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This article is cited in 1 scientific paper (total in 1 paper) Inverse problems for evolutionary quasi-variational hemivariational inequalities with application to mixed boundary value problems Zijia Pengab, Guangkun Yangac, Zhenhai Liudb, S. Migórskief a College of Mathematics and Physics, Guangxi Minzu University, Nanning, Guangxi, P. R. China b Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, Guangxi Minzu University, Nanning, Guangxi, P. R. China c Center for Applied Mathematics of Guangxi, Guangxi Minzu University, Nanning, Guangxi, P. R. China d Center for Applied Mathematics of Guangxi, Yulin Normal University, P. R. China e College of Sciences, Beibu Gulf University, Qinzhou, Guangxi, P. R. China f Jagiellonian University in Krakow, Krakow, Poland
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     § 1. IntroductionIn this paper, we investigate the following class of evolutionary quasi-variational hemivariational inequalities in which the constraint set is solution-dependent: find $u\in\mathcal{W}\cap G(u)$ such that
$$
\begin{equation}
\langle Lu + \mathscr{T}(a,u)-h, v-u \rangle + J^0(\gamma u; \gamma v - \gamma u) + \varphi(v)-\varphi(u)\geqslant 0 \quad \text{for all } \ v \in G(u).
\end{equation}
\tag{QVHI}
$$
Here, $L\colon \mathcal{W}\subset\mathcal{V} \to \mathcal{V}^*$ is a linear densely defined maximal monotone operator, the solution space $\mathcal{W}$ is a subspace of a reflexive Banach space $\mathcal{V}$, the governing operator $\mathscr{T} \colon \mathscr{A} \times \mathcal{V} \to \mathcal{V}^*$ depends on a parameter $a\in \mathscr{A}$, $\mathscr{A}$ represents a subset of admissible parameters of a given Banach space $\mathscr{A}_2$, $G \colon \mathcal{V} \to 2^{\mathcal{V}}$ is a constraint set-valued mapping, $J^0(\gamma u;\gamma v-\gamma u)$ stands for the Clarke generalized directional derivative of a locally Lipschitz functional $J\colon \mathcal{X}\to \mathbb{R}$ at the point $\gamma u$ and in the direction $\gamma v-\gamma u$ (see Definition 9), $\gamma\colon \mathcal{V} \to \mathcal{X}$ is a linear continuous operator, $\varphi \colon \mathcal{V} \to \mathbb{R} \cup\{+\infty\}$ is proper convex and lower semicontinuous, and $h \in \mathcal{V}^*$. The problem (QVHI) serves as a direct problem for the following inverse one in which we look for a functional parameter $a^* \in \mathscr{A}$ such that
$$
\begin{equation}
F(a^*) = \inf_{a \in \mathscr{A}} F(a), \qquad F(a) = \inf_{u \in \Theta (a)} f(u) + \alpha \| a\|_{\mathscr{A}_2},
\end{equation}
\tag{IP}
$$
where $\Theta\colon \mathscr{A} \to 2^{\mathcal{W}}$ is a set-valued mapping which with each parameter $a\in \mathscr{A}$ associates the set $\Theta(a)$ of solutions to the problem (QVHI), $\alpha>0$ stands for a regularization parameter and $f$ is a prescribed function. The main results of the paper concern the solvability and compactness of the solution set to the inequality (QVHI), which are established by employing a fixed point argument applied to a variational hemivariational inequality. Next, general existence and compactness results for the inverse problem (IP) have been proved. Finally, the abstract results are illustrated in the study of an identification problem for an initial-boundary value problem of parabolic type with mixed multivalued and non-monotone boundary conditions and a state constraint. The concept of a hemivariational inequality is based on the notion of the Clarke generalized subgradient for locally Lipschitz functions, and it has been originated in the 1980s with the works of Panagiotopoulos, see [36], [37]. The variational-hemivariational inequalities have proved to be an important tool in the study mathematical models described by various types of multivalued and non-monotone relations, and to solve open problems in applied mathematics. For such reasons, the literature on this kind of inequalities is very extensive, we refer to monographs [8], [33], [36], [44], to [5], [9], [12], [21], [22], [24], [25], [31], [34], [38]–[40], [42], [45] for evolutionary variational, hemivariational, and variational hemivariational inequalities, and to [2], [3], [18], [23], [35] for evolutionary quasi-variational and hemivariational inequalities, and the references cited there. The contributions to the field of variational-hemivariational inequalities are numerous. Further, various inequality problems have been widely used in many mathematical models in mechanics, engineering, economics and other fields, see [14], [27], [33], [37], [43], [44]. On the other hand, the literature in this area of inverse problems is also very rich and here we mention some contributions for inequality problems. In [47], Zeng and Migórski studied inverse problems for an elliptic variational-hemivariational inequality with constraint and applied their results to a frictional unilateral contact problem in non-linear elasticity. Gwinner [13] has treated a parameter identification problem for elliptic variational inequalities of second kind. Inspired by this paper, more recently, Khan and Motreanu [19] used the total variation regularization approach to study an inverse problem of discontinuous parameter identification driven by an elliptic quasi-variational inequality. This work was extended by Migórski, Khan, and Zeng [28] to the setting of an elliptic quasi-variational inequality. They developed a general regularization framework to obtain an existence result for a parameter identification inverse problem, and then applied the theoretical framework to a concrete material parameter identification problem for an implicit obstacle problem with the $p$-Laplacian operator. Subsequently, the results of [28] were further generalized in [29] by the same authors to a class of elliptic quasi-hemivariational inequalities. There, an application of a parameter identification problem to mixed boundary value problems is also presented. Recently, Peng and Liu [41] studied an inverse problem governed by a constrained elliptic quasi-variational hemivariational inequality. Although the inverse problem of elliptic inequality parameter identification is relatively mature, the inverse problem of parabolic parameter identification is still in the initial stage of development. For example, [1], [10], [15], [26], [32] have studied parameter identification and coefficient inverse problem of evolutionary variational and hemivariational inequalities. Hasanov and Liu [15] treated the inverse problem for non-linear parabolic variational inequalities, and proposed a method to identify unknown coefficients using least square cost functional minimization. Then, Migórski and Ochal [32] made an extension of [15] providing a method to study an inverse coefficient problem for a parabolic hemivariational inequality. In [48], Zeng, Motreanu, and Khan studied the solvability and optimal control of multivalued evolutionary quasi-variational and hemivariational inequalities. There, the principal operator is set-valued and pseudomonotone, and the second member represents a control in the optimal control problem. In this paper we study a new class of evolutionary quasi-variational hemivariational inequalities. We employ a parametric approach by means of evolutionary variational-hemivariational inequalities. Since the constraint set depends on the unknown solution, the solvability of such inequalities combines both the treatment of a variational-hemivariational inequality and a fixed point argument. The first novelty of the paper is to examine, for the first time, the inverse problems for parabolic quasi-variational hemivariational inequality which extends the earlier results for elliptic problems in [28], [29], and [41]. A new existence result for the direct problem removes an essential compactness hypothesis, which is significant in applications, on the compactness of the operator $\gamma \colon {\mathcal V} \to {\mathcal X}$, an assumption required, for example, in [18], [48]. Furthermore, the fixed point argument we employ in this paper is used in the space ${\mathcal W}$ instead of the space ${\mathcal V}$. The Mosco type convergence conditions in hypothesis $(\mathrm{H}_4)$(iii), (iv) seems to be weaker than the usual one since they are considered in the space ${\mathcal W}$ of solutions. The second novelty of the paper is a new application. We shall provide new results for an initial-boundary value problem of the parabolic type with mixed multivalued and non-monotone boundary conditions and state constraint involving an operator of $p$-Laplace type. This problem, in its weak formulation, leads to an evolutionary quasi-variational hemivariational inequality for which we establish a result on its solvability. Moreover, for this parabolic problem, we prove an existence result for a regularized inverse optimization problem. The setting of the direct and inverse problems for the parabolic inequality which includes spaces, operators, hypotheses of the Mosco type convergence, interior point conditions, is well-designed in a way the abstract results can be used in the application. Since the solution set for the parabolic inequality in the identification problem is not a singleton, the latter leads to a double minimization problem, which needs to analyze various properties of the solution mapping. Finally, we note that the study on the nonstationary constrained quasi-variational hemivariational inequality under consideration is motivated by non-monotone boundary semipermeability conditions, which is met in certain types of heat conduction problems (the behaviour of a semipermeable membrane of finite thickness, a temperature control problems, etc.), see [30] and the references therein. § 2. Notations and essential preliminariesIn this section we briefly recall a basic material from non-linear analysis which is needed in what follows. We refer to [4], [6], [8], [11], [33], [46] for more details. Given a Banach space $Z$, we denote by $\langle\,{\cdot}\,,{\cdot}\,\rangle_Z$ the duality between $Z$ and its dual $Z^*$. The symbol $\|\,{\cdot}\,\|_Z$ stands for the norm in $Z$, and the strong convergence, the weak convergence, and the weak$^*$ convergence are denoted by $\to$, $\rightharpoonup$ and $\stackrel{*}{\rightharpoonup}$, respectively. A set-valued operator $A$ from $Z$ to $2^{Z^*}$ is viewed as a subset of $Z \times Z^*$ with its domain and graph defined by $D(A) = \{ u \in Z \mid Au \ne \varnothing \}$ and $\operatorname{Gr}(A)= \{ (u, u^*) \in Z \times Z^* \mid u^* \in Au \}$, respectively. First, we recall basic definitions and results on single-valued and set-valued operators of monotone type. Definition 1. Let $Z$ be a reflexive Banach space with the dual $Z^*$. An operator $A \colon D(A) \subset Z \to Z^*$ is called (i) monotone if and only if $\langle Au-Av,u-v \rangle_Z\geqslant 0$ for all $u$, $v \in D(A)$; (ii) maximal monotone if and only if it is monotone and $\langle Au-w,u-v \rangle_Z\geqslant0$ for all $u\in D(A)$ entails $Av=w$; (iii) $L$-pseudomonotone (pseudomonotone with respect to $D(L)$ for a linear, densely defined and maximal monotone operator $L\colon D(L)\subset Z\to Z^*$) if and only if, for any sequence $\{u_n\} \subset D(L)$ with $u_n\rightharpoonup u$ in $Z$, $Lu_n\rightharpoonup Lu$ in $Z^*$ and $\limsup \langle Au_n, u_n - u \rangle_Z \leqslant 0$, we have $Au_n\rightharpoonup Au$ in $Z^*$ and $\lim \langle Au_n,u_n \rangle_Z \to \langle Au, u \rangle_Z$. Definition 2. Let $Z$ be a reflexive Banach space. A set-valued operator $A \colon D(A)\subset Z \to 2^{Z^*}$ is called (i) monotone if and only if $\langle u^*-v^*,u-v \rangle_Z \geqslant 0$ for all $u^*\in Au$, $v^*\in Av$, $u, v \in D(A)$; (ii) maximal monotone if and only if it is monotone and has maximal graph, that is $\langle u^*-v^*,u-v \rangle_Z \geqslant 0$ for all $u^*\in Au$, and $u\in D(A)$ entails $v^*\in Av$ and $v\in D(A)$; (iii) $L$-pseudomonotone if and only if
Proposition 3 (see Proposition 2 in [48]). Let $Z$ be a reflexive Banach space. Assume that $A$, $B\colon Z \to 2^{Z^*}$ are $L$-pseudomonotone and bounded, that is, they map bounded sets into bounded sets. Then the set-valued map $A+B\colon Z \to 2^{Z^*}$ is $L$-pseudomonotone. Definition 4 (see Theorem 2.14 in [12]). Let $Z$ be a reflexive Banach space. A set-valued operator $A \colon D(A) \subset Z \to 2^{Z^*}$ is said to be strongly quasi-bounded if, for every $M>0$, there exists $K_M>0$ such that if $u\in D(A)$ and $w\in A(u)$ are such that $\langle w,u \rangle_Z\leqslant M$ and $\| u\|_Z\leqslant M$, then $\| w\|_{Z^*}\leqslant K_M$. The following result gives a sufficient condition for strongly quasi-boundedness of an operator. Proposition 5 (see Proposition 14 in [7]). Let $Z$ be a reflexive Banach space. If the set-valued operator $A \colon D(A) \subset Z \to 2^{Z^*}$ is monotone and satisfies $0\in \operatorname{int} D(A)$, then A is strongly quasi-bounded. Theorem 6 (see Theorem 3.1 in [12]). Let $Z$ be a reflexive and strictly convex Banach space, $L\colon D(L) \subset Z \to Z^*$ be a linear densely defined maximal monotone operator, and $A\colon Z\to2^{Z^*}$ be a bounded $L$-pseudomonotone operator such that where $d\colon \mathbb{R}_+\to\mathbb{R}$ is a function such that $d(r)\to+\infty $ as $r\to+ \infty$. If $B\colon D(B) \subset Z \to 2^{Z^*}$ is a maximal monotone operator which is strongly quasi-bounded and $0 \in B(0)$, then $L+A+B$ is surjective, that is, its range is $Z^*$. Proposition 7 (see Proposition 3.8 in [33]). Let $E$ and $Y$ be metric spaces and $A\colon E\to 2^Y$. Then the following statements are equivalent. (i) $A$ is upper semicontinuous. (ii) For each closed $D\subset Y$, $A^-(D):=\{x\in E\mid A(x)\cap D\neq\varnothing\}$ is closed in $E$. (iii) For each closed $O\subset Y$, $A^+(O):=\{x\in E\mid A(x)\subset O\}$ is open in $E$, that is, if $x\in E$, $\{x_n\}\subset E$ with $x_n\to x$ and $O\subset Y$ is an open set such that $A(x)\subset O$, then we can find $n_0\in\mathbb{N}$, depending on $O$, such that $A(x_n)\subset O$ for all $n\geqslant n_0$. Next, we recall some standard facts on convex and nonsmooth analysis, see [8], [11], and [33]. Definition 8. Let $\varphi\colon Z\to\mathbb{R}\cup\{+\infty\}$ be a proper convex lower semicontinuous functional. The subdifferential of $\varphi$ is the mapping $\partial_c\varphi\colon Z\to 2^{Z^*}$ defined by $\partial_c\varphi(u)=\{z^* \in Z^* \mid \langle z^*, v-u \rangle_Z \leqslant \varphi(v)-\varphi(u) \text{ for all } v\in Z \}$ for $u \in Z$. Definition 9. Let $J \colon Z\to \mathbb{R}$ be a locally Lipschitz functional. The Clarke generalized directional derivative of $J$ at a point $u\in Z$ in a direction $v\in Z$ is defined by while the Clarke generalized gradient of $J$ at $u$ is a subset of the dual space $Z^*$ given by $\partial J(u)=\{\xi\in Z^* \mid J^0(u;v)\geqslant \langle\xi, v\rangle_Z \text{ for all } v\in Z\}$. Proposition 10. Assume $J\colon Z\to \mathbb{R}$ is a locally Lipschitz functional. Then the following assertions hold. (i) For each $u\in Z$, the function $v\mapsto J^0(u;v)$ is positively homogeneous, subadditive, and satisfies $|J^0(u;v)|\leqslant L_{u}\| v\|_Z$ with $L_{u}> 0$ being the Lipschitz constant near $u\in Z$. (ii) The functional $(u,v)\mapsto J^0(u;v)$ is upper semicontinuous. (iii) $J^0(u;v)=\max\left\{\langle\xi, v\rangle_Z\mid \xi\in \partial J(u)\right\}$ for all $u$, $v \in Z$. (iv) $\partial J(u)$ is a non-empty convex weakly* compact subset of $Z^*$, and $\| \xi\|_{Z^*} \leqslant L_u$ for all $\xi\in\partial J(u)$, $u\in Z$. (v) The graph of the generalized gradient $\partial J$ is closed in the topology of $Z\times(w^*\text{-}Z^*)$, that is, if $\{u_n\}\subset Z$ and $\{\xi_n\}\subset Z^*$ are such that $\xi_n\in\partial J(u_n)$ and $u_n\to u$ in $Z$, $\xi_n\stackrel{*}{\rightharpoonup} \xi$ in $Z^*$, then $\xi\in\partial J(u)$. Finally, we recall a compactness theorem and a well-known fixed point result of Kluge. Definition 11. Assume that $1\leqslant p \leqslant +\infty$, $1\leqslant q < +\infty$, $Y$ and $Z$ are Banach spaces such that $Y\subset Z$ with continuous embedding, and $[0,T]$ is a finite time interval. Let $\Psi$ denote a finite partition of $[0,T]$ by a family of disjoint subintervals $\delta_i=(a_i,b_i)$ such that $[0,T]=\bigcup_{i = 1}^n \overline\delta_i$. Let $\mathcal{P}$ denote the family of all such partitions. By Proposition 12 (see Proposition 2 in [16]). Let $1\leqslant p$, $q < +\infty$. Assume $Z_1$, $Z_2$, $Z_3$ are Banach spaces such that $Z_1$ is reflexive, $Z_1\subset Z_2\subset Z_3$, the embedding $Z_1\subset Z_2$ is compact, and the embedding $Z_2\subset Z_3$ is continuous. If a subset $\mathcal{M}_1\subset \mathcal{M}^{p,q}(0,T; Z_1, Z_3)$ is bounded, then it is relatively compact in $L^p(0,T; Z_2)$. Theorem 13 (see Kluge’s fixed point theorem, [20]). Let $K$ be a non-empty closed convex subset of a reflexive Banach space $Z$. Assume that $T \colon K \to 2^K$ is a set-valued map such that, for all $u\in K$, $T(u)$ is non-empty closed convex subset of $Z$, and the graph of $T$ is sequentially weakly closed. If $T(K)$ is bounded, then $T$ has a fixed point. § 3. Existence result for the direct problemIn this section, we will study an evolutionary quasi-variational hemivariational inequality. which is the direct problem for the parameter identification treated in § 4. Let $\mathcal{V}$ and $\mathcal{X}$ be two reflexive Banach spaces with duals $\mathcal{V}^*$ and $\mathcal{X}^*$, respectively. The duality bracket $\langle\,{\cdot}\,,{\cdot}\,\rangle_\mathcal{V}$ between $\mathcal{V}^*$ and $\mathcal{V}$ is shortly denoted by $\langle\,{\cdot}\,,{\cdot}\,\rangle$. Let $\mathcal{W}$ be a subspace of $\mathcal{V}$ and $L\colon \mathcal{W}\subset\mathcal{V} \to \mathcal{V}^*$ be a linear and unbounded operator. We assume that $\mathcal{W}$ is a reflexive Banach space endowed with the graph norm $\| u\|_\mathcal{W}=\| u\|_\mathcal{V}+\| Lu\|_{\mathcal{V}^*}$ for $u\in\mathcal{W}$. Let $\gamma\colon \mathcal{V} \to \mathcal{X}$ be a linear continuous operator with the adjoint $\gamma^* \colon \mathcal{X}^*\to \mathcal{V}^*$. We use the notation $\widetilde{u}=\gamma(u)$ for all $u\in \mathcal{V}$. Let $\varphi \colon \mathcal{V} \to \mathbb{R}\cup\{+\infty\}$ be a proper convex lower semicontinuous functional, and $J \colon \mathcal{X}\to \mathbb{R}$ be a locally Lipschitz functional. Moreover, let $\mathscr{A}_1$ be a Banach space and $\mathscr{A}$ be a subset of $\mathscr{A}_1$. Given $a\in \mathscr{A}$, $h \in {\mathcal{V} ^*}$, $ \mathscr{T} \colon \mathscr{A}_1 \times \mathcal{V} \to \mathcal{V}^*$ and $G \colon \mathcal{V} \to 2^{\mathcal{V}}$, we consider the following quasi-variational hemivariational inequality problem: find $u\in\mathcal{W}\cap {G(u)}$ such that
$$
\begin{equation}
\langle Lu + \mathscr{T}(a,u)-h,v-u \rangle + J^0(\widetilde{u};\widetilde{v}-\widetilde{u}) + \varphi(v)-\varphi(u)\geqslant 0 \quad \text{for all } \ v \in G(u).
\end{equation}
\tag{1}
$$
The assumptions for the data of problem (1) are as follows. Hypothesis $(\mathrm{H}_1)$. Let $D(L)\subset \mathcal{W}$ be such that $L\colon D(L)\subset \mathcal{V} \to \mathcal{V}^*$ is a linear densely defined maximal monotone operator. Also, for each sequence $\{u_n\}\subset D(L)$, if $\{u_n\}$ is bounded in $\mathcal{W}$, then $\{\widetilde{u}_n\}$ is relatively compact in $\mathcal{X}$. Hypothesis $(\mathrm{H}_2)$. $J\colon \mathcal{X}\to \mathbb{R}$ is locally Lipschitz and there exist non-negative constants $c_J$ and $d_J$, and $p>1$ such that (i) $\| \xi\|_{\mathcal{X}^*} \leqslant c_J(1 + \| u\|^{p-1}_\mathcal{X})$ for all $\xi\in \partial J(u)$ and $ u \in \mathcal{X}$; (ii) $\langle \xi_1-\xi_2,u_1-u_2\rangle_\mathcal{X}\geqslant -d_J\| u_1-u_2\|_\mathcal{X}^p$ for all $u_i\in \mathcal{X}$, $\xi_i\in \partial J(u_i)$, $i=1,2$. Hypothesis $(\mathrm{H}_3)$. $\mathscr{T} \colon \mathscr{A}_1 \times \mathcal{V} \to \mathcal{V}^*$ is a single-valued operator such that (i) $\mathscr{T}(a,{\cdot}\,) \colon \mathcal{V}\to \mathcal{V}^*$ is hemicontinuous, bounded and monotone for each $a \in \mathscr{A}_1$; (ii) there exist constants $\alpha_\mathscr{T}>\|\gamma\|^p d_J$, $\beta_\mathscr{T} \geqslant0$ and $\delta_\mathscr{T}\geqslant0$ such that
$$
\begin{equation*}
\langle {\mathscr{T}(a,v),v \rangle} \geqslant \alpha_\mathscr{T} \| v\|_{\mathcal{V}}^p - \beta_\mathscr{T} \| v\|_{\mathcal{V}}^{p-1}-\delta_\mathscr{T}\ \text{ for all } \ a\in\mathscr{A}_1, \ \ v \in {\mathcal V};
\end{equation*}
\notag
$$
(iii) the sum of operators $\mathscr{T}(a,{\cdot}\,)$ and ${\gamma^*}\circ\partial J\circ \gamma (\cdot)$ is monotone for each $a \in \mathscr{A}_1$. Hypothesis $(\mathrm{H}_4)$. $G\colon \mathcal{V} \to 2^{\mathcal{V}}$ is a set-valued operator such that (i) for each $w\in \mathcal{W}$, $G(w)$ is a non-empty closed convex subset of $\mathcal{V}$; (ii) $0 \in \operatorname{int} \bigcap_{w\in \mathcal{W}} G(w)$; (iii) given $\{w_n\}\subset \mathcal{W}$ and $\{u_n\}\subset \mathcal{V}$, $u_n\in G(w_n)$, if $u_n \in \mathcal{W}$ and $w_n \rightharpoonup w$, $u_n\rightharpoonup u$ in $\mathcal{W}$, then $u\in G(w)$; (iv) for each sequence $\{w_n\}\subset \mathcal{W}$ with $w_n \rightharpoonup w$ in $\mathcal{W}$ and for every $v\in \mathcal{W}\cap G(w)$, there exist a subsequence $\{w_{n_k}\}$ of $\{w_n\}$ and a sequence $\{v_{n_k}\}\subset \mathcal{W}$ such that $v_{n_k} \in G(w_{n_k})$ and $v_{n_k} \to v$ in $\mathcal{V}$, as $k \to \infty$; (v) the set $\mathcal{W}\cap G(w)$ as a subset of $\mathcal{V}$ is dense in $G(w)$ for each $w\in \mathcal{W}$. Hypothesis $(\mathrm{H}_5)$. $\varphi \colon \mathcal{V} \to \mathbb{R} \cup\{+\infty\}$ is a proper convex lower semicontinuous functional such that $0 \in \partial_c \varphi(0)$ and Note that the interior point condition (2) plays a significant role in solving evolutionary variational inequality problems, see, for instance, ($\mathrm{H}_\Phi$) in [12] and $\mathrm{H}(\varphi)$ in [48]. If $\varphi \colon \mathcal{V} \to \mathbb{R}$ admits only finite values, then clearly $D(\varphi)=\mathcal{V}$, and thus the interior point condition (2) can be omitted. Everywhere below in this section we suppose that $a \in \mathscr{A}$ is a fixed parameter. To solve problem (1), we first fix $w\in \mathcal{W}$ and consider the following variational hemivariational inequality problem: find $u\in\mathcal{W}\cap G(w)$ such that
$$
\begin{equation}
\langle Lu + \mathscr{T}(a,u)-h,v-u \rangle + J^0(\widetilde{u};\widetilde{v}-\widetilde{u}) + \varphi(v)-\varphi(u)\geqslant 0 \quad \text{for all } \ v \in G(w).
\end{equation}
\tag{3}
$$
Theorem 14. Under hypotheses $(\mathrm{H}_1)$, $(\mathrm{H}_2)$, $(\mathrm{H}_3)$(i), (ii), $(\mathrm{H}_4)$(i), (ii) and $(\mathrm{H}_5)$, for each $w\in\mathcal{W}$, inequality (3) has a solution. Proof. Let $w\in \mathcal{W}$ be fixed. We define a proper convex lower semicontinuous functional $\Phi_w\colon \mathcal{V} \to \mathbb{R} \cup \{+ \infty\} $ by
$$
\begin{equation*}
\Phi_w (u) = \begin{cases} \varphi(u), &u \in G(w), \\ +\infty, &u \notin G(w), \end{cases} \quad \text{for} \quad u \in {\mathcal V}.
\end{equation*}
\notag
$$
Now problem (3) can be reformulated as follows: find $u\in \mathcal{W}$ such that
$$
\begin{equation}
\langle Lu + \mathscr{T}(a,u)-h,v-u \rangle + J^0(\widetilde{u};\widetilde{v}-\widetilde{u}) + \Phi_w(v)-\Phi_w(u)\geqslant 0 \quad \text{for all } \ v \in \mathcal{V}.
\end{equation}
\tag{4}
$$
To solve (4), we consider the following inclusion:
$$
\begin{equation}
Lu + \mathscr{T}(a,u) + \gamma^*\partial J(\widetilde u) + \partial_c \Phi_w (u) \ni h \quad \text{in } \ \mathcal{V}^*.
\end{equation}
\tag{5}
$$
It is well-known that if $u\in\mathcal{W}$ is a solution to (5), then it solves (4).
To solve (5), we define $\mathcal{F}(u):= \mathscr{T}(a,u)+ {\gamma^*}\,\partial J(\widetilde u)$ for $u \in {\mathcal V}$. We claim that $\mathcal{F}\colon \mathcal{V} \to 2^{\mathcal{V}^*}$ is bounded, coercive, and $L$-pseudomonotone. In fact, it follows from $(\mathrm{H}_2)$(i) and $(\mathrm{H}_3)$(i) that $\mathcal{F}\colon\mathcal{V} \to 2^{\mathcal{V}^*}$ is bounded. Next, using hypotheses $(\mathrm{H}_2)$ and $(\mathrm{H}_3)$(ii), we have
$$
\begin{equation*}
\langle \mathcal{F}(u), u \rangle\geqslant \alpha_\mathscr{T} \| u\|_{\mathcal{V}}^p -d_J\|\gamma\|^p \| u\|_{\mathcal{V}}^p-\beta_\mathscr{T} \| u\|_{\mathcal{V}}^{p-1}-c_J \|\gamma\| \, \| u\|_{\mathcal{V}} -\delta_\mathscr{T} \quad \text{for all } u \in \mathcal{V},
\end{equation*}
\notag
$$
which implies that $\mathcal{F}$ is coercive as $ \alpha_\mathscr{T}>d_J\|\gamma\|^p$. By assumptions $(\mathrm{H}_1)$, $(\mathrm{H}_2)$, and Propositions 7 and 10, we can deduce that $\gamma^*\circ\partial J\circ \gamma\colon \mathcal{V}\to2^{\mathcal{V}^*}$ is bounded and $L$-pseudomonotone. Recall that $\mathscr{T}(a,{\cdot}\,)$ is $L$-pseudomonotone since it is pseudomonotone being a monotone and hemicontinuous operator, see Proposition 27.6 in [46]. Thus, by Proposition 3, we infer that $\mathcal{F}$ is $L$-pseudomonotone as the sum of two $L$-pseudomonotone operators.
Having in mind that $\partial_c\Phi_w\colon\mathcal{V}\to 2^{\mathcal{V}^*}$ is a maximal monotone operator, see Theorem 2.43 in [4], we are in a position to show that it is strongly quasi-bounded and $0\in\partial_c\Phi_w(0)$. It follows from Corollary 2.38 in [4] that $\operatorname{int} D(\Phi_w)\subset D(\partial_c\Phi_w)$. Note that if $\operatorname{int}U_1\subset U_2$ for $U_1, U_2\subset \mathcal{V}$, then we have $\operatorname{int} U_1\subset \operatorname{int}U_2$. Thus, $\operatorname{int}D(\Phi_w)\subset \operatorname{int} D(\partial_c\Phi_w)$. From $(\mathrm{H}_4)$(ii) and $(\mathrm{H}_5)$, it follows that $0\in \operatorname{int}D(\Phi_w)$. Consequently, we have $0\in \operatorname{int}D(\partial_c\Phi_w)$, which implies that $\partial_c\Phi_w$ is strongly quasi-bounded by Proposition 5. To verify that $0\in\partial_c\Phi_w(0)$, it suffices to prove that
$$
\begin{equation*}
0=\langle0,u-0\rangle\leqslant\Phi_w(u)-\Phi_w(0) \quad \text{for all } \ u \in \mathcal{V},
\end{equation*}
\notag
$$
which is equivalent to show that
$$
\begin{equation*}
0\leqslant\Phi_w(u)-\varphi(0) \quad \text{for all } \ u\in G(w),
\end{equation*}
\notag
$$
as $\Phi_w(0)=\varphi(0)$ from $(\mathrm{H}_4)$(ii) and $\Phi_w(u)=+\infty$ for $u \notin {G(w)}$. The above inequality is obviously true since $0\in\partial_c\varphi(0)$ and $\Phi_w(u)=\varphi(u)$ for $ u \in G(w)$. Thus, we have $0\in\partial_c\Phi_w(0)$.
Now we apply Theorem 6 to deduce that problem (5) admits at least one solution. Therefore, we conclude that problem (3) has a solution. This completes the proof of Theorem 14. For each $w \in \mathcal{W}$, we denote by $\mathcal{K}(w)$ the solution set of problem (3). This introduces a set-valued mapping $\mathcal{K}\colon \mathcal{W} \to {2^{\mathcal{W}}}$ which, under the hypotheses of Theorem 14, has non-empty values. Obviously, any fixed point of the mapping $\mathcal{K}$ is a solution to problem (1). We have the following Minty type formulation of problem (3). Lemma 15. Under the hypotheses of Theorem 14, hypotheses $(\mathrm{H}_3)$(iii) and $(\mathrm{H}_4)$(v), for each $w\in \mathcal{W}$ fixed, $u\in \mathcal{W}\cap G(w)$ is a solution to problem (3) if and only if for all $v \in \mathcal{W}\cap G(w)$ and all $\xi\in \partial J(\widetilde{v})$. Proof. Let $w\in \mathcal{W}$ and $u\in\mathcal{W}\cap {G(w)}$ be a solution to problem (3). Then, for all $v\in G(w)$, inequality (3) holds. By Proposition 10(iii), we know that
$$
\begin{equation*}
J^0(\widetilde u;\widetilde{v}-\widetilde{u}) = \max\{\langle \xi, \widetilde{v}-\widetilde{u} \rangle_\mathcal{X} \mid \xi\in \partial J(\widetilde{u}) \} \quad \text{for all } v\in\mathcal{V}.
\end{equation*}
\notag
$$
Thus, there exists $\xi_{uv}\in\partial J(\widetilde{u})$ such that $J^0(\widetilde{u};\widetilde{v}-\widetilde{u})= \langle\xi_{uv},\widetilde{v}-\widetilde{u}\rangle_\mathcal{X}$. Now inequality (3) can be rewritten as
$$
\begin{equation*}
\langle Lu + \mathscr{T}(a,u),v-u \rangle +\langle\xi_{uv},\widetilde{v}-\widetilde{u}\rangle_\mathcal{X} +\varphi(v)-\varphi(u)\geqslant \langle h, v-u \rangle \quad \text{for all } \ v\in G(w),
\end{equation*}
\notag
$$
that is,
$$
\begin{equation}
\langle Lu + \mathscr{T}(a,u)+\gamma^*\xi_{uv},v-u \rangle + \varphi(v)-\varphi(u)\geqslant\langle h, v-u \rangle \quad \text{for all } \ v\in G(w).
\end{equation}
\tag{7}
$$
Employing the monotonicity of the operator $L$, we get By $(\mathrm{H}_3)$(iii) we have
$$
\begin{equation}
\langle\mathscr{T}(a,v) +\gamma^*\xi-\mathscr{T}(a,u)-\gamma^*\xi_{uv}, v-u \rangle \geqslant0 \quad \text{for all } \ \xi\in \partial J(\widetilde{v}).
\end{equation}
\tag{9}
$$
Adding (7)–(9), we obtain
$$
\begin{equation*}
\langle Lv + \mathscr{T}(a,v)+\gamma^*\xi,v-u \rangle + \varphi(v)-\varphi(u)\geqslant \langle h, v-u \rangle,
\end{equation*}
\notag
$$
for all $v \in G (w) $ and all $\xi\in \partial J(\widetilde{v})$. Thus, inequality (6) holds.
Conversely, we assume that $u\in\mathcal{W}\cap G(w)$ satisfies inequality (6). Note that $v_\lambda=\lambda z+(1-\lambda)u\in\mathcal{W}\cap G(w)$ for all $z \in\mathcal{W}\cap G(w)$ and $\lambda\in(0,1)$ by $(\mathrm{H}_4)$(i). Inserting $v=v_\lambda$ in (6), using the positive homogeneity of $J^0(v_\lambda;{\cdot}\,)$ and the convexity of $\varphi$, and then dividing both sides of (6) by $\lambda$, we have
$$
\begin{equation}
\langle \lambda Lz+(1-\lambda)Lu + \mathscr{T}(a,v_\lambda)-h,z-u \rangle + J^0(\widetilde v_\lambda;\widetilde{z}-\widetilde{u}) +\varphi(z)-\varphi(u)\geqslant0 .
\end{equation}
\tag{10}
$$
Subsequently, we use $(\mathrm{H}_1)$, $(\mathrm{H}_3)$(i) and Proposition 10(ii), and pass to the upper limit, as $\lambda\to 0$, in the last inequality. For all $z \in \mathcal{W}\cap G(w)$, we get
$$
\begin{equation}
\langle Lu + \mathscr{T}(a,u)-h, z-u \rangle + J^0(\widetilde u;\widetilde{z}-\widetilde{u}) +\varphi(z) -\varphi(u)\geqslant0 .
\end{equation}
\tag{11}
$$
By the density of $\mathcal{W}\cap {G(w)}$ in $G(w)$, from $(\mathrm{H}_4)$(v), for each $v\in G(w)$ we can find a sequence $\{v_n\}\subset \mathcal{W}\cap G(w)$ such that $v_n\to v$ in $\mathcal{V}$. Recall that the convex functional $\varphi$ is locally Lipschitz continuous on $\operatorname{int} D(\varphi)$, see Proposition 5.2.10 in [11], and $\bigcup_{w \in \mathcal{W}}G(w) \subset \operatorname{int} D(\varphi)$ by $(\mathrm{H}_5)$. Thus, $\varphi$ is continuous on $G(w)$ for all $w \in \mathcal{W}$. Inserting $z=v_n$ in (11) and then passing to the upper limit, we obtain (3), which completes the proof of Lemma 15. In the following two lemmas, we shall provide results on the convexity, closedness, and boundedness of the values of the set-valued mapping $\mathcal{K}$. Lemma 16. Under the hypotheses of Lemma 15, the solution set $\mathcal{K}(w)$ is convex and closed in $\mathcal{W}$ for each $w\in \mathcal{W}$. Proof. Let $w\in \mathcal{W}$ and $u_1$, $u_2\in \mathcal{K}(w)$. By Lemma 15, for all $v \in\mathcal{W}\cap G(w)$ and $\xi \in \partial J(\widetilde{v})$, we have
$$
\begin{equation*}
\langle Lv + \mathscr{T}(a,v)-h,v-u_i \rangle +\langle \xi, \widetilde{v}-\widetilde u_i \rangle_\mathcal{X} +\varphi(v)-\varphi(u_i)\geqslant 0,\qquad i = 1,2.
\end{equation*}
\notag
$$
We set $u_\lambda=\lambda u_1+(1-\lambda)u_2$ for $\lambda\in [0,1]$. For all $v \in\mathcal{W}\cap G(w)$ and $\xi \in \partial J(\widetilde{v})$, by the convexity of $\varphi$, we obtain
$$
\begin{equation*}
\langle Lv + \mathscr{T}(a,v)+\gamma^*\xi-h,v-u_\lambda \rangle +\varphi(v)-\varphi(u_\lambda)\geqslant 0.
\end{equation*}
\notag
$$
Invoking Lemma 15, we infer that $u_\lambda$ is a solution to problem (3). So, $\mathcal{K}(w)$ is convex.
Fix $w\in \mathcal{W}$ and let $\{u_n\}\subset \mathcal{K}(w)$ be such that $ u_n \to u$ in $\mathcal{W}$, as $n \to \infty$. In order to show that $\mathcal{K}(w)$ is closed in $\mathcal{W}$, we will prove that $u\in\mathcal{K}(w)$. Since $\mathcal{K}(w)\subset G(w)$, by $(\mathrm{H}_4)$(i), it follows that $u\in G(w)$. Next, we aim to show that $u$ satisfies (3). Using Lemma 15, we have
$$
\begin{equation}
\langle Lv + \mathscr{T}(a,v)-h,v-u_n \rangle + \langle \xi, \widetilde{v}-\widetilde u_n \rangle_\mathcal{X} + \varphi(v)-\varphi(u_n)\geqslant 0
\end{equation}
\tag{12}
$$
for all $v \in\mathcal{W}\cap G(w)$ and all $\xi\in \partial J(\widetilde{v})$. Since $\bigcup_{w\in \mathcal{W}} G(w) \subset \operatorname{int} D(\varphi)$ by $(\mathrm{H}_5)$, we know that $\varphi$ is continuous on $G(w)$. So, passing to the limit in (12), as $n \to \infty$, we deduce that $u$ satisfies (6). Using Lemma 15 again, we have $u\in\mathcal{K}(w)$. This shows that $\mathcal{K}(w)$ is a closed subset of $\mathcal{W}$. Lemma is proved. Lemma 17. Under the hypotheses of Theorem 14, the set $\mathcal{K}(\mathcal{W})$ is bounded in $\mathcal{W}$. Proof. First we claim that $\mathcal{K}(\mathcal{W})$ is bounded in $\mathcal{V}$. In fact, if this were not true, then there would exist sequences $\{u_n\}, \{w_n\} \subset \mathcal{W}$ with $u_n\in \mathcal{K}(w_n)$ such that $\| u_n\|_\mathcal{V}\to +\infty$. From $u_n\in \mathcal{K}(w_n)$, it follows
$$
\begin{equation}
\langle Lu_n+\mathscr{T}(a,u_n)-h,v-u_n\rangle + J^0(\widetilde u_n;\widetilde v -\widetilde u_n) + \varphi(v)-\varphi(u_n)\geqslant 0
\end{equation}
\tag{13}
$$
for all $v\in G(w_n)$. Since $0\in G(w)$ for all $w\in \mathcal{W}$, we can take $v=0$ in (13) to derive
$$
\begin{equation*}
\langle Lu_n,-u_n \rangle \geqslant \langle \mathscr{T}(a,u_n)-h,u_n \rangle +\langle \xi_n,\widetilde u _n \rangle_\mathcal{X}-\varphi(0)+\varphi(u_n),
\end{equation*}
\notag
$$
where $\xi_n \in \partial J(\widetilde u_n)$ satisfies $\langle \xi_n,-\widetilde u_n \rangle_\mathcal{X} = J^0(\widetilde u _n; -\widetilde u _n)$. From $(\mathrm{H}_1)$–$(\mathrm{H}_3)$, we have, for some $\xi\in\partial J(0)$,
$$
\begin{equation}
\begin{aligned} \, \!0 &\geqslant \langle Lu_n,-u_n \rangle \,{\geqslant}\, \langle \mathscr{T}(a,u_n)-h,u_n \rangle+\langle \xi_n-\xi,\widetilde u_n-0 \rangle_\mathcal{X} +\langle \xi,\widetilde u_n \rangle_\mathcal{X}\,{-}\,\varphi(0)\,{+}\,\varphi(u_n) \nonumber \\ &\geqslant\alpha_\mathscr{T} \| u_n\|_{\mathcal{V}}^p - \beta_\mathscr{T} \| u_n\|_{\mathcal{V}}^{p-1} -\delta_\mathscr{T} -\| h\|_{\mathcal{V}^*}\| u_n\|_{\mathcal{V}} -d_J\|\widetilde u_n\|^p_\mathcal{X} \nonumber \\ &\qquad-c_J \|\gamma\|\, \| u\|_{\mathcal{V}}-\varphi(0)+\varphi(u_n) \nonumber \\ &=\alpha_\mathscr{T} \| u_n\|_{\mathcal{V}}^p-d_J\|\gamma\|^p \| u\|^p_\mathcal{V} -\beta_\mathscr{T} \|u_n\|_{\mathcal{V}}^{p-1} -\| h\|_{\mathcal{V}^*}\| u_n\|_{\mathcal{V}}-c_J \|\gamma\| \, \|u\|_{\mathcal{V}}-\delta_\mathscr{T}, \end{aligned}
\end{equation}
\tag{14}
$$
where $\varphi(u_n)\geqslant \varphi(0)$. Here, we have used the fact that $0\in\partial_c \varphi(0)$. However, (14) leads a contradiction as $\alpha_\mathscr{T}>{d_J}\|\gamma\|^p$ and $\|{u_n}\|_\mathcal{V}\to +\infty$. Hence $\mathcal{K}(\mathcal{W})$ is a bounded subset of $\mathcal{V}$.
On the other hand, by hypothesis $(\mathrm{H}_4)$(ii), there is a ball $B_\mathcal{V}(0,\varepsilon_0)$ with $\varepsilon_0 > 0$ such that $B_\mathcal{V}(0,\varepsilon_0)\subset G(w)$ for all $w\in \mathcal{W}$. From the condition $0\in \operatorname{int} D(\varphi)$, it follows that $\varphi$ is locally Lipschitz continuous at $0$, see Proposition 5.2.10 in [11]. Hence there exists $0<\varepsilon_1<\varepsilon_0$ such that $\varphi$ is Lipschitz continuous on $B_\mathcal{V}(0,\varepsilon_1)$. Thus, $\varphi$ is bounded on $B_\mathcal{V}(0,\varepsilon_1)$. Since $\langle Lu, u\rangle\geqslant0$, from inequality (13), we get
$$
\begin{equation*}
\langle Lu, v\rangle \geqslant \langle \mathscr{T}(a,u)-h, u-v\rangle -J^0(\widetilde u;\widetilde v -\widetilde u) -\varphi(v)+\varphi(u).
\end{equation*}
\notag
$$
Next, we use $(\mathrm{H}_2)$(i), $(\mathrm{H}_3)$, the boundedness of $\varphi$ on $B_\mathcal{V}(0,\varepsilon_1)$, the boundedness of $\mathcal{K}(\mathcal{W})$ in $\mathcal{V}$, and $\varphi(u)\geqslant 0$ to deduce that there exists a positive constant $M>0$ such that
$$
\begin{equation*}
\inf_{v \in B_\mathcal{V}(0,\varepsilon_1)} \bigl( \langle \mathscr{T}(a,u)-h,u-v\rangle - J^0(\widetilde u;\widetilde v -\widetilde u) -\varphi(v)+\varphi(u)\bigr) \geqslant -M \quad \text{for all } \ u \in \mathcal{K}(\mathcal{W}).
\end{equation*}
\notag
$$
From the two last inequalities, it follows that
$$
\begin{equation*}
\inf_{v \in B_\mathcal{V}(0,\varepsilon_1)} \langle Lu,v\rangle \geqslant-M \quad \text{for all } \ u \in \mathcal{K}(\mathcal{W}).
\end{equation*}
\notag
$$
Consequently, we can further deduce that
$$
\begin{equation*}
|\langle Lu,v\rangle|\leqslant M \quad \text{for all } \ v \in B_\mathcal{V}(0,\varepsilon_1), \ \ u\in \mathcal{K}(\mathcal{W}).
\end{equation*}
\notag
$$
Therefore, for all $u\in \mathcal{K}(\mathcal{W})$, we have
$$
\begin{equation*}
\| Lu\|_{\mathcal{V}^*}=\sup_{\|z\|_\mathcal{V}=1} |\langle Lu,z\rangle| = \sup_{\| v\|_\mathcal{V}= \varepsilon_1/2} \biggl|\biggl\langle Lu,\frac{2v}{\varepsilon_1}\biggr\rangle \biggr| \leqslant \frac{2}{\varepsilon_1} \sup_{\|v\|_\mathcal{V}\leqslant\varepsilon_1} |\langle Lu,{v}\rangle|\leqslant \frac{2M}{\varepsilon_1}.
\end{equation*}
\notag
$$
We conclude that the solution set $\mathcal{K}(\mathcal{W})$ is bounded in $\mathcal{W}$. Lemma 17 is proved. The main existence and compactness result for the quasi-variational hemivariational inequality problem (1) reads as follows. Theorem 18. Under hypotheses $(\mathrm{H}_1)$–$(\mathrm{H}_5)$, the solution set to problem (1) is non-empty and sequentially weakly compact in $\mathcal{W}$. Proof. First, we will prove that set of solutions to problem (1) is non-empty. We shall apply Theorem 13 to the set-valued mapping $\mathcal{K} \colon \mathcal{W} \to {2^{\mathcal{W}}}$. From Theorem 14 and Lemmas 16 and 17, we know that $\mathcal{K}$ has non-empty convex closed values in $\mathcal{W}$, and $\mathcal{K}(\mathcal{W})$ is a bounded subset of $\mathcal{W}$. In order to deduce that $\mathcal{K}$ has a fixed point in $\mathcal{W}$, it remains to check that the graph $\operatorname{Gr}(\mathcal{K})$ of the map $\mathcal{K}$ is sequentially weakly closed in $\mathcal{W}\times\mathcal{W}$.
To this end, let $\{(w_n,u_n)\}\subset \operatorname{Gr}(\mathcal{K})$ satisfy $(w_n,u_n)\rightharpoonup (w,u) $ in $\mathcal{W}\times\mathcal{W}$ as $n\to\infty$. It follows from $(\mathrm{H}_4)$(iii) that $u\in G(w)$. We shall prove that $u$ satisfies problem (3). From $(\mathrm{H}_4)$(iv), we can find a sequence $\{ z_{n_k}\}\subset\mathcal{W}$ with $z_{n_k}\in G(w_{n_k})$ such that $z_{n_k}\to u$ in $\mathcal{V}$ as $k\to\infty$. The fact that $u_n\in \mathcal{K}(w_n)$ means that inequality (13) holds for all $v\in G(w_n)$. We insert $v=z_{n_k}$ in (13) to get
$$
\begin{equation}
\langle \mathscr{T}(a,u_{n_k}),u_{n_k}-z_{n_k}\rangle \leqslant \langle Lu_{n_k}-h,z_{n_k}- u_{n_k}\rangle + J^0(\widetilde u_{n_k};\widetilde z_{n_k} -\widetilde u_{n_k}) +\varphi(z_{n_k})-\varphi(u_{n_k}).
\end{equation}
\tag{15}
$$
Combining the convergences $u_{n_k}\rightharpoonup u$ in $\mathcal{V}$, $Lu_{n_k}\rightharpoonup Lu$ in $\mathcal{V}^*$, and the monotonicity of $L$, we obtain
$$
\begin{equation*}
0 \leqslant \liminf_{k \to \infty} \langle L{u_{n_k} - Lu, u_{n_k} - u} \rangle = \liminf_{k \to \infty} \langle Lu_{n_k}, u_{n_k} \rangle - \langle Lu, u \rangle,
\end{equation*}
\notag
$$
which gives
$$
\begin{equation}
\limsup_{k \to \infty} \langle Lu_{n_k},z_{n_k}-u_{n_k}\rangle =\lim_{k \to \infty} \langle Lu_{n_k},z_{n_k}\rangle- \liminf_{k \to \infty} \langle Lu_{n_k}, u_{n_k} \rangle \leqslant 0.
\end{equation}
\tag{16}
$$
From the hypothesis $(\mathrm{H}_1)$, we have $\widetilde u_{n_k} \to \widetilde u$ in $\mathcal{X}$. So, using the upper semicontinuity of the functional $(u,v)\mapsto J^0(u;v)$, we have
$$
\begin{equation}
\limsup_{k \to \infty} J^0(\widetilde u_{n_k}; \widetilde z_{n_k} -\widetilde u_{n_k}) \leqslant J^0(\widetilde u; \widetilde u-\widetilde u) = 0.
\end{equation}
\tag{17}
$$
Recall that $\varphi$ is continuous on $G(w)$ as $\bigcup_{w \in \mathcal{W}} G(w) \subset \operatorname{int} D(\varphi)$ due to $(\mathrm{H}_5)$. Hence
$$
\begin{equation}
\limsup_{k \to \infty} (\varphi(z_{n_k})-\varphi(u_{n_k})) = \lim_{k \to \infty} \varphi(z_{n_k}) - \liminf_{k \to \infty}\varphi(u_{n_k})\leqslant 0.
\end{equation}
\tag{18}
$$
We exploit (16)–(18) and pass to the upper limit in (15) to get
$$
\begin{equation*}
\limsup_{k \to \infty} \langle \mathscr{T}(a,u_{n_k}),u_{n_k}-z_{n_k}\rangle\leqslant0.
\end{equation*}
\notag
$$
The boundedness of the operator $\mathscr{T}$ allows us to suppose that, up to a subsequence, $\mathscr{T}(a,u_{n_k})\rightharpoonup u^*$ in $\mathcal{V}^*$ for some $u^* \in \mathcal{V}^*$. Hence,
$$
\begin{equation}
\begin{aligned} \, \limsup_{k \to \infty } \langle \mathscr{T}(a,u_{n_k}),u_{n_k}-u\rangle &= \limsup_{k \to \infty} \langle \mathscr{T}(a,u_{n_k}),u_{n_k}-z_{n_k}\rangle \nonumber \\ &\qquad+\lim_{k \to \infty}\langle \mathscr{T}(a,u_{n_k}),z_{n_k}-u\rangle\leqslant0. \end{aligned}
\end{equation}
\tag{19}
$$
From (19) and the pseudomonotonicity of $\mathscr{T}(a, {\cdot}\,)$ for all $a \in \mathscr{A}$ (recall $\mathscr{T}(a, {\cdot}\,)$ is monotone and hemicontinuous), it follows that
$$
\begin{equation}
\mathscr{T}(a,u)=u^*\quad\text{and}\quad \langle\mathscr{T}(a,u_n), u_n\rangle \to\langle\mathscr{T}(a,u), u\rangle.
\end{equation}
\tag{20}
$$
Let $v\in \mathcal{W}\cap G(w)$. By $(\mathrm{H}_4)$(iv), up to a subsequence of $\{w_{n_k}\}$, there is a sequence $\{v_{n_k}\}\subset \mathcal{W}$ with $v_{n_k}\in G(w_{n_k})$ such that $v_{n_k}\to v$ in $\mathcal{V}$. Let $n=n_k$ and choose $v=v_{n_k}$ in (13). Using $(\mathrm{H}_1)$, $(\mathrm{H}_5)$, (20) and Proposition 10(ii) and passing to the upper limit, we obtain
$$
\begin{equation}
\begin{aligned} \, 0&\leqslant \limsup_{k \to \infty} \langle Lu_{n_k},v_{n_k} \rangle-\liminf_{k \to \infty} \langle Lu_{n_k}, u_{n_k} \rangle+\limsup_{k \to \infty} \langle \mathscr{T}(a,u_{n_k})-h,v_{n_k}-u_{n_k} \rangle \nonumber \\ &\qquad +\limsup_{k \to \infty}J^0(\widetilde u_{n_k};\widetilde v_{n_k}-\widetilde u_{n_k})+ \limsup_{k \to \infty }\varphi(v_{n_k})-\liminf_{k \to \infty}\varphi(u_{n_k}) \nonumber \\ &\leqslant\langle Lu + \mathscr{T}(a,u)-h,v-u \rangle +J^0(\widetilde u;\widetilde{v}-\widetilde u)+\varphi(v) -\varphi(u). \end{aligned}
\end{equation}
\tag{21}
$$
Furthermore, we can deduce that the last inequality holds for all $v\in G(w)$, by using a similar density argument as in the proof of Lemma 15. Hence we have $(w,u) \in \operatorname{Gr}(\mathcal{K})$. Therefore, the graph $\operatorname{Gr}(\mathcal{K})$ of $\mathcal{K}$ is sequentially weakly closed in $\mathcal{W}\times \mathcal{W}$. Now, we can use Theorem 13 with $Z=\mathcal{W}$ and $T=\mathcal{K}$ to derive that $\mathcal{K}$ has a fixed point $u$ which is evidently a solution to problem (1).
We proceed with the proof that the set of solutions to problem (1) is sequentially weakly compact in $\mathcal{W}$. Let $\{u_n\}$ be a sequence of solutions to problem (1). Then, $u_n\in\mathcal{W}\cap G(u_n)$ and
$$
\begin{equation}
\langle Lu_n + \mathscr{T}(a,u_n)-h,v-u_n \rangle+J^0(\widetilde u_n;\widetilde{v}-\widetilde u_n) +\varphi(v)-\varphi(u_n)\geqslant0
\end{equation}
\tag{22}
$$
for all $v\in G(u_n)$. By Lemma 17, $\{u_n\}$ is bounded in $\mathcal{W}$. Using the reflexivity of $\mathcal{W}$, we find, along with a subsequence, that $u_n\rightharpoonup u$ in $\mathcal{W}$ with some $u \in \mathcal{W}$. Since $u_n\in G(u_n)$, using hypothesis $(\mathrm{H}_4)$(iii), we have $u\in G(u)$.
For any $v\in\mathcal{W}\cap G(u)$, by assumption $(\mathrm{H}_4)$(iv), we can find a sequence $\{v_{n_k}\}\subset\mathcal{W}$ such that $v_{n_k}\in G(u_{n_k})$ and $v_{n_k}\to v$ in $\mathcal{V}$. Taking $n=n_k$ and $v=v_{n_k}$ in (22), we have
$$
\begin{equation}
\langle Lu_{n_k} + \mathscr{T}(a,u_{n_k})-h,v_{n_k}-u_{n_k} \rangle+J^0(\widetilde u_{n_k}; \widetilde{v}_{n_k}-\widetilde u_{n_k}) +\varphi(v_{n_k})-\varphi(u_{n_k})\geqslant0
\end{equation}
\tag{23}
$$
for all $v\in \mathcal{W}\cap G(u_{n_k})$. By the same argument as (21), we have
$$
\begin{equation*}
\langle Lu + \mathscr{T}(a,u)-h,v-u \rangle+J^0(\widetilde u;\widetilde{v}-\widetilde u) +\varphi(v)-\varphi(u)\geqslant0 \quad \text{for all } \ v\in \mathcal{W}\cap G(u).
\end{equation*}
\notag
$$
Furthermore, this inequality also holds for all $v\in G(u)$ by a similar density argument we have used in the proof of Lemma 15. So, $u\in\mathcal{W}\cap G(u)$ is a solution to problem (1). Consequently, the set of solutions to problem (1) is sequentially weakly compact in $\mathcal{W}$. Theorem 18 is proved. § 4. Inverse problem for parameter identificationThe goal of this section is to investigate an inverse problem to identify the functional parameter in problem (1). In this context, the quasi-variational hemivariational inequality (1) is considered to be a direct problem. Let $\Theta\colon \mathscr{A} \to 2^{\mathcal{W}}$ be a set-valued mapping which associates, with each parameter $a\in \mathscr{A}$, the set $\Theta(a)$ of solutions to problem (1). We consider an inverse problem of parameter identification as the following regularized optimization problem: find $ a^* \in \mathscr{A}$ such that
$$
\begin{equation}
F(a^*) = \inf_{a \in \mathscr{A}} F(a) \quad \text{with } \ F(a) = \inf_{u \in \Theta (a)} f(u) + \alpha \| a\|_{\mathscr{A}_2},
\end{equation}
\tag{24}
$$
where $\alpha>0$ is a regularization parameter and $f$ is a functional defined on $\mathcal{W}$. For the inverse problem (24) we need the following assumptions. Hypothesis $(\mathrm{H}_6)$. $\mathscr{A}_2$ is a Banach space, and the set $\mathscr{A}\subset \mathscr{A}_1\cap \mathscr{A}_2$ is bounded in $\mathscr{A}_1$ and weakly*-closed in $\mathscr{A}_2$. Hypothesis $(\mathrm{H}_7)$. $f\colon \mathcal{W} \to \mathbb{R}$ is bounded from below and weakly lower semicontinuous. Hypothesis $(\mathrm{H}_8)$. $\mathscr{T}(a,{\cdot}\,)\colon \mathcal{V}\,{\to}\, \mathcal{V}^*$ is continuous for each $a\in \mathscr{A}$. Given $\{a_n\}\,{\subset}\, \mathscr{A}$, $a \in \mathscr{A}$, $\{u_n\}$, $\{v_n\} \subset \mathcal{V}$, if $\{a_n\}$ is bounded in $\mathscr{A}_1$ and $ a_n \stackrel{*}{\rightharpoonup} a$ in $\mathscr{A}_2$, $v_n\to v$ in $\mathcal{V}$, and $\{u_n\}$ is bounded in $\mathcal{V}$, then The proof of the following result is similar to that of Lemma 17. Lemma 19. Under the hypotheses of Theorem 18, the set $\Theta(a)$ of solutions to problem (1) is bounded in $\mathcal{W}$ uniformly with respect to $a\in\mathscr{A}$. Next, analogously as in the proof to Lemma 15, we can obtain the following Minty formulation for problem (1). Lemma 20. Under the hypotheses of Theorem 14 and assumption $(\mathrm{H}_4)$(v), $u\in \mathcal{W}\cap G(u)$ is a solution to problem (1) if and only if We are now in a position to provide the solvability of the inverse problem and the weak compactness of its solution set. Theorem 21. Under the hypotheses of $(\mathrm{H}_1)$–$(\mathrm{H}_8)$, the set of solutions to the inverse problem (24) is non-empty and sequentially weakly* compact in $\mathscr{A}_2$. Proof. First, we claim that, for each $a\in\mathscr{A}$, there exists $u^*\in \Theta(a)$ such that $F(a) = f(u^*) + \alpha\| a\|_{\mathscr{A}_2}$. In fact, by $(\mathrm{H}_7)$, there exists a sequence $\{u_n\}\subset \Theta(a)$ such that
$$
\begin{equation*}
\lim_{n\to \infty}f(u_n) = \inf_{u \in \Theta (a)} f(u).
\end{equation*}
\notag
$$
Since the set $\Theta(a)$ is sequentially weakly compact in $\mathcal{W}$ by Theorem 18, we have $u_{n_k}\rightharpoonup u^*$ in $\mathcal{W}$ for some $u^*\in \Theta(a)$. Using the weak lower semicontinuity of $f$ from $(\mathrm{H}_7)$, we deduce that $f(u^*)= \inf_{u \in \Theta (a)} f(u)$, which proves the claim. Consequently, the function $F\colon\mathscr{A}\to\mathbb{R}$ given in problem (24) is well-defined.
Next, using $(\mathrm{H}_7)$ once more, we can find a minimizing sequence $\{a_n\}\subset \mathscr{A}$ and a constant $\varrho\in \mathbb{R}$ such that Let $u^*_n\in\Theta(a_n)$ be the corresponding sequence of solutions such that
$$
\begin{equation}
F(a_n) = f(u^*_n) + \alpha \| a_n\|_{\mathscr{A}_2} \quad \text{for } \ n = 1,2,\dots\,.
\end{equation}
\tag{26}
$$
In view of (25), (26) and the fact that $f$ is bounded from below, we can deduce that the sequence $\{a_n\}$ is bounded in $\mathscr{A}_2$. It follows from $(\mathrm{H}_6)$ that there exists $a^*\in \mathscr{A}$ such that, passing to a subsequence, we have
$$
\begin{equation}
a_{n_k}\stackrel{*}{\rightharpoonup} a^* \quad \text{in } \mathscr{A}_2 \ \text{ as } \ k\to \infty.
\end{equation}
\tag{27}
$$
Our aim is to show that $a^*$ is a solution to problem (24). Since $u^*_{n_k}\in\Theta(a_{n_k})$, it follows from the uniform boundedness of Lemma 19 that $\{u^*_{n_k}\}$ is bounded in $\mathcal{W}$. Thus, passing to a subsequence again (we use the same symbol for simplicity), we get
$$
\begin{equation}
u^*_{n_k}\rightharpoonup u^* \quad \text{in } \ \mathcal{W} \ \text{ as } \ k\to \infty, \ \text{for some } \ u^* \in\mathcal{W}.
\end{equation}
\tag{28}
$$
Since $u^*_{n_k}\in G(u^*_{n_k})$, we further get $u^*\in G(u^*)$ by $(\mathrm{H}_4)$(iii). Now from Lemma 20 we have
$$
\begin{equation}
\langle Lv + \mathscr{T}(a_{n_k},v) - h, v-u^*_{n_k} \rangle + J^0(\widetilde{u}^*_{n_k}, \widetilde{v}-\widetilde{u}^*_{n_k}) + \varphi(v)-\varphi(u^*_{n_k})\geqslant 0
\end{equation}
\tag{29}
$$
for all $v\in\mathcal{W}\cap G(u^*_{n_k})$. By $(\mathrm{H}_4)$(iv), for each $v\in \mathcal{W} \cap G(u^*)$, there exits a subsequence $\{v_{n_k}\}\subset\mathcal{W}$ with $v_{n_k}\in G(u^*_{n_k})$ such that $v_{n_k}\to v$ in $\mathcal{V}$. We insert $v=v_{n_k}$ into (29) to get
$$
\begin{equation}
\!\langle Lv_{n_k} + \mathscr{T}(a_{n_k},v_{n_k}),v_{n_k}-u^*_{n_k} \rangle +J^0(\widetilde{u}^*_{n_k}, \widetilde{v}_{n_k}-\widetilde{u}^*_{n_k}) +\varphi( v_{n_k})-\varphi(u^*_{n_k})\,{\geqslant}\, \langle h,v_{n_k}-u^*_{n_k} \rangle.
\end{equation}
\tag{30}
$$
Subsequently, we recall that $\{a_{n_k}\}$ is bounded in $\mathscr{A}_1$, $a_{n_k}\stackrel{*}{\rightharpoonup} a^*$ in $\mathscr{A}_2 $, $v_{n_k}\to v$ and $u^*_{n_k}\rightharpoonup u^*$ in $\mathcal{V}$. Since $\{v_{n_k}-u^*_{n_k}\}$ is bounded in $\mathcal{V}$, using $(\mathrm{H}_8)$, we deduce that
$$
\begin{equation}
\limsup_{k \to \infty } \langle \mathscr{T}(a_{n_k},v_{n_k}) - \mathscr{T}(a^*,v_{n_k}),v_{n_k} -u^*_{n_k} \rangle \leqslant 0.
\end{equation}
\tag{31}
$$
From $(\mathrm{H}_1)$, we know that $\widetilde{u}^*_{n_k}\to \widetilde u^*$ in $\mathcal{X}$, so by Proposition 10(ii), we have
$$
\begin{equation}
\limsup_{k \to \infty }J^0(\widetilde{u}^*_{n_k},\widetilde{v}_{n_k}-\widetilde{u}^*_{n_k})\leqslant J^0(\widetilde{u}^*,\widetilde{v}-\widetilde{u}^*).
\end{equation}
\tag{32}
$$
Consequently, we employ $(\mathrm{H}_5)$, (31), (32), the continuity of $\mathscr{T}(a,{\cdot}\,)$ assumed in $(\mathrm{H}_8)$, and pass to the upper limit in (30). For all $v\in \mathcal{W}\cap G(u^*)$, we obtain
$$
\begin{equation*}
\begin{aligned} \, \langle h, v-u^* \rangle &\leqslant \limsup_{k \to \infty} [\langle Lv_{n_k} + \mathscr{T}(a_{n_k},v_{n_k}),v_{n_k}-u^*_{n_k} \rangle \\ &\qquad + J^0(\widetilde{u}^*_{n_k},\widetilde{v}_{n_k}-\widetilde{u}^*_{n_k})+ \varphi(v_{n_k}) - \varphi(u^*_{n_k})] \\ &\leqslant\limsup_{k \to \infty} \langle \mathscr{T}(a_{n_k},v_{n_k})-\mathscr{T}(a^*,v_{n_k}),v_{n_k}-u^*_{n_k} \rangle \\ &\qquad+\limsup_{k \to \infty} \langle \mathscr{T}(a^*,v_{n_k}),v_{n_k}-u^*_{n_k} \rangle \\ &\qquad +\limsup_{k \to \infty} \langle Lv_{n_k},v_{n_k}-u^*_{n_k} \rangle + \limsup_{k \to \infty}J^0(\widetilde{u}^*_{n_k},\widetilde{v}_{n_k}-\widetilde{u}^*_{n_k}) \\ &\qquad +\limsup_{k \to \infty}\varphi(v_{n_k})-\liminf_{k \to \infty} \varphi(u^*_{n_k}) \\ &\leqslant\langle Lv + \mathscr{T}(a^*,v),v-u^* \rangle +J^0(\widetilde u^*,\widetilde v-\widetilde u^*)+\varphi( v) -\varphi(u^*). \end{aligned}
\end{equation*}
\notag
$$
Furthermore, by a density argument similar to that used in the proof of Lemma 15, we infer that the last inequality holds for all $v\in G(u^*)$. Thus, using Lemma 20, we obtain $u^*\in\Theta(a^*)$.
Combining $(\mathrm{H}_7)$, (24)–(28) and $u^*\in\Theta(a^*)$, we have
$$
\begin{equation}
\begin{aligned} \, \varrho &=\inf_{a \in \mathscr{A}} F(a) \leqslant F(a^*) = \inf_{u\in\Theta(a^*)}f(u) + \alpha\|a^*\|_{\mathscr{A}_2} \leqslant f(u^*)+ \alpha\|a^*\|_{\mathscr{A}_2} \nonumber \\ &\leqslant \liminf_{k \to \infty} f(u^*_{n_k})+ \liminf_{k \to \infty }\alpha\| a_{n_k}\|_{\mathscr{A}_2} \leqslant \liminf_{k \to \infty}(f(u^*_{n_k}) + \alpha\| a_{n_k}\|_{\mathscr{A}_2}) \nonumber \\ &=\liminf_{k \to \infty }F(a_{n_k}) =\varrho. \end{aligned}
\end{equation}
\tag{33}
$$
Therefore, $F(a^*)=\varrho$. This means that $a^*\in\mathscr{A}$ is a solution to problem (24).
Finally, let $\{a^*_n\}\subset \mathscr{A}$ be a sequence of solutions to problem (24), that is,
$$
\begin{equation}
F({a_n^*})= \inf_{a \in \mathscr{A}} F(a) \quad \text{for } \ n = 1, 2,\dots\,.
\end{equation}
\tag{34}
$$
Having in mind that $f$ is bounded from below, we deduce that $\{a^*_n\}$ is bounded in $\mathscr{A}_2$. Then, we can show that each weakly* accumulation point of $\{a^*_n\}$ in $\mathscr{A}_2$ is a solution to problem (24) by an analogous reasoning as in the first part of the proof. We conclude that the set of solutions to problem (24) is sequentially weakly* compact in $\mathscr{A}_2$. This completes the proof of Theorem 21. § 5. Parabolic mixed boundary value problemIn this section, we study an initial-boundary value problem of parabolic type with mixed boundary conditions and state constraint, which, in its weak formulation, leads to an evolutionary quasi-variational hemivariational inequality. Let $0<T<+\infty$, $2\leqslant p< + \infty$, and $\Omega$ be a bounded domain in $\mathbb{R}^N$ with Lipschitz continuous boundary $\Gamma$. Assume that $\Gamma=\overline \Gamma_a\cup \overline \Gamma_b \cup \overline \Gamma_c$, where $\Gamma_a$, $\Gamma_b$, $\Gamma_c$ are mutually disjoint and relatively open subsets of $\Gamma$ such that meas$(\Gamma_a)>0$. For brevity, we denote $Q=\Omega\times(0,T)$, $\Sigma_a=\Gamma_a\times(0,T)$, $\Sigma_b=\Gamma_b\times(0,T)$, and $\Sigma_c = \Gamma_c \times(0,T)$. We consider an initial-boundary problem: find a function $u\colon Q \to\mathbb{R}$ such that
$$
\begin{equation}
\begin{gathered} \, u' -\operatorname{div}(a(x,t)|\nabla u|^{p - 2}\, \nabla u)= h(x,t) \quad \text{in } \ Q, \\ u(0) = 0 \quad \text{in }\ \Omega,\qquad u = 0 \quad \text{on } \ \Sigma_a, \\ -\frac{\partial u}{\partial \nu_a} \in \partial_c g(x, t, u) \quad \text{on } \ \Sigma_b, \qquad -\frac{\partial u}{\partial \nu_a} \in \partial j(x, t, u) \quad \text{on } \ \Sigma_c, \end{gathered}
\end{equation}
\tag{35}
$$
and the following state constraint holds:
$$
\begin{equation}
\biggl( \int_0^T \int_\Omega |\nabla u(x,t)|^p \, dx\, dt \biggr)^{1/p} \leqslant \ell(u).
\end{equation}
\tag{36}
$$
Here, the notation $\partial u/\partial \nu_a$ stands for the conormal derivative with respect to the $p$-Laplace operator $-\operatorname{div}(a(x,t)|\nabla \cdot{|^{p - 2}}\,\nabla \cdot\,)$, $\nu$ denotes the unit outward normal vector on the boundary, and $h$ is a prescribed function. The hypotheses on the data are following. Hypothesis $(\mathrm{H}_g)$. $g\colon\Sigma_b\times \mathbb{R}\to \mathbb{R}$ is such that (i) $g(\,{\cdot}\,,s)$ is measurable on $\Sigma_b$ for each $s\in \mathbb{R}$ and there exists $v_1\in L^p(\Sigma_b)$ such that $g(\,{\cdot}\,,v_1(\,{\cdot}\,))\in L^1(\Sigma_b)$, (ii) $g(x,t,{\cdot}\,)$ is a non-negative proper convex lower semicontinuous function, and $g(x,t,0)=0$ for a.e. $(x,t)\in \Sigma_b$, (iii) there exist positive constants $p_1 \leqslant p$ and $d_g$ such that the inequality $|g(x,t,s)| \leqslant d_g(1\,{+}\,|s|^{p_1})$ holds for all $s\in \mathbb{R}$ and a.e. $(x,t)\in \Sigma_b$. Hypothesis $(\mathrm{H}_j)$. $j\colon\Sigma_c\times \mathbb{R}\to \mathbb{R}$ is such that (i) $j(\,{\cdot}\,,s)$ is measurable on $\Sigma_c$ for each $s\in \mathbb{R}$ and $j(x,t,{\cdot}\,)$ is locally Lipschitz for a.e. $(x,t)\in \Sigma_c$, and there exists $v_2\in L^p(\Sigma_c)$ such that $j(\,{\cdot}\,,v_2(\,{\cdot}\,))\in L^1(\Sigma_c)$, (ii) $|\partial j(x,t,s)|\leqslant c_j(1+|s|^{p-1})$ for all $s\in \mathbb{R}$, a.e. $(x,t)\in \Sigma_c$ with $c_j>0$, (iii) $(\xi_1-\xi_2) (s_1-s_2) \geqslant -d_j |s_1-s_2|^p$ for all $s_i\in \mathbb{R}$, $\xi_i\in \partial j(x,t,s_i)$, a.e. $(x,t)\in \Sigma_c$, $i=1$, $2$ with $d_j\geqslant0$, (iv) $j(x,t,{\cdot}\,)$ or $-j(x,t,{\cdot}\,)$ is regular in the sense of Clarke for a.e. $(x,t)\in \Sigma_c$. Hypothesis $(\mathrm{H}_\ell)$. $\ell\colon L^p(Q) \to \mathbb{R}$ is a continuous functional, and there exists a positive constant $c_\ell$ such that $\ell(u)\geqslant c_\ell$ for all $u\in L^p(Q)$. Note that problem (35) has been studied in [30] in a very particular case when $\Sigma_b = \varnothing$ and without constraint (36). We refer to [30] for physical motivations and examples of non-convex functions $j$ satisfying $(\mathrm{H}_j)$. To study the initial-boundary value problem (35), (36), we introduce the following spaces. Let $V = \{u \in W^{1,p}(\Omega)\mid u = 0 \text{ on } \Gamma_a \}$ be the Sobolev space endowed with the norm $\|u\|_V = \| \nabla u\|_{L^p(\Omega)}$. We denote by $\mathcal{V}$ the vector-valued space $L^p(0,T,V)$. Then, $\mathcal{V}^* = L^q(0,T,V^*)$, where $q$ is the conjugate exponent of $p$, and $V^*$ is the dual of $V$. Moreover, we use the reflexive Banach space $\mathcal{W}$ defined by $\mathcal{W}=\{u\in \mathcal{V}\mid u'\in \mathcal{V}^*\}$ and endowed with the graph norm $\| u\|_\mathcal{W}= \| u\|_\mathcal{V}+\| u'\|_{\mathcal{V}^*}$, where the time derivative $u'=\partial u/\partial t$ is understood in the sense of vector-valued distributions. It is well-known that the embedding $\mathcal{W}\subset C([0,T], H)$ is continuous with $H=L^2(\Omega)$. Let $1\leqslant r,s<+\infty$. Assume that $Z_1$, $Z_3$ are Banach spaces such that $Z_1$ is reflexive, $Z_1\subset L^p(\Omega)\subset Z_3$, and the embedding $Z_1\subset L^p(\Omega)$ is compact. For example, $Z_1=W_0^{1,p}(\Omega)$ and $Z_3=W^{-1,q}(\Omega)$. Consider the Banach space
$$
\begin{equation*}
\mathcal{M}^{s,r}(0,T;Z_1,Z_3)=L^s(0,T;Z_1)\cap \mathrm{BV}^r(0,T;Z_3)
\end{equation*}
\notag
$$
with the norm
$$
\begin{equation*}
\|\,{\cdot}\,\| _{\mathcal{M}^{s,r}}=\|\,{\cdot}\,\|_{L^s(0,T;Z_1)} +\|\,{\cdot}\,\|_{\mathrm{BV}^r(0,T;Z_3)}.
\end{equation*}
\notag
$$
In what follows, we denote $\mathcal{M}^{s,r}(0,T;Z_1,Z_3)$ by $\mathcal{M}^{s,r}$, for simplicity. According to Proposition 12, we know that the embedding $\mathcal{M}^{s,r}\subset L^s(0,T;L^p(\Omega))$ is continuous and compact. Let $c(p) > 0$ be the largest constant such that the following monotonicity inequality holds:
$$
\begin{equation}
(|\xi|^{p-2}\xi-|\eta|^{p-2}\eta)\cdot(\xi-\eta)\geqslant c(p)|\xi-\eta|^p \quad \text{for all } \ \xi, \eta\in\mathbb{R}^N,\ \ \xi \neq \eta.
\end{equation}
\tag{37}
$$
Given positive constants $c_1$, $c_2$ and $c_3$ with $c_1c(p)>d_j\| \gamma\|^p$, the set of admissible parameters $\mathscr{A}$ is defined by
$$
\begin{equation}
\mathscr{A}=\{a\in L^p(Q)\mid 0<c_1 \leqslant a(x,t)\leqslant c_2 \text{ a.e. in } Q, \, \| a\|_{{\mathcal M}^{s,r}} \leqslant c_3\}.
\end{equation}
\tag{38}
$$
Following (36), we introduce the constraint set $G(u)$ given by
$$
\begin{equation}
G(u)=\{v\in \mathcal{V}\mid \| v\|_\mathcal{V}\leqslant\ell(u)\}.
\end{equation}
\tag{39}
$$
We also introduce some operators and functionals as follows. Let $\overline{\gamma}\colon {V}\to L^p(\Gamma_c)$ be the trace operator, which is known to be linear, continuous and compact. Define $\mathcal{X}=L^p(\Sigma_c)$ and denote by $\gamma\colon \mathcal{V}\to \mathcal{X}$ the Nemytskii operator of $\overline{\gamma}$, which is linear and continuous but not compact. Let the operator $L \colon \mathcal{W}\subset \mathcal{V} \to \mathcal{V}^*$ be defined by $\langle Lu,v\rangle=\langle u',v \rangle$ for $u\in \mathcal{W}$ and $v\in \mathcal{V}$. It is well-known that $L$ is a linear densely defined maximal monotone operator on $D(L)=\{u\in\mathcal{W}\mid u(0)=0\}$, see [46], Proposition 32.10 and Theorem 32.L. We define the operator $\mathscr{T} \colon \mathscr{A}_1 \times \mathcal{V} \to \mathcal{V}^*$ by
$$
\begin{equation}
\langle \mathscr{T}(a,u), v \rangle =\int_0^T \int_\Omega \bigl(a(x,t)|\nabla u(x,t)|^{p-2} \nabla u(x,t), \nabla v(x,t) \bigr)_{\mathbb{R}^N}\, dx\, dt
\end{equation}
\tag{40}
$$
for all $a\in \mathscr{A}$ and $u, v\in\mathcal{V}$. Consider the functional $\varphi \colon \mathcal{V} \to \mathbb{R}$ given by
$$
\begin{equation}
\varphi(u)=\int_0^T \int_{\Gamma_b} g(x,t,\widetilde{u}(x,t)) \, d\Gamma \, dt \quad \text{for } \ u\in \mathcal{V},
\end{equation}
\tag{41}
$$
and define the functional $J \colon \mathcal{X} \to \mathbb{R}$ as follows
$$
\begin{equation}
J(z)=\int_0^T \int_{\Gamma_c} j(x,t,z(x,t)) \, d\Gamma \, dt \quad \text{for } \ z\in\mathcal{X}.
\end{equation}
\tag{42}
$$
Under assumption $(\mathrm{H}_j)$(i)–(iii), the functional $J$ is Lipschitz continuous on each bounded subset of $\mathcal{X}$, hence it is also locally Lipschitz, and
$$
\begin{equation}
J^0(w;v)\leqslant\int_0^T \int_{\Gamma_c} j^0(x,t, w(x,t);v(x,t)) \, d\Gamma \, dt \quad \text{for all } \ w, v\in\mathcal{X}.
\end{equation}
\tag{43}
$$
If, in addition, $(\mathrm{H}_j)$(iv) holds, then (43) becomes equality; see, for example, Theorem 4.15 in [33]. Combining (40)–(42), we obtain the weak formulation of (35) as follows:
$$
\begin{equation}
\begin{aligned} \, &\langle Lu+\mathscr{T}(a,u)-h,v-u\rangle +\varphi(v)-\varphi(u) \nonumber \\ &\qquad +\int_{\Sigma_c} j^0\bigl(x,t,\widetilde u(x,t); \widetilde v(x,t)-\widetilde u(x,t)\bigr) \, d\Gamma\, dt\geqslant 0 \quad\text{for all } \ v\in \mathcal{V}. \end{aligned}
\end{equation}
\tag{44}
$$
Next, for $u\in \mathcal{V}$, we introduce the indicator functional $I_{G(u)}\colon \mathcal{V}\to \mathbb{R}\cup\{+\infty\}$ by
$$
\begin{equation}
I_{G(u)} (v) := \begin{cases} 0, &v \in G(u), \\ +\infty, &v \notin G(u). \end{cases}
\end{equation}
\tag{45}
$$
Now the weak formulation of (35) with the state constraint (36) is given by: find $u\in\mathcal{W}\cap G(u)$ such that
$$
\begin{equation*}
\begin{aligned} \, &\langle Lu+\mathscr{T}(a,u)-h,v-u\rangle +\varphi(v)-\varphi(u)+I_{G(u)} (v)-I_{G(u)}(u) \\ &\qquad + \int_{\Sigma_c} j^0(x,t,\widetilde u(x,t); \widetilde v(x,t)-\widetilde u(x,t)) \, d\Gamma\, dt \geqslant 0 \quad \text{for all } \ v\in \mathcal{V}. \end{aligned}
\end{equation*}
\notag
$$
The latter is equivalent to the following quasi-variational hemivariational inequality: find $u\in\mathcal{W}\cap{G(u)}$ such that
$$
\begin{equation}
\begin{aligned} \, &\langle Lu+\mathscr{T}(a,u)-h,v-u\rangle +\varphi(v)-\varphi(u) \nonumber \\ &\qquad +\int_{\Sigma_c} j^0(x,t,\widetilde u(x,t); \widetilde v(x,t)-\widetilde u(x,t)) \, d\Gamma \, dt \geqslant 0 \quad \text{for all } \ v \in G(u). \end{aligned}
\end{equation}
\tag{46}
$$
With the data defined above, problems (1) and (46) are also equivalent in view of hypothesis $(\mathrm{H}_j)$(iv). Theorem 22. If hypotheses $(\mathrm{H}_g)$, $(\mathrm{H}_j)$, and $(\mathrm{H}_\ell)$ hold, $\mathscr{A}$ is given by (38), and $h\in \mathcal{V}^*$, then the solution set of problem (46) is non-empty sequentially weakly compact in $\mathcal{W}$, and uniformly bounded with respect to $a\in\mathscr{A}$. Proof. By Theorem 18 and Lemma 19, it suffices to verify hypotheses $(\mathrm{H}_1)$–$(\mathrm{H}_5)$. In fact, $(\mathrm{H}_1)$ is satisfied since $\mathcal{W}$ is continuously and compactly embedded into $\mathcal{X}$. By $(\mathrm{H}_j)$, it is easy to verify that $(\mathrm{H}_2)$ holds with $c_J=c_j \max\{1, m(\Sigma_c)^{(p-1)/p}\}$ and $d_J=d_j$, see [33], Theorems 4.15(vi) and 4.20. Next, from the definitions (38) and (40), we deduce that
$$
\begin{equation}
|\langle \mathscr{T}(a,u),v \rangle |\leqslant c_2\|\nabla u\|^{p-1}_{L^p(Q)}\|\nabla v\|_{L^p(Q)}=c_2\| u\|^{p-1}_\mathcal{V}\| v\|_\mathcal{V},
\end{equation}
\tag{47}
$$
which implies that the operator $\mathscr{T}\colon \mathscr{A}\times\mathcal{V}\to \mathcal{V}^*$ is bounded. We claim that $\mathscr{T}(a,{\cdot}\,)$ is continuous for each $a\in\mathscr{A}$. Indeed, assume that $u_n\to u$ in $\mathcal{V}$, and then, by the converse Lebesgue-dominated convergence, see [33], Theorem 2.39, up to a sequence, we have
$$
\begin{equation}
\nabla u_n(x,t)\to \nabla u(x,t) \quad \text{for a.e. } \ (x,t) \in Q,
\end{equation}
\tag{48}
$$
and there exists $\overline w \in L^p(Q)$ such that $|\nabla u_n(x,t)| \leqslant \overline w(x,t)$ for a.e. $(x,t)\in Q$. Note that, for all $v\in \mathcal{V}$,
$$
\begin{equation}
|\langle\mathscr{T}(a,u_n)-\mathscr{T}(a,u),v\rangle| \leqslant c_2\| v\|_\mathcal{V} \bigl\| |\nabla u_n|^{p-2} \nabla u_n-|\nabla u|^{p-2} \nabla u\bigr\|_{L^q(Q)}.
\end{equation}
\tag{49}
$$
Hence
$$
\begin{equation}
\|\mathscr{T}(a,u_n)-\mathscr{T}(a,u)\|_{\mathcal{V}^*}\leqslant c_2 \bigl\| |\nabla u_n|^{p-2} \nabla u_n-|\nabla u|^{p-2} \nabla u\bigr\|_{L^q(Q)}.
\end{equation}
\tag{50}
$$
On the other hand, we readily see that
$$
\begin{equation}
\begin{aligned} \, &\bigl| |\nabla u_n(x,t)|^{p-2}\nabla u_n(x,t)-|\nabla u(x,t)|^{p-2} \nabla u(x,t) \bigr| \nonumber \\ &\qquad\leqslant |\overline w(x,t)|^{p-1}+|\nabla u(x,t)|^{p-1}:=z(x,t) \end{aligned}
\end{equation}
\tag{51}
$$
for a.e. $(x,t)\in Q$. Obviously, $z\in L^q(Q)$, and so, so by the Lebesgue-dominated convergence theorem,
$$
\begin{equation*}
\lim_{n\to \infty} \bigl\| |\nabla u_n|^{p-2} \nabla u_n-|\nabla u|^{p-2} \nabla u\bigr\|_{L^q(Q)}=0,
\end{equation*}
\notag
$$
which, together with (50), implies that $\mathscr{T}(a,u_n)\to\mathscr{T}(a,u)$ in $\mathcal{V}^*$. Therefore, for each $a\in\mathscr{A}$, the operator $\mathscr{T}(a,{\cdot}\,)$ is continuous, and thus hemicontinuous. Using the monotonicity inequality (37), we can readily see that $\mathscr{T}(a,{\cdot}\,)$ is strongly monotone with a positive constant $c_1c(p)$. This guarantees that $\mathscr{T}$ is coercive and the operator $\mathscr{T}(a,{\cdot}\,)+{\gamma^*}\circ\partial J\circ \gamma (\,{\cdot}\,)$ is monotone since $c_1c(p)>d_j\| \gamma\|^p$. Thus, $(\mathrm{H}_3)$ is verified.
We proceed with checking $(\mathrm{H}_4)$. By definition (39), it is immediate that $G(w)$ is a convex closed subset of $\mathcal{V}$. Moreover, $B_{\mathcal{V}}(0,c_\ell/2)\subset G(w)$ for all $w \in \mathcal{W}$ since $\ell(w)\geqslant c_\ell$ by $(\mathrm{H}_\ell)$. Thus, $0\in \operatorname{int} \bigcap_{w\in \mathcal{W}} G(w)$. In order to verify $(\mathrm{H}_4)$(iii), suppose that the sequences $\{w_n\}$, $\{u_n\}\subset \mathcal{W}$ with $u_n\in G(w_n)$ satisfy $w_n\rightharpoonup w$ in $\mathcal{W}$ and $u_n\rightharpoonup u$ in $\mathcal{W}$. We have $\| u_n\|_\mathcal{V}\leqslant\ell(w_n)$ with $u_n\in G(w_n)$. Since $\mathcal{W}\subset L^p(Q)$ is compact, we get $w_n\to w$ in $L^p(Q)$. Now by definition (39), assumption $(\mathrm{H}_\ell)$, and the weak lower semicontinuity of the norm, we have
$$
\begin{equation*}
\| u\|_\mathcal{V}\leqslant\liminf_{n \to \infty}\| u_n\|_\mathcal{V}\leqslant \liminf_{n \to \infty}\ell(w_n)=\ell(w),
\end{equation*}
\notag
$$
and hence $u\in G(w)$. Thus, $(\mathrm{H}_4)$(iii) holds. Next, consider a sequence $\{w_n\}\subset\mathcal{W}$ such that $w_n \rightharpoonup w$ in $\mathcal{W}$. For any $v\in \mathcal{W}\cap G(w)$, we put We have $v_n\in \mathcal{W}$, and $v_n\to v$ in $\mathcal{V}$ as the embedding $\mathcal{W}\subset L^p(Q)$ is compact and $\ell\colon L^p(Q) \to \mathbb{R}$ is continuous. It is readily to see that $\|v_n\|_\mathcal{V} \leqslant\ell(w_n)$ as $\|v\|_\mathcal{V}\leqslant\ell(w)$ since $v\in G(w)$. Hence, $(\mathrm{H}_4)$(iv) is verified. We use the definition of $G(w)$ in (39) and the density of $D(L)$ in $\mathcal{V}$ to deduce that $\mathcal{W}\cap G(w)$ is dense in $G(w)$. Therefore, $(\mathrm{H}_4)$(v) is verified.
It remains to verify $(\mathrm{H}_5)$. In fact, by a standard argument, we can deduce that $\varphi\colon \mathcal{V}\to \mathbb{R}$ is a proper convex lower semicontinuous functional. Since $g$ is non-negative and $g(x,t,0)=0$, we get $0\in \partial_c\varphi(0)$. Moreover, the interior point condition (2) follows from the property $D(\varphi)=\mathcal{V}$. Thus, $(\mathrm{H}_5)$ holds. This completes the proof of Theorem 22. Next, given $z\in{L^p(Q;\mathbb{R}^N)}$ and $\alpha>0$, we consider the following inverse problem for the quasi-variational hemivariational inequality: find $a^* \in \mathscr{A}$ such that
$$
\begin{equation}
F(a^*) = \inf_{a \in \mathscr{A}} F(a) \quad \text{with } \ F(a) = \inf_{u \in \Theta(a)} \|\nabla u-z\|_{L^p(Q)}^p + \alpha \| a\|_{\mathcal{M}^{s,r}},
\end{equation}
\tag{52}
$$
where $\Theta(a)$ denotes the solution set of problem (46). Theorem 23. Under $(\mathrm{H}_g)$, $(\mathrm{H}_j)$, and $(\mathrm{H}_\ell)$ and $h \in \mathcal{V}^*$, the solution set of problem (52) is non-empty and sequentially weakly* compact in $\mathcal{M}^{s,r}$. Proof. Let $\mathscr{A}_1 =L^p(Q)$ and $\mathscr{A}_2=\mathcal{M}^{s,r}$. Then $\mathscr{A}\subset \mathscr{A}_1\cap\mathscr{A}_2$. To prove the theorem it suffices to verify the conditions $(\mathrm{H}_1)$–$(\mathrm{H}_8)$ and apply Theorem 21. In fact, $(\mathrm{H}_1)$–$(\mathrm{H}_5)$ were already verified in Theorem 22. It remains to verify $(\mathrm{H}_6)$–$(\mathrm{H}_8)$.
We first consider $(\mathrm{H}_6)$. By definition (38), the set $\mathscr{A}$ is bounded in $\mathscr{A}_1$. Suppose that $a_n\stackrel{*}{\rightharpoonup}a$ in $\mathcal{M}^{s,r}$ with $a_n\in \mathscr{A}$ for $n=1,2,\dots$ . We shall show that $a\in \mathscr{A}$. By Proposition 12, we have $\mathcal{M}^{s,r} \subset L^{s}(0,T;L^p(\Omega))$ compactly, and thus, $a_n\to a$ in $L^k(Q)$ with $k=\min\{s,p\}$. Then, $a_n(x,t)\to a(x,t)$ for a.e. $(x,t)\in Q$, along a subsequence. Consequently, it follows from $a_n\in\mathscr{A}$ that $c_1\leqslant a(x,t) \leqslant c_2$ for a.e. $(x,t)\in Q$. Moreover, we have
$$
\begin{equation*}
\| a\|_{\mathscr{A}_2} \leqslant\liminf_{n \to \infty}\| a_n\|_{\mathscr{A}_2} \leqslant c_3.
\end{equation*}
\notag
$$
Therefore, $a\in\mathscr{A}$. This implies that $\mathscr{A}$ is weakly*-closed in $\mathscr{A}_2$, and completes the verification of $(\mathrm{H}_6)$. Next, the condition $(\mathrm{H}_7)$ can be easily verified by taking $f(u)=\|\nabla u-z\|_{L^p(Q)}^p$.
Finally, for $(\mathrm{H}_8)$, we have proved in Theorem 22 that $\mathscr{T}(a,{\cdot}\,)\colon \mathcal{V}\to \mathcal{V}^*$ is continuous for each $a\in\mathscr{A}$. Let $\{a_n\}\subset\mathscr{A}$, $a\in\mathscr{A}$, $\{v_n\},\{u_n\}\subset\mathcal{V}$, and assume that $\{a_n\}$ is bounded in $\mathscr{A}_1$, $\{u_n\}$ is bounded in $\mathcal{V}$, $a_n\stackrel{*}{\rightharpoonup} a$ in $\mathscr{A}_2 $ and $v_n\to v$ in $\mathcal{V}$. We have
$$
\begin{equation}
\langle\mathscr{T}(a_n,v_n)-\mathscr{T}(a,v_n), u_n\rangle \leqslant \| u_n\|_\mathcal{V} \biggl( \int_Q |a_n(x,t)-a(x,t)|^q |\nabla v_n(x,t)|^p\,dx\, dt\biggr)^{1/q}.
\end{equation}
\tag{53}
$$
Recall that $v_n\to v$ in $\mathcal{V}$ and $a_n\to a$ in $L^k(Q)$ with $k=\min\{s,p\}$, as before. Passing to subsequences, we have $a_n(x,t)\to a(x,t)$, $\nabla v_n(x,t)\to \nabla v(x,t)$ for a.e. $(x,t)\in Q$, and there exists $\overline{w}\in L^p(Q)$ such that $|\nabla v_n(x,t)| \leqslant \overline{w}(x,t)$. Since $a_n$, $a\in \mathscr{A}$, we get
$$
\begin{equation*}
|a_n(x,t)-a(x,t)|^q |\nabla v_n(x,t){|^p}\leqslant 2^q\,c_2^q\,\overline{w}(x,t)^p := z(x,t) \quad \text{a.e. } \ (x,t)\in Q,
\end{equation*}
\notag
$$
where $z\in L^1(Q)$. Thus, by the Lebesgue-dominated convergence theorem, we have
$$
\begin{equation*}
\lim_{n\to \infty} \int_Q |a_n(x,t)-a(x,t)|^q |\nabla v_n(x,t)|^p\, dx\, dt = 0.
\end{equation*}
\notag
$$
Passing the upper limit in (53), since $\{u_n\}$ is bounded in $\mathcal{V}$, we conclude that
$$
\begin{equation}
\limsup_{n\to \infty} \langle\mathscr{T}(a_n,v_n)-\mathscr{T}(a,v_n), u_n\rangle \leqslant0.
\end{equation}
\tag{54}
$$
Hence $(\mathrm{H}_8)$ holds, which completes the proof of Theorem 23. 
Citation in format AMSBIB
\Bibitem{PenYanLiu24} |
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