \in (0, T)$, and $w_1(0) = w_2 (0) = 0$. We add these two inequalities to obtain

$$\begin{aligned}&\langle w'_1(t) - w'_2(t), w_2(t) - w_1(t) \rangle _{V^* \times V} + \langle A(t, w_1(t)) - A(t, w_2(t)), w_2(t) - w_1(t)\rangle _{V^* \times V} \\&\qquad + j^0(t, \zeta _1(t), Mw_1(t); Mw_2(t) - Mw_1(t)) + j^0(t, \zeta _2(t), Mw_2(t); Mw_1(t) - Mw_2(t)) \\&\qquad + \varphi (t, \eta _1(t), Mw_2(t)) - \varphi (t, \eta _1(t), Mw_1 (t)) + \varphi (t, \eta _2(t), Mw_1(t)) - \varphi (t, \eta _2(t), Mw_2 (t)) \\&\quad \ge \langle \xi _1(t), w_1(t) - w_2(t) \rangle _{V^* \times V} - \langle \xi _2(t), w_2(t) - w_1(t) \rangle _{V^* \times V} \end{aligned}$$

for a.e. $t \in (0,T)$. Next, we integrate the above inequality on (0, t), and use hypotheses H(A)(4), H(j)(5) and $H(\varphi )$(4) to get

$$\begin{aligned}&\frac{1}{2}\Vert w_1(t) - w_2(t) \Vert ^2_H - \frac{1}{2}\Vert w_1(0) - w_2(0) \Vert ^2_H + m_A \, \int _{0}^{t} \, \Vert w_1(s) - w_2(s) \Vert _V^2 \, ds \\&\quad \le {\overline{m}}_{2j} \Vert A_M\Vert \int _{0}^{t} \Vert \zeta _1 (s) - \zeta _2(s) \Vert _Z \Vert w_1(s) - w_2(s)\Vert _V \, ds \\&\qquad + m_{2j} \Vert A_M\Vert ^2 \int _{0}^{t}\Vert w_1(s) - w_2(s)\Vert _V^2 \, ds \\&\qquad + \, \beta _{\varphi }\Vert A_M\Vert \int _{0}^{t} \Vert \eta _1(s)-\eta _2(s)\Vert _Y \, \Vert w_1(s)-w_2(s)\Vert _V \, ds \\&\qquad + \int _{0}^{t}\Vert \xi _1(s) - \xi _2(s) \Vert _{V^*} \Vert w_1(s) - w_2(s) \Vert _V \, ds \end{aligned}$$

for all $t \in [0,T]$. Next, using hypothesis $(H_2)$ and the Hölder inequality, we have

$$\begin{aligned}&(m_A - m_{2j} \Vert A_M\Vert ^2) \, \Vert w_1 - w_2 \Vert _{L^2(0, t; V)}^2 \le {\overline{m}}_{2j} \Vert A_M\Vert \Vert \zeta _1 - \zeta _2 \Vert _{L^2(0, t; Z)} \Vert w_1 - w_2 \Vert _{L^2(0, t; V)}\\&\quad + \beta _{\varphi }\Vert A_M\Vert \Vert \eta _1 - \eta _2 \Vert _{L^2(0, t; Y)} \Vert w_1 - w_2 \Vert _{L^2(0, t; V)} +\Vert \xi _1 - \xi _2 \Vert _{L^2(0, t; V^*)} \Vert w_1 - w_2 \Vert _{L^2(0, t; V)} \end{aligned}$$

for all $t \in [0,T]$. Hence, by $(H_2)$, the inequality (3.14) follows.

Step 4 In this part of the proof we use a fixed point argument. We define the operator $\Lambda :L^2(0,T; V^* \times Y \times Z) \rightarrow L^2(0,T;V^* \times Y \times Z)$ by

$$\begin{aligned} \Lambda (\xi , \eta , \zeta ) = ({R}_1 w_{\xi \eta \zeta }, {R_2} w_{\xi \eta \zeta }, {S} w_{\xi \eta \zeta }) \quad \mathrm{for\ all} \ \ (\xi , \eta , \zeta ) \in L^2(0,T;V^* \times Y \times Z), \end{aligned}$$

where $w_{\xi \eta \zeta } \in {{\mathcal {W}}}$ denotes the unique solution to Problem 14 corresponding to $(\xi , \eta , \zeta )$.

From hypothesis H(R, S), inequality (3.14) and by the Hölder inequality, we find a constant $c > 0$ such that

$$\begin{aligned}&\Vert \Lambda (\xi _1,\eta _1,\zeta _1)(t) - \Lambda (\xi _2, \eta _2,\zeta _2)(t) \Vert ^2_{V^* \times Y \times Z} \\&\quad = \Vert ({R}_1 w_1)(t) - ({R}_1 w_2)(t) \Vert _{V^*}^2 + \Vert ({R_2} w_1)(t) - ({R_2} w_2)(t) \Vert _Y^2 + \Vert ({S} w_1)(t) - ({S} w_2)(t) \Vert _Z^2 \\&\quad \le \Big ( c_{R_1} \int _0^t \Vert w_1(s) - w_2(s) \Vert _V \, ds \Big )^2 + \Big ( c_{R_2} \int _0^t \Vert w_1(s) - w_2(s) \Vert _V \, ds \Big )^2 \\&\qquad + \Big ( c_{S} \int _0^t \Vert w_1(s) - w_2(s) \Vert _V \, ds \Big )^2 \le c \, \Vert w_1 - w_2 \Vert _{L^2(0, t; V)}^2 \\&\quad \le c \, \Big ( \Vert \zeta _1 - \zeta _2 \Vert ^2_{L^2(0, t; Z)} + \Vert \eta _1 - \eta _2 \Vert ^2_{L^2(0, t; Y)} + \Vert \xi _1 - \xi _2 \Vert ^2_{L^2(0, t; V^*)} \Big ), \end{aligned}$$

which entails

$$\begin{aligned}&\Vert \Lambda (\xi _1, \eta _1, \zeta _1)(t) - \Lambda (\xi _2, \eta _2, \zeta _2)(t) \Vert ^2_{V^* \times Y \times Z} \nonumber \\&\quad \le c \, \int _{0}^{t} \, \Vert (\xi _1, \eta _1, \zeta _1)(s) - (\xi _2,\eta _2, \zeta _2)(s) \Vert ^2_{V^* \times Y \times Z} \, ds \end{aligned}$$

(3.15)

for a.e. $t \in (0,T)$. By Lemma 10, we deduce that there exists a unique fixed point $(\xi ^*,\eta ^*,\zeta ^*)$ of $\Lambda$, i.e.,

$$\begin{aligned} (\xi ^*,\eta ^*,\zeta ^*) \in L^2(0,T;V^* \times Y \times Z) \ \ \mathrm{and}\ \ \Lambda (\xi ^*, \eta ^*, \zeta ^*) = (\xi ^*, \eta ^*, \zeta ^*). \end{aligned}$$

Step 5 Let $(\xi ^*,\eta ^*, \zeta ^*) \in L^2(0,T;V^* \times Y \times Z)$ be the unique fixed point of the operator $\Lambda$. We define $w_{\xi ^* \eta ^* \zeta ^*} \in {{\mathcal {W}}}$ to be the unique solution to Problem 14 corresponding to $(\xi ^*,\eta ^*, \zeta ^*)$. From the definition of the operator $\Lambda$, we have

$$\begin{aligned} \xi ^* = {R}_1(w_{\xi ^* \eta ^* \zeta ^*}), \ \ \eta ^* = {R_2}(w_{\xi ^* \eta ^* \zeta ^*}) \ \ \text{ and } \ \ \zeta ^* = {S}(w_{\xi ^* \eta ^* \zeta ^*}) . \end{aligned}$$

Finally, we use these relations in Problem 14, and conclude that $w_{\xi ^* \eta ^* \zeta ^*}$ is the unique solution to Problem 11. This completes the proof of the theorem. $\square$

Next, we consider the constrained variational–hemivariational inequality with the nonhomogeneous initial condition $w_0 \in V$, $w_0 \not = 0$.

Problem 19

Find $w \in {{\mathcal {W}}}$ such that $w(t) \in K$ for a.e. $t \in (0, T)$ and

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \langle w'(t) + A(t, w(t)) + (R_1 w)(t) - f(t), v - w(t) \rangle _{V^* \times V} \\ \ \ \qquad + \, j^0 (t, (Sw)(t), M w(t); Mv - Mw(t))\\ \ \ \qquad + \, \varphi (t, (R_2w)(t), Mv) - \varphi (t, (R_2w)(t), Mw(t)) \ge 0 \\ \qquad \qquad \qquad \qquad \ \ \text{ for } \text{ all } \ \ v \in K, \ \text{ a.e. } \ \ t \in (0,T), \\ w(0) = w_0 . \end{array}\right. } \end{aligned}$$

Theorem 20

Under hypotheses H(A), H(j), $H(\varphi )$, H(R, S), H(K), H(M), $(H_1)$, and $(H_2)$, Problem 19 has a unique solution.

Proof

Let $\widetilde{w}(t) = w(t) - w_0$ and ${\widetilde{K}} = \{v-w_0 \mid v\in K\} \subset V$. We define operators ${\widetilde{A}} :(0,T) \times V \rightarrow V^*$ and ${\widetilde{M}} :V \rightarrow X$ by

$$\begin{aligned}&{\widetilde{A}}(t, v) = A(t, v+w_0) \ \ \text{ for } \ \ v \in V, \ \text{ a.e. } \ t \in (0, T), \end{aligned}$$

(3.16)

$$\begin{aligned}&{\widetilde{M}} v = M (v+w_0) \ \ \text{ for } \ \ v \in V. \end{aligned}$$

(3.17)

With this notation, Problem 19 can be reformulated as follows.

Problem 21

Find $\widetilde{w}\in {\mathcal {W}}$ such that $\widetilde{w}(t) \in {\widetilde{K}}$ for a.e. $t \in (0, T)$ and

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \langle \widetilde{w}'(t) + \widetilde{A}(t, \widetilde{w}(t)) + (R_1 \widetilde{w})(t) - f(t), v - \widetilde{w}(t) \rangle _{V^* \times V} \\ \ \ \qquad + \, j^0 (t, (S(\widetilde{w}+w_0))(t), \widetilde{M} \widetilde{w}(t); \widetilde{M}v - \widetilde{M}\widetilde{w}(t))\\ \ \ \qquad + \, \varphi (t, (R_2(\widetilde{w}+w_0))(t), \widetilde{M}v) - \varphi (t, (R_2(\widetilde{w}+w_0))(t), \widetilde{M}\widetilde{w}(t)) \ge 0 \\ \qquad \qquad \qquad \qquad \ \ \text{ for } \text{ all } \ \ v \in {\widetilde{K}}, \ \text{ a.e. } \ \ t \in (0,T), \\ \widetilde{w}(0) = 0 . \end{array}\right. } \end{aligned}$$

We will apply Theorem 13 to obtain that Problem 21 has a unique solution $\widetilde{w} \in {{{\mathcal {W}}}}$. It is sufficient to show that the operators ${\widetilde{A}}$ and ${\widetilde{M}}$ defined by (3.16) and (3.17) satisfy the hypotheses H(A) and H(M).

We first verify that ${\widetilde{A}}$ satisfies H(A). The conditions H(A)(1), (2) are obvious. For $v\in V$, a.e. $t\in (0,T)$, we have

$$\begin{aligned} \Vert {\widetilde{A}}(t,v)\Vert _{V^*}=\Vert A(t,v+w_0)\Vert _{V^*}\le a_0(t)+a_1\Vert v+w_0\Vert _V\le a_0(t)+a_1\Vert w_0\Vert _V+a_1 \Vert v\Vert _V. \end{aligned}$$

Hence, H(A)(3) holds with $\widetilde{a}_0(t)=a_0(t)+a_1\Vert w_0\Vert _V$ and $\widetilde{a}_1=a_1$. Moreover, for $v_1$, $v_2\in V$, a.e. $t\in (0,T)$, we have

$$\begin{aligned}&\langle {\widetilde{A}}(t,v_1)-{\widetilde{A}}(t,v_2),v_1-v_2\rangle _{V^*\times V}\\&\quad =\langle A(t,v_1+w_0)-A(t,v_2+w_0),v_1-v_2\rangle _{V^*\times V}\\&\quad =\langle A(t,v_1+w_0)-A(t,v_2+w_0),(v_1+w_0)-(v_2+w_0)\rangle _{V^*\times V}\\&\quad \ge m_A\Vert (v_1+w_0)-(v_2+w_0)\Vert _V^2=m_A\Vert v_1-v_2\Vert _V^2. \end{aligned}$$

Hence, H(A)(4) holds with $m_{\widetilde{A}}=m_A$.

Next, we check that ${\widetilde{M}}$ satisfies hypothesis H(M) with $A_{\widetilde{M}}=A_M$ and ${\widetilde{M}}0=A_Mw_0+M0$. From the linearity of the operator $A_M :V \rightarrow X$, $A_M v = M v - M0$ for $v \in V$ and (3.17), we have

$$\begin{aligned} A_{\widetilde{M}}v = {\widetilde{M}}v - {\widetilde{M}}0 = M(v + w_0) - M w_0 = A_M(v+w_0) + M0 -(A_M w_0 + M0) = A_M v \end{aligned}$$

for $v \in V$, and

$$\begin{aligned} {\widetilde{M}}0 = Mw_0 = A_Mw_0 + M0, \end{aligned}$$

which means that ${\widetilde{M}}$ is an affine operator. Furthermore, for $v\in V$, a.e. $t\in (0,T)$, we get

$$\begin{aligned} (\widetilde{{\mathcal {M}}}v)(t)={\widetilde{M}}(v(t))=M(v(t)+w_0)={\mathcal {M}}(v+w_0)(t). \end{aligned}$$

It follows from the compactness of ${\mathcal {M}}$ that $\widetilde{{\mathcal {M}}}$ is compact. Therefore, ${\widetilde{M}}$ satisfies H(M).

It follows from Theorem 13 that Problem 21 has a unique solution $\widetilde{w} \in {{{\mathcal {W}}}}$. Therefore, $w \in {{{\mathcal {W}}}}$ given by $w(t) = \widetilde{w}(t) + w_0$ is the unique solution to Problem 19. This completes the proof. $\square$

4 Unilateral Frictional Contact Problem

In this section, we study a unilateral viscoelastic frictional contact problem to which our main results of Sect. 3 can be applied. We give the classical formulation of the contact problem, provide its variational formulation, and prove a result on its unique weak solvability.

Consider the following physical setting. A viscoelastic body occupies a bounded domain $\Omega \subset {\mathbb {R}}^{d}$, $d=1$, 2, 3. The boundary of $\Omega$, denoted by $\Gamma$, is assumed to be Lipschitz continuous. We denote by ${\varvec{\nu }}$ the outward unit normal at $\Gamma$. We suppose that $\Gamma$ consists of three mutually disjoint and measurable parts $\overline{\Gamma }_{D}$, $\overline{\Gamma }_{N}$ and $\overline{\Gamma }_{C}$ such that $\mathrm{meas}\,(\Gamma _{D}) > 0$. The symbol ${\mathbb {S}}^{d}$ denotes the space of $d\times d$ symmetric matrices. The standard inner products and norms on ${\mathbb {R}}^{d}$ and ${\mathbb {S}}^{d}$ are given by

$$\begin{aligned}&{\varvec{u}}\cdot {\varvec{v}}=u_{i}v_{i},~\Vert {\varvec{u}}\Vert =({\varvec{u}}\cdot {\varvec{u}})^{1/2}~~~\mathrm{for}~~{\varvec{u}}=(u_{i}), {\varvec{v}}=(v_{i})\in {\mathbb {R}}^{d},\\&{\varvec{\sigma }}\cdot {\varvec{\tau }}=\sigma _{i}\tau _{i},~\Vert {\varvec{\sigma }}\Vert =({\varvec{\sigma }}\cdot {\varvec{\sigma }})^{1/2}~~~\mathrm{for}~~{\varvec{\sigma }}= (\sigma _{ij}),{\varvec{\tau }}=(\tau _{ij})\in {\mathbb {S}}^{d}, \end{aligned}$$

respectively. For a vector field ${\varvec{v}}$, $v_{\nu }$ and ${\varvec{v}}_{\tau }$ denote its normal and tangential components on the boundary defined by

$$\begin{aligned} v_{\nu }={\varvec{v}}\cdot {\varvec{\nu }}\ \ \text{ and } \ \ {\varvec{v}}_{\tau }={\varvec{v}}-v_{\nu }{\varvec{\nu }}. \end{aligned}$$

Given a tensor ${\varvec{\sigma }}$, the symbols $\sigma _{\nu }$ and ${\varvec{\sigma }}_{\tau }$ denote its normal and tangential components on the boundary, i.e.,

$$\begin{aligned} \sigma _{\nu }=({\varvec{\sigma }}{\varvec{\nu }})\cdot {\varvec{\nu }}\ \ \text{ and } \ \ {\varvec{\sigma }}_{\tau }={\varvec{\sigma }}{\varvec{\nu }}-\sigma _{\nu }{\varvec{\nu }}. \end{aligned}$$

The classical model for the contact process on the finite time interval is the following.

Problem 22

Find a displacement field ${\varvec{u}}:\Omega \times (0, T) \rightarrow {\mathbb {R}}^d$ and a stress field ${\varvec{\sigma }}:\Omega \times (0, T) \rightarrow {\mathbb {S}}^d$ such that for all $t\in (0,T)$,

$$\begin{aligned} {\varvec{\sigma }}(t)&={{\mathscr {A}}}{\varvec{\varepsilon }}({{\varvec{u}}}'(t)) +{{\mathscr {B}}}{\varvec{\varepsilon }}({{\varvec{u}}}(t)) +\int _0^t{{\mathscr {C}}}(t-s){\varvec{\varepsilon }}({{\varvec{u}}}'(s))\,ds \quad&\mathrm{in}&\Omega , \end{aligned}$$

(4.1)

$$\begin{aligned} {{\varvec{u}}}''(t)&=\mathrm{Div}\,{\varvec{\sigma }}(t)+{\varvec{f}}_0(t)\quad&\mathrm{in}&\Omega , \end{aligned}$$

(4.2)

$$\begin{aligned} {\varvec{u}}(t)&={\varvec{0}}&\mathrm{on}&\Gamma _D, \end{aligned}$$

(4.3)

$$\begin{aligned} {\varvec{\sigma }}(t){\varvec{\nu }}&={\varvec{f}}_N(t)\quad&\mathrm{on}&\Gamma _N, \end{aligned}$$

(4.4)

$$\begin{aligned} \nonumber u'_{\nu }(t)&\le g, \ \sigma _{\nu }(t)+\eta (t) \le 0, \ (u'_{\nu }(t)-g)(\sigma _{\nu }(t)+\eta (t))=0, \nonumber \\ \eta (t)&\in k(u_{\nu }(t))\, \partial j_{\nu }(u'_{\nu }(t)) \quad&\mathrm{on}&\Gamma _C, \end{aligned}$$

(4.5)

$$\begin{aligned} \Vert {\varvec{\sigma }}_\tau (t)\Vert&\le F_b\Big (\int _0^t\Vert {\varvec{u}}_\tau (s)\Vert \,ds\Big ),\quad&\nonumber \\ -{\varvec{\sigma }}_\tau&=F_b\Big (\int _0^t\Vert {\varvec{u}}_\tau (s)\Vert \,ds\Big ) \frac{{{\varvec{u}}}_\tau '(t)}{\Vert {{\varvec{u}}}_\tau '(t)\Vert }\ \ \ \mathrm{if}\ {{\varvec{u}}}_\tau '(t)\ne {\varvec{0}}\quad&\mathrm{on}&\Gamma _C, \end{aligned}$$

(4.6)

and

$$\begin{aligned} {\varvec{u}}(0)={\varvec{u}}_0,\qquad {{\varvec{u}}}'(0)={\varvec{w}}_0\quad \ \mathrm{in}\quad \Omega . \end{aligned}$$

(4.7)

In this problem, equation (4.1) represents the constitutive law for viscoelastic materials with long memory in which ${\mathscr {A}}$ is the viscosity operators, ${\mathscr {B}}$ represents the elasticity operator and ${\mathscr {C}}$ is the relaxation tensor, and ${\varvec{\varepsilon }}({\varvec{u}})$ denotes the linearized strain tensor defined by

$$\begin{aligned} {\varvec{\varepsilon }}({\varvec{u}}) = (\varepsilon _{ij}({\varvec{u}})), \ \ \ \varepsilon _{ij} ({\varvec{u}}) = \frac{1}{2} (u_{i,j} + u_{j,i}) \ \ \text{ in } \ \ \Omega . \end{aligned}$$

Equation (4.2) is the equation of motion in which $\mathrm{Div} {\varvec{\sigma }}= (\sigma _{ij,j})$ and ${\varvec{f}}_{0}$ denotes the density of the body forces. The displacement homogeneous boundary condition (4.3) means that the body is fixed on $\Gamma _{D}$, while (4.4) is the traction boundary condition with surface tractions of density ${\varvec{f}}_N$ acting on $\Gamma _{N}$. Condition (4.5) is the Signorini unilateral contact boundary condition for the normal velocity in which $g > 0$ and $\partial j_\nu$ denotes the Clarke subgradient of a prescribed function $j_\nu$. Condition $\eta (t) \in k(u_{\nu }(t))\, \partial j_{\nu }(u'_{\nu }(t))$ on $\Gamma _C$ represents the normal damped response condition where k is a given damper coefficient depending on the normal displacement. Condition (4.6) is a version of the Coulomb law of dry friction in which $F_b$ denotes the friction bound. The latter may depend on the total (accumulated slip) represented by

$$\begin{aligned} \int _0^t \Vert {\varvec{u}}_{\tau }({\varvec{x}},s) \Vert \, ds \end{aligned}$$

at the point $x \in \Gamma _C$ in the time interval [0, t]. Examples and details on mechanical interpretation of conditions (4.5) and (4.6) can be found in [18, 28] and references therein. Finally, conditions as in (4.7) are the initial conditions in which ${\varvec{u}}_0$ and ${\varvec{w}}_0$ represent the initial displacement and the initial velocity, respectively.

To provide the weak formulation of Problem 22, we use the following spaces

$$\begin{aligned} {\mathcal {H}}=L^{2}(\Omega ;{\mathbb {S}}^{d}), \ \ V=\{\, {\varvec{v}}\in H^{1}(\Omega ;{\mathbb {R}}^{d})\mid {\varvec{v}}={\mathbf{0}}~\mathrm{on}~\Gamma _{D}\, \}. \end{aligned}$$

The space ${\mathcal {H}}$ is a Hilbert space endowed with the inner product

$$\begin{aligned} \langle {\varvec{\sigma }}, {\varvec{\varepsilon }}\rangle _{{\mathcal {H}}}= \int _{\Omega }\sigma _{ij}({\varvec{x}})\varepsilon _{ij}({\varvec{x}})\, dx \ \ \text{ for } \text{ all } \ \ {\varvec{\sigma }}, {\varvec{\varepsilon }}\in {{{\mathcal {H}}}}, \end{aligned}$$

and the associated norm $\Vert \cdot \Vert _{{\mathcal {H}}}$. The inner product and the corresponding norm on V are given by

$$\begin{aligned} ({\varvec{u}},{\varvec{v}})_{V} = ({\varvec{\varepsilon }}({\varvec{u}}), {\varvec{\varepsilon }}({\varvec{v}}))_{{\mathcal {H}}},~~\Vert {\varvec{v}}\Vert _{V}=\Vert {\varvec{\varepsilon }}({\varvec{v}})\Vert _{{\mathcal {H}}} \ \ \text{ for } \text{ all } \ \ {\varvec{u}},\, {\varvec{v}}\in V. \end{aligned}$$

Recall, see e.g. [28], that, since $\mathrm{meas}(\Gamma _{D}) > 0$, V is a Hilbert space. Moreover, the continuity of the trace operator $\gamma :V\rightarrow L^{2}(\Gamma _C;{\mathbb {R}}^{d})$ implies

$$\begin{aligned} \Vert {\varvec{v}}\Vert _{L^{2}(\Gamma ;{\mathbb {R}}^{d})} \le \Vert \gamma \Vert \Vert {\varvec{v}}\Vert _{V} \ \ \text{ for } \text{ all } \ \ {\varvec{v}}\in V, \end{aligned}$$

where $\Vert \gamma \Vert$ denotes the norm of the trace operator in ${{{\mathcal {L}}}}(V,L^2(\Gamma _C;{\mathbb {R}}^d))$. We define a space of fourth order tensor fields

$$\begin{aligned} Q_\infty =\{\, {\varvec{\sigma }}=(\sigma _{ijkl}) \mid \sigma _{ijkl}=\sigma _{jikl} =\sigma _{klij}\in L^{\infty }(\Omega ), 1\le i,j,k,l\le d \, \} \end{aligned}$$

which is a real Banach space with the norm

$$\begin{aligned} \Vert {\varvec{\sigma }}\Vert _{Q_\infty } =\sum _{1\le i,j,k,l\le d} \Vert \sigma _{ijkl}\Vert _{L^{\infty }(\Omega )} \ \ \text{ for } \text{ all } \ \ {\varvec{\sigma }}\in Q_\infty . \end{aligned}$$

Our hypotheses on Problem 22 read as follows.

$\underline{H({{\mathscr {A}}})}:$    ${{\mathscr {A}}} :\Omega \times {\mathbb {S}}^{d}\rightarrow {\mathbb {S}}^{d}$ is such that

• (1) ${{\mathscr {A}}}(\cdot ,{\varvec{\varepsilon }})$ is measurable on $\Omega$ for all ${\varvec{\varepsilon }}\in {\mathbb {S}}^{d}$.

• (2) there exists $L_{{{\mathscr {A}}}}>0$ such that $\Vert {\mathscr {A}}({\varvec{x}},{\varvec{\varepsilon }}_{1})- {\mathscr {A}}({\varvec{x}},{\varvec{\varepsilon }}_{2})\Vert \le L_{{{\mathscr {A}}}} \Vert {\varvec{\varepsilon }}_{1}-{\varvec{\varepsilon }}_{2}\Vert$ for all

  ${\varvec{\varepsilon }}_{1}$, ${\varvec{\varepsilon }}_{2}\in {\mathbb {S}}^{d}$, a.e. ${\varvec{x}}\in \Omega$.

• (3) $\text{ there } \text{ exists } \ m_{{\mathscr {A}}}>0 \ \text{ such } \text{ that } \ \ ({\mathscr {A}}({\varvec{x}},{\varvec{\varepsilon }}_{1})- {\mathscr {A}}({\varvec{x}},{\varvec{\varepsilon }}_{2}))\cdot ({\varvec{\varepsilon }}_{1}-{\varvec{\varepsilon }}_{2})\ge m_{{\mathscr {A}}} \Vert {\varvec{\varepsilon }}_{1}-{\varvec{\varepsilon }}_{2}\Vert ^{2}$

  $\ \ \qquad \text{ for } \text{ all } \ {\varvec{\varepsilon }}_{1}, {\varvec{\varepsilon }}_{2}\in {\mathbb {S}}^{d}, \ \text{ a.e. } \ {\varvec{x}}\in \Omega$.

• (4) ${{\mathscr {A}}}({\varvec{x}}, {\mathbf{0}}) = {\mathbf{0}}$ for a.e. ${\varvec{x}}\in \Omega$.

$\underline{H({{\mathscr {B}}})}:$    ${\mathscr {B}}:\Omega \times {\mathbb {S}}^{d}\rightarrow {\mathbb {S}}^{d}$ is such that

• (1) ${\mathscr {B}}(\cdot ,{\varvec{\varepsilon }})$ is measurable on $\Omega$ for all ${\varvec{\varepsilon }}\in {\mathbb {S}}^{d}$.

• (2) there exists $L_{{\mathscr {B}}}>0$ such that $\Vert {\mathscr {B}}({\varvec{x}},{\varvec{\varepsilon }}_{1})- {\mathscr {B}}({\varvec{x}},{\varvec{\varepsilon }}_{2})\Vert \le L_{{\mathscr {B}}} \Vert {\varvec{\varepsilon }}_{1}-{\varvec{\varepsilon }}_{2}\Vert$ for all

  ${\varvec{\varepsilon }}_{1}$, ${\varvec{\varepsilon }}_{2}\in {\mathbb {S}}^{d}$, a.e. ${\varvec{x}}\in \Omega$.

• (3) ${{\mathscr {B}}}({\varvec{x}}, {\mathbf{0}}) = {\mathbf{0}}$ for a.e. ${\varvec{x}}\in \Omega$.

$\underline{H({{\mathscr {C}}})}:$    ${\mathscr {C}}\in C([0,T]; Q_\infty )$.

$\underline{H({k})}:$    $k :\Gamma _C \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is such that

• (1) $k(\cdot , r)$ is measurable on $\Gamma _{C}$ for all $r \in {\mathbb {R}}$.

• (2) there exist $k_1$, $k_2$ such that $0 < k_1 \le k({\varvec{x}},t)\le k_2$ for all $r \in {\mathbb {R}}$, a.e. ${\varvec{x}}\in \Gamma _{C}$.

• (3) there exists $L_{k}>0$ such that $|k({\varvec{x}},r_{1})-k({\varvec{x}},r_{2})|\le L_{k}|r_{1}-r_{2}|$ for all $r_{1}$, $r_{2}\in {\mathbb {R}}$, a.e. ${\varvec{x}}\in \Gamma _{C}$.

$\underline{H({j_\nu })}:$    $j_\nu :\Gamma _C \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is such that

• (1) $j_{\nu }(\cdot ,r)$ is measurable on $\Gamma _{C}$ for all $r \in {\mathbb {R}}$ and there exists ${e}\in L^{2}(\Gamma _{C})$ such that $j_{\nu }(\cdot ,{e}(\cdot )) \in L^{1}(\Gamma _{C})$.

• (2) $j_{\nu }({\varvec{x}},\cdot )$ is locally Lipschitz on ${\mathbb {R}}$ for a.e. ${\varvec{x}}\in \Gamma _{C}$.

• (3) there are $\overline{c}_{0}$, $\overline{c}_{1} \ge 0$ such that $|\partial j_{\nu }({\varvec{x}},r)|\le \overline{c}_{0} + \overline{c}_{1}| r |$ for all $r \in {\mathbb {R}}$, a.e. ${\varvec{x}}\in \Gamma _{C}$.

• (4) there exists $\alpha _{j_{\nu }} \ge 0$ such that $j_{\nu }^0({\varvec{x}},r_{1}; r_{2}-r_{1}) + j_{\nu }^0({\varvec{x}},r_{2};r_{1}-r_{2}) \le \alpha _{j_{\nu }}|r_{1}-r_{2}|^{2}$ for all $r_{1}$, $r_{2}\in {\mathbb {R}}$, a.e. ${\varvec{x}}\in \Gamma _{C}$.

$\underline{H({F_b})}:$    ${F}_b :\Gamma _C \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is such that

• (1) $F_b(\cdot , r)$ is measurable on $\Gamma _{C}$ for all $r \in {\mathbb {R}}$.

• (2) there exists $L_{F_b}>0$ such that $|F_b({\varvec{x}},r_{1}) - F_b({\varvec{x}},r_{2})|\le L_{F_b}|r_{1}-r_{2}|$ for all $r_{1}$, $r_{2}\in {\mathbb {R}}$, a.e. ${\varvec{x}}\in \Gamma _{C}$.

• (3) $F_{b}({\varvec{x}},r)=0$ for all $r\le 0$, $F_{b}({\varvec{x}},r) \ge 0$ for all $r\ge 0$ for a.e. ${\varvec{x}}\in \Gamma _{C}$.

$\underline{(H_3)}:$    ${\varvec{f}}_0 \in L^2(0,T;L^2(\Omega ;{\mathbb {R}}^d))$, ${\varvec{f}}_N \in L^2(0,T;L^2(\Gamma _N;{\mathbb {R}}^d))$, ${\varvec{u}}_0$, ${\varvec{w}}_0\in V$.

Further, we introduce the set of admissible velocity fields U defined by

$$\begin{aligned} U=\{\, {\varvec{v}}\in V \mid v_{\nu } \le g \ \ \text{ on } \ \ \Gamma _{C}\, \}, \end{aligned}$$

and an element ${\varvec{f}}\in V^*$ by

$$\begin{aligned} \langle {\varvec{f}},{\varvec{v}}\rangle _{V^* \times V}= \langle {\varvec{f}}_{0}, {\varvec{v}}\rangle _{L^{2}(\Omega ;{\mathbb {R}}^{d})} +\langle {\varvec{f}}_{N},{\varvec{v}}\rangle _{L^{2}(\Gamma _{N};{\mathbb {R}}^{d})} \end{aligned}$$

(4.8)

for all ${\varvec{v}}\in V$.

We now turn to the weak formulation of Problem 22. Let $({\varvec{u}}, {\varvec{\sigma }})$ be a smooth solution to this problem which means that the data are smooth functions such that all the derivatives and all the conditions are satisfied in the usual sense at each point. Let ${\varvec{v}}\in U$ and $t\in (0,T)$. We multiply (4.2) by ${\varvec{v}}-{\varvec{u}}'(t)$, use an the integration by parts formula, and apply the boundary conditions (4.3) and (4.4) to obtain

$$\begin{aligned}&\int _\Omega {\varvec{u}}''(t)\cdot ({\varvec{v}}-{\varvec{u}}'(t)) \,dx +\int _\Omega {\varvec{\sigma }}(t)\cdot \big ( {\varvec{\varepsilon }}({\varvec{v}}) -{\varvec{\varepsilon }}({\varvec{u}}'(t)) \big ) \, dx \\&\quad =\int _\Omega {\varvec{f}}_0(t)\cdot ({\varvec{v}}-{\varvec{u}}'(t))dx +\int _{\Gamma _N} {\varvec{f}}_N(t)\cdot ({\varvec{v}}-{\varvec{u}}'(t))\, d\Gamma +\int _{\Gamma _C} {\varvec{\sigma }}(t){\varvec{\nu }}\cdot ({\varvec{v}}-{\varvec{u}}'(t))\, d\Gamma . \end{aligned}$$

From (4.5) and the definition of the Clarke subgradient, we have

$$\begin{aligned}&\sigma _{\nu }(t) (v_\nu - u_\nu '(t)) = (\sigma _{\nu }(t) + \eta (t))(v_\nu - g) - (\sigma _{\nu }(t) + \eta (t))(u_\nu '(t) - g) \nonumber \\&\quad - \eta (t) (v_\nu - u_\nu '(t)) \ge - k(u_\nu (t)) j_\nu ^0(u_\nu '(t); v_\nu - u_\nu '(t)) \ \ \text{ on } \ \ \Gamma _C. \end{aligned}$$

(4.9)

On the other hand, the friction law (4.6) can be equivalently written as

$$\begin{aligned} {\varvec{\sigma }}_{\tau }(t) \cdot ({\varvec{v}}_\tau - {\varvec{u}}_\tau '(t)) \ge - F_{b}\Big (\int _0^t\Vert {\varvec{u}}_{\tau }(s)\Vert \,ds\Big ) \big ( \Vert {\varvec{v}}_\tau \Vert - \Vert {\varvec{u}}_\tau '(t)\Vert \big ) \ \ \text{ on } \ \ \Gamma _C.\nonumber \\ \end{aligned}$$

(4.10)

Combining (4.9), (4.10) and the decomposition formula, see [28, (6.33)], we obtain

$$\begin{aligned}&F_{b}\Big (\int _0^t\Vert {\varvec{u}}_{\tau }(s)\Vert \,ds\Big ) \big ( \Vert {\varvec{v}}_{\tau }\Vert -\Vert {\varvec{u}}'_{\tau }(t)\Vert \big ) +k(u_{\nu }(t)) \, j_{\nu }^0(u'_{\nu }(t); v_{\nu }-u'_{\nu }(t))\\&\quad +\, {\varvec{\sigma }}(t){\varvec{\nu }}\cdot ({\varvec{v}}-{\varvec{u}}'(t))\ge 0 \ \ \text{ on } \ \ \Gamma _C. \end{aligned}$$

The latter and the definition (4.8) imply

$$\begin{aligned}&\int _\Omega {\varvec{u}}''(t)\cdot ({\varvec{v}}-{\varvec{u}}'(t))\, dx +\langle {\varvec{\sigma }}(t), {\varvec{\varepsilon }}({\varvec{v}})-{\varvec{\varepsilon }}({\varvec{u}}'(t)) \rangle _{{\mathcal {H}}} \\&\quad + \int _{\Gamma _{C}} F_{b}\Big (\int _0^t\Vert {\varvec{u}}_{\tau }(s)\Vert \,ds\Big ) (\Vert {\varvec{v}}_{\tau }\Vert - \Vert {\varvec{u}}'_{\tau }(t)\Vert )\, d\Gamma \\&\quad +\int _{\Gamma _{C}}k(u_{\nu }(t)j_{\nu }^0(u'_{\nu }(t); v_{\nu }-u'_{\nu }(t))\, d\Gamma \ge \langle {\varvec{f}}, {\varvec{v}}- {\varvec{u}}\rangle _{V^* \times V}. \end{aligned}$$

Finally, exploiting (4.1), we are lead to the following variational formulation of Problem 22.

Problem 23

Find ${\varvec{u}}:(0,T)\rightarrow V$ such that ${\varvec{u}}(0)={\varvec{u}}_0$, ${\varvec{u}}'(0)={\varvec{w}}_0$ and

$$\begin{aligned}&\int _\Omega {\varvec{u}}''(t)\cdot ({\varvec{v}}-{\varvec{u}}'(t)) \, dx +\langle {\mathscr {A}}({\varvec{\varepsilon }}({\varvec{u}}'(t))), {\varvec{\varepsilon }}({\varvec{v}})-{\varvec{\varepsilon }}({\varvec{u}}'(t)) \rangle _{{\mathcal {H}}} \\&\quad + \, \langle {\mathscr {B}}({\varvec{\varepsilon }}({\varvec{u}}(t))) + \int _0^t {\mathscr {C}}(t-s){\varvec{\varepsilon }}({\varvec{u}}'(s))\, ds, {\varvec{\varepsilon }}({\varvec{v}})-{\varvec{\varepsilon }}({\varvec{u}}'(t)) \rangle _{{\mathcal {H}}} \\&\quad + \int _{\Gamma _{C}} F_{b}\Big ( \int _0^t\Vert {\varvec{u}}_{\tau }(s)\Vert \, ds \Big ) (\Vert {\varvec{v}}_{\tau }\Vert - \Vert {\varvec{u}}'_{\tau }(t)\Vert ) \, d\Gamma \\&\quad +\int _{\Gamma _{C}} k(u_{\nu }(t)) \, j_{\nu }^0(u'_{\nu }(t); v_{\nu }-u'_{\nu }(t))\, d\Gamma \ge \langle {\varvec{f}}, {\varvec{v}}- {\varvec{u}}\rangle _{V^*\times V} \end{aligned}$$

for all ${\varvec{v}}\in U$, a.e. $t \in (0,T)$.

The following result concerns the unique solvability and regularity of solution to Problem 23.

Theorem 24

Assume hypotheses $H({{\mathscr {A}}})$, $H({{\mathscr {B}}})$, $H({{\mathscr {C}}})$, H(k), $H(j_\nu )$, $H(F_b)$, $(H_3)$ and the following smallness condition

$$\begin{aligned} m_{{\mathscr {A}}} > \alpha _{j_{\nu }}k_2\Vert \gamma \Vert ^{2}. \end{aligned}$$

Then Problem 23 has a unique solution ${\varvec{u}}\in C([0, T]; V)$, ${\varvec{u}}' \in {{{\mathcal {W}}}}$ with ${\varvec{u}}'(t) \in U$ for a.e. $t \in (0, T)$.

Proof

We will apply Theorem 20 with the following functional framework: $X=Y=Z=L^2(\Gamma _C)$, $K=U$, and $M=\gamma$. Let the operator $A :V \rightarrow V^*$, and functions $\varphi :Y\times X\rightarrow {\mathbb {R}}$ and $j :Z\times X\rightarrow {\mathbb {R}}$ be defined by

$$\begin{aligned}&\langle A{\varvec{w}}, {\varvec{v}}\rangle _{V^* \times V} = \langle {\mathscr {A}}({\varvec{\varepsilon }}({\varvec{w}})), {\varvec{\varepsilon }}({\varvec{v}}) \rangle _{{\mathcal {H}}} \ \ \text{ for } \ \ {\varvec{w}}, {\varvec{v}}\in V, \\&\varphi (y,x)= \int _{\Gamma _{C}} F_{b}(y) \Vert x \Vert \, d\Gamma \ \ \text{ for } \ \ y \in Y, x \in X, \\&j(z,x)= \int _{\Gamma _{C}} k(z)j_{\nu }(x)\, d\Gamma \ \ \text{ for } \ \ z \in Z, x \in X. \end{aligned}$$

Note that A, $\varphi$ and j are now independent of t. Moreover, we introduce operators $R_1 :{{\mathcal {V}}} \rightarrow {{\mathcal {V}}}^*$, $R_2 :{{\mathcal {V}}} \rightarrow L^2(0,T;Y)$, and $S :{{\mathcal {V}}} \rightarrow L^2(0,T;Z)$ given by

$$\begin{aligned}&\langle (R_1{\varvec{w}})(t),{\varvec{v}}\rangle _{V^* \times V} = \langle {\mathscr {B}}({\varvec{\varepsilon }}({\varvec{u}}_0)+ \int _0^t{\varvec{\varepsilon }}({\varvec{w}}(s))\, ds), {\varvec{\varepsilon }}({\varvec{v}}) \rangle _{{\mathcal {H}}} \\&\quad + \, \left\langle \int _0^t {{\mathscr {C}}}(t-s) {\varvec{\varepsilon }}({\varvec{w}}(s))\, ds, {\varvec{\varepsilon }}({\varvec{v}}) \right\rangle _{{{\mathcal {H}}}} \ \ \text{ for } \ \ {\varvec{w}}\in {{{\mathcal {V}}}}, {\varvec{v}}\in V, \, t \in (0, T), \\&(R_2{\varvec{w}})(t)= \int _0^t \Vert \int _0^s{\varvec{w}}_\tau (r)dr+{\varvec{u}}_{0\tau }\Vert \, ds \ \ \text{ for } \ \ {\varvec{w}}\in {{{\mathcal {V}}}}, t \in (0, T), \\&(S{\varvec{w}})(t)= \int _0^t w_\nu (s)\,ds + u_{0\nu } \ \ \text{ for } \ \ {\varvec{w}}\in {{{\mathcal {V}}}}, t \in (0, T). \end{aligned}$$

Let ${\varvec{w}}(t)={\varvec{u}}'(t)$ for all $t\in (0,T)$. Then, with the above notation, Problem 23 can be formulated as follows.

Problem 25

Find ${\varvec{w}}\in {\mathcal {W}}$ such that ${\varvec{w}}(t) \in U$ for a.e. $t \in (0, T)$, ${\varvec{w}}(0)={\varvec{w}}_0$ and

$$\begin{aligned}&\langle {\varvec{w}}'(t) + A({\varvec{w}}(t)) + (R_1 {\varvec{w}})(t) - {\varvec{f}}(t), {\varvec{v}}- {\varvec{w}}(t) \rangle _{V^* \times V} \\&\quad + \, j^0 ((S{\varvec{w}})(t), M {\varvec{w}}(t); M{\varvec{v}}- M{\varvec{w}}(t))\\&\quad + \, \varphi ((R_2{\varvec{w}})(t), M{\varvec{v}}) - \varphi ((R_2{\varvec{w}})(t), M{\varvec{w}}(t)) \ge 0 \end{aligned}$$

for all ${\varvec{v}}\in U$, a.e. $t \in (0,T)$.

It is clear that the set $K= U$ is a closed and convex subset of V with ${\varvec{0}}\in K$, i.e., H(K) holds. The hypotheses H(A), H(j), $H(\varphi )$, H(R, S), H(M), $(H_1)$, and $(H_2)$ can be verified as in the proof in [18, Theorem 13]. We deduce from Theorem 20 that Problem 25 has a solution ${\varvec{w}}\in {\mathcal {W}}$ such that ${\varvec{w}}(t) \in U$ for a.e. $t \in (0, T)$. From the relation ${\varvec{u}}(t) = \int _0^t {\varvec{w}}(s) \, ds + {\varvec{u}}_0$ for all $t \in (0, T)$, we conclude that ${\varvec{u}}\in C([0, T]; V)$, ${\varvec{u}}' \in {{{\mathcal {W}}}}$ with ${\varvec{u}}'(t) \in U$ for a.e. $t \in (0, T)$. This completes the proof. $\square$

A couple of functions $({\varvec{u}},{\varvec{\sigma }})$ which satisfies Problem 23 and (4.1) is called a weak solution to Problem 22. Under the assumptions of Theorem 24, Problem 22 has a unique weak solution. Further, this solution has the regularity ${\varvec{u}}\in C([0, T]; V)$, ${\varvec{u}}' \in {{\mathcal {W}}}$ with ${\varvec{u}}'(t) \in U$ for a.e. $t \in (0, T)$, ${\varvec{\sigma }}\in L^2(0, T; {{{\mathcal {H}}}})$, and $\mathrm{Div} \, {\varvec{\sigma }}\in {{\mathcal {V}}}^*$.

Finally, we note that the main contribution of this paper consists in extending [18, Theorem 9] from the case of no constraints (i.e., $K = V$) to the case of a nonempty, closed and convex set of constraints $K \subset V$. Moreover, we have used the recent existence result for an equilibrium problem involving bifunctions which is a novelty for variational–hemivariational inequalities. It is reasonable to conjecture that Problem 11 has solutions without the smallness assumption in $(H_2)$, however, the uniqueness is likely to be lost.