1 Introduction

Variational inequalities constitute one of the most vibrant branches of applied mathematics that have been extended in a multitude of directions. One of the notable generalizations of variational inequalities that have attracted a great deal of attention is the so-called hemivariational inequality, pioneered in the 1980s by P.D. Panagiotopoulos [39, 40]. During the last several decades, hemivariational inequalities have been explored extensively in novel engineering applications. In contrast to variational inequalities, which emerge from convex energy principles, the hemivariational inequalities are connected to nonsmooth and nonconvex energy functionals and naturally involve generalized derivatives of nonsmooth functionals, see [5]. For some of the recent developments in hemivariational inequalities and their applications, we refer the reader to [1, 7, 8, 11, 15,16,17, 26,27,28,29,30, 38, 41,42,44], and the cited references. The necessity to understand time-dependent applied models involving nonsmooth and nonconvex energy principles gives rise to evolutionary hemivariational inequalities. A prototypical example is the study of nonstationary fluid flow problems modeled by nonmonotone and set-valued frictional laws, see [34,35,36].

In this paper, our focus is on a new class of evolutionary quasi–variational–hemivariational inequalities (in short, (QVHVI)), where the underlying constraint set depends on the unknown solution. In contrast to the classical variational and hemivariational inequalities where the constraints sets remain fixed, their quasi-variants pose novel theoretical and computational challenges due to the peculiar dependence on the unknown solution. On the other hand, since the quasi-variants appear very frequently in applied models, they have received significant attention in recent years, see [9, 13, 14, 18, 20,21,22,23,24, 27, 31,32,33] and the cited references therein.

This paper is devoted to a thorough study of an evolutionary quasi–variational–hemivariational inequality described in Sect. 3. Besides some auxiliary results, we give two novel results. The first result of this paper proves the solvability and weak sequential compactness of the solution set for the (QVHVI) given in Problem 1. In proving this result, the central role is played by Kluge’s fixed point theorem, Minty’s formulation, and some techniques from the nonsmooth analysis. Our second result addresses an optimal control problem associated with the above (QVHVI). Here, our approach is based on using minimizing sequences in conjunction with the Kuratowski-type continuity properties.

We organize the contents of this paper into four sections. Section 2 presents the necessary background material. In Sect. 3, we investigate the existence and compactness of the solution set of (QVHVI) in Problem 1. Section 4 is devoted to a nonlinear optimal control problem associated with (QVHVI). The paper concludes with some remarks.

2 Mathematical Background

We start with a brief discussion on some classes of set-valued operators of monotone type. An elaborate presentation can be found in [4, 37, 45].

Definition 1

Let E be a reflexive Banach space with the dual \(E^*\) and let \(A:D(A)\subset E\rightarrow 2^{E^*}\) be a set-valued mapping with the domain \(D(A)= \{ u \in E \mid Au \not = \emptyset \}\). We say that

  1. (i)

    A is monotone, if

    $$\begin{aligned} \langle u^*-v^*,u-v\rangle _E\ge 0\ \,{\mathrm{for~all }}\,\, u^*\in Au, v^*\in Av \ \,{\mathrm{and }}\,u, v\in D(A); \end{aligned}$$
  2. (ii)

    A is maximal monotone, if it is monotone and has a maximal graph, that is,

    $$\begin{aligned} \langle u^*-v^*,u-v\rangle _E\ge 0\ \,{\mathrm{for~all }}\, u^*\in Au \ {\mathrm{and }}\, u\in D(A) \end{aligned}$$

    implies \(v\in D(A)\) and \(v^*\in Av\);

  3. (iii)

    A is pseudomonotone with respect to \(D({\mathcal {L}})\) (or \({\mathcal {L}}\)-pseudomonotone) for a linear, closed, densely defined, and maximal monotone operator \(\mathcal L:D({\mathcal {L}})\subset E\rightarrow E^*\) if

    1. (a)

      for each \(u\in E\), the set Au is nonempty, bounded, closed, and convex in \(E^*\);

    2. (b)

      A is upper semicontinuous from any finite-dimensional subspace of E to \(E^*\) endowed with the weak topology;

    3. (c)

      for any sequences \(\{u_n\}\subset D({\mathcal {L}})\) and \(\{u_n^*\}\subset E^*\) with

      $$\begin{aligned} \left\{ \begin{array}{lll} u_n\rightarrow u \text{ weakly } \text{ in } E, \\ \mathcal Lu_n\rightarrow \mathcal Lu \text{ weakly } \text{ in } E^*, \\ u_n^*\in Au_n \text{ for } \text{ all } n\in {\mathbb {N}},\\ u_n^*\rightarrow u^* \text{ weakly } \text{ in } E^*, \\ \limsup _{n\rightarrow \infty }\langle u_n^*,u_n-u\rangle _E\le 0, \end{array}\right. \end{aligned}$$

      we have \(u^*\in Au\) and \(\displaystyle \lim _{n\rightarrow \infty }\langle u_n^*,u_n\rangle _E=\langle u^*,u\rangle _E\).

The sum of set-valued pseudomonotone operators is pseudomonotone (see, e.g., [4, Theorem 2.124]). The same holds for pseudomonotone operators with respect to \(D({\mathcal {L}})\).

Proposition 2

Assume that \(A:E\rightarrow 2^{E^*}\) and \(B:E\rightarrow 2^{E^*}\) are pseudomonotone with respect to \(D({\mathcal {L}})\) and bounded in the sense of mapping bounded sets to bounded sets. Then, the set-valued sum \(A+B:E\rightarrow 2^{E^*}\) is pseudomonotone with respect to \(D({\mathcal {L}})\).

Proof

Since the sets Au and Bu are nonempty, bounded, closed, and convex in \(E^*\), the same remains true for \((A+B)u=Au+Bu\). The values of A and B being compact sets in the weak topology of \(E^*\) permit the use of [19, Theorem 1.2.14] to guarantee that \(A+B\) is upper semicontinuous from any finite-dimensional subspace of E to \(E^*\) endowed with the weak topology.

Let the sequences \(\{u_n\}\subset D({\mathcal {L}})\), \(\{u_n^*\}\subset E^*\) and \(\{v_n^*\}\subset E^*\) satisfy \(u_n\rightarrow u\) weakly in E, \(\mathcal Lu_n\rightarrow \mathcal Lu\) weakly in \(E^*\), \(u_n^*\in Au_n\) and \(v_n^*\in Bu_n\) for all \(n\in {\mathbb {N}}\), \(u_n^*+v_n^*\rightarrow w^*\) weakly in \(E^*\), and \(\limsup _{n\rightarrow \infty }\langle u_n^*+v_n^*,u_n-u\rangle _E\le 0\). We claim that \(\limsup _{n\rightarrow \infty }\langle u_n^*,u_n-u\rangle _E\le 0\) and \(\limsup _{n\rightarrow \infty }\langle v_n^*,u_n-u\rangle _E\le 0\). Arguing by contradiction, along a subsequence, we suppose

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle u_n^*,u_n-u\rangle _E=\lim _{k\rightarrow \infty }\langle u_{n_k}^*,u_{n_k}-u\rangle _E=:a>0. \end{aligned}$$

Then, the inequality \(\displaystyle \limsup _{k\rightarrow \infty }\langle v_{n_k}^*,u_{n_k}-u\rangle _E\le -a\) is true. Consequently, it follows that \(\lim _{n\rightarrow \infty }\langle v_{n_k}^*,u_{n_k}\rangle _E=\langle v^*,u\rangle _E\) because \(B:E\rightarrow 2^{E^*}\) is pseudomonotone with respect to \(D({\mathcal {L}})\). The preceding inequality leads to the contradiction \(0\le -a\), which proves the claim.

Using that \(A:E\rightarrow 2^{E^*}\) and \(B:E\rightarrow 2^{E^*}\) are bounded and the space \(E^*\) is reflexive, up to subsequences, it holds that \(u_n^*\rightarrow u^*\) weakly in \(E^*\) and \(v_n^*\rightarrow v^*\) weakly in \(E^*\) for \(u^*,v^*\in E^*\) with \(u^*+v^*=w^*\). Then, the pseudomonotonicity of A and B entails \(u^*\in Au\), \(v^*\in Bu\), \( \lim _{n\rightarrow \infty }\langle u_n^*,u_n\rangle _{E^*\times E}=\langle u^*,u\rangle _E\) and \(\lim _{n\rightarrow \infty }\langle v_n^*,u_n\rangle _E=\langle v^*,u\rangle _E\). It turns out that \(w^*\in Au+Bu\) and \(\lim _{n\rightarrow \infty }\langle u_n^*+v_n^*,u_n\rangle _E=\langle w^*,u\rangle _E\), thus completing the proof. \(\square \)

Next, we focus on some tools from convex analysis and nonsmooth analysis. Let E be a Banach space with its dual \(E^*\) and let \(\varphi :E \rightarrow {\mathbb {R}} \cup \{ +\infty \}\) be a proper, convex, and lower semicontinuous function. The subdifferential of \(\varphi \) is the mapping \(\partial _C \varphi :E \rightarrow 2^{E^*}\) defined by

$$\begin{aligned} \partial _C \varphi (u) = \left\{ \, u^*\in E^* \mid \langle u^*, v -u \rangle _E \le \varphi (v)-\varphi (u) \ {\mathrm{for~all}} \ v \in E \, \right\} . \end{aligned}$$

We recall the following basic result (see, e.g., [6, Theorem 6.3.19]).

Theorem 3

Let E be a real Banach space and \(\varphi :E\rightarrow {\mathbb {R}}\cup \{+\infty \}\) be a proper, convex, and lower semicontinuous function. Then, \(\partial _C\varphi :E\rightarrow 2^{E^*}\) is a maximal monotone operator.

Let \(J :E \rightarrow {\mathbb {R}}\) be a locally Lipschitz function and \(u,v\in E\). The generalized directional derivative of J at u in the direction v is defined as

$$\begin{aligned} J^0(u;v): = \limsup \limits _{w\rightarrow u,\, t\downarrow 0 } \frac{J(w+t v)-J(w)}{t}, \end{aligned}$$

and the generalized gradient \(\partial J:E\rightarrow 2^{E^*}\) of \(J :E \rightarrow {\mathbb {R}}\) is defined by

$$\begin{aligned} \partial J(u) =\{\, \xi \in E^{*} \,\mid \, J^0 (u; v)\ge \langle \xi , v\rangle _E \ \ {\mathrm{for~all}} \ \ v \in E \, \}. \end{aligned}$$

The following result summarizes some basic results on generalized gradients, (see [37, Proposition 3.23]).

Proposition 4

Let \(J:E \rightarrow {\mathbb {R}}\) be locally Lipschitz with constant \(L_{u}>0\) near \(u\in E\). Then, we have:

  1. (a)

    the function \(v\mapsto J^0(u;v)\) is positively homogeneous, is subadditive, and satisfies

    $$\begin{aligned} |J^0(u;v)|\le L_{u}\Vert v\Vert _E \ \ {\mathrm{for~all }}\,\, v\in E; \end{aligned}$$
  2. (b)

    \((u,v)\mapsto J^0(u;v)\) is upper semicontinuous;

  3. (c)

    \(\partial J(u)\) is a nonempty, convex, and weakly \(^{*}\) compact subset of \(E^{*}\) with

    $$\begin{aligned} \Vert \xi \Vert _{E^{*}}\le L_{u} \text{ for } \text{ all }~\xi \in \partial J(u); \end{aligned}$$
  4. (d)

    for all \(v\in E\), \(J^0(u;v)=\max \big \{\langle \xi , v\rangle _E \mid \xi \in \partial J(u)\big \}\).

Besides, we mention the concept of strongly quasi-boundedness for set-valued operators, see [10, Definition 2.14].

Definition 5

Let E be a reflexive Banach space with the dual \(E^*\). A set-valued map \(A:D(A)\subset E\rightarrow 2^{E^*}\) is called strongly quasi-bounded if for each \(M>0\) there exists \(K_M>0\) such that for any \(u\in D(A)\) and \(u^*\in Au\) with

$$\begin{aligned} \langle u^*,u\rangle _E\le M \ \ {\mathrm{and }} \ \Vert u\Vert _E\le M, \end{aligned}$$

it follows that \(\Vert u^*\Vert _{E^*}\le K_M\).

The following condition ensuring strongly quasi-boundedness was given by Browder–Hess [3, Proposition 14].

Proposition 6

Let E be a reflexive Banach space with its dual \(E^*\). If \(A:D(A)\subset E\rightarrow 2^{E^*}\) is a monotone operator such that \(0\in \mathrm{int}D(A)\), then A is strongly quasi-bounded.

We also recall from [10, Theorem 3.1] the following surjectivity result:

Theorem 7

Let E be a reflexive, strictly convex Banach space, let \(\mathcal L:D({\mathcal {L}})\subset E\rightarrow E^*\) be a linear, closed, densely defined, and maximal monotone operator, and let \(A:E\rightarrow 2^{E^*}\) be a bounded and \({\mathcal {L}}\)-pseudomonotone operator such that

$$\begin{aligned} \langle Au,u\rangle _E\ge r(\Vert u\Vert _E)\Vert u\Vert _E\ \ {\mathrm{for~all }} \ u\in E, \end{aligned}$$

where \(r:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\) satisfies \(r(s)\rightarrow +\infty \) as \(s\rightarrow +\infty \). If \(B:D(B)\subset E\rightarrow 2^{E^*}\) is a maximal monotone operator which is strongly quasi-bounded and \(0\in B(0)\), then \({\mathcal {L}}+A+B\) is surjective, that is, its range is \(E^*\).

We conclude this section by recalling Kluge’s fixed point theorem [25].

Theorem 8

Let C be a nonempty, closed, and convex subset of a reflexive Banach space Z. Assume that \(\Psi :C\rightarrow 2^C\) is a set-valued map such that for every \(u\in C\) the set \(\Psi (u)\) is nonempty, closed, and convex, and the graph of \(\Psi \) is sequentially weakly closed. If \(\Psi (C)\) is bounded, then \(\Psi \) has a fixed point.

3 Existence Results

Let \({\mathcal {V}}\) and \({\mathcal {E}}\) be reflexive Banach spaces with duals \({\mathcal {V}}^*\) and \(\mathcal {E}^*,\) let \({\mathcal {X}}\) be a nonempty subset of \({\mathcal {V}}\), and let \(\gamma :\mathcal V\rightarrow {\mathcal {E}}\) be a bounded linear operator. Let \(\langle \cdot ,\cdot \rangle _{{\mathcal {V}}}\) be the duality pairing of \({\mathcal {V}}^*\) and \({\mathcal {V}}\) and let \(\langle \cdot ,\cdot \rangle _{{\mathcal {E}}}\) be the duality pairing of \({\mathcal {E}}^*\) and \({\mathcal {E}}\). Given a linear, closed, densely defined, and maximal monotone operator \({\mathcal {L}}:D(\mathcal L)\rightarrow {\mathcal {V}}^*,\) two set-valued operators \({\mathcal {T}}:{\mathcal {V}}\rightarrow 2^{{\mathcal {V}}^*}\) and \({\mathcal {K}}:{\mathcal {X}}\rightarrow 2^{{\mathcal {X}}},\) a proper, convex, and lower semicontinuous function \(\varphi :{\mathcal {V}}\rightarrow {\mathbb {R}}\cup \{+\infty \}\), a locally Lipschitz function \(J:{\mathcal {E}}\rightarrow {\mathbb {R}}\), and an element \(f\in {\mathcal {V}}^*\), we formulate the following evolutionary problem in the form of (QVHVI):

Problem 1

Find \(u\in {\mathcal {K}}(u)\cap D({\mathcal {L}})\) such that for some \(u^*\in {\mathcal {T}}(u)\), we have

$$\begin{aligned} \langle \mathcal Lu+u^*-f,v-u\rangle _{\mathcal {V}}+J^0(\gamma u;\gamma (v-u))+\varphi (v)-\varphi (u)\ge 0,\quad \text {for all}\ v\in {\mathcal {K}}(u), \end{aligned}$$

where \(J^0\) is the generalized derivative of J to be defined shortly.

We now formulate the necessary assumptions on the data of (QVHVI):

\({\underline{H(0)}}\): \({\mathcal {V}}\) is a reflexive, strictly convex Banach space, \({\mathcal {E}}\) is a reflexive Banach space, and \({\mathcal {X}}\) is a nonempty, closed, and convex subset of \({\mathcal {V}}\).

\({\underline{H({\mathcal {L}})}}\): \({\mathcal {L}}:D({\mathcal {L}})\subset {\mathcal {V}}\rightarrow {\mathcal {V}}^*\) is a linear, closed, densely defined, and maximal monotone operator.

\({\underline{H({\mathcal {T}})}}\): \({\mathcal {T}}:\mathcal V\rightarrow 2^{{\mathcal {V}}^*}\) is bounded and pseudomonotone and there exist constants \(m_{\mathcal {T}}>0\), \(d_{\mathcal {T}}\ge 0\) and \(p>1\) such that

$$\begin{aligned} \inf _{u^*\in {\mathcal {T}} u}\langle u^*,u\rangle _{\mathcal {V}}\ge m_{\mathcal {T}}\Vert u\Vert _{{\mathcal {V}}}^{p}-d_{\mathcal {T}},\quad \text {for all}\ u\in {\mathcal {V}}. \end{aligned}$$

\({\underline{H(\gamma )}}\): \(\gamma :{\mathcal {V}}\rightarrow {\mathcal {E}}\) is a linear and compact operator.

\({\underline{H(J)}}\): \(J:{\mathcal {E}}\rightarrow {\mathbb {R}}\) is a locally Lipschitz function for which there exist constants \(p>1\), \(\theta \in [1,p]\) and \(c_J>0\) with \(m_\mathcal T\kappa (\theta )>c_J\Vert \gamma \Vert ^p\) and

$$\begin{aligned} \Vert \xi \Vert _{{\mathcal {E}}^*}\le c_J\left( 1+\Vert z\Vert _\mathcal E^{\theta -1}\right) \end{aligned}$$

for all \(\xi \in \partial J(u)\) and \(u\in {\mathcal {E}}\), where

$$\begin{aligned} \kappa (\theta ):=\left\{ \begin{array}{ll} 1&{}\text{ if }~\theta =p\\ +\infty &{}\text{ if } \theta <p \end{array} \right. . \end{aligned}$$

\({\underline{H({\mathcal {K}})}}\): \({\mathcal {K}}:\mathcal X\rightarrow 2^{{\mathcal {X}}}\) is such that

  1. (i)

    For each \(w\in {\mathcal {X}}\) the set \({\mathcal {K}}(w)\) is nonempty, closed, and convex in \({\mathcal {V}}\).

  2. (ii)

    \(\displaystyle 0\in \text{ int }\bigcap _{w\in \mathcal X}{\mathcal {K}}(w).\)

  3. (iii)

    If the sequences \(\{w_n\}\subset {\mathcal {X}}\) and \(\{u_n\}\subset {\mathcal {K}}(w_n)\cap D({\mathcal {L}})\) satisfy

    $$\begin{aligned} w_n\rightarrow w \text{ weakly } \text{ in } ~{\mathcal {V}}, u_n\rightarrow u~\text{ weakly } \text{ in }~ {\mathcal {V}}~\text{ and }~\mathcal Lu_n\rightarrow \mathcal Lu~\text{ weakly } \text{ in }~\mathcal V^* \end{aligned}$$

    for some \(w\in {\mathcal {X}}\) and \(u\in D({\mathcal {L}})\), then \(u\in {\mathcal {K}}(w)\).

  4. (iv)

    For every sequence \(\{w_n\}\subset {\mathcal {X}}\) with \(w_n\rightarrow w\) weakly in \({\mathcal {V}}\) and for every \(v\in {\mathcal {K}}(w)\), there exist a subsequence \(\{w_{n_k}\}\) of \(\{w_n\}\) and a sequence \(\{v_{n_k}\}\subset {\mathcal {X}}\) with \(v_{n_k}\in \mathcal K(w_{n_k})\) such that \(v_{n_k}\rightarrow v\) in \({\mathcal {V}}\) as \(k\rightarrow \infty \).

\({\underline{H(\varphi )}}\): \(\varphi :{\mathcal {V}}\rightarrow {\mathbb {R}}\cup \{+\infty \}\) is a proper, convex, and lower semicontinuous function satisfying

$$\begin{aligned} \bigcup _{w\in {\mathcal {X}}}{\mathcal {K}}(w)\subset \text{ int }D(\varphi ), \quad 0\in \text{ int }D(\varphi ),\quad 0\in \partial _C\varphi (0). \end{aligned}$$

A fruitful approach for quasi-variational inequalities is defining a variational selection and finding its fixed point. To elucidate, we first consider, each \(w\in {\mathcal {X}}\), the following parametric problem:

Problem 2

Find \(u\in {\mathcal {K}}(w)\cap D({\mathcal {L}})\) such that for some \(u^*\in {\mathcal {T}}(u)\), we have

$$\begin{aligned}&\langle \mathcal Lu+u^*-f,v-u\rangle _{\mathcal {V}}+J^0(\gamma u;\gamma (v-u))\nonumber \\&+\varphi (v)-\varphi (u)\ge 0,\quad \text {for all}\ v\in {\mathcal {K}}(w). \end{aligned}$$
(1)

This permits us to define the set-valued mapping \({\mathcal {S}}:{\mathcal {X}}\rightarrow 2^{{\mathcal {X}}}\) that assigns to each \(w\in {\mathcal {X}}\) the solution set \({\mathcal {S}}(w)\) of Problem 2 corresponding to w. This set-valued map is the so-called variational selection for the (QVHVI) defined in Problem 1. Evidently, any fixed of \({\mathcal {S}}\) is a solution of Problem 1.

We will need the following technical result that circumvents difficulties related to the unboundedness of \({\mathcal {L}}\).

Lemma 9

Assume that H(0), \(H({\mathcal {L}})\), \(H({\mathcal {T}})\), \(H(\gamma )\), H(J), \(H(\varphi )\), and \(H({\mathcal {K}})\)(i)–(ii) hold. If \(\{w_n\}\subset {\mathcal {X}}\) and \(\{u_n\}\subset D({\mathcal {L}})\) are sequences such that \(\{u_n\}\) is bounded in \({\mathcal {V}}\) and \(u_n\in {\mathcal {S}}(w_n)\) for each \(n\in {\mathbb {N}}\), then \(\{\mathcal Lu_n\}\) is bounded in \({\mathcal {V}}^*\).

Proof

By assumption, there exists \(u_n\in {\mathcal {K}}(w_n)\) such that for some \(u_n^*\in \mathcal Tu_n\), we have

$$\begin{aligned} \langle \mathcal Lu_n+u_n^*-f,v-u_n\rangle _{\mathcal {V}}+J^0(\gamma u_n;\gamma (v-u_n))+\varphi (v)-\varphi (u_n)\ge 0\quad \text {for all}\ v\in {\mathcal {K}}(w_n). \end{aligned}$$
(2)

By hypothesis \(H({\mathcal {K}})\)(ii), there is an arbitrarily small neighborhood C of 0 in \({\mathcal {V}}\) satisfying \(C\subset {\mathcal {K}}(w_n)\) for all \(n\in {\mathbb {N}}\). We will show that

$$\begin{aligned} \inf _{n\in {\mathbb {N}},v\in C}\langle \mathcal {L} u_n,v\rangle _{\mathcal {V}}>-\infty , \end{aligned}$$
(3)

which proves our result (see similar ideas in part (I) of the proof of [45, Proposition 32.33] and in Step 4 of the proof of [20, Theorem 5.1]). Inequality (2) and the monotonicity of \({\mathcal {L}}\) yield

$$\begin{aligned} \langle \mathcal Lu_n,v\rangle _{{\mathcal {V}}} \ge \langle u_n^*-f,u_n-v\rangle _{{\mathcal {V}}} -J^0(\gamma u_n;\gamma (v-u_n))+\varphi (u_n)-\varphi (v)\quad \text {for all}\ v\in C. \end{aligned}$$
(4)

Thanks to hypothesis \(H({\mathcal {T}})\) and the fact that \(\{u_n\}\) is bounded in \({\mathcal {V}}\), the sequence \(\{u_n^*\}\) is bounded in \({\mathcal {V}}^*\). We deduce from the boundedness of \(\{u_n\}\) and C, hypothesis H(J) and Proposition 4 that \(J^0(\gamma u_n;\gamma (v-u_n))\) is bounded. By condition \(H(\varphi )\), it follows that \(\varphi \) is bounded from below by an affine function (see, e.g., [2, Proposition 1.10]). Furthermore, \(\varphi \) is bounded on C because C can be chosen arbitrarily small in \(\text{ int } D(\varphi )\). We conclude that

$$\begin{aligned} \inf _{n\in {\mathbb {N}},v\in C}\left[ \langle u_n^*-f,u_n-v\rangle _{{\mathcal {V}}}-J^0(\gamma u_n;\gamma (v-u_n))+\varphi (u_n)-\varphi (v)\right] >-\infty , \end{aligned}$$

whence (3) follows via (4), which completes the proof. \(\square \)

The following result provides useful information on the set-valued mapping \({\mathcal {S}}\).

Lemma 10

Assume that H(0), \(H({\mathcal {L}})\), \(H({\mathcal {T}})\), \(H(\gamma )\), H(J), \(H(\varphi )\), and \(H({\mathcal {K}})\mathrm{(i)-(ii)}\) remain valid. Then, the following statements hold:

  1. (i)

    \({\mathcal {S}}(w)\ne \emptyset \), for all \(w\in {\mathcal {X}}\).

  2. (ii)

    The set \({\mathcal {S}}(w)\) is closed for all \(w\in {\mathcal {X}}\).

  3. (iii)

    The set \({\mathcal {S}}({\mathcal {X}})\) is bounded in \({\mathcal {V}}\).

Proof

(i). Fix \(w\in {\mathcal {X}}\) and consider the function \(\Phi :{\mathcal {V}}\rightarrow {\mathbb {R}}\cup \{+\infty \}\) defined by

$$\begin{aligned} \Phi (u)=\left\{ \begin{array}{lll} \varphi (u)&{} \text{ if }~ u\in {\mathcal {K}}(w),\\ +\infty &{} \text{ otherwise }. \end{array}\right. \end{aligned}$$

Problem 2 reads as follows: Find \(u\in {\mathcal {K}}(w)\cap D({\mathcal {L}})\) such that for some \(u^*\in {\mathcal {T}}(u)\), we have

$$\begin{aligned} \langle \mathcal Lu+u^*-f,v-u\rangle _{\mathcal {V}}+J^0(\gamma u;\gamma (v-u))+\Phi (v)-\Phi (u)\ge 0,\quad \text {for all}\ v\in {\mathcal {V}}. \end{aligned}$$
(5)

For the solvability of (5), we reformulate it as the inclusion problem of finding \(u\in {\mathcal {K}}(w)\cap D({\mathcal {L}})\) such that

$$\begin{aligned} f\in \mathcal Lu+\mathcal Tu+\gamma ^*\partial J(\gamma u)+\partial _C\Phi (u). \end{aligned}$$
(6)

Let \({\mathcal {F}}:{\mathcal {V}}\rightarrow 2^{{\mathcal {V}}^*}\) be defined by

$$\begin{aligned} {\mathcal {F}}(u):=\mathcal Tu+\gamma ^*\partial J(\gamma u)&\text{ for } \text{ all } u\in {\mathcal {V}}. \end{aligned}$$

The boundedness of \({\mathcal {T}}\) and hypotheses H(J) and \(H(\gamma )\) confirm that \({\mathcal {F}}:{\mathcal {V}}\rightarrow 2^{\mathcal V^*}\) is bounded.

Next, we claim that \({\mathcal {F}}\) is pseudomonotone with respect to \(D({\mathcal {L}})\). First, we prove that \(u\mapsto \gamma ^*\partial J(\gamma u)\) is strongly–weakly upper semicontinuous. Let B be a weakly closed set in \({\mathcal {V}}^*\) and let \(u_n\rightarrow u\) in \(\mathcal V\) as \(n\rightarrow \infty \) so that for every \(n\in {\mathbb {N}}\), there exists \(\xi _n\in \partial J(\gamma u_n)\) satisfying \(\gamma ^*\xi _n\in \gamma ^*\partial J(\gamma u_n)\cap B\). The boundedness of \(\partial J\) implies that the sequence \(\{\xi _n\}\) is bounded in \({\mathcal {E}}^*\). We may assume that \(\xi _n\rightarrow \xi \) weakly in \({\mathcal {E}}^*\) for some \(\xi \in {\mathcal E^*}\). Because of the weak closedness of B and linearity of \(\gamma ^*\), we have \(\gamma ^*\xi \in B\). For each \(n\in {\mathbb {N}}\), we have

$$\begin{aligned} \langle \xi _n,\gamma v\rangle _{{\mathcal {E}}}\le J^0(\gamma u_n;\gamma v)&\text{ for } \text{ all } v\in {\mathcal {V}}. \end{aligned}$$

By the upper semicontinuity of \(J^0\) and compactness of \(\gamma \), we get

$$\begin{aligned} \langle \xi ,\gamma v\rangle _{\mathcal E}=\lim _{n\rightarrow \infty }\langle \xi _n,\gamma v\rangle _{{\mathcal {E}}}\le \limsup _{n\rightarrow \infty }J^0(\gamma u_n;\gamma v)\le J^0(\gamma u;\gamma v)&\text{ for } \text{ all } v\in {\mathcal {V}}, \end{aligned}$$

implying that \(\gamma ^*\xi \in \gamma ^*\partial J(\gamma u)\cap B\) (see, e.g., [37, Proposition 3.8]).

Now, let \(u_n\rightarrow u\) weakly in \({\mathcal {V}}\) and \(u_n^*=\gamma ^*\xi _n\rightarrow u^*\) weakly in \({\mathcal {V}}^*\) with \(\xi _n\in \partial J(\gamma u_n)\) for every \(n\in {\mathbb {N}}\). The compactness of \(\gamma :{\mathcal {V}}\rightarrow {\mathcal {E}}\) implies \(\gamma u_n\rightarrow \gamma u\) strongly in \({\mathcal {E}}\). The boundedness of \(\partial J\) ensures along a relabeled subsequence \(\xi _n\rightarrow \xi \) weakly in \({\mathcal {E}}^*\) for some \(\xi \in {\mathcal {E}}^*\). From the strong–weak closedness of the graph of \(\partial J\), we infer that \(\xi \in \partial J(\gamma u)\), thus \(\gamma ^*\xi \in \gamma ^*\partial J(\gamma u)\), and \(\lim _{n\rightarrow \infty }\langle u_n^*,u_n\rangle _{{\mathcal {V}}}=\langle u^*,u\rangle _{{\mathcal {V}}}\) since \(\gamma \) is compact. It turns out that \(u\mapsto \gamma ^*\partial J(\gamma u)\) is pseudomonotone, so pseudomonotone with respect to \(D({\mathcal {L}})\). Combining with hypothesis \(H({\mathcal {T}})\), we can apply Proposition 2 to deduce that \({\mathcal {F}}\) is pseudomonotone with respect to \(D({\mathcal {L}})\).

The next step is to prove that \({\mathcal {F}}\) is coercive. For this, \(H({\mathcal {T}})\) and H(J) ensure that for any \(u\in {\mathcal {V}}\), we have

$$\begin{aligned} \langle \mathcal Fu,u\rangle _{{\mathcal {V}}}&\ge m_\mathcal T\Vert u\Vert _{{\mathcal {V}}}^p -d_{\mathcal {T}}-\Vert \gamma \Vert c_J\left( 1+\Vert \gamma u\Vert _{\mathcal {E}}^{\theta -1}\right) \Vert u\Vert _{{\mathcal {V}}}\nonumber \\&= m_{\mathcal {T}}\Vert u\Vert _{{\mathcal {V}}}^p-\Vert \gamma \Vert ^\theta c_J\Vert u\Vert _{{\mathcal {V}}}^\theta -d_\mathcal T-\Vert \gamma \Vert c_J\Vert u\Vert _{{\mathcal {V}}}. \end{aligned}$$
(7)

If \(\theta <p\), from (7) it is obvious that \({\mathcal {F}}\) is coercive. When \(\theta =p\), the assumption \(m_{\mathcal {T}}>\Vert \gamma \Vert ^p c_J\) in \(H({\mathcal {T}})\) and (7) render that \({\mathcal {F}}\) is coercive.

We claim that \(\partial _C\Phi :{\mathcal {V}}\rightarrow 2^{{\mathcal {V}}^*}\) is strongly quasi-bounded. As noted in Theorem 3, \(\partial _C\Phi :{\mathcal {V}}\rightarrow 2^{{\mathcal {V}}^*}\) is a maximal monotone operator. We have \(0\in \text{ int }D(\partial _C\Phi )\) owing to hypotheses \(H(\varphi )\) and \(H({\mathcal {K}})\mathrm{(ii)}\) in conjunction with \(\text{ int }D(\Phi )\subset \text{ int }D(\partial _C\Phi )\). Then, Proposition 6 establishes the validity of the claim.

We are now in a position to apply Theorem 7 ensuring that the inclusion problem (6) has at least a solution. As easily noticed from (1), this guarantees the existence of solutions to Problem 2.

(ii) Given \(w\in {\mathcal {X}}\), let \(\{u_n\}\subset \mathcal S(w)\) satisfy \(u_n\rightarrow u\) in \({\mathcal {V}}\) as \(n\rightarrow \infty \). Then, \(u\in {\mathcal {K}}(w)\), thanks to the closedness of \({\mathcal {K}}(w)\) and there exists \(u_n^*\in {\mathcal {T}}(u_n)\) such that

$$\begin{aligned}&\langle \mathcal Lu_n+u_n^*-f,v-u_n\rangle _{\mathcal {V}}+J^0(\gamma u_n; \gamma (v-u_n))\nonumber \\&+\varphi (v)-\varphi (u_n)\ge 0\quad \text {for all}\ v\in {\mathcal {K}}(w). \end{aligned}$$
(8)

By Lemma 9, the sequence \(\{\mathcal Lu_n\}\) is bounded in \({\mathcal {V}}^*\). Therefore, up to a subsequence, we have \(\mathcal Lu_n\rightarrow \mathcal Lu\) weakly in \({\mathcal {V}}^*\) because the graph of the linear and maximal monotone operator \({\mathcal {L}}\) is weakly closed. The boundedness of \({\mathcal {T}}\) allows us to suppose that \(u_n^*\rightarrow u^*\) weakly in \({\mathcal {V}}^*\) for some \(u^*\in \mathcal V^*\). Inserting \(v=u\) in (8) gives

$$\begin{aligned}&\limsup _{n\rightarrow \infty }\langle u_n^*,u_n-u\rangle _{{\mathcal {V}}} \le \limsup _{n\rightarrow \infty }\langle \mathcal Lu_n-f,u-u_n\rangle _{\mathcal {V}}\\&+\limsup _{n\rightarrow \infty }J^0(\gamma u_n;\gamma (u-u_n))+\varphi (u) -\liminf _{n\rightarrow \infty }\varphi (u_n)\le 0, \end{aligned}$$

where the lower semicontinuity of \(\varphi \) and the upper semicontinuity of \(J^0\) have been used. At this point, the pseudomonotonicity of \({\mathcal {T}}\) entails \(u^*\in {\mathcal {T}}(u)\) and \(\langle u_n^*,u_n\rangle _{{\mathcal {V}}}\rightarrow \langle u^*,u\rangle _{{\mathcal {V}}}\). Letting \(n\rightarrow \infty \) in (8) leads to \(u\in {\mathcal {S}}(w)\).

(iii) The proof will be carried out by a contrapositive argument. Assume that there are sequences \(\{u_n\}\subset D(\mathcal L)\) and \(\{w_n\}\subset {\mathcal {X}}\) with \(u_n\in {\mathcal {S}}(w_n)\) for each \(n\in {\mathbb {N}}\) such that \(\Vert u_n\Vert _{{\mathcal {V}}}\rightarrow +\infty \) as \(n\rightarrow \infty \). There exists \(u_n^*\in \mathcal Tu_n\) with

$$\begin{aligned}&\langle \mathcal Lu_n+u_n^*-f,v-u_n\rangle _{\mathcal {V}} +J^0(\gamma u_n;\gamma (v-u_n))\\&+\varphi (v)-\varphi (u_n)\ge 0\quad \text {for all}\ v\in {\mathcal {K}}(w_n). \end{aligned}$$

Since \(0\in {\mathcal {K}}(w)\) for all \(w\in {\mathcal {X}}\), we can set \(v=0\) to derive

$$\begin{aligned} \langle \mathcal Lu_n,-u_n\rangle _{\mathcal {V}}\ge \langle u_n^*-f,u_n\rangle _{\mathcal {V}}+\langle \xi _n, \gamma u_n\rangle _{{\mathcal {E}}}+\varphi (u_n)-\varphi (0), \end{aligned}$$

where \(\xi _n\in \partial J(\gamma u_n)\) verifies \(\langle \xi _n,-\gamma (u_n)\rangle _{{\mathcal {E}}}=J^0(\gamma u_n;\gamma (-u_n))\). The monotonicity of \({\mathcal {L}}\), hypotheses \(H({\mathcal {T}})\) and H(J), and the existence of an affine bound from below for the function \(\varphi \) (see, e.g., [2, Proposition 1.10]) yield

$$\begin{aligned} 0&\ge \langle u_n^*-f,u_n\rangle _{\mathcal {V}}+\langle \xi _n, \gamma u_n\rangle _{{\mathcal {E}}}+\varphi (u_n)-\varphi (0)\\&\ge m_{\mathcal {T}}\Vert u_n\Vert _{{\mathcal {V}}}^{p}-d_{\mathcal {T}} -\Vert f\Vert _{{\mathcal {V}}^*}\Vert u_n\Vert _{{\mathcal {V}}}-\Vert \gamma \Vert c_J\left( 1 +\Vert \gamma u_n\Vert _{\mathcal {E}}^{\theta -1}\right) \Vert u_n\Vert _{{\mathcal {V}}} \\&\quad -c_\varphi \Vert u_n\Vert -d_\varphi -\varphi (0), \end{aligned}$$

with positive constants \(c_\varphi ,d_\varphi \), which is a contradiction as \(\Vert u_n\Vert _{{\mathcal {V}}}\rightarrow \infty \). Thus, the set \({\mathcal {S}}({\mathcal {X}})\) is bounded in \({\mathcal {V}}\). \(\square \)

The next lemma gives the Minty’s formulation for Problem 2.

Lemma 11

Besides the conditions of Lemma 10, additionally assume that the map \(u\in {\mathcal {V}}\mapsto \mathcal Tu+\gamma ^*\partial J(\gamma u)\in 2^{{\mathcal {V}}^*}\) is monotone. Then, the following assertions are valid:

  1. (i)

    For each \(w\in {\mathcal {X}}\), \(u\in {\mathcal {K}}(w)\cap D({\mathcal {L}})\) is a solution to Problem 2, if and only if,

    $$\begin{aligned} \langle \mathcal Lv+v^*-f,v-u\rangle _{\mathcal V}+\langle \xi ,\gamma (v-u)\rangle _{\mathcal E}+\varphi (v)-\varphi (u)\ge 0 \end{aligned}$$
    (9)

    for all \(v^*\in \mathcal Tv\), all \(\xi \in \partial J(\gamma v)\) and all \(v\in {\mathcal {K}}(w)\cap D({\mathcal {L}})\).

  2. (ii)

    For each \(w\in {\mathcal {X}}\), the set \({\mathcal {S}}(w)\) is convex.

Proof

(i) We fix \(w\in {\mathcal {X}}\), and assume that \(u\in {\mathcal {K}}(w)\cap D({\mathcal {L}})\) is a solution to Problem 2. Then, with an element \(u^*\in {\mathcal {T}}(u)\), we have

$$\begin{aligned} \langle \mathcal Lu+u^*-f,v-u\rangle _{{\mathcal {V}}}+J^0(\gamma u;\gamma (v-u))+\varphi (v)-\varphi (u)\ge 0\quad \text {for all}\ v\in {\mathcal {K}}(w). \end{aligned}$$

Notice that \(J^0(\gamma u;\gamma (v-u))=\max \{\langle \xi , \gamma (v-u)\rangle _{{\mathcal {E}}}\,:\, \xi \in \partial J(\gamma u)\}\) whenever \(v\in {\mathcal {V}}\) (see Proposition 4(d)).

Let \(z\in {\mathcal {K}}(w)\cap D({\mathcal {L}})\). The monotonicity of \({\mathcal {L}}\) and \(u\mapsto \mathcal Tu+\gamma ^*\partial J(\gamma u)\) provide

$$\begin{aligned} \langle \mathcal Lz+z^*-f,z-u\rangle _{{\mathcal {V}}}+\langle \xi , \gamma (z-u)\rangle _{{\mathcal {E}}}+\varphi (z)-\varphi (u)\ge 0, \end{aligned}$$

for all \(z^*\in \mathcal Tz\) and \(\xi \in \partial J(\gamma z)\). We thus arrive at (9).

For the converse, we assume that \(u\in {\mathcal {K}}(w)\cap D({\mathcal {L}})\) solves (9). For any \(z\in {\mathcal {K}}(w) \cap D(\mathcal L)\) and \(t\in (0,1)\), we note that \(v_t=t z+(1-t)u\) is admissible as test function in (9), which for some \(v_t^*\in \mathcal Tv_t\) yields

$$\begin{aligned} \langle t\mathcal Lz+(1-t)\mathcal Lu+v_t^*-f,z-u\rangle _{\mathcal V}+J^0(\gamma v_t;\gamma (z-u))+\varphi (z)-\varphi (u)\ge 0\nonumber \\ \end{aligned}$$
(10)

due to the positive homogeneity of \(J^0(v_t;\cdot )\) and the convexity of \(\varphi \).

The boundedness of \({\mathcal {T}}\) and the reflexivity of \({\mathcal {V}}^*\) ensure that along a sequence \(v_t^*\rightarrow z^*\) weakly in \({\mathcal {V}}^*\) as \(t\rightarrow 0\) for some \(z^*\in {\mathcal {V}}^*\). The graph of \(\mathcal T\) is strongly–weakly closed owing to the pseudomonotonicity and boundedness of \({\mathcal {T}}\), thereby \(z^*\in \mathcal Tu\). Due to the compactness of \(\gamma \) and upper semicontinuity of \(J^0\), we can pass to the upper limit as \(t\rightarrow 0\) in (10), so that for each \(z\in {\mathcal {K}}(w) \cap D({\mathcal {L}})\), there is \(z^*\in {\mathcal {T}}(u)\) with

$$\begin{aligned} \langle \mathcal Lu+z^*-f,z-u\rangle _{{\mathcal {V}}} +J^0(\gamma u;\gamma (z-u))+\varphi (z)-\varphi (u)\ge 0. \end{aligned}$$

The density of \(D({\mathcal {L}})\) in \({\mathcal {V}}\), in conjunction with the continuity of \(\varphi \) on \({\mathcal {K}}(w)\), hypothesis \(H(\varphi )\), the fact that \(J^0\) is upper semicontinuous, and the assumption that \({\mathcal {T}}\) is strongly–weakly closed, for each \(z\in {\mathcal {K}}(w)\), there is an element \(z^*\in {\mathcal {T}}(u)\) for which we have

$$\begin{aligned} \langle \mathcal Lu+z^*-f,z-u\rangle _{{\mathcal {V}}} +J^0(\gamma u;\gamma (z-u))+\varphi (z)-\varphi (u)\ge 0. \end{aligned}$$
(11)

We are now prepared to prove that \(u\in {\mathcal {S}}(w)\). For this, assume that u is not a solution to Problem 2. Then, for each \(u^*\in {\mathcal {T}}(u)\), there exists \(v\in {\mathcal {K}}(w)\) such that

$$\begin{aligned} \langle \mathcal Lu+ u^*-f,v-u\rangle _{{\mathcal {V}}} +J^0(\gamma u;\gamma (v-u))+\varphi (v)-\varphi (u)<0. \end{aligned}$$

Denote \({\mathcal {R}}(u)={\mathcal {T}}(u)+\gamma ^*\partial J(\gamma u)\). Then, by Proposition 4, for each \(v^*\in {\mathcal {R}}(u)\) there exists \(v\in {\mathcal {K}}(w)\) satisfying

$$\begin{aligned} \langle v^*,v-u\rangle _{{\mathcal {V}}}< \varphi (u)-\varphi (v) +\langle f-\mathcal Lu,v-u\rangle _{{\mathcal {V}}}. \end{aligned}$$
(12)

Given \(v\in {\mathcal {K}}(w)\), we introduce the set

$$\begin{aligned} {\mathcal {Q}}_v:=\left\{ v^*\in {\mathcal {R}}(u)\,\mid \,\langle v^*,v-u\rangle _{{\mathcal {V}}} <\varphi (u)-\varphi (v)+\langle f-\mathcal Lu,v-u\rangle _{{\mathcal {V}}}\right\} , \end{aligned}$$

which is weakly open in \({\mathcal {V}}^*\). As already observed, \(\{{\mathcal {Q}}_v\}_{v\in {\mathcal {K}}(w)}\) is an open covering of \({\mathcal {R}}(u)\). Since \({\mathcal {R}}(u)\) is weakly compact in \({\mathcal {V}}^*\), a finite sub-covering \(\{{\mathcal {Q}}_{v_1},\mathcal Q_{v_2},\ldots ,{\mathcal {Q}}_{v_n}\}\) can be found corresponding to the points \(v_1,v_2,\ldots ,v_n\in {\mathcal {K}}(w)\). Let \(\kappa _1,\kappa _2,\ldots ,\kappa _n\) be a subordinate partition of unity on \({\mathcal {R}}(u)\) (see, e.g., [12, Lemma 7.3]), that is, for each \(i=1,2,\ldots ,n\), \(\kappa _i:{\mathcal {R}}(u) \rightarrow [0,1]\) is a weakly continuous function with the support in \({\mathcal {Q}}_{v_i}\) and

$$\begin{aligned} \sum _{i=1}^n\kappa _i(v^*)=1 \text{ for } \text{ all }~ v^*\in {\mathcal {R}}(u). \end{aligned}$$

We also introduce the map \({\mathcal {N}}:{\mathcal {R}}(u)\rightarrow {\mathcal {K}}(w)\) by

$$\begin{aligned} {\mathcal {N}}(v^*)=\sum _{i=1}^n\kappa _i(v^*)(v_i)\ \ \text{ for } \text{ all } v^*\in {\mathcal {R}}(u), \end{aligned}$$

which is weakly continuous complying with \(\kappa _i\) for \(i=1,2,\ldots ,n\). By (12) and the convexity of \(\varphi \), for any \(v^*\in {\mathcal {R}}(u)\), we have

$$\begin{aligned} \langle v^*,{\mathcal {N}}(v^*)-u\rangle _{{\mathcal {V}}}&= \sum _{i=1}^n\kappa _i(v^*)\langle v^*,v_i-u\rangle _{{\mathcal {V}}}\nonumber \\&<\varphi (u)-\varphi \left( \sum _{i=1}^n\kappa _i(v^*)v_i\right) +\left\langle f-\mathcal Lu, \sum _{i=1}^n\kappa _i(v^*)v_i-u\right\rangle _{\mathcal {V}}\nonumber \\&=\varphi (u)-\varphi ({\mathcal {N}}(v^*))+\langle f-\mathcal Lu,{\mathcal {N}}(v^*)-u\rangle _{\mathcal {V}}. \end{aligned}$$
(13)

Let us define the maps \(\Upsilon :{\mathcal {K}}(w)\rightarrow 2^{\mathcal R(u)}\) by

$$\begin{aligned} \Upsilon (v):=\left\{ v^*\in {\mathcal {R}}(u)\,\mid \,\langle v^*,v-u\rangle _{{\mathcal {V}}} \ge \varphi (u)-\varphi (v)+\langle f-\mathcal Lu,v-u\rangle _{\mathcal {V}}\right\} \end{aligned}$$

for all \(v\in {\mathcal {K}}(w)\) and \(\Psi :{\mathcal {R}}(u)\rightarrow 2^{{\mathcal {R}}(u)}\) by

$$\begin{aligned} \Psi (v^*):=\Upsilon ({\mathcal {N}}(v^*))\ \ \text{ for } \text{ all } v^*\in \mathcal R(u). \end{aligned}$$

By means of (11), it is seen that \(\Upsilon \) has nonempty, weakly compact, and convex values. We prove that \(\Upsilon \) is upper semicontinuous from the norm topology of \({\mathcal {V}}\) to the weak topology of \({\mathcal {V}}^*\). Making use of a classical result (see, e.g., [37, Proposition 3.8]), this amounts to check that for each weakly closed set B in \({\mathcal {V}}^*\), the set

$$\begin{aligned} \Upsilon ^-(B):=\left\{ v\in {\mathcal {K}}(w)\,\mid \,\Upsilon (v)\cap B\ne \emptyset \right\} , \end{aligned}$$

is closed in \({\mathcal {V}}\). Let \(\{v_n\}\subset \Upsilon ^-(B)\) satisfy \(v_n\rightarrow v\) in \({\mathcal {V}}\) as \(n\rightarrow \infty \), for some \(v\in {\mathcal {V}}\). For each \(n\in {\mathbb {N}}\), we can find \(v_n^*\in {\mathcal {R}}(u)\cap B\) such that

$$\begin{aligned} \langle v_n^*,v_n-u\rangle _{{\mathcal {V}}}\ge \varphi (u)-\varphi (v_n) +\langle f-\mathcal Lu,v_n-u\rangle _{\mathcal {V}}. \end{aligned}$$
(14)

Through the weak compactness of \({\mathcal {R}}(u)\), we may suppose that \(v_n^*\rightarrow v^*\in B\) weakly in \({\mathcal {V}}^*\) as \(n\rightarrow \infty \), for some \(v^*\in {\mathcal {R}}(u)\). Now pass to the upper limit as \(n\rightarrow \infty \) in (14) and use the lower semicontinuity of \(\varphi \) to get

$$\begin{aligned} \langle v^*,v-u\rangle _{{\mathcal {V}}}\ge \varphi (u)-\varphi (v) +\langle f-\mathcal Lu,v-u\rangle _{\mathcal {V}}, \end{aligned}$$

which means that \(v^*\in \Upsilon (v)\cap B\), thus proving the desired property for \(\Upsilon \). As a consequence, through the weak continuity of \({\mathcal {N}}\) we infer that \(\Psi \) is also strongly–weakly upper semicontinuous (refer, e.g., to [19, Theorem 1.2.8]).

The preceding arguments enable us to apply Tychonov’s fixed point principle (see, e.g., [12, Theorem 8.6]) to the map \(\Psi \). Hence, there exists \(v^*\in {\mathcal {R}}(u)\) such that

$$\begin{aligned} \langle v^*,{\mathcal {N}}(v^*)-u\rangle _{{\mathcal {V}}}\ge \varphi (u) -\varphi ({\mathcal {N}}(v^*))+\langle f-\mathcal Lu, \mathcal N(v^*)-u\rangle _{{\mathcal {V}}}. \end{aligned}$$

This contradicts (13), thus establishing that \(u\in {\mathcal {K}}(w)\) solves Problem 2.

(ii) Let \(u_1,u_2\in {\mathcal {S}}(w)\) and \(t\in (0,1)\). On the basis of part (i), for \(i=1,2\) we have

$$\begin{aligned} \langle \mathcal Lv+v^*-f,v-u_i\rangle _\mathcal V+\langle \xi ,\gamma (v-u_i)\rangle _{\mathcal E}+\varphi (v)-\varphi (u_i)\ge 0, \end{aligned}$$

for all \(v^*\in \mathcal Tv\), \(\xi \in \partial J(\gamma v)\) and \(v\in {\mathcal {K}}(w)\cap D({\mathcal {L}})\). Set \(u_t=tu_1+(1-t)u_2\). Then, the convexity of \(\varphi \) implies

$$\begin{aligned} \langle \mathcal Lv+v^*-f+ \gamma ^*\xi ,v-u_t\rangle _\mathcal V+\varphi (v)-\varphi (u_t)\ge 0, \end{aligned}$$

for all \(v^*\in \mathcal Tv\), \(\xi \in \partial J(\gamma v)\) and \(v\in {\mathcal {K}}(w)\cap D({\mathcal {L}})\). Invoking part (i), we infer that \(u_t\) is a solution to Problem 2, so the set \({\mathcal {S}}(w)\) is convex. \(\square \)

The above preparation permits us to give the following existence result for Problem 1.

Theorem 12

Assume that H(0), \(H({\mathcal {L}})\), \(H({\mathcal {T}})\), \(H(\gamma )\), H(J), \(H(\varphi )\), and \(H({\mathcal {K}})\) are fulfilled. If, in addition, \(u\in {\mathcal {V}}\mapsto \mathcal Tu+\gamma ^*\partial J(\gamma u)\in 2^{{\mathcal {V}}^*}\) is monotone, then the set of solutions to Problem 1 is nonempty and sequentially weakly compact in \({\mathcal {V}}\).

Proof

To prove that the set of solutions to Problem 1 is nonempty, we apply Theorem 8 to the set-valued mapping \({\mathcal {S}}:{\mathcal {X}}\rightarrow 2^{{\mathcal {X}}}\). We first check that the graph \(Gr({\mathcal {S}})\) of \({\mathcal {S}}\) is sequentially weakly closed in \({\mathcal {V}}\times {\mathcal {V}}\). Let \(\{(w_n,u_n)\}\subset Gr({\mathcal {S}})\) satisfy \((w_n,u_n)\rightarrow (w,u)\) weakly in \({\mathcal {V}}\times {\mathcal {V}}\) for some \((w,u)\in \mathcal V\times {\mathcal {V}}\). The fact that \(u_n\in {\mathcal {S}}(w_n)\) means that there is \(u_n^*\in \mathcal Tu_n\) such that

$$\begin{aligned} \langle \mathcal Lu_n+u_n^*-f,v-u_n\rangle _{\mathcal {V}}+J^0(\gamma u_n;\gamma (v-u_n))+\varphi (v)-\varphi (u_n)\ge 0\quad \text {for all} \ v\in {\mathcal {K}}(w_n). \end{aligned}$$
(15)

The boundedness of \({\mathcal {T}}\) allows us to suppose that up to a subsequence \(u_n^*\rightarrow u^*\) weakly in \({\mathcal {V}}^*\) as \(n\rightarrow \infty \), for some \(u^*\in {\mathcal {V}}^*\).

It follows from Lemma 9 that the sequence \(\{\mathcal Lu_n\}\) is bounded in \({\mathcal {V}}^*\). Note that \({\mathcal {L}}\) is linear and maximal monotone, so it is weakly closed graph. This combined with the convergence \(u_n\rightarrow u\) weakly in \({\mathcal {V}}\) entails \(u\in D({\mathcal {L}})\) and \({\mathcal {L}}(u_n)\rightarrow {\mathcal {L}}(u)\) weakly in \({\mathcal {V}}^*\). Then, hypothesis \(H({\mathcal {K}})\)(iii) implies \(u\in {\mathcal {K}}(w)\). In turn, condition \(H({\mathcal {K}})\)(iv) provides a sequence \(\{z_n\}\subset {\mathcal {V}}\) with \(z_n\in {\mathcal {K}}(w_n)\) and \(z_n\rightarrow u\) as \(n\rightarrow \infty \).

Upon inserting \(v=z_n\) in (15), we find

$$\begin{aligned}&\langle \mathcal Lu_n-f,z_n-u_n\rangle _{\mathcal {V}} +\langle u_n^*,z_n-u\rangle _{\mathcal {V}}+J^0(\gamma u_n; \gamma (z_n-u_n))\nonumber \\&+\varphi (z_n)-\varphi (u_n)\ge \langle u_n^*,u_n-u\rangle _{{\mathcal {V}}}. \end{aligned}$$
(16)

The monotonicity of \({\mathcal {L}}\) ensures

$$\begin{aligned} 0\le \liminf _{n\rightarrow \infty }\langle {\mathcal {L}} u_n-\mathcal Lu,u_n-u\rangle _{{\mathcal {V}}}=\liminf _{n\rightarrow \infty }\langle {\mathcal {L}} u_n,u_n\rangle _{{\mathcal {V}}}-\langle {\mathcal {L}} u,u\rangle _{\mathcal V}, \end{aligned}$$

which gives

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle {\mathcal {L}} u_n,z_n-u_n\rangle _{\mathcal V}\le \limsup _{n\rightarrow \infty }\langle {\mathcal {L}} u_n,z_n\rangle _{{\mathcal {V}}}-\liminf _{n\rightarrow \infty } \langle {\mathcal {L}} u_n,u_n\rangle _{{\mathcal {V}}}\le 0. \end{aligned}$$
(17)

The compactness of \(\gamma \) and the upper semicontinuity of \((u,v)\mapsto J^0(u;v)\) yield

$$\begin{aligned} \limsup _{n\rightarrow \infty }J^0(\gamma u_n;\gamma (z_n-u_n))\le 0. \end{aligned}$$
(18)

By \(H(\varphi )\), we have \(\bigcup _{w\in \mathcal X}K(w)\subset \text{ int }D(\varphi )\), so \(\varphi \) is continuous on \(\bigcup _{w\in {\mathcal {X}}}K(w)\), resulting in

$$\begin{aligned} \limsup _{n\rightarrow \infty } [\varphi (z_n)-\varphi (u_n)]\le \limsup _{n\rightarrow \infty } \varphi (z_n)-\liminf _{n\rightarrow \infty } \varphi (u_n)\le 0. \end{aligned}$$
(19)

Letting \(n\rightarrow \infty \) in (16) and using (17)–(19), we get

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle u_n^*,u_n-u\rangle _{{\mathcal {V}}}\le 0. \end{aligned}$$

The latter combined with the pseudomonotonicity of \({\mathcal {T}}\) results in

$$\begin{aligned} u^*\in \mathcal Tu\quad \text{ and } \quad \langle u_n^*,u_n\rangle _{{\mathcal {V}}}\rightarrow \langle u^*,u\rangle _{{\mathcal {V}}}. \end{aligned}$$
(20)

Let \(v\in {\mathcal {K}}(w)\). By hypothesis \(H({\mathcal {K}})\)(iv), up to a subsequence of \(\{w_n\}\), there is a sequence \(\{v_n\}\subset {\mathcal {V}}\) with \(v_n\in {\mathcal {K}}(w_n)\) and \(v_n \rightarrow v\) as \(n\rightarrow \infty \). Taking \(v=v_n\) in (15), it turns out from (17)–(20) that \((w,u)\in Gr({\mathcal {S}})\) since

$$\begin{aligned} 0&\le \limsup _{n\rightarrow \infty }\left[ \langle \mathcal Lu_n +u_n^*-f,v_n-u_n\rangle _{\mathcal {V}}+J^0(\gamma u_n;\gamma (v_n-u_n)) +\varphi (v_n)-\varphi (u_n)\right] \\&\le \limsup _{n\rightarrow \infty }\langle \mathcal Lu_n +u_n^*-f,v_n-u_n\rangle _{\mathcal {V}}+\limsup _{n\rightarrow \infty }J^0(\gamma u_n; \gamma (v_n-u_n)) \\&\quad +\limsup _{n\rightarrow \infty }\varphi (v_n)-\liminf _{n\rightarrow \infty }\varphi (u_n)\\&\le \langle \mathcal Lu+u^*-f,v-u\rangle _{\mathcal {V}}+J^0(\gamma u;\gamma (v-u))+\varphi (v)-\varphi (u). \end{aligned}$$

Therefore, the graph \(Gr({\mathcal {S}})\) of \({\mathcal {S}}\) is sequentially weakly closed in \({\mathcal {V}}\times {\mathcal {V}}\).

The other requirements needed to apply Theorem 8 for \({\mathcal {S}}\) are fulfilled according to Lemmas 10 and 11 (ii). We infer from Theorem 8 that \({\mathcal {S}}\) has at least a fixed point in \({\mathcal {X}}\); thus, there exists a solution to Problem 1.

Now, we proceed to show that the set of solutions to Problem 1 is sequentially weakly compact. Let \(\{u_n\}\) be a sequence of solutions to Problem 1. Then, \(u_n\in {\mathcal {K}}(u_n)\), and for some \(u_n^*\in \mathcal T(u_n)\), we have

$$\begin{aligned} \langle \mathcal Lu_n+u_n^*-f,v-u_n\rangle _{\mathcal {V}}+J^0(\gamma u_n;\gamma (v-u_n))+\varphi (v)-\varphi (u_n)\ge 0 \end{aligned}$$
(21)

for all \(v\in {\mathcal {K}}(u_n)\). The reasoning in the proof of Lemma 10(iii) reveals that the sequence \(\{u_n\}\) is bounded. Without loss of generality, we may assume that \(u_n\rightarrow u\) weakly in \({\mathcal {V}}\) for some \(u\in {\mathcal {V}}\). Since \(u_n\in {\mathcal {S}}(u_n)\), by Lemma 9 the sequence \(\{\mathcal Lu_n\}\) is bounded in \({\mathcal {V}}^*\). The reflexivity of \({\mathcal {V}}^*\) and because \({\mathcal {L}} \) is linear and closed, we find along a subsequence that \(\mathcal Lu_n\rightarrow \mathcal Lu\) weakly in \({\mathcal {V}}^*\) as \(n\rightarrow \infty \). Then, using hypothesis \(H({\mathcal {K}})\)(iii), it is true that \(u\in {\mathcal {K}}(u)\).

By virtue of condition \(H({\mathcal {K}})\)(iv), possibly for a subsequence of \(\{u_n\}\), there is \(\{z_n\}\subset {\mathcal {X}}\) such that \(z_n\in {\mathcal {K}}(u_n)\) and \(z_n\rightarrow u\). As \({\mathcal {T}}\) is a bounded mapping, the sequence \(\{u_n^*\}\) is bounded in \(\mathcal V^*\), so we can suppose that \(u_n^* \rightarrow u^*\) weakly in \({\mathcal {V}}\) for some \(u^*\in {\mathcal {V}}^*\).

Insert \(v=z_n\) in (21) and use (17)–(19) to deduce that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle u_n^*,u_n-u\rangle _{{\mathcal {V}}}\le 0. \end{aligned}$$

Due to the pseudomonotonicity of \({\mathcal {T}}\), we derive \(u^*\in {\mathcal {T}} u\) and \(\lim _{n\rightarrow \infty }\langle u_n^*,u_n\rangle _{\mathcal {V}}= \langle u^*,u\rangle _{\mathcal {V}}\).

For any \(v\in {\mathcal {K}}(u)\), by assumption \(H({\mathcal {K}})\)(iv) we are able to find a sequence \(\{v_n\}\) with \(v_n\in {\mathcal {K}}(u_n)\) and \(v_n\rightarrow v\) in \({\mathcal {V}}\). Passing to the limit in (21) with \(v_n\) as test element yields

$$\begin{aligned} \langle \mathcal Lu+u^*-f,v-u\rangle _{\mathcal {V}}+J^0(\gamma u;\gamma (v-u))+\varphi (v)-\varphi (u)\ge 0, \end{aligned}$$

so u is a solution to Problem 1. Consequently, the set of solutions to Problem 1 is sequentially weakly compact. \(\square \)

4 An Optimal Control Problem

In this section, we focus on an optimal control problem associated with the (QVHVI). In the following, we continue to adhere to the notation used in Problem 1. Additionally, let \({\mathcal {W}}\) be a reflexive Banach space which is compactly embedded in \({\mathcal {V}}^*\) and let \({\mathcal {B}}\) be a nonempty and weakly closed subset of \({\mathcal {W}}\). Consider the set-valued mapping \(\Pi :{\mathcal {V}}^*\rightarrow 2^{{\mathcal {X}}}\) defined by

$$\begin{aligned} \Pi (f):=\left\{ u\in D({\mathcal {L}})\,\mid \,u \text{ is } \text{ a } \text{ solution } \text{ to } \text{ Problem }~9~\text {corresponding to} f\in \mathcal V^*\right\} . \end{aligned}$$

We also need to introduce functions \(g:{\mathcal {X}}\rightarrow {\mathbb {R}}\) and \(h:{\mathcal {B}}\rightarrow {\mathbb {R}}\) that satisfy the following:

H(g): \(g:{\mathcal {X}}\rightarrow {\mathbb {R}}\) is bounded from below and sequentially weakly lower semicontinuous.

H(h): \(h:{\mathcal {B}}\rightarrow {\mathbb {R}}\) is bounded from below, sequentially weakly lower semicontinuous and coercive, that is, \(h(f)\rightarrow +\infty \) as \(\Vert f\Vert _{\mathcal {W}}\rightarrow +\infty \).

Our focus is on the following nonlinear optimal control problem:

Problem 3

Find \(f\in {\mathcal {B}}\) such that

$$\begin{aligned} F(f)=\inf _{l\in {\mathcal {B}}}F(l) \text{ with } F(l):=\inf _{u\in \Pi (l)}g(u)+h(l), \end{aligned}$$
(22)

where \(\Pi (l)\) is the set of solutions to Problem 1 corresponding to \(l\in {\mathcal {V}}^*\).

To give an existence result for the above control problem, we begin with the following:

Lemma 13

Under the assumptions of Theorem 12, the following statements hold:

  1. (i)

    \(\Pi :{\mathcal {V}}^*\rightarrow 2^{{\mathcal {X}}}\) is a bounded map.

  2. (ii)

    If \(\{f_n\}\subset {\mathcal {V}}^*\) is such that \(f_n\rightarrow f\) in \({\mathcal {V}}^*\) as \(n\rightarrow \infty \) for some \(f\in {\mathcal {V}}^*\), then

    $$\begin{aligned} w\text{-- }\limsup _{n\rightarrow \infty }\Pi (f_n)\subset \Pi (f), \end{aligned}$$

    where \(w\text{-- }\limsup _{n\rightarrow \infty }\Pi (f_n)\) stands for the sequential Kuratowski upper limit of \(\{\Pi (f_n)\}\) with respect to the weak topology of \({\mathcal {V}}\), namely,

    $$\begin{aligned}&\text {w}-\limsup _{n\rightarrow \infty }\Pi (f_n)\\&:=\bigg \{u\in {\mathcal {X}} \mid \exists \, u_{n_k}\in \Pi (f_{n_k}), u_{n_k}\rightarrow u\ \text{ weakly } \text{ in } \ {\mathcal {V}}\ {\mathrm{as}}\ k\rightarrow \infty \bigg \}. \end{aligned}$$

Proof

(i) To prove the claim, we begin by assuming that there exists a bounded set \({\mathcal {C}}\) in \({\mathcal {V}}^*\) such that \(\Pi ({\mathcal {C}})\) is unbounded. This allows us to find sequences \(\{f_n\}\subset \mathcal C\) and \(\{u_n\}\subset {\mathcal {V}}\) with \(u_n\in \Pi (f_n)\) for all \(n\in {\mathbb {N}}\) satisfying \( \Vert u_n\Vert _{\mathcal {V}} \rightarrow \infty \) as \(n\rightarrow \infty \). We have \(u_n\in {\mathcal {K}}(u_n)\) and \(u_n^*\in {\mathcal {T}}(u_n)\) such that

$$\begin{aligned}&\langle \mathcal Lu_n+u^*_n-f_n,v-u_n\rangle _{\mathcal V}+\varphi (v)-\varphi (u_n)\\&+J^0(\gamma u_n;\gamma (v-u_n))\ge 0\quad \text {for all}\ v\in {\mathcal {K}}(u_n). \end{aligned}$$

By assumption \(H({\mathcal {K}})\)(ii), we have \(0\in {\mathcal {K}}(w)\) for all \(w\in {\mathcal {X}}\), and hence, we can insert \(v=0\) in the above inequality. By means of \(H(\mathcal L)\), \(H({\mathcal {T}})\), \(H(\gamma )\), H(J), \(H(\varphi )\), and the boundedness of \({\mathcal {C}}\), we get

$$\begin{aligned} 0&=\langle {\mathcal {L}}0,-u_n\rangle _{{\mathcal {V}}} \ge \langle \mathcal Lu_n,-u_n\rangle _{{\mathcal {V}}}\nonumber \\&\ge \langle u^*_n-f_n,u_n\rangle _{{\mathcal {V}}} -\varphi (0)+\varphi (u_n)-J^0(\gamma u_n;-\gamma u_n)\nonumber \\&\ge m_{\mathcal {T}}\Vert u_n\Vert _{{\mathcal {V}}}^{p}-d_{\mathcal {T}}-M_{{\mathcal {C}}} \Vert u_n\Vert _{{\mathcal {V}}}-\Vert \gamma \Vert c_J\left( 1 +\Vert \gamma u_n\Vert _{\mathcal {E}}^{\theta -1}\right) \Vert u_n\Vert _{{\mathcal {V}}} \nonumber \\&\quad -c_\varphi \Vert u_n\Vert -d_\varphi -\varphi (0), \end{aligned}$$

with a constant \(M_{\mathcal {C}}>0\). Using condition H(J), in the limit as \(n\rightarrow \infty \), leads to a contradiction. We conclude that \(\Pi :{\mathcal {V}}^*\rightarrow 2^{{\mathcal {X}}}\) is a bounded map.

(ii) Let \(u\in w\text{-- }\limsup _{n\rightarrow \infty }\) \(\Pi (f_n)\). Let \(\{u_{n_k}\}\subset {\mathcal {V}}\) with \(u_{n_k}\in \Pi (f_{n_k})\) be such that

$$\begin{aligned} u_{n_k}\rightarrow u\ \, \text{ weakly } \text{ in } {\mathcal {V}}\ \, \text{ as } k\rightarrow \infty . \end{aligned}$$
(23)

Then, \(u_{n_k}\in {\mathcal {K}}(u_{n_k})\), and for some \(u_{n_k}^*\in {\mathcal {T}}(u_{n_k})\), we have

$$\begin{aligned}&\langle \mathcal Lu_{n_k}+u_{n_k}^*,v-u_{n_k}\rangle _{\mathcal V}+\varphi (v)-\varphi (u_{n_k})+J^0(\gamma u_{n_k};\gamma (v-u_{n_k})) \nonumber \\&\qquad \ge \langle f_{n_k},v-u_{n_k}\rangle _{\mathcal {V}}\quad \text {for all}\ v\in {\mathcal {K}}(u_{n_k}). \end{aligned}$$
(24)

The boundedness of \({\mathcal {T}}\) allows us to assume that \(u_{n_k}^*\rightarrow u^*\) weakly in \({\mathcal {V}}^*\) for some \(u^*\in {\mathcal {V}}^*\). The convergence in (23) and condition \(H({\mathcal {K}})\)(iii) imply that \(u\in \mathcal K(u)\). Lemma 9 and the fact that \({\mathcal {L}}\) is linear and graph closed imply \(u\in D({\mathcal {L}})\) and \(\mathcal Lu_{n_k}\rightarrow \mathcal Lu\) weakly in \({\mathcal {V}}^*\) as \(k\rightarrow \infty \). Besides, assumption \(H({\mathcal {K}})\)(iv) provides a sequence \(\{z_k\}\subset {\mathcal {V}}\) with \(z_k\in {\mathcal {K}}(u_{n_k})\) such that \(z_k\rightarrow u\) in \({\mathcal {V}}\). Inserting \(v=z_k\) in (24) yields

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle u_{n_k}^*,u_{n_k}-u\rangle _{\mathcal V}\le 0, \end{aligned}$$

and the pseudomonotonicity of \({\mathcal {T}}\) further ensures that

$$\begin{aligned} u^*\in \mathcal Tu \text{ and } \langle u_{n_k}^*,u_{n_k}\rangle _{{\mathcal {V}}}\rightarrow \langle u^*,u\rangle _{{\mathcal {V}}}. \end{aligned}$$

Let \(z\in {\mathcal {K}}(u)\). Hypothesis \(H({\mathcal {K}})\)(iv) guarantees the existence, up to a subsequence, of \(v_k\in {\mathcal {K}}(u_{n_k})\) with \(v_k\rightarrow z\) as \(k\rightarrow \infty \). Set \(v=v_k\) in (24). Passing to the limit on the pattern of Step 1 in the proof of Theorem 12 leads to

$$\begin{aligned} \langle \mathcal Lu+u^*,z-u\rangle _{\mathcal V}+\varphi (z)-\varphi (u)+J^0(\gamma u;\gamma (z-u))\ge \langle f,z-u\rangle _{\mathcal {V}}. \end{aligned}$$

Hence, u is a solution to Problem 1 associated with f, that is, \(u\in \Pi (f)\). \(\square \)

The following is the existence result for Problem 3:

Theorem 14

Besides the assumptions of Theorem 12, additionally assume the conditions H(g) and H(h). Then, the set of solutions to Problem 3 is nonempty and sequentially weakly compact.

Proof

We claim that for each \(f\in {\mathcal {V}}^*\), there exists \(u\in \Pi (f)\) such that \(F(f)=g(u)+h(f)\). To this end, we recall from Theorem 12 that the set \(\Pi (f)\) is nonempty and sequentially weakly compact in \({\mathcal {V}}\). Then, by the Weierstrass minimization theorem, assumption H(g) entitles the existence of \(u\in \Pi (f)\) satisfying

$$\begin{aligned} g(u)=\inf _{v\in \Pi (f)}g(v), \end{aligned}$$

which proves the claim. Consequently, the function \(F:{\mathcal {V}}^*\rightarrow {\mathbb {R}}\) given in (22) is well defined.

Denote \(\theta :=\inf _{f\in {\mathcal {B}}}F(f)\). Let \(\{f_n\}\subset {\mathcal {B}}\) be a minimizing sequence for F, that is,

$$\begin{aligned} \lim _{n\rightarrow \infty } F(f_n)=\theta \in [-\infty ,+\infty ). \end{aligned}$$
(25)

Let \(u_n\in \Pi (f_n)\) be a sequence such that

$$\begin{aligned} F(f_n)=g(u_n)+h(f_n)\ \, \text{ for } \text{ each } n\in {\mathbb {N}}, \end{aligned}$$
(26)

which exists by the arguments used above. Note that the sequence \(\{f_n\}\) is bounded in \({\mathcal {W}}\). If not, we can assume that, along a subsequence, we have

$$\begin{aligned} \Vert f_n\Vert _{{\mathcal {W}}}\rightarrow +\infty , \ \ \text{ as }\ \ n\rightarrow \infty . \end{aligned}$$
(27)

However, by (26) and hypothesis H(g), we have \(F(f_n)\ge M_g+h(f_n)\), with \(M_g\in {\mathbb {R}}\). Applying (27) and the coercivity of h in H(h), we obtain a contradiction to (25):

$$\begin{aligned} \lim _{n\rightarrow \infty }F(f_n)\ge M_g+\lim _{n\rightarrow \infty }h(f_n)=+\infty . \end{aligned}$$

Therefore, the sequence \(\{f_n\}\) must be bounded in \({\mathcal {W}}\).

Due to the reflexivity of \({\mathcal {W}}\), there exists \(f^*\in {\mathcal {W}}\) such that, along a subsequence,

$$\begin{aligned} f_n\rightarrow f^*\ \,\text{ weakly } \text{ in } {\mathcal {H}} \ \ \text{ as }\ \ n\rightarrow \infty , \end{aligned}$$
(28)

with \(f^*\in {\mathcal {B}}\), by the sequential weak closedness of \({\mathcal {B}}\). Since \(\{u_n\}\subset \Pi (\{f_n\})\), it follows from Lemma 13(i) that \(\{u_n\}\) is bounded in \({\mathcal {V}}\). Without any loss of generality, we may assume

$$\begin{aligned} u_n\rightarrow {\widetilde{u}} \ \ \text{ weakly } \text{ in }~ {\mathcal {V}}~ \text{ for } \text{ some }~ {\widetilde{u}}\in {\mathcal {X}}. \end{aligned}$$
(29)

Note that \({\mathcal {W}}\) is compactly embedded in \({\mathcal {V}}^*\); thus, by (28), \(f_n\rightarrow f^*\) in \({\mathcal {V}}^*\). Then, from (29) and Lemma 13(ii), we infer that \({\widetilde{u}}\in \Pi (f^*)\). Relying on the sequential weak lower semicontinuity of h and g, in conjunction with (25) and (26), we obtain

$$\begin{aligned} \theta =\inf _{f\in {\mathcal {B}}}F(f)\le F(f^*)&=\inf _{u\in \Pi (f^*)}g(u)+h(f^*)\nonumber \\&\le g({\widetilde{u}})+h(f^*)\nonumber \\&\le \liminf _{n\rightarrow \infty }g(u_n)+\liminf _{n\rightarrow \infty } h(f_n)\nonumber \\&\le \liminf _{n\rightarrow \infty }\big [g(u_n)+h(f_n)\big ]\nonumber \\&= \liminf _{n\rightarrow \infty }F(f_n)=\theta . \end{aligned}$$
(30)

This establishes that \(f\in {\mathcal {B}}\) is a solution to Problem 3.

It remains to show that the set of solutions to Problem 3 is sequentially weakly compact. Let \(\{f_n\}\) be a sequence of solutions to Problem 3. Hypotheses H(g) and H(h) determine the coercivity of the cost function F; thus, the sequence \(\{f_n\}\) is bounded in \({\mathcal {W}}\). Therefore, we can find \(f^*\in {\mathcal {W}}\) such that, for a subsequence, convergence (28) holds. There is a sequence \(u_n\in \Pi (f_n)\) that verifies (26). From Lemma 13(i), we may assume that (29) is valid with some \(\widetilde{u}\in \Pi (f^*)\). Carrying out the same reasoning as in (30), we conclude that \(f^*\) is a solution to Problem 3. The proof is thus complete. \(\square \)

5 Concluding Remarks

We studied a new class of evolutionary quasi–variational–hemivariational inequalities. The main contribution is a new existence result for the considered inequality problem and the solvability of an associated optimal control problem. It is of genuine interest to study the impact of data contamination on the optimal control problem by developing a regularization framework for the stable approximation. We plan to address this in future work.