Abstract Quasi-Variational Evolution Inclusions on Real Hilbert Spaces with Conservative Quantities

Abstract

In the present paper, we consider a conservative evolution inclusion on a real Hilbert space with quasi-variational structures for not only time-dependent subdifferentials of convex functions but also inner products. Actually, this paper has three main purposes. The first one is to make it clear how to construct a real Hilbert space which enables us to handle the evolution equation with a mass-conservative property for the consideration. The second one is to show the existence of strong solutions to the Cauchy problem of the evolution inclusion on the Hilbert space, which is constructed in the first main purpose to handle the mass-conservative evolution inclusion. The third one is to investigate the relation between two evolution inclusions.

1 Introduction

Throughout this paper, we consider real Hilbert spaces V with an inner product $(\cdot ,\cdot )_V$ and H with an inner product $(\cdot ,\cdot )_H$, in which the embedding $V \subset H$ is dense, continuous and compact. Besides, we consider a nonempty closed subset A of a real Banach space X with a norm $\Vert \cdot \Vert _X$. For any $\tilde{v} \in A$, we denote by $V(\tilde{v})=V$ a real Hilbert space with the inner product $(\cdot ,\cdot )_{V(\tilde{v})}$, and by $V^*(\tilde{v})$ a dual space of $V(\tilde{v})$ as well as by $V^*$ a dual space of V and $\langle \cdot ,\cdot \rangle _{V^*,\,V}$ a duality pair between $V^*$ and V. Then, for each $\tilde{v} \in A$ the dual space $V^*(\tilde{v})$ also becomes a real Hilbert space with an inner product $(\cdot ,\cdot )_{V^*(\tilde{v})}$ given by

$$\begin{aligned} \begin{array}{r} (\xi _1^*,\xi _2^*)_{V^*(\tilde{v})}:=\left\langle \xi _1^*,F(\tilde{v})^{-1}\xi _2^* \right\rangle _{V^*,\,V}, \quad \forall \xi _1^*,\,\xi _2^* \in V^*=V^*(\tilde{v}),\\ (\text{ cf. } \text{ See } \text{ Lemma } \text{1 }) \end{array} \end{aligned}$$

where $F(\tilde{v})^{-1}$ is the duality map from $V(\tilde{v})$ onto $V^*(\tilde{v})$.

In what follows, we fix an element $\eta _0 \in V$ with $\Vert \eta _0\Vert _H=1$ and consider the following Cauchy problem of evolution inclusion with quasi-variational structures, denoted by (M):={(1)–(5)}:

$$\begin{aligned}{} & {} u'(t)+u^*(t)-\langle u^*(t),\eta _0\rangle _{V^*,\,V} \eta _0+g(t,u(t),v(t))=0\nonumber \\{} & {} \quad \text{ in } \quad V^*(v(t)),\quad \text{ a.a. }~t \in (0,T), \end{aligned}$$

(1)

$$\begin{aligned}{} & {} u^*(t) \in \partial _{V^*(v(t))} \phi (t,u,v(t);u(t)),\quad \text{ a.a. }~t \in (0,T), \end{aligned}$$

(2)

$$\begin{aligned}{} & {} \langle g(t,u(t),v(t)),\eta _0 \rangle _{V^*,\,V}=0,\quad \text{ a.a. }~t \in (0,T), \end{aligned}$$

(3)

$$\begin{aligned}{} & {} v(t)=S(u;t,0)v_0 \quad \text{ in } \quad A, \quad \forall t \in [0,T], \end{aligned}$$

(4)

$$\begin{aligned}{} & {} u(0)=u_0 \quad \text{ in } \quad V^*, \end{aligned}$$

(5)

where for any $(t,\tilde{u},\tilde{v}) \in [0,T] \times C([0,T];V^*) \times A$ the operator $\partial _{V^*(\tilde{v})} \phi (t,\tilde{u},\tilde{v})$ is the subdifferential of $\phi (t,\tilde{u},\tilde{v})$ with respect to the inner product $(\cdot ,\cdot )_{V^*(\tilde{v})}$, that is,

$$\begin{aligned} \begin{array}{l} \hat{\xi }^* \in \partial _{V^*(\tilde{v})} \phi \bigl (t,\tilde{u},\tilde{v};\xi ^*\bigr )\\ ~\Longleftrightarrow ~\left\{ \begin{array}{l} \bigl (\hat{\xi }^*,\bar{\xi }^*-\xi ^*\bigr )_{V(\tilde{v})} \le \phi \bigl (t,\tilde{u},\tilde{v};\bar{\xi }^*\bigr )- \phi \bigl (t,\tilde{u},\tilde{v};\xi ^*\bigr ),\\ \forall \bar{\xi }^* \in D\left( \phi (t,\tilde{u},\tilde{v})\right) : =\bigl \{\tilde{\xi }^* \in V^*;\,\phi \bigl (t,\tilde{u},\tilde{v};\tilde{\xi }^*\bigr )< \infty \bigr \}. \end{array} \right. \end{array} \end{aligned}$$

From (1) and (3), we have

$$\begin{aligned} \langle u'(t),\eta _0 \rangle _{V^*,\,V}=0,\quad \text{ a.a. }~t \in (0,T), \end{aligned}$$

hence,

$$\begin{aligned} \langle u(t),\eta _0 \rangle _{V^*,\,V}=\langle u_0,\eta _0 \rangle _{V^*,\,V},\quad 0 \le \forall t \le T. \end{aligned}$$

(6)

Equality (6) implies that the system (M) has a conservative quantity in time. Hence, it is meaningful that we show that the system (M) has at least one strong solution under suitable assumptions by using the general theory of evolution inclusions with quasi-variational structures as far as possible, which is established in [6, 9].

Actually, as one of the typical examples, in [3,4,5, 8] the following initial-boundary value problem (T):={(7)–(13)} of the tumor invasion system with indirect chemotaxis effect is considered, which comes from the field of mathematical biology and was originally proposed in [1, 3]:

$$\begin{aligned}{} & {} u_t=\nabla \cdot (D(v,w)\nabla u^*)-\nabla \cdot (u \nabla v),\quad \text{ a.e. } \text{ in } \quad \varOmega \times (0,T), \end{aligned}$$

(7)

$$\begin{aligned}{} & {} u^* \in \partial _{\mathbb {R}} \hat{\beta } (v;u),\quad \text{ a.e. } \text{ in } \quad \varOmega \times (0,T),\end{aligned}$$

(8)

$$\begin{aligned}{} & {} v_t=d_v\varDelta v+awz,\quad \text{ a.e. } \text{ in } \quad \varOmega \times (0,T),\end{aligned}$$

(9)

$$\begin{aligned}{} & {} w_t=-awz, \quad \text{ a.e. } \text{ in } \quad \varOmega \times (0,T),\end{aligned}$$

(10)

$$\begin{aligned}{} & {} z_t=d_z\varDelta z-bz+cu, \quad \text{ a.e. } \text{ in } \quad \varOmega \times (0,T), \end{aligned}$$

(11)

$$\begin{aligned}{} & {} \{D(v,w)\nabla u^*-u\nabla v\} \cdot \nu =\nabla v \cdot \nu =\nabla z \cdot \nu =0, \quad \text{ a.e. } \text{ on } \quad \varGamma \times (0,T), \end{aligned}$$

(12)

$$\begin{aligned}{} & {} u(0)=u_0,~v(0)=v_0,~w(0)=w_0,~z(0)=z_0,\quad \text{ a.e. } \text{ in } \quad \varOmega , \end{aligned}$$

(13)

where $\varOmega$ is a bounded domain in $\mathbb {R}^N~(N=1,2,3)$ with a smooth boundary $\varGamma$; $\nu$ is an outer unit normal vector on $\varGamma$; D(v, w) is a strictly positive and bounded function from $\mathbb {R} \times \mathbb {R}$ into $\mathbb {R}$; $\hat{\beta }(v)$ is a proper l.s.c. convex function from $\mathbb {R}$ into $\mathbb {R} \cup \{\infty \}$ and $\partial _{\mathbb {R}} \hat{\beta }(v)$ is the subdifferential of $\hat{\beta }(v)$ on $\mathbb {R}$; $a,~b,~c,~d_v,$ and $d_z$ are positive constants; $u_0,~v_0,~w_0$ and $z_0$ are initial data. The problem (T) contains two quasi-variational structures. One comes from the function D(v, w), which depends on a pair of unknown functions (v, w). Actually, in [8] for each pair (v, w) a real Hilbert space $V(v,w)=H^1(\varOmega )$ whose inner product is given by

$$\begin{aligned} (\xi _1,\xi _2)_{V(v,w)}:= & {} \displaystyle {\int _\varOmega D(v,w) \nabla \xi _1 \cdot \nabla \xi _2 \hbox {d}x+\left( \int _\varOmega \xi _1 \hbox {d}x\right) \left( \int _\varOmega \xi _2 \hbox {d}x\right) }\\{} & {} \forall \xi _1,\xi _2 \in H^1(\varOmega ), \end{aligned}$$

is considered, and it is shown that Eq. (7) and the constraint condition (8) can be expressed by an evolution inclusion on $V^*(v,w)$ (cf. See (1) and (2)), which is the dual space of V(v, w). The other comes from proper, l.s.c., convex functions $\phi (v)$ on $V^*(v,w)$, which is defined by

$$\begin{aligned}{} & {} \phi (v;\xi ^*)\\{} & {} \quad :=\left\{ \begin{array}{ll} \displaystyle {\int _\varOmega \hat{\beta }(v(x);\xi ^*(x))\hbox {d}x}, &{}\quad \text{ if } \quad \xi ^* \in D(v,w):=\left\{ \xi ^* \in L^2(\varOmega );\, \hat{\beta } (v;\xi ^*) \in L^1(\varOmega )\right\} ,\\ \infty , &{}\quad \text{ if } \quad \xi ^* \in V^*(v,w) \setminus D(v,w). \end{array} \right. \end{aligned}$$

Moreover, from (7), (12) and (13) we get

$$\begin{aligned} \int _\varOmega u(t)\, \hbox {d}x=\int _\varOmega u_0\, \hbox {d}x,\quad 0 \le \forall t \le T, \end{aligned}$$

hence the total mass over $\varOmega$ of the unknown function u is conservative in time. Hence, the main purpose of the present paper is to construct a general framework of abstract evolution inclusions with conservative properties by extending the argumentations in [8, 9] in order to treat the problem (T) as one of the examples of (M).

At the end of this section, we give the assumptions for all prescribed data in the problem (M). First of all, in order to give the quasi-variational structure for inner products of $V^*$, we assume (A1), which was proposed in [2] for time-dependent structures and sometimes called the Damlamian Condition.

• (A1) A family $\{(\cdot ,\cdot )_{V(\tilde{v})}\,;\,\tilde{v} \in A\}$ of inner products on V is uniformly equivalent to $(\,\cdot ,\,\cdot \,)_V$, that is, there exist constants $C_1>0$ and $C_2>0$ such that

  $$\begin{aligned} C_1\Vert \xi \Vert _V \le \Vert \xi \Vert _{V(\tilde{v})} \le C_2 \Vert \xi \Vert _V,\quad \forall \tilde{v} \in A,~\forall \xi \in V, \end{aligned}$$

  where we put $\Vert \xi \Vert _V:=\sqrt{(\xi ,\xi )_V}$ and $\Vert \xi \Vert _{V(\tilde{v})}:=\sqrt{(\xi ,\xi )_{V(\tilde{v})}}$.

In (A2), we assume some properties of the operators S(u; t, s) in (4), which decide the dynamics of the unknown function v.

• (A2) A class $\{\{S(\tilde{u}\,;t,s)\,;\,0 \le s \le t \le \bar{T}\}\,;\,0 \le \bar{T} \le T,~\tilde{u} \in C([0,\bar{T}];V^*)\}$ of families of operators $S(\tilde{u}\,;t,s):A \longmapsto A~(0 \le s \le t \le \bar{T} \le T)$ satisfies the conditions (a)–(f):

  • (a) Assume that a sequence $\{(\tilde{u}_m,\tilde{v}_m)\}_{m \in \mathbb {N}} \subset C([0,\bar{T}];V^*) \times A$ and a pair $(\tilde{u},\tilde{v}) \in C([0,\bar{T}];V^*) \times A$ satisfy

    $$\begin{aligned} (\tilde{u}_m,\tilde{v}_m) \longrightarrow (\tilde{u},\tilde{v}) \quad \text{ in } \quad C([0,\bar{T}];V^*) \times X \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

    Then, for any $s \in [0,\bar{T}]$ we have

    $$\begin{aligned} {\varvec{S}}(\tilde{u}_m,\tilde{v}_m,s) \longrightarrow {\varvec{S}}(\tilde{u},\tilde{v},s) \quad \text{ in } \quad C([s,\bar{T}];X) \quad \text{ as } \quad m \rightarrow \infty , \end{aligned}$$

    where the operator ${\varvec{S}}:C([0,\bar{T}];V^*) \times A \times [0,\bar{T}] \longmapsto C([s,\bar{T}];V^*)$ is defined by

    $$\begin{aligned}{} & {} \forall (\bar{u},\bar{v},s) \in C([0,\bar{T}];V^*) \times A \times [0,\bar{T}],\\{} & {} ({\varvec{S}}(\bar{u},\bar{v},s))(t):=S(\bar{u};t,s)\bar{v} \quad \text{ in } \quad X,\quad s \le \forall t \le \bar{T}. \end{aligned}$$

  • (b) $S(\tilde{u};t,t)$ is the identity operator on A for all $t \in [0,\bar{T}]$.

  • (c) $S(\tilde{u}\,;t,s)=S(\tilde{u}\,;t,\tau ) \circ S(\tilde{u}\,;\tau ,s)$ on A for all $s,\,t,\,\tau \in [0,\bar{T}]$ with $s \le \tau \le t$.

  • (d) The following equality holds for all $\tau \in [0,\bar{T}]$:

    $$\begin{aligned} S(\sigma _\tau \tilde{u};t,s)=S(\tilde{u};t+\tau ,s+\tau ) \quad \text{ on } \quad A, \quad 0 \le \forall s \le \forall t \le \bar{T}-\tau , \end{aligned}$$

    where $\sigma _\tau \tilde{u}$ is a $\tau -$shift function of $\tilde{u}$ defined by

    $$\begin{aligned} (\sigma _\tau \tilde{u})(t):=\left\{ \begin{array}{ll} \tilde{u}(t+\tau ) \quad &{}\text{ if } \quad t \in [0,\bar{T}-\tau ],\\ \tilde{u}(\bar{T}) \quad &{}\text{ if } \quad t \in (\bar{T}-\tau ,\bar{T}]. \end{array} \right. \end{aligned}$$

  • (e) We have ${\varvec{S}}(\tilde{u},\tilde{v},0) \in W^{1,1}(0,\bar{T};X)$ for all $\tilde{v} \in A$, where the function ${\varvec{S}}(\tilde{u},\tilde{v},0)$ is the same one that is given in (a) above.

  • (f) For any $\tilde{u}_1,\tilde{u}_2 \in C([0,\bar{T}];V^*)$, we assume that there exists a time $\bar{T}_0 \in (0,\bar{T}]$ such that

    $$\begin{aligned} \tilde{u}_1(t)=\tilde{u}_2(t) \quad \text{ in } \quad V^*,\quad 0 \le \forall t \le \bar{T}_0. \end{aligned}$$

    Then, we have

    $$\begin{aligned} S(\tilde{u}_1;t,0)=S(\tilde{u}_2;t,0) \quad \text{ on } \quad A,\quad 0 \le \forall t \le \bar{T}_0. \end{aligned}$$

In order to give a quasi-variational structure of convex functions $\phi (t,\tilde{u},\tilde{v})$ in (1), we prepare a class $\mathscr {C}$ of families of proper l.s.c. convex functions on $V^*$, which is given by

$$\begin{aligned} \mathscr {C}:=\left\{ \left\{ \phi (t,\tilde{u},\tilde{v});\,0 \le t \le T\right\} ;\,\tilde{u} \in C([0,T];V^*),~\tilde{v} \in A\right\} , \end{aligned}$$

and denote by $D(\phi (t,\tilde{u},\tilde{v}))$ the effective domain of $\phi (t,\tilde{u},\tilde{v})$, that is,

$$\begin{aligned} D(\phi (t,\tilde{u},\tilde{v})):=\left\{ \xi ^* \in V^*;\,\phi (t,\tilde{u},\tilde{v};\xi ^*)< \infty \right\} . \end{aligned}$$

Then, we assume (A3).

• (A3) The following condition is satisfied:

  • (a) There exists a proper l.s.c. convex function $\phi$ on $V^*$ such that the following properties are satisfied:

    • (a1) The following inequality holds for all $(t,\tilde{u},\tilde{v}) \in [0,T] \times C([0,T];V^*) \times A$:

      $$\begin{aligned} \phi (\xi ^*) \le \phi (t,\tilde{u},\tilde{v};\xi ^*), \quad \forall \xi ^* \in V^*. \end{aligned}$$

    • (a2) For any $r \ge 0$ a level set $\left\{ \xi ^* \in V^*\,;\,\Vert \xi ^*\Vert _{V^*} \le r,~|\phi (\xi ^*)| \le r \right\}$ is relatively compact in $V^*$.

    • (a3) $D(\phi ) \subset H$, where $D(\phi ):=\{\xi ^* \in V^*\,;\,\phi (\xi ^*)<\infty \}$ is the effective domain of the function $\phi$. Moreover, there exists a constant $C_3>0$ such that

      $$\begin{aligned} |\phi (\xi ^*)| \le C_3,\quad \forall \xi ^* \in D(\phi ). \end{aligned}$$

    • (a4) $\phi$ is continuous on $D(\phi )$ with respect to the strong topology of H.

    From (a1) and (a3), we get

    $$\begin{aligned} D(\phi (t,\tilde{u},\tilde{v})) \subset H, \quad \forall (t,\tilde{u},\tilde{v}) \in [0,T] \times C([0,T];V^*) \times A. \end{aligned}$$

  • (b) There exists a constant $C_4>0$ such that

    $$\begin{aligned}{} & {} |\phi (t,\tilde{u},\tilde{v};\xi _1^*)-\phi (t,\tilde{u},\tilde{v};\xi _2^*)| \le C_4 \Vert \xi _1^*-\xi _2^*\Vert _H, \\{} & {} \forall (t,\tilde{u},\tilde{v}) \in [0,T] \times C([0,T];V^*) \times A, \quad \forall \xi _1^*,\, \xi _2^* \in D(\phi (t,\tilde{u},\tilde{v})). \end{aligned}$$

    Hence, the function $\phi (t,\tilde{u},\tilde{v})$ is continuous on $D(\phi (t,\tilde{u},\tilde{v}))$ with respect to the strong topology of H.

  • (c) For any $\tilde{u}_1,\,\tilde{u}_2 \in C([0,T];V^*)$, we assume that there exists $T_0 \in [0,T]$ such that

    $$\begin{aligned} \tilde{u}_1(t)=\tilde{u}_2(t)\quad \text{ in } \quad V^*,\quad 0 \le \forall t \le T_0. \end{aligned}$$

    Then, we have

    $$\begin{aligned} \phi (t,\tilde{u}_1,\tilde{v})=\phi (t,\tilde{u}_2,\tilde{v}) \quad \text{ on } \quad V^*, \quad \forall (t,\tilde{v}) \in [0,T] \times A. \end{aligned}$$

  • (d) Assume that a sequence $\{(\tilde{u}_m,\tilde{v}_m)\}_{m \in \mathbb {N}} \subset C([0,T];V^*) \times A$ and a pair $(\tilde{u},\tilde{v}) \in C([0,T];V^*) \times A$ satisfy

    $$\begin{aligned} (\tilde{u}_m,\tilde{v}_m) \longrightarrow (\tilde{u},\tilde{v}) \quad \text{ in } \quad C([0,T];V^*) \times X \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

    Then, for any $t \in [0,T]$ the following convergence holds as $m \rightarrow \infty$:

    $$\begin{aligned} \phi (t,\tilde{u}_m,\tilde{v}_m) \longrightarrow \phi (t,\tilde{u},\tilde{v}) \quad \text{ on } \quad V^*(\tilde{v}) \end{aligned}$$

    in the following sense, which is stronger than the Mosco convergence in [12]:

    • (i) For any $\xi ^* \in D(\phi (t,\tilde{u},\tilde{v}))$ there exists a sequence $\{\xi _m^*\}_{m \in \mathbb {N}} \subset H$ such that

      $$\begin{aligned}{} & {} \xi _m^* \longrightarrow \xi ^* \quad \text{ in } \quad H \quad \text{ as } \quad m \rightarrow \infty ,\\{} & {} \lim _{m \rightarrow \infty } \phi \big (t,\tilde{u}_m,\tilde{v}_m;\xi _m^* \big )=\phi \big (t,\tilde{u},\tilde{v};\xi ^* \big ). \end{aligned}$$

    • (ii) For any subsequence $\{(\tilde{u}_{m_k},\tilde{v}_{m_k})\}_{k \in \mathbb {N}}$ of $\{(\tilde{u}_m,\tilde{v}_m)\}_{m \in \mathbb {N}}$, we have

      $$\begin{aligned} \phi \big (t,\tilde{u},\tilde{v};\xi ^* \big ) \le \liminf _{k \rightarrow \infty } \phi \big (t,\tilde{u}_{m_k},\tilde{v}_{m_k};\xi _k^*\big ) \end{aligned}$$

      whenever a sequence $\{\xi _k^*\}_{k \in \mathbb {N}} \subset V^*$ and a function $\xi ^* \in V^*$ satisfy

      $$\begin{aligned} \xi _k^* \longrightarrow \xi ^* \quad \text{ weakly } \text{ in } \quad V^* (\tilde{v}) \quad \text{ as } \quad k \rightarrow \infty . \end{aligned}$$

• (A4) A single-valued perturbation $g:[0,T] \times D(\phi ) \times A \longmapsto V^*$ satisfies the following properties:

  • (a) $\langle g(t,z,\tilde{v}),\eta _0 \rangle _{V^*,\,V}=0$ for all $(t,z,\tilde{v}) \in [0,T] \times D(\phi ) \times A$.

  • (b) There exists a function $\ell :A \longmapsto \mathbb {R}$ and a constant $C_5>0$ such that for any $r \ge 0$ a level set $\{\tilde{v} \in A;\, \ell (\tilde{v}) \le r\}$ is compact in X and

    $$\begin{aligned} \Vert g (t,z,\tilde{v})\Vert _{V^*} \le \ell (\tilde{v}) \sqrt{|\phi (z)|+C_5},\quad \forall (t,z,\tilde{v}) \in [0,T] \times D(\phi ) \times A, \end{aligned}$$

    where $\phi$ is the same function that is given in (a) in (A3).

  • (c) Assume that a sequence $\{\tilde{u}_m\}_{m \in \mathbb {N}} \subset C([0,T];V^*)$ and a function $\tilde{u} \in C([0,T];V^*)$ satisfy

    $$\begin{aligned} \tilde{u}_m \longrightarrow \tilde{u} \quad \text{ in } \quad C([0,T];V^*) \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

    Then, for any $\tilde{v} \in A$ we have

    $$\begin{aligned} {\varvec{G}}(\tilde{u}_m,\tilde{v}) \longrightarrow {\varvec{G}}(\tilde{u},\tilde{v}) \quad \text{ weakly } \text{ in } \quad L^2(0,T;V^*) \quad \text{ as } \quad m \rightarrow \infty , \end{aligned}$$

    where the function ${\varvec{G}}(\tilde{u},\tilde{v}) \in L^2(0,T;V^*)$ is defined by

    $$\begin{aligned} ({\varvec{G}}(\tilde{u},\tilde{v}))(t):=g(t,\tilde{u}(t),S(\tilde{u};t,0)\tilde{v}) \quad \text{ in } \quad V^*,\quad \forall t \in [0,T]. \end{aligned}$$

• (A5) There exist a number $c_0 \in \mathbb {R}$ and $D(c_0) \subset H$ such that $c_0\eta _0 \in D(c_0)$ and

  $$\begin{aligned} D(c_0)= & {} D(\phi (t,\tilde{u},\tilde{v})) \cap \left\{ \xi ^* \in V^*;\,\langle \xi ^*,\eta _0 \rangle _{V^*,\,V}=c_0 \right\} ,\\{} & {} \forall (t,\tilde{u},\tilde{v}) \in [0,T] \times C([0,T];V^*) \times A, \end{aligned}$$

  which implies that the set $D(c_0)$ is independent of the choice of $(t,\tilde{u},\tilde{v})$. Using the set $D(c_0)$, we define a class of initial data $\mathscr {D}(c_0)$ by

  $$\begin{aligned} \mathscr {D}(c_0):=\left\{ (u,v) \in D(c_0) \times A;\, \begin{array}{l} u \in D(\phi (0,\tilde{u},v))~\text{ for } \text{ all }~\tilde{u} \in C([0,T];V^*)\\ \text{ with }~\tilde{u}(0)=u \end{array} \right\} . \end{aligned}$$

  We assume $(u_0,v_0) \in \mathscr {D}(c_0)$. Then, we see that the conservative quantity (6) of the problem (M) coincides with the number $c_0$, that is,

  $$\begin{aligned} \langle u(t),\eta _0 \rangle _{V^*,\,V}=\langle u_0,\eta _0\rangle _{V^*,\,V}=c_0,\quad 0 \le \forall t \le T. \end{aligned}$$

Next, we fix an initial data $(u_0,v_0) \in \mathscr {D}(c_0)$, and define three subsets $\mathscr {W}(u_0) \subset \mathscr {V}(u_0) \subset \mathscr {U}(u_0)$ by the following ways:

$$\begin{aligned} \mathscr {U}(u_0):= & {} \left\{ \tilde{u} \in C([0,T];V^*);\, \begin{array}{l} \tilde{u}(0)=u_0,\\ \langle \tilde{u}(t),\eta _0 \rangle _{V^*,\,V}=c_0,\quad 0 \le \forall t \le T,\\ \displaystyle {\sup _{0\,\le \,t\,\le \,T} \Vert \tilde{u}(t)\Vert _{V^*}+\int _0^T \phi (\tilde{u}(t))dt<\infty } \end{array}\right\} ,\\ \mathscr {V}(u_0):= & {} \left\{ \tilde{u} \in \mathscr {U}(u_0);\, \sup _{0\,\le \,t\,\le \,T} \Vert \tilde{u}(t)\Vert _{V^*}+\sup _{0\,\le \,t\,\le \,T} \phi (\tilde{u}(t))<\infty \right\} ,\\ \mathscr {W}(u_0):= & {} \left\{ \tilde{u} \in \mathscr {U}(u_0);\, \Vert \tilde{u}'\Vert _{L^2(0,T;V^*)}+\sup _{0\,\le \,t\,\le \,T} \Vert \tilde{u}(t)\Vert _{V^*}+\sup _{0\,\le \,t\,\le \,T} \phi (\tilde{u}(t))<\infty \right\} . \end{aligned}$$

Moreover, for any $R \ge 0$ we define sets $\mathscr {W}(u_0,R) \subset \mathscr {V}(u_0,R)$ and $\mathscr {U}(u_0,R)$ by the following ways:

$$\begin{aligned} \mathscr {U}(u_0,R):= & {} \left\{ \tilde{u} \in \mathscr {U}(u_0);\, \sup _{0\,\le \,t\,\le \,T} \Vert \tilde{u}(t)\Vert _{V^*}+\int _0^T \phi (\tilde{u}(t))dt \le R \right\} ,\\ \mathscr {V}(u_0,R):= & {} \left\{ \tilde{u} \in \mathscr {U}(u_0);\, \sup _{0\,\le \,t\,\le \,T} \Vert \tilde{u}(t)\Vert _{V^*}+\sup _{0\,\le \,t\,\le \,T} \phi (\tilde{u}(t)) \le R\right\} ,\\ \mathscr {W}(u_0,R):= & {} \left\{ \tilde{u} \in \mathscr {U}(u_0);\, \Vert \tilde{u}'\Vert _{L^2(0,T;V^*)}+\sup _{0\,\le \,t\,\le \,T} \Vert \tilde{u}(t)\Vert _{V^*}+\sup _{0\,\le \,t\,\le \,T} \phi (\tilde{u}(t)) \le R \right\} . \end{aligned}$$

Using the class $\mathscr {C}$ and the dynamical system $\{S(\tilde{u};t,s);\,0 \le s \le t \le T\}$, we define a class $\mathscr {X}(u_0,v_0)$, which gives the quasi-variational structure of convex functions, by

$$\begin{aligned} \mathscr {X}(u_0,v_0)=\left\{ \left\{ \varphi (t,\tilde{u},v_0);\,0 \le t \le T\right\} ;\,\tilde{u} \in \mathscr {U}(u_0) \right\} , \end{aligned}$$

where $\varphi (t,\tilde{u},v_0)$ is defined by

$$\begin{aligned} \varphi (t,\tilde{u},v_0):=\phi (t,\tilde{u},S(\tilde{u};t,0)v_0), \quad \forall (t,\tilde{u}) \in [0,T] \times \mathscr {U}(u_0). \end{aligned}$$

For the family $\left\{ \varphi (t,\tilde{u},v_0)\,;\,0 \le t \le T\right\} \in \mathscr {X}(u_0,v_0)$, we assume (A6), which was proposed in [10, 11] for time-dependent structures and sometimes called the Kenmochi Condition for $\{\varphi (t,\tilde{u},v_0);\,0 \le t \le T\}$ on $\mathscr {X}(u_0,v_0)$.

• (A6) For any $\{\varphi (t,\tilde{u},v_0)\,;\,0 \le t \le T\} \in \mathscr {X}(u_0,v_0)$, the following condition is satisfied: for any $r>0$ there exist nonnegative functions $\alpha _r(\tilde{u}) \in L^2(0,T)$ and $\beta _r(\tilde{u}) \in L^1(0,T)$ such that the following property is satisfied:

  $$\begin{aligned} \left( \begin{array}{l} \text{ for } \text{ any }\, s,\,t \in [0,T]\, \text{ and }\, \xi ^*(\tilde{u},s) \in D(\varphi (s,\tilde{u},v_0))\, \text{ with }\\ \Vert \xi ^*(\tilde{u},s)\Vert _{V^*(S(\tilde{u};\,t,0)v_0)} \le r~\text{ there } \text{ exists }\, \xi ^*(\tilde{u},s,t) \in D(\varphi (t,\tilde{u},v_0))\, \text{ such } \text{ that }\\ \begin{array}{cl} \text{(K1) }&{}\Vert \xi ^*(\tilde{u},s,t)-\xi ^*(\tilde{u},s)\Vert _H \le \left( \sqrt{|\varphi (s,\tilde{u},v_0;\xi ^*(\tilde{u},s))|}+1 \right) \displaystyle {\left| \int _s^t \alpha _r (\tilde{u};\tau )d\tau \right| },\\ \text{(K2) }&{}|\varphi (t,\tilde{u},v_0;\xi ^*(\tilde{u},s,t)) -\varphi (s,\tilde{u},v_0;\xi ^*(\tilde{u},s))| \\ &{}\quad \le \left( |\varphi (s,\tilde{u},v_0;\xi ^*(\tilde{u},s))|+1 \right) \displaystyle {\left| \int _s^t \beta _r (\tilde{u};\tau )d\tau \right| }. \end{array} \end{array} \right) \end{aligned}$$

Moreover, we assume that the following uniform estimates are satisfied:

• (A7) There exists a sufficiently substantial number $R_*>0$, which depends on $\Vert u_0\Vert _{V^*}$ and $\varphi (0,u_0,v_0)$, such that for any $R \ge R_*$ the following properties are satisfied:

  • (a) There exists a family $\{M(r,R);\,0<r<\infty \}$ such that

    $$\begin{aligned} \forall r>0,\quad \sup _{\tilde{u}\, \in \, \mathscr {U}(u_0,\,R)} \left( \Vert \alpha _r(\tilde{u})\Vert _{L^2(0,T)}+\Vert \beta _r(\tilde{u})\Vert _{L^1(0,T)}\right) \le M(r,R). \end{aligned}$$

  • (b) For any $r>0,~\varepsilon >0$ and $R \ge R_*$ there exists a constant $\delta _{r,\varepsilon ,R}>0$ such that

    $$\begin{aligned}{} & {} \sup _{ \tilde{u} \, \in \, \mathscr {U}(u_0,\,R)} \Biggl \{ \sup _{0\,\le \,t\,\le \,T} \int _t^{\min \{T,\,t+\delta _{r,\varepsilon ,R}\}} \left( |\alpha _r(\tilde{u};s)|^2+\beta _r (\tilde{u};s)\right. \\{} & {} \qquad \left. +\Vert ({\varvec{S}}(\tilde{u},v_0,0))'(s)\Vert _X \right) ds \Biggr \} \le \varepsilon . \end{aligned}$$

• (A8) There exist a family $\{h(\tilde{u}) \in W^{1,2}(0,T;V^*)\,;\,\tilde{u} \in \mathscr {U}(u_0)\}$ and a constant $C_6>0$ such that

  $$\begin{aligned} \left\langle h(\tilde{u};t),\eta _0\right\rangle _{V^*,\,V}=c_0,\quad 0 \le \forall t \le T, \end{aligned}$$

  where $c_0$ is the same number that is given in (A5), and

  $$\begin{aligned}{} & {} \sup _{\tilde{u}\,\in \,\mathscr {U}(u_0)} \left\{ \Vert h'(\tilde{u})\Vert _{L^2(0,T;V^*)}^2 +\sup _{0\,\le \,t\,\le \,T} \Vert h(\tilde{u};t)\Vert _{V^*}\right. \\{} & {} \qquad \left. +\sup _{0\,\le \,t\,\le \,T} |\varphi (t,\tilde{u},v_0;h(\tilde{u};t))| \right\} \le C_6. \end{aligned}$$

• (A9) There exists a constant $C_7>0$ such that

  $$\begin{aligned} \sup _{\tilde{u}\,\in \,\mathscr {U}(u_0)} \left( \sup _{0\,\le \,t\,\le \,T} \ell (({\varvec{S}}(\tilde{u},v_0,0))(t)) +\int _0^T \Vert ({\varvec{S}}(\tilde{u},v_0,0))'(t)\Vert _Xdt\right) \le C_7, \end{aligned}$$

  where $\ell$ is the same function that is given in (A4).

2 A Real Hilbert Space $V_0^*$

We devote this section to construct a real Hilbert space $V_0^*$, which enables us to treat an evolution inclusion {(1), (2)} on $V^*$ with the conservative property (6).

2.1 Structures of the Dual Spaces $V^*(\tilde{v})$

In this subsection, first of all we make the structure of $V^*$ clear. Since the embedding $V \subset H$ is continuous, we see that there exists a constant $C_6>0$ such that

$$\begin{aligned} \Vert \xi \Vert _H \le C_6 \Vert \xi \Vert _V,\quad \forall \xi \in V. \end{aligned}$$

(14)

Moreover, we see from [13, Proposition 21.35] that the embedding $H \subset V^*$ is also dense and compact. Now, we denote by $F:V \longmapsto V^*$ a duality map and by $\langle \cdot , \cdot \rangle _{V^*,V}$ a duality pair between $V^*$ and V. Then, $V^*$ becomes a real Hilbert space whose inner product is given by

$$\begin{aligned} (\xi _1^*,\xi _2^*)_{V^*}=\left\langle \xi _1^*,F^{-1}\xi _2^* \right\rangle _{V^*,\,V},\quad \forall \xi _1^*,~\xi _2^* \in V^*, \end{aligned}$$

and the Gelfand triplet of $V \subset H \subset V^*$ gives the following equality;

$$\begin{aligned} \langle z,\xi \rangle _{V^*,\,V}=(z,\xi )_H,\quad \forall z \in H,~\forall \xi \in V. \end{aligned}$$

(15)

Using (a3) in (A3) and (15), we can rewrite the conservative quantity (6) into

$$\begin{aligned} (u(t),\eta _0)_H=(u_0,\eta _0)_H,\quad 0 \le \forall t \le T, \end{aligned}$$

and replace the conservative condition for $\tilde{u} \in \mathscr {U}(u_0)$ by

$$\begin{aligned} \langle \tilde{u}(t),\eta _0 \rangle _{V^*,\,V}=(\tilde{u}(t),\eta _0)_H=c_0,\quad 0 \le \forall t \le T. \end{aligned}$$

Moreover, for each $\tilde{v} \in A$ we denote by $V(\tilde{v})$, $V^*(\tilde{v})$ and $\langle \cdot ,\cdot \rangle _{V^*(\tilde{v}),\,V(\tilde{v})}$ a real Hilbert space V with the inner product $(\cdot , \cdot )_{V(\tilde{v})}$, which is given in (A1), the dual space of $V(\tilde{v})$ and the duality pair between $V^*(\tilde{v})$ and $V(\tilde{v})$, respectively. Using the duality map $F(\tilde{v}):V(\tilde{v}) \longmapsto V^*(\tilde{v})$, the dual space $V^*(\tilde{v})$ also becomes a real Hilbert space whose inner product is given by

$$\begin{aligned} (\xi _1^*,\xi _2^*)_{V^*(\tilde{v})}:=\bigl \langle \xi _1^*,F(\tilde{v})^{-1}\xi _2^* \bigr \rangle _{V^*(\tilde{v}),\,V(\tilde{v})}, \quad \forall \xi _1^*,\, \xi _2^* \in V^*(\tilde{v}). \end{aligned}$$

At the beginning of this section, we show Lemma 1 which is originally given in [7, Section 3]. Although the result of Lemma 1 brings the quasi-variational structure on $V^*$, it is not directly used in order to analyze the problem (M).

Lemma 1

We have $V^*=V^*(\tilde{v})$ for all $\tilde{v} \in A$, and the following equality holds:

$$\begin{aligned} \left\langle \xi ^*,\xi \right\rangle _{V^*(\tilde{v}),\,V(\tilde{v})}=\left\langle \xi ^*,\xi \right\rangle _{V^*,\,V},\quad \forall \xi ^* \in V^*,~\forall \xi \in V. \end{aligned}$$

Moreover, we have the following inequality, which implies that the family of inner products $\{(\cdot ,\cdot )_{V^*(\tilde{v})};\,\tilde{v} \in A\}$ on $V^*$ is uniformly equivalent to $(\cdot ,\cdot )_{V^*}$:

$$\begin{aligned} C_1\Vert \xi ^*\Vert _{V^*(\tilde{v})} \le \Vert \xi ^*\Vert _{V^*} \le C_2 \Vert \xi ^*\Vert _{V^*(\tilde{v})}, \quad \forall \xi ^* \in V^*,~\forall \tilde{v} \in A, \end{aligned}$$

where $C_1$ and $C_2$ are the same constants that are given in (A1).

Proof

In the following argumentation, we fix an element $\tilde{v} \in A$, and for any $\xi ^* \in V^*$, we define a linear function $L\xi ^*:V(\tilde{v}) \longmapsto \mathbb {R}$ by

$$\begin{aligned} (L\xi ^*)(\xi ):=\langle \xi ^*,\xi \rangle _{V^*,\,V},\quad \forall \xi \in V(\tilde{v})=V. \end{aligned}$$

(16)

Then, we see that the function $L \xi ^*$ is independent of the choice of $\tilde{v} \in A$. From (A1), we get

$$\begin{aligned} |(L \xi ^*)(\xi )| \le \frac{1}{C_1} \Vert \xi ^*\Vert _{V^*} \Vert \xi \Vert _{V(\tilde{v})}, \quad z \in V(\tilde{v}), \end{aligned}$$

hence,

$$\begin{aligned} C_1\Vert L \xi ^*\Vert _{V^*(\tilde{v})} \le \Vert \xi ^*\Vert _{V^*}, \end{aligned}$$

(17)

which implies that $L \xi ^*$ is bounded on $V(\tilde{v})$. Hence, we get $L \xi ^* \in V^*(\tilde{v})$ and see that the linear operator $L:V^* \longmapsto V^*(\tilde{v})$ is well defined. Moreover, we see from (16) that L is injective on $V^*$ and the following equality holds:

$$\begin{aligned} \left\langle L \xi ^*,\xi \right\rangle _{V^*(\tilde{v}),\,V(\tilde{v})}=\langle \xi ^*,\xi \rangle _{V^*,\,V}, \quad \forall \xi \in V(\tilde{v}). \end{aligned}$$

(18)

Next, for any $\tilde{\xi }^* \in V^*(\tilde{v})$ we define a linear function $\bar{\xi }^*:V \longmapsto \mathbb {R}$ by

$$\begin{aligned} \bar{\xi }^*(\xi ):=\bigl \langle \tilde{\xi }^*,\xi \bigr \rangle _{V^*(\tilde{v}),\,V(\tilde{v})},\quad \forall \xi \in V=V(\tilde{v}). \end{aligned}$$

(19)

Using (A1) again, we get

$$\begin{aligned} \bigl |\bar{\xi }^*(\xi )\bigr | \le C_2 \bigl \Vert \tilde{\xi }^*\bigr \Vert _{V^*(\tilde{v})} \bigl \Vert \xi \bigr \Vert _V, \quad \tilde{\xi } \in V, \end{aligned}$$

hence,

$$\begin{aligned} \bigl \Vert \bar{\xi }^*\bigr \Vert _{V^*} \le C_2\bigl \Vert \tilde{\xi }^*\bigr \Vert _{V^*(\tilde{v})}, \end{aligned}$$

(20)

which implies that $\bar{\xi }^*$ is bounded on V. Hence, from (19) we get $\bar{\xi }^* \in V^*$ and

$$\begin{aligned} \bigl \langle L \bar{\xi }^*,\xi \bigr \rangle _{V^*(\tilde{v}),\,V(\tilde{v})}= \bigl \langle \bar{\xi }^*,\xi \bigr \rangle _{V^*,\,V}=\bigl \langle \tilde{\xi }^*,\xi \bigr \rangle _{V^*(\tilde{v}),\,V(\tilde{v})}, \quad \forall \xi \in V=V(\tilde{v}). \end{aligned}$$

Hence we see that for any $\tilde{\xi }^* \in V^*(\tilde{v})$ there exists an element $\bar{\xi }^* \in V^*$ such that $L \bar{\xi }^*=\tilde{\xi }^*$, which implies that the operator L is surjective.

As a result, since the operator L is bijective, we identify $L\xi ^*$ with $\xi ^*$ and see from (17), (18) and (20) that this lemma holds. $\square$

Using the element $\eta _0 \in V$ which appears in (1) and (3), we classify $V^*(\tilde{v})$ in Lemma 2. For any number $c \in \mathbb {R}$, we define a subset $V^*(c)$ of $V^*(\tilde{v})$ by

$$\begin{aligned} V^*(c):=\left\{ \xi ^*+c\eta _0 \in V^*(\tilde{v});\, \xi ^* \in V^*(\tilde{v})\quad \text{ and } \quad \bigl \langle \xi ^*,\eta _0 \bigr \rangle _{V^*,\,V}=0 \right\} . \end{aligned}$$

Owing to Lemma 1, we see that the subset $V^*(c)$ of $V^*(\tilde{v})$ is independent of the choice of $\tilde{v} \in A$, which is one of the key points in this paper.

Lemma 2

The family $\{V^*(c);\,c \in \mathbb {R}\}$ satisfies the following properties:

• (a) $V^*(c)$ is $*$-weakly closed and convex in $V^*(\tilde{v})$ for all $c \in \mathbb {R}$ and $\tilde{v} \in A$. Especially, $V^*(0)$ is a linear subspace of $V^*(\tilde{v})$ for all $\tilde{v} \in A$.

• (b) $V^*(c_1) \cap V^*(c_2)=\emptyset$ whenever $c_1,~c_2 \in \mathbb {R}$ satisfy $c_1 \ne c_2$, and

  $$\begin{aligned} V^*(\tilde{v})=\bigcup _{c \, \in \, \mathbb {R}} V^*(c). \end{aligned}$$

Proof

In the following argumentation, we fix an element $\tilde{v} \in A$. Since for any $c \in \mathbb {R}$ we easily see from the definition of $V^*(c)$ that $V^*(c)$ is $*$-weakly closed and convex in $V^*(\tilde{v})$, we omit the proof of (a) here and show (b) in the following.

We define an equivalence relation $\sim$ on $V^*(\tilde{v})$ by

$$\begin{aligned} \xi _1^* \sim \xi _2^* \qquad \text{ if } \text{ and } \text{ only } \text{ if } \qquad \langle \xi _1^*,\eta _0 \rangle _{V^*,\,V}=\langle \xi _2^*,\eta _0 \rangle _{V^*,\,V}. \end{aligned}$$

From (15), we get

$$\begin{aligned} \langle c\eta _0,\eta _0 \rangle _{V^*,\,V}=c (\eta _0,\eta _0)_H=c, \quad \forall c \in \mathbb {R}. \end{aligned}$$

(21)

Applying the general set theory, the dual space $V^*(\tilde{v})$ can be classified by

$$\begin{aligned} V^*(\tilde{v})=\bigcup _{c \, \in \, \mathbb {R}} [c\eta _0],\qquad [c_1\eta _0] \cap [c_2\eta _0]=\emptyset ,~ \quad \forall c_1,c_2 \in \mathbb {R}~\text{ with }~c_1\ne c_2, \end{aligned}$$

(22)

where $[c\eta _0]$ is an equivalence class containing the element $c\eta _0 \in V(\tilde{v})$ as a representative element.

Next, let $\xi ^*$ be any element in $[c\eta _0]$. Since we have $\langle \xi ^*,\eta _0 \rangle _{V^*,\,V}=c$, we see from (21) that the following equality holds:

$$\begin{aligned} \langle \xi ^*-c\eta _0,\eta _0 \rangle _{V^*,\,V}=0, \end{aligned}$$

which implies $\xi ^*=(\xi ^*-c\eta _0)+c\eta _0 \in V^*(c)$. Hence, we get $[c\eta _0] \subset V^*(c)$.

Conversely, let $\xi ^*$ be any element in $V^*(c)$. Then, we see that there exists an element $\tilde{\xi }^* \in V^*(\tilde{v})$ such that $\xi ^*=\tilde{\xi }^*+c\eta _0$ and $\langle \tilde{\xi },\eta \rangle _{V^*,\,V}=0$. Hence, from (21) again we get

$$\begin{aligned} \bigl \langle \xi ^*,\eta _0 \bigr \rangle _{V^*,\,V}=\bigl \langle \tilde{\xi }^*+c \eta _0,\eta _0 \bigr \rangle _{V^*,\,V}= \bigl \langle \tilde{\xi }^*,\eta _0 \bigr \rangle _{V^*,\,V}+c (\eta _0,\eta _0)_H=c, \end{aligned}$$

which implies $\xi ^* \in [c\eta _0]$. Hence, we get $V^*(c) \subset [c\eta _0]$.

From the results in the argumentation above, we get $[c\eta _0]=V^*(c)$ and (22). Hence, we see that (b) holds. $\square$

Next, we define a projection operator $P:V^* \longmapsto V^*$ by

$$\begin{aligned} P \xi ^*:=\xi ^*-\left\langle \xi ^*,\eta _0\right\rangle _{V^*,\,V}\eta _0,\quad \forall \xi ^* \in V^*. \end{aligned}$$

(23)

For any number $c \in \mathbb {R}$, we denote by $P_c$ the restriction of the projection P on $V^*(c)$. Then, the operators P and $P_c$ satisfy properties in Lemma 3.

Lemma 3

The following properties hold:

• (a) The projection P is linear and bounded on $V^*$, and $P(V^*)=V^*(0)$.

• (b) For any number $c \in \mathbb {R}$, the restriction $P_c:V^*(c) \longmapsto V^*(0)$ is bijective and continuous with respect to the strong topology of $V^*$. Especially, the operator $P_0$ is the identity on $V^*(0)$.

Proof

(a) From (23), we get $P \xi ^*=\xi ^*$ for all $\xi ^* \in V^*(0)$, hence, $V^*(0) \subset P(V^*)$. On the other hand, from the following equation:

$$\begin{aligned} \left\langle P \xi ^*,\eta _0\right\rangle _{V^*,\,V}=\left\langle \xi ^*,\eta _0\right\rangle _{V^*,\,V} -\left\langle \xi ^*,\eta _0\right\rangle _{V^*,\,V} (\eta _0,\eta _0)_H=0, \quad \forall \xi ^* \in V^*, \end{aligned}$$

we get $P(V^*) \subset V^*(0)$. Hence, we get the relation $P(V^*)=V^*(0)$.

Moreover, we see from (15) to (23) that P is linear and the following inequality holds for all $\xi ^* \in V^*$ and $\xi \in V$:

$$\begin{aligned} \bigl |\left\langle P \xi ^*,\xi \right\rangle _{V^*,\,V}\bigr |\le & {} \bigl | \left\langle \xi ^*,\xi \right\rangle _{V^*,\,V}\bigr | +\bigl | \left\langle \xi ^*,\eta _0\right\rangle _{V^*,\,V}\bigr | \bigl | \left\langle \eta _0,\xi \right\rangle _{V^*,\,V}\bigr |\nonumber \\\le & {} \left( 1+\Vert \eta _0\Vert _V \Vert \eta _0\Vert _{V^*}\right) \left\| \xi ^*\right\| _{V^*} \left\| \xi \right\| _V, \end{aligned}$$

(24)

which implies the operator P is bounded with an estimate

$$\begin{aligned} \Vert P\Vert _{\mathscr {L}(V^*,\,V^*)} \le 1+\Vert \eta _0\Vert _V \Vert \eta _0\Vert _{V^*}. \end{aligned}$$

(b) We fix any $c \in \mathbb {R}$. We see from (a) that the operator $P_c:V^*(c) \longmapsto V^*(0)$ is continuous with respect to the strong topology of $V^*$. Using (a) and $V^*(c) \subset V^*$, we get $P_c(V^*(c)) \subset V^*(0)$. On the other hand, for any $\xi ^* \in V^*(0)$ we put $\tilde{\xi }^*:=\xi ^*+c\eta _0 \in V^*$. Then, we get

$$\begin{aligned} \bigl \langle \tilde{\xi }^*,\eta _0 \bigr \rangle _{V^*,\,V}=\bigl \langle \xi ^*,\eta _0 \bigr \rangle _{V^*,\,V}+c\Vert \eta _0\Vert _H^2=c, \end{aligned}$$

hence,

$$\begin{aligned} \xi ^*=\tilde{\xi }^*-\bigl \langle \tilde{\xi }^*,\eta _0\bigr \rangle _{V^*\,V}\eta _0=P_c \tilde{\xi }^*, \end{aligned}$$

which implies $\tilde{\xi }^* \in V^*(c)$ and $\xi ^* \in P_c (V^*(c))$, that is, $V^*(0) \subset P_c (V^*(c))$. Hence, we get the relation $V^*(0)=P_c(V^*(c))$, which implies that the operator $P_c:V^*(c) \longmapsto V^*(0)$ is surjective.

In order to show that $P_c$ is injective, we assume that there exist elements $\xi _1^*,\,\xi _2^* \in V^*(c)$ such that

$$\begin{aligned} P_c \xi _1^*=P_c \xi _2^* \quad \text{ in } \quad V^*. \end{aligned}$$

Then, we have

$$\begin{aligned} \bigl \langle P_c \xi _1^*,z \bigr \rangle _{V^*,\,V}=\bigl \langle P_c \xi _2^*,z \bigr \rangle _{V^*,\,V}, \end{aligned}$$

hence,

$$\begin{aligned} \bigl \langle \xi _1^*-c\eta _0,z \bigr \rangle _{V^*,\,V}= \bigl \langle \xi _2^*-c\eta _0,z \bigr \rangle _{V^*,\,V}\quad \forall z \in V, \end{aligned}$$

which implies $\xi _1^*=\xi _2^*$ in $V^*$. Hence, the operator $P_c$ is injective on $V^*(c)$. $\square$

Corollary 1

For any $c \in \mathbb {R}$, the inverse $P_c^{-1}:V^*(0) \longmapsto V^*(c)$ is contraction with respect to the strong topology of $V^*$.

Proof

We see from (23) that the inverse $P_c^{-1}:V^*(0) \longmapsto V^*(c)$ is given by

$$\begin{aligned} P_c^{-1} \xi ^*=\xi ^*+c\eta _0,\quad \forall \xi ^* \in V^*(0). \end{aligned}$$

(25)

Then, we get

$$\begin{aligned} \left\| P_c^{-1}\xi ^*\right\| _{V^*} \le \left\| \xi ^*\right\| _{V^*}+c\Vert \eta _0\Vert _{V^*}, \quad \forall \xi ^* \in V^*(0). \end{aligned}$$

(26)

For any $\xi _1^*,\, \xi _2^* \in V^*(0)$, we get

$$\begin{aligned} \bigl | \bigl \langle P_c^{-1} \xi _1^*-P_c^{-1} \xi _2^*,\xi \bigr \rangle _{V^*,\,V}\bigr | =\bigl |\bigl \langle \xi _1^*-\xi _2^*,\xi \bigr \rangle _{V^*,\,V}\bigr | \le \bigl \Vert \xi _1^*-\xi _2^*\bigr \Vert _{V^*}\Vert \xi \Vert _V,\quad \forall \xi \in V, \end{aligned}$$

hence,

$$\begin{aligned} \bigl \Vert P_c^{-1} \xi _1^*-P_c^{-1} \xi _2^*\bigr \Vert _{V^*} \le \bigl \Vert \xi _1^*-\xi _2^*\bigr \Vert _{V^*}, \end{aligned}$$

(27)

which implies that $P_c^{-1}$ is contraction with respect to the strong topology of $V^*$. $\square$

2.2 A Structure of $V_0^*$ and a Relation with $V^*$

We define a nonempty closed subspace $H_0$ of H by

$$\begin{aligned} H_0:=\{\lambda \eta _0 \in H\,;\lambda \in \mathbb {R}\}^\bot =\{ z \in H\,;\,(z,\eta _0)_H=0\}. \end{aligned}$$

(28)

Then, $H_0$ becomes a real Hilbert space with an inner product

$$\begin{aligned} (z_1,z_2)_{H_0}:=(z_1,z_2)_H,\quad \forall z_1, z_2 \in H_0. \end{aligned}$$

Moreover, we define $V_0:=V \cap H_0$, which also becomes a real Hilbert space with an inner product

$$\begin{aligned} (\eta _1,\eta _2)_{V_0}:=(\eta _1,\eta _2)_V,\quad \forall \eta _1, \eta _2 \in V_0. \end{aligned}$$

(29)

Lemma 4

The imbedding $V_0 \subset H_0$ is dense and compact.

Proof

We show that $V_0$ is dense in $H_0$. Since V is dense in H and $H_0 \subset H$, we see that for any element $z \in H_0$ there exists a sequence $\{\xi _m\}_{m \in \mathbb {N}} \subset V$ such that

$$\begin{aligned} \xi _m \longrightarrow z \quad \text{ in } \quad H \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

(30)

We consider a sequence $\{z_m\}_{m \in \mathbb {N}} \subset V_0$ which is defined by

$$\begin{aligned} \forall m \in \mathbb {N},\quad z_m:=P\xi _m=\xi _m-(\xi _m,\eta _0)_H \eta _0. \end{aligned}$$

Using the equality $(z,\eta _0)_H=0$, we get

$$\begin{aligned} \Vert z_m-z\Vert _{H_0}= & {} \Vert (\xi _m-z)-(\xi _m-z,\eta _0)_H \eta _0\Vert _H\nonumber \\\le & {} \Vert \xi _m-z\Vert _H+|(\xi _m-z,\eta _0)_H| \Vert \eta _0\Vert _H \le 2\Vert \xi _m-z\Vert _H. \end{aligned}$$

(31)

From (30) and (31), we get

$$\begin{aligned} z_m \longrightarrow z \quad \text{ in } \quad H_0 \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

Finally, we show that the imbedding $V_0 \subset H_0$ is compact. Since the imbedding $V \subset H$ is compact and $V_0 \subset V$, we see that for any sequence $\{\xi _m\}_{m \in \mathbb {N}} \subset V_0$ there exists a subsequence $\{\xi _{m_k}\}_{k \in \mathbb {N}} \subset \{\xi _m\}_{m \in \mathbb {N}}$ and an element $z \in H$ such that

$$\begin{aligned} \xi _{m_k} \longrightarrow z \quad \text{ in } \quad H \quad \text{ as } \quad k \rightarrow \infty . \end{aligned}$$

(32)

Moreover, we have

$$\begin{aligned} (\xi _{m_k},\eta _0)_H=0,\quad \forall k \in \mathbb {N}. \end{aligned}$$

(33)

From (32) and (33), we get

$$\begin{aligned} \lim _{k \rightarrow \infty } (\xi _{m_k},\eta _0)_H=(z,\eta _0)_H=0, \end{aligned}$$

which implies $z \in H_0$. Hence, this lemma is proved. $\square$

We denote by $V_0^*$ the dual space of $V_0$ and by $\langle \cdot ,\cdot \rangle _{V_0^*,V_0}$ a duality pair between $V_0^*$ and $V_0$. Using the duality map $F_0:V_0 \longmapsto V_0^*$, the dual space $V_0^*$ also becomes a real Hilbert space with the inner product $(\cdot ,\cdot )_{V_0^*}$ defined by

$$\begin{aligned} \left( \eta _1^*,\eta _2^*\right) _{V_0^*}:=\left\langle \eta _1^*,F_0^{-1} \eta _2^* \right\rangle _{V_0^*,\,V_0}, \quad \forall \eta _1^*, \eta _2^* \in V_0^*. \end{aligned}$$

For any $\xi ^* \in V^*(0)$, we define a function $J \xi ^*:V_0 \longmapsto \mathbb {R}$ by

$$\begin{aligned} \left( J \xi ^*\right) (\eta ):=\left\langle \xi ^*,\eta \right\rangle _{V^*,V},\quad \forall \eta \in V_0. \end{aligned}$$

(34)

In order to make the relation between $V_0^*$ and $V^*(0)$ clear, we show Lemma 5.

Lemma 5

For any $\xi ^* \in V^*(0)$, we have $J \xi ^* \in V_0^*$, especially, $Jz=z$ for all $z \in H_0$. Moreover, the operator $J:V^*(0) \longmapsto V_0^*$ is linear, bijective, bounded on $V^*(0)$, and there exists a constant $C_9>0$ such that

$$\begin{aligned} \left\| J \xi ^*\right\| _{V_0^*} \le \left\| \xi ^*\right\| _{V^*} \le C_9 \left\| J \xi ^*\right\| _{V_0^*}, \quad \forall \xi ^* \in V^*(0). \end{aligned}$$

(35)

Hence, we have $J(V^*(0))=V_0^*$.

Proof

It is clear that $J \xi ^*$ is linear on $V_0$. Because of $\Vert \eta \Vert _{V_0}=\Vert \eta \Vert _V$ for all $\eta \in V_0$ (cf. (29)), we get

$$\begin{aligned} \bigl |\bigl (J \xi ^*\bigr )(\eta )\bigr | \le \Vert \xi ^*\Vert _{V^*} \Vert \eta \Vert _{V_0},\quad \forall \eta \in V_0, \end{aligned}$$

which implies that $J \xi ^*$ is bounded on $V_0$, that is, $J \xi ^* \in V_0^*$ and the following inequality holds:

$$\begin{aligned} \Vert J \xi ^*\Vert _{V_0^*} \le \Vert \xi ^*\Vert _{V^*},\quad \forall \xi ^* \in V^*(0). \end{aligned}$$

(36)

Moreover, from (15) to (34) we have the following equalities:

$$\begin{aligned}{} & {} \bigl \langle J \xi ^*,\eta \bigr \rangle _{V_0^*,\,V_0}=\bigl \langle \xi ^*,\eta \bigr \rangle _{V^*,\,V},\quad \forall \eta \in V_0, \quad (\text{ cf }.~(34)) \end{aligned}$$

(37)

$$\begin{aligned}{} & {} \begin{array}{ll} (z,\eta )_{H_0}=(z,\eta )_H=\langle z,\eta \rangle _{V^*,\,V}=\langle J z,\eta \rangle _{V_0^*,\,V_0}&\quad \forall z \in H_0, \quad \forall \eta \in V_0.\quad (\text{ cf }.~(15)) \end{array}\nonumber \\ \end{aligned}$$

(38)

which gives the Gelfand triplet $V_0 \subset H_0 \subset V_0^*$. From Lemma 4 and (38), we can identify Jz with z and get the equality:

$$\begin{aligned} Jz=z\quad \text{ in } \quad H_0,\quad \forall z \in H_0. \end{aligned}$$

(39)

We see from (23) and (37) that the following equality holds:

$$\begin{aligned} \bigl |\bigl \langle \xi ^*,\xi \bigr \rangle _{V^*,\,V}\bigr |= & {} \bigl |\bigr \langle \xi ^*,P \xi \bigr \rangle _{V^*,\,V} +(\xi ,\eta _0)_H \bigl \langle \xi ^*,\eta _0 \bigr \rangle _{V^*,\,V}\bigr |\nonumber \\= & {} \bigl |\langle J \xi ^*,P \xi \bigr \rangle _{V_0^*,\,V_0}\bigr | \le \bigl \Vert J \xi ^*\bigr \Vert _{V_0^*} \Vert P \xi \Vert _{V_0}, \quad \forall \xi ^* \in V^*(0),~\xi \in V.\qquad \quad \end{aligned}$$

(40)

Using (14) and (23), we get the following inequality:

$$\begin{aligned} \Vert P \xi \Vert _V \le \Vert \xi \Vert _V+\Vert \xi \Vert _H \Vert \eta _0\Vert _H \Vert \eta _0\Vert _V \le (1+C_6\Vert \eta _0\Vert _V)\Vert \xi \Vert _V,\quad \forall \xi \in V. \end{aligned}$$

(41)

We see from (40) and (41) that the following inequality holds:

$$\begin{aligned} \Vert \xi ^*\Vert _{V^*} \le (1+C_6 \Vert \eta _0\Vert _V) \Vert J \xi ^*\Vert _{V_0^*},\quad \forall \xi ^* \in V^*(0). \end{aligned}$$

(42)

Hence, we see from (36) and (42) that (35) holds, which implies that the operator J is injective on $V^*(0)$.

Next, for any $\eta ^* \in V_0^*$ we consider a function $\tilde{\eta }^*:V \longmapsto \mathbb {R}$, which is defined by

$$\begin{aligned} \tilde{\eta }^* (\xi ):=\langle \eta ^*,P\xi \rangle _{V_0^*\,,V_0},\quad \forall \xi \in V. \end{aligned}$$

(43)

Using Lemma 3, we get the following two equalities and one inequality:

$$\begin{aligned} \tilde{\eta }^*(\alpha \xi _1+\beta \xi _2)= & {} \alpha \langle \eta ^*,P\xi _1\rangle _{V_0^*,V_0}+\beta \langle \eta ^*,P\xi _2 \rangle _{V_0^*,V_0} =\alpha \tilde{\eta }^*(\xi _1)+\beta \tilde{\eta }^*(\xi _2),\\{} & {} \forall \alpha ,\beta \in \mathbb {R},~\forall \xi _1,\xi _2 \in V,\\ |\langle \tilde{\eta }^*,\xi \rangle _{V^*,\,V}|\le & {} \Vert \eta ^*\Vert _{V_0^*} \Vert P\xi \Vert _V \le \left( 1+C_6\Vert \eta _0\Vert _V\right) \Vert \eta ^*\Vert _{V_0^*} \Vert \xi \Vert _V, \quad \forall \xi \in V,\\ \tilde{\eta }^*(\eta _0)= & {} \langle \tilde{\eta }^*,P\eta _0\rangle _{V_0^*,\,V_0}=\langle \tilde{\eta }^*,0\rangle _{V_0^*,\,V_0}=0, \end{aligned}$$

which imply $\tilde{\eta }^* \in V^*(0)$. Moreover, from (34) and (43) we get

$$\begin{aligned} \langle J \tilde{\eta }^*,\eta \rangle _{V_0^*,\,V_0}=\langle \tilde{\eta }^*,\eta \rangle _{V^*,\,V} =\langle \eta ^*,P\eta \rangle _{V_0^*,\,V_0}=\langle \eta ^*,\eta \rangle _{V_0^*,\,V_0},\quad \forall \eta \in V_0, \end{aligned}$$

which implies $J\tilde{\eta }^*=\eta$ in $V_0^*$. Hence, we see that the operator J is surjective. $\square$

As a result of Lemma 5, we get Lemma 6.

Lemma 6

We have $H_0=J(H \cap V^*(0))$, hence, $V_0=J(V \cap V^*(0))$.

Proof

At first, we show $H_0=J(H \cap V^*(0))$. Let z be any element in $H_0 \subset H$. Using the Gelfand triplet (15), we get

$$\begin{aligned} (z,\eta _0)_H=\langle z,\eta _0 \rangle _{V^*,\,V}=0, \end{aligned}$$

which implies $z \in H \cap V^*(0)$. From (39) we get $z=Jz \in J(H \cap V^*(0))$.

Conversely, let $\tilde{z}$ be any element in $J(H \cap V^*(0)) \subset V_0^*$. Then, we see that there exists an element $z \in H \cap V^*(0)$ such that the following equality holds for all $\eta \in V_0$:

$$\begin{aligned} \langle \tilde{z},\eta \rangle _{V_0^*,\,V_0}=\langle Jz,\eta \rangle _{V_0^*,\,V_0}= \langle z,\eta \rangle _{V^*,\,V}=(z,\eta )_H=(z,\eta )_{H_0}. \end{aligned}$$

From Lemma 5, we get $\tilde{z}=Jz=z \in H_0$.

Repeating the argumentation similar to the derivation of $H_0=J(H \cap V^*(0))$ above, we get $V_0=J(V \cap V^*(0))$. $\square$

Now we give Proposition 1, which gives an expression of an element $\xi ^* \in V^*$ and is originally obtained in [8, Lemma 1.1] and [9, Proposition 6.1].

Proposition 1

For any $\xi ^* \in V^*$, we have the following equality:

$$\begin{aligned} \left\langle \xi ^*,\xi \right\rangle _{V^*,\,V}=\left\langle (J \circ P)\xi ^*,P \xi \right\rangle _{V_0^*,\,V_0}+ \left\langle \xi ^*,\eta _0 \right\rangle _{V^*,\,V}\left( \xi ,\eta _0\right) _H,\quad \forall \xi \in V. \end{aligned}$$

Proof

From (23) to (37) in the proof of Lemma 5, we get

$$\begin{aligned} \left\langle \xi ^*,\xi \right\rangle _{V^*,\,V}= & {} \left\langle \xi ^*,P \xi \right\rangle _{V^*,\,V}+\left( \xi ,\eta _0\right) _H \left\langle \xi ^*,\eta _0 \right\rangle _{V^*,\,V}\nonumber \\= & {} \left\langle P\xi ^*,P \xi \right\rangle _{V^*,\,V}+\left\langle \xi ^*,\eta _0\right\rangle _{V^*,\,V} \left\langle \eta _0,P \xi \right\rangle _{V^*,\,V}+\left( \xi ,\eta _0\right) _H \left\langle \xi ^*,\eta _0 \right\rangle _{V^*,\,V}\nonumber \\= & {} \left\langle (J \circ P)\xi ^*,P \xi \right\rangle _{V_0^*,\,V_0}+\left\langle \xi ^*,\eta _0\right\rangle _{V^*,\,V} \left\langle \eta _0,P \xi \right\rangle _{V^*,\,V}+\left( \xi ,\eta _0\right) _H \left\langle \xi ^*,\eta _0\right\rangle _{V^*,\,V}.\nonumber \\ \end{aligned}$$

(44)

Since we see from the Gelfand triplet (15) that the following equality holds:

$$\begin{aligned} \left\langle \eta _0,P \xi \right\rangle _{V^*,\,V}= & {} \left\langle \eta _0,\xi \right\rangle _{V^*,\,V}- \left\langle \xi ,\eta _0 \right\rangle _{V^*,\,V} \left\langle \eta _0,\eta _0 \right\rangle _{V^*,\,V}\\= & {} (\eta _0,\xi )_H-(\xi ,\eta _0)_H\Vert \eta _0\Vert _H^2=0,\quad \forall \xi \in V, \end{aligned}$$

we see from (44) that this lemma holds. $\square$

3 Quasi-Variational Structures on $V_0^*$

3.1 Quasi-Variational Inner Products on $V_0^*$

The spaces $V^*,~V(\tilde{v}),~V^*(\tilde{v}),~H_0$ and $V_0$ are the same ones that are constructed in Section 2. For any $\tilde{v} \in A$, we denote by $V_0(\tilde{v})$ a real Hilbert space with an inner product

$$\begin{aligned} (\eta _1,\eta _2)_{V_0(\tilde{v})}:=(\eta _1,\eta _2)_{V(\tilde{v})},\quad \forall \eta _1,\,\eta _2 \in V_0(\tilde{v})=V_0. \end{aligned}$$

We see from (A1) that the family $\{(\cdot ,\cdot )_{V_0(\tilde{v})};\,\tilde{v} \in A\}$ of inner products on $V_0$ is uniformly equivalent to $(\cdot ,\cdot )_{V_0}$, that is, the following inequalities hold:

$$\begin{aligned} C_1 \Vert \eta \Vert _{V_0} \le \Vert \eta \Vert _{V_0(\tilde{v})} \le C_2 \Vert \eta \Vert _{V_0},\quad \forall \eta \in V_0. \end{aligned}$$

Since for any $\tilde{v} \in A$ the embedding $V(\tilde{v}) \subset H$ is dense and compact, we see that for any $\tilde{v} \in A$ the embedding $V_0(\tilde{v}) \subset H_0$ is also dense and compact by repeating the argumentation similar to Lemma 4. In the following argumentation, for any $\tilde{v} \in A$ we denote the dual space of $V_0(\tilde{v})$, the duality pair between $V_0^*(\tilde{v})$ and $V_0(\tilde{v})$ and the duality map by $V_0^*(\tilde{v})$, $\langle \,\cdot \,,\,\cdot \,\rangle _{V_0^*(\tilde{v}),\,V_0(\tilde{v})}$ and $F_0(\tilde{v}):V_0(\tilde{v}) \longmapsto V_0^*(\tilde{v})$, respectively. Then, we see that for any $\tilde{v} \in A$ the dual space $V_0^*(\tilde{v})$ becomes a real Hilbert space with an inner product $(\,\cdot ,\,\cdot \,)_{V_0^*(\tilde{v})}$, which is given by

$$\begin{aligned} \big (\eta _1^*,\eta _2^* \big )_{V_0^*(\tilde{v})}:=\bigl \langle \eta _1^*,F_0(\tilde{v})^{-1}\eta _2^* \bigr \rangle _{V_0^*(\tilde{v}),\,V_0(\tilde{v})}, \quad \forall \eta _1^*,\,\eta _2^* \in V_0^*(\tilde{v}). \end{aligned}$$

(45)

From (37), (38) and (39), we have the following Gelfand triplet for $V_0 \subset H_0 \subset V_0^*$:

$$\begin{aligned} \begin{array}{l} \left\langle Jz,\eta \right\rangle _{V_0^*,\,V_0}=\langle z,\eta \rangle _{V^*,\,V}=(z,\eta )_H=(z,\eta )_{H_0}=(Jz,\eta )_{H_0},\\ \quad \forall z \in H \cap V^*(0),~\forall \eta \in V_0, \end{array} \end{aligned}$$

which implies from Lemma 6

$$\begin{aligned} \langle z,\eta \rangle _{V_0^*,\,V_0}=(z,\eta )_{H_0},\quad \forall z \in H_0,~\forall \eta \in V_0. \end{aligned}$$

First of all, we give Lemma 7, which implies the uniform equivalence of a family $\{(\cdot ,\cdot )_{V_0^*(\tilde{v})};\,\tilde{v} \in A\}$ of inner products on $V_0^*$ defined by (45). Actually, we easily see that Lemma 7 holds by repeating the same argumentation that is given in the proof of Lemma 1; hence, we omit its proof here.

Lemma 7

For any $\tilde{v} \in A$, we have $V_0^*(\tilde{v})=V_0^*$ and the following equality:

$$\begin{aligned} \langle \eta ^*,\eta \rangle _{V_0^*(\tilde{v}),\,V_0(\tilde{v})}=\langle \eta ^*,\eta \rangle _{V_0^*,\,V_0},\quad \forall \eta ^* \in V_0^*,~\forall \eta \in V_0. \end{aligned}$$

Moreover, we have

$$\begin{aligned} C_1 \Vert \eta ^*\Vert _{V_0^*(\tilde{v})} \le \Vert \eta ^*\Vert _{V_0^*} \le C_2 \Vert \eta ^*\Vert _{V_0^*(\tilde{v})}, \quad \forall \tilde{v} \in A,~\forall \eta ^* \in V_0^*. \end{aligned}$$

where $C_1>0$ and $C_2>0$ are the same constants that are given in (A1).

Using Proposition 1 with Lemmas 1 and 7, we have Proposition 2.

Proposition 2

The following equality holds for all $\tilde{v} \in A$:

$$\begin{aligned} F_0(\tilde{v})=J \circ P \circ F(\tilde{v}) \quad \text{ on } \quad V_0. \end{aligned}$$

Proof

Using the definition of the duality maps of $F(\tilde{v})$ and $F_0(\tilde{v})$, we see from Proposition 1 and Lemmas 1 and 7 that the following equality holds for all $\eta \in V_0$:

$$\begin{aligned} \left\langle F_0(\tilde{v}) \eta ,\eta \right\rangle _{V_0^*,\,V_0}= & {} \Vert \eta \Vert _{V_0(\tilde{v})}^2=\Vert \eta \Vert _{V(\tilde{v})}^2 =\left\langle F(\tilde{v})\eta ,\eta \right\rangle _{V^*,\,V}\\= & {} \left\langle (J \circ P \circ F(\tilde{v})) \eta ,\eta \right\rangle _{V_0^*,~V_0}, \end{aligned}$$

which implies that this proposition holds. $\square$

Under the above settings, we assume that (A10) is satisfied, which gives the uniform Lipschitz continuous dependence of $F(\tilde{v}) \in \mathscr {L}(V,\,V^*)$ on $\tilde{v} \in A$.

• (A10) There exists a constant $C_{10}>0$ such that

  $$\begin{aligned} \Vert F(\tilde{v}_1)-F(\tilde{v}_2)\Vert _{\mathscr {L}(V,\,V^*)} \le C_{10} \Vert \tilde{v}_1-\tilde{v}_2\Vert _X, \quad \forall \tilde{v}_1,\,\tilde{v}_2 \in A. \end{aligned}$$

Then, we see that Lemma 8 holds.

Lemma 8

(cf. [7, Lemma 3.2]) There exists a constant $C_{11}>0$, which depends on $C_i~(i=1,2,10)$, such that the following inequality holds:

$$\begin{aligned} \left| \Vert \eta ^*\Vert _{V_0^*(\tilde{v}_1)}^2-\Vert \eta ^*\Vert _{V_0^*(\tilde{v}_2)}^2 \right|\le & {} C_{11} \Vert \tilde{v}_1-\tilde{v}_2\Vert _X \Vert \eta ^*\Vert _{V_0^*(\tilde{v}_1)}^2,\\{} & {} \forall \tilde{v}_1,\,\tilde{v}_2 \in A,~\forall \eta ^* \in V_0^*. \end{aligned}$$

Proof

We see from Lemma 7 that the following inequality holds:

$$\begin{aligned} \left| \left\langle \tilde{\eta }^*,F_0(\tilde{v})^{-1}\eta ^*\right\rangle _{V_0^*,\,V_0}\right| =\bigl |(\eta ^*,\tilde{\eta }^*)_{V_0^*(\tilde{v})}\bigr | \le \frac{1}{C_1^2} \cdot \Vert \eta ^*\Vert _{V_0^*} \Vert \tilde{\eta }^*\Vert _{V_0^*}, \end{aligned}$$

hence,

$$\begin{aligned} \bigl \Vert F_0(\tilde{v})^{-1}\bigr \Vert _{\mathscr {L}(V_0^*,\,V_0)} \le \frac{1}{C_1^2},\quad \forall \tilde{v} \in A. \end{aligned}$$

(46)

Using (24), Lemma 5 and Proposition 2, we get

$$\begin{aligned} \Vert (F_0(\tilde{v}_1)-F_0(\tilde{v}_2))\eta \Vert _{V_0^*}= & {} \Vert ((J \circ P \circ F(\tilde{v}_1))-(J \circ P \circ F(\tilde{v}_2)))\eta \Vert _{V_0^*}\\\le & {} \Vert ((P \circ F(\tilde{v}_1))-(P \circ F(\tilde{v}_2))) \eta \Vert _{V^*}\\\le & {} \left( 1+\Vert \eta _0\Vert _{V^*} \Vert \eta _0\Vert _V \right) \Vert (F(\tilde{v}_1)-F(\tilde{v}_2))\eta \Vert _{V^*}\\\le & {} C_{10} \left( 1+\Vert \eta _0\Vert _{V^*} \Vert \eta _0\Vert _V \right) \Vert \tilde{v}_1-\tilde{v}_2\Vert _X \Vert \eta \Vert _{V_0}, \end{aligned}$$

which implies

$$\begin{aligned} \Vert F_0(\tilde{v}_1)-F_0(\tilde{v}_2)\Vert _{\mathscr {L}(V_0,\,V_0^*)} \le C_{10} \left( 1+\Vert \eta _0\Vert _{V^*} \Vert \eta _0\Vert _V \right) \Vert \tilde{v}_1-\tilde{v}_2\Vert _X. \end{aligned}$$

(47)

Hence, from (46), (47) and Lemma 7 we have

$$\begin{aligned}{} & {} \bigl | \Vert \eta ^*\Vert _{V_0^*(\tilde{v}_1)}^2-\Vert \eta ^*\Vert _{V_0^*(\tilde{v}_2)}^2 \bigr |\\{} & {} \quad =\bigl | \left\langle \eta ^*,F_0(\tilde{v}_1)^{-1}\eta ^* \right\rangle _{V_0^*(\tilde{v}_1),\,V_0(\tilde{v}_1)}- \left\langle \eta ^*,F_0(\tilde{v}_2)^{-1}\eta ^*\right\rangle _{V_0^*(\tilde{v}_2),\,V_0(\tilde{v}_2)} \bigr |\\{} & {} \quad =\bigl | \left\langle \eta ^*,\bigl (F_0(\tilde{v}_1)^{-1} \circ \bigl (F_0(\tilde{v}_1)-F_0(\tilde{v}_2)\bigr ) \circ F_0(\tilde{v}_2)^{-1} \bigr ) \eta ^* \right\rangle _{V_0^*,\,V_0} \bigr |\\{} & {} \quad \le \left\| \left( F_0(\tilde{v}_1)^{-1} \circ \left( F_0(\tilde{v}_1)-F_0(\tilde{v}_2) \right) \circ F_0(\tilde{v}_2)^{-1} \right) \eta ^* \right\| _{V_0} \Vert \eta ^*\Vert _{V_0^*}\\{} & {} \quad \le \left\| F_0(\tilde{v}_1)^{-1}\right\| _{\mathscr {L}(V_0^*,V_0)} \left\| F_0(\tilde{v}_1)-F_0(\tilde{v}_2)\right\| _{\mathscr {L}(V_0,V_0^*)} \left\| F_0(\tilde{v}_2)^{-1}\right\| _{\mathscr {L}(V_0^*,V_0)} \Vert \eta ^*\Vert _{V_0^*}^2\\{} & {} \quad \le \frac{C_2^2C_{10}}{C_1^2} \cdot \left( 1+\Vert \eta _0\Vert _{V^*} \Vert \eta _0\Vert _V \right) \Vert \tilde{v}_1-\tilde{v}_2\Vert _X \Vert \eta ^*\Vert _{V_0^*(\tilde{v}_1)}^2, \end{aligned}$$

which implies that this lemma holds. $\square$

3.2 A Dynamical System

Let the number $c_0>0$ be the same one that is given in (A5). For any $\bar{T} \in (0,T]$, we define an operator ${\varvec{J}}_{\bar{T}}:C([0,\bar{T}];V_0^*) \longmapsto C([0,\bar{T}];V^*)$ by

$$\begin{aligned} \forall \tilde{w} \in C([0,\bar{T}];V_0^*),\quad ({\varvec{J}}_{\bar{T}}\tilde{w})(t):=(J \circ P_{c_0})^{-1}\tilde{w}(t) \quad \text{ in } \quad V^*,\quad 0 \le \forall t \le \bar{T}. \end{aligned}$$

(48)

Using Lemma 3, Corollary 1 and Lemma 5, we get Lemma 9.

Lemma 9

For any $\bar{T} \in (0,T]$ the operator ${\varvec{J}}_{\bar{T}}:C([0,\bar{T}];V_0^*) \longmapsto C([0,\bar{T}];V^*)$ is injective and continuous with respect to the strong topologies.

Proof

At first, we assume that $\tilde{w}_1,~\tilde{w}_2$ in $C([0,\bar{T};V_0^*)$ satisfy

$$\begin{aligned} {\varvec{J}}_{\bar{T}}\tilde{w}_1={\varvec{J}}_{\bar{T}}\tilde{w}_2 \quad \text{ in } \quad C([0,\bar{T}];V^*). \end{aligned}$$

Then, from (48) we have

$$\begin{aligned} \left( J \circ P_{c_0}\right) ^{-1}\tilde{w}_1(t)=\left( J \circ P_{c_0}\right) ^{-1}\tilde{w}_2(t) \quad \text{ in } \quad V^*,\quad 0 \le \forall t \le \bar{T}. \end{aligned}$$

From Lemma 3, Corollary 1 and Lemma 5, we get

$$\begin{aligned} \tilde{w}_1(t)=\tilde{w}_2(t) \quad \text{ in } \quad V_0^*,\quad 0 \le \forall t \le \bar{T}, \end{aligned}$$

which implies $\tilde{w}_1=\tilde{w}_2$ in $C([0,\bar{T}];V_0^*)$, that is, the operator ${\varvec{J}}_{\bar{T}}$ is injective.

Next, using (27) in the proof of Corollary 1 and Lemma 5 again, we get

$$\begin{aligned} \left\| {\varvec{J}}_{\bar{T}}\tilde{w}_1-{\varvec{J}}_{\bar{T}} \tilde{w}_2\right\| _{C([0,\bar{T}];V^*)}= & {} \max _{0\,\le \,t\,\le \,\bar{T}} \left\| P_{c_0}^{-1} \left( J^{-1}\tilde{w}_1(t)\right) -P_{c_0}^{-1} \left( J^{-1}\tilde{w}_2(t) \right) \right\| _{V^*}\\\le & {} \max _{0\,\le \,t\,\le \,\bar{T}} \left\| J^{-1}\tilde{w}_1(t)-J^{-1}\tilde{w}_2(t) \right\| _{V^*}\\\le & {} C_9 \max _{0\,\le \,t\,\le \,\bar{T}} \left\| \tilde{w}_1(t)-\tilde{w}_2(t) \right\| _{V_0^*}\\= & {} C_9 \left\| \tilde{w}_1-\tilde{w}_2\right\| _{C([0,\bar{T}];V_0^*)}, \end{aligned}$$

which implies that the operator ${\varvec{J}}_{\bar{T}}$ is Lipschitz continuous from $C([0,\bar{T}];V_0^*)$ into $C([0,\bar{T}];V^*)$ with respect to their strong topologies. $\square$

Now, for any $\bar{T} \in (0,T]$ and $\tilde{w} \in C([0,\bar{T}];V_0^*)$ we define a family of operators $\{S_0(\tilde{w};t,s);0 \le s \le t \le \bar{T}\}$ on A by

$$\begin{aligned} S_0(\tilde{w}\,;t,s):=S\left( {\varvec{J}}_{\bar{T}}\tilde{w}\,;t,s\right) \quad \text{ on } \quad A,\quad 0 \le \forall s \le \forall t \le \bar{T}. \end{aligned}$$

(49)

As a direct consequence of (A2) and Lemma 9, we get some properties which are clearly stated in Lemma 10. Since their proofs are quite standard, we omit them in this paper.

Lemma 10

The following properties are satisfied:

• (a) Assume that a sequence $\{(\tilde{w}_m,\tilde{v}_m)\}_{m \in \mathbb {N}} \subset C([0,\bar{T}];V_0^*) \times A$ and a pair $(\tilde{w},\tilde{v}) \in C([0,\bar{T}];V_0^*) \times A$ satisfy

  $$\begin{aligned} (\tilde{w}_m,\tilde{v}_m) \longrightarrow (\tilde{w},\tilde{v}) \quad \text{ in } \quad C([0,\bar{T}];V_0^*) \times X \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

  Then, for any $s \in [0,\bar{T}]$ we have

  $$\begin{aligned} {\varvec{S}}_0(\tilde{w}_m,\tilde{v}_m,s) \longrightarrow {\varvec{S}}_0(\tilde{w},\tilde{v},s) \quad \text{ in } \quad C([s,\bar{T}];X) \quad \text{ as } \quad m \rightarrow \infty , \end{aligned}$$

  where the operator ${\varvec{S}}_0:C([0,\bar{T}];V_0^*) \times A \times [0,\bar{T}] \longmapsto C([s,\bar{T}];X)$ is defined by

  $$\begin{aligned}{} & {} \forall (\bar{w},\bar{v},s) \in C([0,\bar{T}];V_0^*) \times A \times [0,\bar{T}],\\{} & {} \quad ({\varvec{S}}_0(\bar{w},\bar{v},s))(t):=S_0(\bar{w};t,s)\bar{v} \quad \text{ in } \quad X,\quad \forall t \in [s,\bar{T}]. \end{aligned}$$

• (b) $S_0(\tilde{w};t,t)$ is the identity operator on A for all $t \in [0,\bar{T}]$.

• (c) $S_0(\tilde{w}\,;t,s)=S_0(\tilde{w}\,;t,\tau ) \circ S_0(\tilde{w}\,;\tau ,s)$ on A for all $s,t,\tau \in [0,\bar{T}]$ with $s \le \tau \le t$.

• (d) The following equality holds for all $\tau \in [0,\bar{T}]$:

  $$\begin{aligned} S_0(\sigma _\tau \tilde{w};t,s)=S_0(\tilde{w};t+\tau ,s+\tau )\quad \text{ on } \quad A,\quad 0 \le \forall s \le \forall t \le \bar{T}-\tau . \end{aligned}$$

• (e) We have ${\varvec{S}}_0(\tilde{w},\tilde{v},0) \in W^{1,1}(0,\bar{T};X)$ for all $\tilde{v} \in A$.

• (f) For any $\tilde{w}_1,\,\tilde{w}_2 \in C([0,\bar{T}];V_0^*)$, we assume that there exists $\bar{T}_0 \in (0,\bar{T}]$ such that

  $$\begin{aligned} \tilde{w}_1(t)=\tilde{w}_2(t) \quad \text{ in } \quad V_0^*,\quad \forall t \in [0,\bar{T}_0]. \end{aligned}$$

  Then, we have

  $$\begin{aligned} S_0(\tilde{w}_1;t,0)=S_0(\tilde{w}_2;t,0) \quad \text{ on } \quad A,\quad \forall t \in [0,\bar{T}_0]. \end{aligned}$$

3.3 Quasi-Variational Convex Functions on $V_0^*$

At the beginning of this subsection, for any pair $(\tilde{w},\tilde{v}) \in C([0,T];V_0^*) \times A$ we define a family of functions $\left\{ \phi _{c_0} (t,\tilde{w},\tilde{v})\,;\,0 \le t \le T\right\}$ on $V_0^*$ by

$$\begin{aligned} \phi _{c_0} (t,\tilde{w},\tilde{v}\,;\eta ^*):=\left\{ \begin{array}{ll} \phi \left( t,{\varvec{J}}_T\tilde{w},\tilde{v}\,;(J \circ P_{c_0})^{-1}\eta ^*\right) , &{}\quad \text{ if } \quad \eta ^* \in D_0:=(J \circ P_{c_0}) D(c_0),\\ \infty , &{} \quad \text{ if } \quad \eta ^* \in V_0^* \setminus D_0, \end{array} \right. \end{aligned}$$

(50)

and consider a family $\mathscr {C}_0$, which is defined by

$$\begin{aligned} \mathscr {C}_0:=\left\{ \left\{ \phi _{c_0} (t,\tilde{w},\tilde{v});\,0 \le t \le T \right\} ;\, \tilde{w} \in C([0,T];V_0^*),~\tilde{v} \in A\right\} . \end{aligned}$$

Then, we get Lemmas 11–14.

Lemma 11

For any $(t,\tilde{w},\tilde{v}) \in [0,T] \times C([0,T];V_0^*) \times A$, the function $\phi _{c_0}(t,\tilde{w},\tilde{v})$ is proper, l.s.c. and convex on $V_0^*$.

Proof

Since from (A5) we have $0=(J \circ P_{c_0})(c_0 \eta _0) \in D_0$, we see that $\phi _{c_0}(t,\tilde{w},\tilde{v})$ is proper because of $D_0 \ne \emptyset$.

Next, we show that $\phi _{c_0}(t,\tilde{w},\tilde{v})$ is convex on $V_0^*$. Using Lemma 3, Corollary 1 and Lemma 5, we see that for any elements $\eta _1^*,~\eta _2^* \in D_0$ there exist elements $\xi _1^*,~\xi _2^* \in D(c_0)$, respectively, which are uniquely determined, such that

$$\begin{aligned} \eta _1^*=(J \circ P_{c_0})\xi _1^*,\quad \eta _2^*=(J \circ P_{c_0})\xi _2^*. \end{aligned}$$

Since $D(c_0)$ is convex, we get the following equality for all $\lambda \in [0,1]$:

$$\begin{aligned} \lambda \xi _1^*+(1-\lambda ) \xi _2^*=\lambda (J \circ P_{c_0})^{-1} \eta _1^*+(1-\lambda )(J \circ P_{c_0})^{-1} \eta _2^* \in D(c_0), \end{aligned}$$

hence,

$$\begin{aligned} (J \circ P_{c_0}) \left( \lambda (J \circ P_{c_0})^{-1} \eta _1^*+(1-\lambda ) (J \circ P_{c_0})^{-1} \eta _2^* \right) \in D_0. \end{aligned}$$

Using (23) and Lemma 5, we get the following equality for all $\lambda \in [0,1]$:

$$\begin{aligned}{} & {} (J \circ P_{c_0}) \left( \lambda (J \circ P_{c_0})^{-1} \eta _1^*+(1-\lambda ) (J \circ P_{c_0})^{-1} \eta _2^* \right) \nonumber \\{} & {} \quad = J \left( \left( \lambda (J \circ P_{c_0})^{-1} \eta _1^*+(1-\lambda ) (J \circ P_{c_0})^{-1} \eta _2^* \right) -c_0\eta _0\right) \nonumber \\{} & {} \quad = J \left( \lambda \left\{ (J \circ P_{c_0})^{-1} \eta _1^*-c_0\eta _0\right\} +(1-\lambda ) \left\{ (J \circ P_{c_0})^{-1} \eta _2^*-c_0\eta _0\right\} \right) \nonumber \\{} & {} \quad = \lambda J \left( (J \circ P_{c_0})^{-1} \eta _1^*-c_0\eta _0\right) +(1-\lambda ) J \left( (J \circ P_{c_0})^{-1} \eta _2^*-c_0\eta _0\right) \nonumber \\{} & {} \quad = \lambda (J \circ P_{c_0}) \left( (J \circ P_{c_0})^{-1} \eta _1^*\right) +(1-\lambda ) (J \circ P_{c_0}) \left( (J \circ P_{c_0})^{-1} \eta _2^*\right) \nonumber \\{} & {} \quad =\lambda \eta _1^*+(1-\lambda ) \eta _2^*. \end{aligned}$$

(51)

Since $\phi (t,{\varvec{J}}_T\tilde{w},\tilde{v})$ is convex, from (51) we get

$$\begin{aligned}{} & {} \phi _{c_0} \left( t,\tilde{w},\tilde{v};\lambda \eta _1^*+(1-\lambda )\eta _2^*\right) \\{} & {} \quad =\phi \left( t,{\varvec{J}}_T\tilde{w},\tilde{v};\lambda (J \circ P_{c_0})^{-1} \eta _1^* +(1-\lambda ) (J \circ P_{c_0})^{-1} \eta _2^*\right) \\{} & {} \quad \le \lambda \phi \left( t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1} \eta _1^*\right) +(1-\lambda ) \phi \left( t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1} \eta _2^*\right) \\{} & {} \quad =\lambda \phi _{c_0} (t,\tilde{w},\tilde{v};\eta _1^*)+(1-\lambda ) \phi _{c_0} (t,\tilde{w},\tilde{v};\eta _2^*), \end{aligned}$$

which implies that $\phi _{c_0}(t,\tilde{w},\tilde{v})$ is convex on $V_0^*$.

Finally, we show that $\phi _{c_0}(t,\tilde{w},\tilde{v})$ is l.s.c. on $V_0^*$. In order to do this, we show that for any $r \in \mathbb {R}$ the level set $K(r):=\{\tilde{\eta }^* \in V_0^*\,;\,\phi _{c_0} (t,\tilde{w},\tilde{v}\,;\tilde{\eta }^*) \le r\}$ is closed in $V_0^*$. We consider a sequence $\{\eta _m^*\}_{m \in \mathbb {N}} \subset K(r)$ and a function $\eta ^* \in V_0^*$ satisfying

$$\begin{aligned} \eta _m^* \longrightarrow \eta ^* \quad \text{ in } \quad V_0^* \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

(52)

Using Corollary 1 and Lemma 5 again, we see from (52) that the following convergence holds:

$$\begin{aligned} (J \circ P_{c_0})^{-1} \eta _m^* \longrightarrow (J \circ P_{c_0})^{-1} \eta ^* \quad \text{ in } \quad V^* \quad \text{ as } \quad m \rightarrow \infty , \end{aligned}$$

(53)

hence,

$$\begin{aligned} \phi \left( t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1} \eta ^*\right)\le & {} \liminf _{m \rightarrow \infty } \phi \left( t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1} \eta _m^*\right) \nonumber \\= & {} \liminf _{m \rightarrow \infty } \phi _{c_0} (t,\tilde{w},\tilde{v};\eta _m^*) \le r. \end{aligned}$$

(54)

Since $V^*(c_0)$ is closed in $V^*$ and $\{(J \circ P_{c_0})^{-1} \eta _m^*\}_{m \in \mathbb {N}} \subset V^*(c_0)$, from (53) we get

$$\begin{aligned} (J \circ P_{c_0})^{-1} \eta ^* \in D(c_0). \end{aligned}$$

(55)

We see from (54) to (55) that $\eta ^* \in D_0$ and $\phi _{c_0} (t,\tilde{w},\tilde{v}\,;\eta ^*) \le r$. As a result, we get $\eta ^* \in K(r)$ which implies that the level set K(r) is closed in $V_0^*$. $\square$

Lemma 12

There exists a proper l.s.c. convex function $\phi _{c_0}$ on $V_0^*$ such that the following properties are satisfied:

• (a) The following inequality holds for all $(t,\tilde{w},\tilde{v}) \in [0,T] \times C([0,T];V_0^*) \times A$:

  $$\begin{aligned} \phi _{c_0} (\eta ^*) \le \phi _{c_0} (t,\tilde{w},\tilde{v};\eta ^*),\quad \forall \eta ^* \in V_0^*. \end{aligned}$$

• (b) For any $r \ge 0$ the level set $\{\tilde{\eta }^* \in V_0^*\,;\,\Vert \tilde{\eta }^*\Vert _{V_0^*} \le r,~ |\phi _{c_0} (\tilde{\eta }^*)| \le r \}$ is relatively compact in $V_0^*$.

Proof

Using $\phi$ in (c) in (A3), we define a function $\phi _{c_0}:V_0^* \longmapsto \mathbb {R} \cup \{\infty \}$ by

$$\begin{aligned} \phi _{c_0} (\eta ^*):=\left\{ \begin{array}{ll} \phi \left( (J \circ P_{c_0})^{-1}\eta ^*\right) ,\quad &{}\text{ if }\quad \eta ^* \in D(\phi _{c_0}):= (J \circ P_{c_0})(D(\phi ) \cap V^*(c_0)),\\ \infty ,&{}\text{ if } \quad \eta ^* \in V_0^* \setminus D(\phi _{c_0}). \end{array} \right. \end{aligned}$$

Since from (a) in (A3) and (A5) we have $D(c_0) \subset D(\phi ) \cap V^*(c_0)$, we see that $\phi _{c_0}$ is proper. Repeating the argumentation, which is similar to the proof of Lemma 11, we can show that $\phi _{c_0}$ is convex and l.s.c. on $V_0^*$. So, we omit its proof in this proof.

At first, we show (a). Using (a) in (A3) again, we see that $D_0 \subset D(\phi _{c_0})$ and the following inequality holds for all triplet $(t,\tilde{w},\tilde{v}) \in [0,T] \times C([0,T];V_0^*) \times A$:

$$\begin{aligned} \phi _{c_0}(\eta ^*)= & {} \phi \left( (J \circ P_{c_0})^{-1}\eta ^*\right) \le \phi \left( t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1}\eta ^*\right) \\= & {} \phi _{c_0} (t,\tilde{w},\tilde{v};\eta ^*),\quad \forall \eta ^* \in D_0, \end{aligned}$$

which implies that (a) holds.

Next, we show (b). For any $r \ge 0$, we consider the level set $B_r$, which is given by

$$\begin{aligned} B_r:=\{ \tilde{\eta }^* \in V_0^*;\,\Vert \tilde{\eta }^*\Vert _{V_0^*} \le r,~|\phi _{c_0}(\tilde{\eta }^*)| \le r\}. \end{aligned}$$

From (26) in the proof of Corollary 1 and Lemma 5, we see that the following inequalities hold for all $\tilde{\eta }^* \in B_r$:

$$\begin{aligned} \left\| (J \circ P_{c_0})^{-1}\tilde{\eta }^* \right\| _{V^*}= & {} \left\| P_{c_0}^{-1}\left( J^{-1}\tilde{\eta }^*\right) \right\| _{V^*} \le \left\| J^{-1}\tilde{\eta }^*\right\| _{V^*}+c_0\Vert \eta _0\Vert _{V^*}\\\le & {} C_9 \left\| \tilde{\eta }^*\right\| _{V_0^*}+c_0\Vert \eta _0\Vert _{V^*} \le C_9 r+c_0\Vert \eta _0\Vert _{V^*}, \end{aligned}$$

and

$$\begin{aligned} \left| \phi \left( (J \circ P_{c_0})^{-1}\tilde{\eta }^*\right) \right| =|\phi _{c_0}(\tilde{\eta }^*)| \le r \end{aligned}$$

because of $(J \circ P_{c_0})^{-1} \tilde{\eta }^* \in D(\phi ) \cap V^*(c_0)$. Taking a constant $c(r)>0$ by

$$\begin{aligned} c(r):=\max \{r,~C_9 r+c_0\Vert \eta _0\Vert _{V^*}\}, \end{aligned}$$

we get

$$\begin{aligned} \left\{ (J \circ P_{c_0})^{-1} \tilde{\eta }^* \in V^*;\, \tilde{\eta }^* \in B_r \right\} \subset \left\{ \xi ^* \in V^*;\, \Vert \xi ^*\Vert _{V^*} \le c(r),~|\phi (\xi ^*)| \le c(r) \right\} . \end{aligned}$$

We see from (a) in (A3) that the set $\{ (J \circ P_{c_0})^{-1}\tilde{\eta }^* \in V^*\,;\, \tilde{\eta }^* \in B_r\}$ is relatively compact in $V^*$. Hence, for any sequence $\{\tilde{\eta }_m^*\}_{m \in \mathbb {N}} \subset B_r$ there exist a subsequence $\{\tilde{\eta }_{m_k}^*\}_{k \in \mathbb {N}}$ of $\{\tilde{\eta }_m^*\}_{m \in \mathbb {N}}$ and an element $\xi ^* \in V^*$ such that

$$\begin{aligned} (J \circ P_{c_0})^{-1} \tilde{\eta }_{m_k}^* \longrightarrow \xi ^* \quad \text{ in } \quad V^* \quad \text{ as } \quad k \rightarrow \infty . \end{aligned}$$

(56)

Since we have the following equality:

$$\begin{aligned} \left\langle (J \circ P_{c_0})^{-1}\tilde{\eta }_{m_k}^*,\eta _0 \right\rangle _{V^*,\,V}=c_0, \quad \forall k \in \mathbb {N}, \end{aligned}$$

from (56) we get

$$\begin{aligned} \langle \xi ^*,\eta _0 \rangle _{V^*,\,V}=c_0, \quad \text{ hence }, \quad \xi ^* \in V^*(c_0). \end{aligned}$$

Using Lemmas 3 and 5, from (56) we get

$$\begin{aligned} \tilde{\eta }_{m_k}^*=(J \circ P_{c_0}) \left( (J \circ P_{c_0})^{-1}\tilde{\eta }_{m_k}^*\right) \longrightarrow (J \circ P_{c_0}) \xi ^* \quad \text{ in } \quad V_0^* \quad \text{ as } \quad k \rightarrow \infty . \end{aligned}$$

Hence, we see that $B_r$ is relatively compact in $V_0^*$. $\square$

We get Lemma 13 as a consequence from (d) in (A3) and Lemma 10 whose proof is omitted here.

Lemma 13

Assume that functions $\tilde{w}_1,\,\tilde{w}_2 \in C([0,T];V_0^*)$ satisfy

$$\begin{aligned} \tilde{w}_1(t)=\tilde{w}_2(t) \quad \text{ in } \quad V_0^*,\quad \forall t \in [0,T]. \end{aligned}$$

Then, we have

$$\begin{aligned} \phi _{c_0} (t,\tilde{w}_1,\tilde{v})=\phi _{c_0} (t,\tilde{w}_2,\tilde{v}) \quad \text{ on } \quad V_0^*, \quad \forall t \in [0,T]. \end{aligned}$$

Next, we show a convergence of time-dependent convex functions in $\mathscr {C}_0$, which comes from (d) in (A3) and is much stronger than Mosco convergence in [12].

Lemma 14

Assume that a sequence $\{(\tilde{w}_m,\tilde{v}_m)\}_{m \subset \mathbb {N}} \subset C([0,T];V_0^*) \times A$ and a pair $(\tilde{w},\tilde{v}) \in C([0,T];V_0^*) \times A$ satisfy

$$\begin{aligned} (\tilde{w}_m,\tilde{v}_m) \longrightarrow (\tilde{w},\tilde{v}) \quad \text{ in } \quad C([0,T];V_0^*) \times X \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

(57)

Then, for any $t \in [0,T]$ we have

$$\begin{aligned} \phi _{c_0} (t,\tilde{w}_m,\tilde{v}_m) \longrightarrow \phi _{c_0} (t,\tilde{w},\tilde{v}) \quad \text{ on } \quad V_0^* \quad \text{ as } \quad m \rightarrow \infty \end{aligned}$$

in the following sense:

• (i) For any $\eta ^* \in D_0$, there exists a sequence $\{\eta _m^*\}_{m \in \mathbb {N}} \subset H_0$ such that

  $$\begin{aligned}{} & {} \eta _m^* \longrightarrow \eta ^* \quad \text{ in } \quad H_0 \quad \text{ as } \quad m \rightarrow \infty , \end{aligned}$$

  (58)
  $$\begin{aligned}{} & {} \lim _{m \rightarrow \infty } \phi _{c_0} (t,\tilde{w}_m,\tilde{v}_m;\eta _m^*)=\phi _{c_0} (t,\tilde{w},\tilde{v};\eta ^*). \end{aligned}$$

  (59)

• (ii) For any subsequence $\{(\tilde{w}_{m_k},\tilde{v}_{m_k})\}_{k \in \mathbb {N}}$ of $\{(\tilde{w}_m,\tilde{v}_m)\}_{m \in \mathbb {N}}$ we have

  $$\begin{aligned} \phi _{c_0} \left( t,\tilde{w},\tilde{w}\,;\eta ^*\right) \le \liminf _{k \rightarrow \infty } \phi _{c_0} \left( t,\tilde{w}_{m_k},\tilde{v}_{m_k};\eta _k^*\right) \end{aligned}$$

  (60)

  whenever a sequence $\{\eta _k^*\}_{k \in \mathbb {N}} \subset V_0^*$ and an element $\eta ^* \in V_0^*$ satisfy

  $$\begin{aligned} \eta _k^* \longrightarrow \eta ^* \quad \text{ weakly } \text{ in } \quad V_0^* \quad \text{ as } \quad k \rightarrow \infty . \end{aligned}$$

  (61)

Proof

We show (i). Using Lemma 9, we see from (57) that the following convergence holds as $m \rightarrow \infty$:

$$\begin{aligned} \left( {\varvec{J}}_T\tilde{w}_m,\tilde{v}_m\right) \longrightarrow \left( {\varvec{J}}_T\tilde{w},\tilde{v}\right) \quad \text{ in } \quad C([0,T];V^*) \times X. \end{aligned}$$

Since we have

$$\begin{aligned} D(c_0)=(J \circ P_{c_0})^{-1}D_0 \subset D\left( \varphi \left( t,{\varvec{J}}_T\tilde{w},\tilde{v}\right) \right) , \end{aligned}$$

we see from (e) in (A3) that for any $\eta ^* \in D_0$ there exists a sequence $\{\xi _m^*\}_{m \in \mathbb {N}} \subset H$ such that

$$\begin{aligned} \xi _m^* \in D\left( \phi \left( t,{\varvec{J}}_T\tilde{w}_m,\tilde{v}_m\right) \right) ,\quad \forall m \in \mathbb {N}, \end{aligned}$$

(62)

and the following convergences hold as $m \rightarrow \infty$:

$$\begin{aligned}{} & {} \xi _m^* \longrightarrow (J \circ P_{c_0})^{-1}\eta ^* \quad \text{ in } \quad H, \end{aligned}$$

(63)

$$\begin{aligned}{} & {} \lim _{m \rightarrow \infty } \phi \left( t,{\varvec{J}}_T\tilde{w}_m,\tilde{v}_m;\xi _m^*\right) = \phi \left( t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1}\eta ^*\right) . \end{aligned}$$

(64)

Moreover, we see that for any $m \in \mathbb {N}$ there exist elements $\tilde{\xi }_m^* \in \overline{D(c_0)}$ and $\hat{\xi }_m^* \in D(c_0)$ such that

$$\begin{aligned} \bigl \Vert \xi _m^*-\tilde{\xi }_m^*\bigr \Vert _H=\inf \left\{ \bigl \Vert \xi _m^*-\bar{\xi }^*\bigr \Vert _H\,;\, \bar{\xi }^* \in D(c_0) \right\} \le \bigl \Vert \xi _m^*-(J\circ P_{c_0})^{-1}\eta ^* \bigr \Vert _H \end{aligned}$$

(65)

because of $(J \circ P_{c_0})^{-1} \eta ^* \in D(c_0)$, and

$$\begin{aligned} \bigl \Vert \hat{\xi }_m^*-\tilde{\xi }_m^*\bigr \Vert _H < \frac{1}{m}. \end{aligned}$$

(66)

From (65) to (66), we get the following inequality for all $m \in \mathbb {N}$:

$$\begin{aligned}{} & {} \bigl \Vert \hat{\xi }_m^*-(J \circ P_{c_0})^{-1}\eta ^*\bigr \Vert _H\nonumber \\{} & {} \quad \le \bigl \Vert \hat{\xi }_m^*-\tilde{\xi }_m^*\bigr \Vert _H+ \bigl \Vert \tilde{\xi }_m^*-\xi _m^*\bigr \Vert _H+\bigl \Vert \xi _m^*-(J \circ P_{c_0})^{-1}\eta ^*\bigr \Vert _H\nonumber \\{} & {} \quad \le \frac{1}{m}+2\bigl \Vert \xi _m^*-(J \circ P_{c_0})^{-1} \eta ^*\bigr \Vert _H. \end{aligned}$$

(67)

Hence, we see from (63) and (67) that the following convergence holds:

$$\begin{aligned} \hat{\xi }_m^* \longrightarrow (J \circ P_{c_0})^{-1} \eta ^* \quad \text{ in } \quad H \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

(68)

Now, we put

$$\begin{aligned} \eta _m^*:=(J \circ P_{c_0}) \hat{\xi }_m^*,\quad \forall m \in \mathbb {N}. \end{aligned}$$

Using Lemma 5, we have

$$\begin{aligned} \Vert \eta _m^*-\eta ^*\Vert _{H_0}= & {} \bigl \Vert (J \circ P_{c_0}) \hat{\xi }_m^*-(J \circ P_{c_0}) (J \circ P_{c_0})^{-1} \eta ^*\bigr \Vert _{H_0}\nonumber \\= & {} \bigl \Vert P_{c_0} \hat{\xi }_m^*-P_{c_0} (J \circ P_{c_0})^{-1} \eta ^*\bigr \Vert _H \le \bigl \Vert \hat{\xi }_m^*-(J\circ P_{c_0})^{-1} \eta ^* \bigr \Vert _H, \end{aligned}$$

(69)

hence, see from (62), (68) and (69) that (58) holds. Moreover, from (b) in (A3), (65) and (66) we get

$$\begin{aligned}{} & {} \bigl |\phi _{c_0}\bigl (t,\tilde{w}_m,\tilde{v}_m;\eta _m^*\bigr )-\phi _{c_0}\bigl (t,\tilde{w},\tilde{v};\eta ^*\bigr )\bigr |\\{} & {} \quad =\bigl |\phi \bigl (t,{\varvec{J}}_T\tilde{w}_m,\tilde{v}_m;(J \circ P_{c_0})^{-1} \eta _m^*\bigr )- \phi \bigl (t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1}\eta ^*\bigr )\bigr |\\{} & {} \quad =\bigl |\phi \bigl (t,{\varvec{J}}_T\tilde{w}_m,\tilde{v}_m;\hat{\xi }_m^*\bigr )- \phi \bigl (t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1}\eta ^*\bigr )\bigr |\\{} & {} \quad \le \bigl |\phi \bigl (t,{\varvec{J}}_T\tilde{w}_m,\tilde{v}_m;\hat{\xi }_m^*\bigr ) -\phi \bigl (t,{\varvec{J}}_T\tilde{w}_m,\tilde{v}_m;\xi _m^*\bigr )\bigr |\\{} & {} \qquad +\bigl |\phi \bigl (t,{\varvec{J}}_T\tilde{w}_m,\tilde{v}_m;\xi _m^*\bigr )- \phi \bigl (t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1}\eta ^*\bigr )\bigr |\\{} & {} \quad \le C_4 \bigl \Vert \hat{\xi }_m^*-\xi _m^*\bigr \Vert _H+ \bigl |\phi \bigl (t,{\varvec{J}}_T\tilde{w}_m,\tilde{v}_m;\xi _m^*\bigr ) -\phi \bigl (t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1}\eta ^*\bigr )\bigr |\\{} & {} \quad \le C_4 \left( \frac{1}{m}+\bigl \Vert \xi _m^*-(J \circ P_{c_0})^{-1}\eta ^*\bigr \Vert _H \right) \\{} & {} \qquad +\bigl |\phi \bigl (t,{\varvec{J}}_T\tilde{w}_m,\tilde{v}_m;\xi _m^*\bigr ) -\phi \bigl (t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1}\eta ^*\bigr )\bigr |, \end{aligned}$$

which implies from (63) and (64) that (59) holds. Hence, (i) is proved.

In order to show (ii), we consider a sequence $\{\eta _k^*\}_{k \in \mathbb {N}}$ and an element $\eta ^*$ in $V_0^*$ satisfying (61). From Proposition 1, we get

$$\begin{aligned} \left\langle (J \circ P_{c_0})^{-1}\eta _k^*-(J \circ P_{c_0})^{-1}) \eta ^*,z \right\rangle _{V^*,\,V} =\left\langle \eta _k^*-\eta ^*,Pz \right\rangle _{V_0^*,\,V_0},\quad \forall z \in V, \end{aligned}$$

which implies

$$\begin{aligned} (J \circ P_{c_0})^{-1} \eta _k^* \longrightarrow (J \circ P_{c_0})^{-1} \eta ^* \quad \text{ weakly } \text{ in } \quad V^* \quad \text{ as } \quad k \rightarrow \infty . \end{aligned}$$

Using (d) in (A3), we see that the following inequality holds:

$$\begin{aligned} \phi _{c_0} (t,\tilde{w},\tilde{v};\eta ^*)= & {} \phi \left( t,{\varvec{J}}_T\tilde{w},\tilde{v};(J \circ P_{c_0})^{-1}\eta ^*\right) \\\le & {} \liminf _{k \rightarrow \infty } \phi \left( t,{\varvec{J}}_T\tilde{w}_{n_k},\tilde{v}_{n_k};(J \circ P_{c_0})^{-1} \eta _k^*\right) \\= & {} \liminf _{k \rightarrow \infty } \phi _{c_0} \left( t,\tilde{w}_{n_k},\tilde{v}_{n_k};\eta _k^*\right) , \end{aligned}$$

hence, (60) holds. $\square$

Next, using the subsets $\mathscr {W}(u_0) \subset \mathscr {V}(u_0) \subset \mathscr {U}(u_0)$, we define subsets

$$\begin{aligned} \mathscr {W}_0(u_0) \subset \mathscr {V}_0(u_0) \subset \mathscr {U}_0(u_0) \end{aligned}$$

by the following ways:

$$\begin{aligned} \mathscr {U}_0(u_0):= & {} {\varvec{J}}_T^{-1}\bigl (\mathscr {U}(u_0)\bigr ),\quad \mathscr {V}_0(u_0):={\varvec{J}}_T^{-1}\bigl (\mathscr {V}(u_0)\bigr ),\nonumber \\ \mathscr {W}_0(u_0):= & {} {\varvec{J}}_T^{-1}\bigl (\mathscr {W}(u_0)\bigr ), \end{aligned}$$

(70)

and for any $R \ge 0$ sets $\mathscr {W}_0(u_0,R) \subset \mathscr {V}_0(u_0,R)$ and $\mathscr {U}_0(u_0,R)$ by

$$\begin{aligned} \mathscr {U}_0(u_0,R):= & {} \left\{ \tilde{w} \in \mathscr {U}_0(u_0);\, \sup _{0\,\le \,t\,\le \,T} \Vert \tilde{w}(t)\Vert _{V_0^*}+\int _0^T \phi _{c_0} (\tilde{w}(t))dt \le R \right\} ,\\ \mathscr {V}_0(u_0,R):= & {} \left\{ \tilde{w} \in \mathscr {U}_0(u_0);\, \sup _{0\,\le \,t\,\le \,T} \Vert \tilde{w}(t)\Vert _{V_0^*}+\sup _{0\,\le \,t\,\le \,T} \phi _{c_0} (\tilde{u}(t)) \le R\right\} ,\\ \mathscr {W}_0(u_0,R):= & {} \left\{ \tilde{w} \in \mathscr {U}_0(u_0);\, \begin{array}{l} {\Vert \tilde{w}'\Vert _{L^2(0,T;V_0^*)}+\sup \limits _{0\,\le \,t\,\le \,T} \Vert \tilde{w}(t)\Vert _{V^*}}\\ {+\sup \limits _{0\,\le \,t\,\le \,T} \phi _{c_0} (\tilde{u}(t)) \le R} \end{array} \right\} . \end{aligned}$$

Moreover, using the family $\mathscr {C}_0$, for each $t \in [0,T]$ and $\tilde{w} \in \mathscr {U}_0(u_0)$ we define a proper l.s.c. convex function on $V_0^*$ by

$$\begin{aligned} \varphi _{c_0}(t,\tilde{w},v_0):=\phi _{c_0}(t,\tilde{w},S_0(\tilde{w}\,;t,0)v_0), \end{aligned}$$

(71)

and consider a family $\mathscr {X}_0(u_0,v_0)$ by

$$\begin{aligned} \mathscr {X}_0(u_0,v_0):=\left\{ \left\{ \varphi _{c_0} (t,\tilde{w},v_0);\,0 \le t \le T \right\} ;\, \tilde{w} \in \mathscr {U}_0(u_0)\right\} . \end{aligned}$$

From (A6), we get Lemma 15, which guarantees the Kenmochi condition for $\left\{ \varphi _{c_0} (t,\tilde{w},v_0)\,;\,0 \le t \le T \right\}$ on $\mathscr {X}_0(u_0,v_0)$.

Lemma 15

For any $\{\varphi _{c_0} (t,\tilde{w},v_0)\,;\,0 \le t \le T\} \in \mathscr {X}_0(u_0,v_0)$, the following condition is satisfied: for any $r>0$ there exist nonnegative functions $\bar{\alpha }_r(\tilde{w}) \in L^2(0,T)$ and $\bar{\beta }_r(\tilde{w}) \in L^1(0,T)$ such that the following property is satisfied:

$$\begin{aligned} \left( \begin{array}{l} \text{ for } \text{ any }\, s,\,t \in [0,T]\, \text{ and }\, \eta ^* \in D_0\, \text{ with }\, \Vert \eta ^*\Vert _{V_0^*(S_0(\tilde{w};\,t,0)v_0)} \le r\\ \text{ there } \text{ exists }\, \eta ^*(\tilde{w},s,t) \in D_0\, \text{ such } \text{ that }\\ \text{(K1) } \quad \displaystyle {\bigl \Vert \eta ^*(\tilde{w},s,t)-\eta ^* \bigr \Vert _{H_0} \le \left( \sqrt{|\varphi _{c_0} (s,\tilde{w},v_0;\eta ^*)|}+1 \right) \displaystyle {\left| \int _s^t \bar{\alpha }_r (\tilde{w};\tau )d\tau \right| }},\\ \text{(K2) }\quad \bigl |\varphi _{c_0} (t,\tilde{w},v_0;\eta ^*(\tilde{w},s,t))-\varphi _{c_0} (s,\tilde{w},v_0;\eta ^*)\bigr |\\ \qquad \qquad \le \left( |\varphi _{c_0} (s,\tilde{w},v_0;\eta ^*)|+1 \right) \displaystyle {\left| \int _s^t \bar{\beta }_r (\tilde{w};\tau )d\tau \right| }. \end{array}\right) \end{aligned}$$

Proof

For any element $\eta ^* \in D_0$ with $\Vert \eta ^*\Vert _{V_0^*(S_0(\tilde{w};\,t,0)v_0)} \le r$, we have $(J \circ P_{c_0})^{-1} \eta ^* \in D(\varphi (s,{\varvec{J}}_T\tilde{w},v_0))$ and see from (25) in the proof of Corollary 1 and Lemmas 5 and 7 that the following inequality holds:

$$\begin{aligned}{} & {} \bigl \Vert (J \circ P_{c_0})^{-1}\eta ^*\bigr \Vert _{V^*(S({\varvec{J}}_T\tilde{w};\,t,0)v_0)} =\bigl \Vert J^{-1}\eta ^*+c_0 \eta _0\bigr \Vert _{V^*(S({\varvec{J}}_T\tilde{w};\,t,0)v_0)}\\{} & {} \quad \le \bigl \Vert J^{-1}\eta ^*\bigr \Vert _{V^*(S({\varvec{J}}_T\tilde{w};\,t,0)v_0)} +c_0\Vert \eta _0\Vert _{V^*(S({\varvec{J}}_T\tilde{w};\,t,0)v_0)}\\{} & {} \quad \le C_9\Vert \eta ^*\Vert _{V_0^*(S_0(\tilde{w};\,t,0)v_0)}+\frac{c_0}{C_1} \cdot \Vert \eta _0\Vert _{V^*} \le rC_7+\frac{c_0}{C_1} \cdot \Vert \eta _0\Vert _{V^*}=:c(r). \end{aligned}$$

Using (A6), we see that there exist nonnegative functions $\alpha _{c(r)}({\varvec{J}}_T\tilde{w}) \in L^2(0,T)$, $\beta _{c(r)}({\varvec{J}}_T\tilde{w}) \in L^1(0,T)$ and an element $\xi ^*(\tilde{w},s,t) \in D(\varphi (t,{\varvec{J}}_T\tilde{w},v_0))$ such that

$$\begin{aligned}{} & {} \bigl \Vert \xi ^*(\tilde{w},s,t)-(J \circ P_{c_0})^{-1} \eta ^*\bigr \Vert _H\nonumber \\{} & {} \quad \le \left( \sqrt{|\varphi _{c_0} (s,\tilde{w},v_0;\eta ^*)|}+1 \right) \left| \int _s^t \alpha _{c(r)} \left( {\varvec{J}}_T\tilde{w};\tau \right) d\tau \right| , \end{aligned}$$

(72)

$$\begin{aligned}{} & {} \left| \varphi \bigl (t,{\varvec{J}}_T\tilde{w},v_0;\xi ^*(\tilde{w},s,t)\bigr ) -\varphi \bigl (s,{\varvec{J}}_T\tilde{w},v_0;(J \circ P_{c_0})^{-1}\eta ^*\bigr )\right| \nonumber \\{} & {} \quad \le \left( |\varphi _{c_0} (s,\tilde{w},v_0;\eta ^*)|+1 \right) \left| \int _s^t \beta _{c(r)} \bigl ({\varvec{J}}_T\tilde{w};\tau \bigr )d\tau \right| . \end{aligned}$$

(73)

At first, in order to verify (K1) we do the following argumentation. For each $\xi ^*(\tilde{w},s,t) \in D(\varphi (t,{\varvec{J}}_T\tilde{w},v_0))$ we choose an element $\tilde{\xi }^*(\tilde{w},s,t) \in \overline{D(c_0)}$ such that

$$\begin{aligned} \bigl \Vert \xi ^*(\tilde{w},s,t)-\tilde{\xi }^*(\tilde{w},s,t)\bigr \Vert _H= & {} \inf \left\{ \bigl \Vert \xi ^*(\tilde{w},s,t)-\bar{\xi }^*\Vert _H;\, \bar{\xi }^* \in D(c_0)\right\} \nonumber \\\le & {} \left\| \xi ^*(\tilde{w},s,t)-(J \circ P_{c_0})^{-1} \eta ^*\right\| _H. \end{aligned}$$

(74)

Moreover, for each $\tilde{\xi }^*(\tilde{w},s,t) \in \overline{D(c_0)}$, which satisfies (74), we choose an element $\hat{\xi }^*(\tilde{w},s,t) \in D(c_0)$ such that

$$\begin{aligned} \bigl \Vert \hat{\xi }^*(\tilde{w},s,t)-\tilde{\xi }^*(\tilde{w},s,t)\bigr \Vert _H \le \left( \sqrt{|\varphi _{c_0} (s,\tilde{w},v_0\,;\eta ^*)|}+1 \right) |t-s|. \end{aligned}$$

(75)

Defining $\eta ^*(\tilde{w},s,t) \in D_0$ by

$$\begin{aligned} \eta ^*(\tilde{w},s,t):=(J \circ P_{c_0})\hat{\xi }(\tilde{w},s,t), \end{aligned}$$

we see from (72), (74) and (75) that the following inequality holds:

$$\begin{aligned}{} & {} \bigl \Vert \eta ^*(\tilde{w},s,t)-\eta ^*\bigr \Vert _{H_0}\nonumber \\{} & {} \quad =\bigl \Vert (J \circ P_{c_0})^{-1} \eta ^*(\tilde{w},s,t)-(J \circ P_{c_0})^{-1}\eta ^* \bigr \Vert _H\nonumber \\{} & {} \quad \le \bigl \Vert \hat{\xi }^*(\tilde{w},s,t)-\tilde{\xi }^*(\tilde{w},s,t)\bigr \Vert _H +\bigl \Vert \tilde{\xi }^*(\tilde{w},s,t)-\xi ^*(\tilde{w},s,t)\bigr \Vert _H\nonumber \\{} & {} \qquad +\bigl \Vert \xi ^*(\tilde{w},s,t)-(J \circ P_{c_0})^{-1}\eta ^* \bigr \Vert _H\nonumber \\{} & {} \quad \le \left( \sqrt{|\varphi _{c_0} (s,\tilde{w},v_0;\eta ^*)|}+1 \right) \left| \int _s^t\left\{ 2 \alpha _{c(r)} \left( {\varvec{J}}_T\tilde{w};\tau \right) +1 \right\} d\tau \right| . \end{aligned}$$

(76)

Next, we verify (K2). From (b) in (A3) and (72) to (75), we get

$$\begin{aligned}{} & {} \left| \varphi _{c_0} \bigl (t,\tilde{w},v_0;\eta ^*(\tilde{w},s,t)\bigr )-\varphi _{c_0} \bigl (s,\tilde{w},v_0;\eta ^*\bigr )\right| \nonumber \\{} & {} \quad =\bigl |\varphi \bigl (t,{\varvec{J}}_T\tilde{w},v_0;\hat{\xi }^*(\tilde{u},s,t)\bigr ) -\varphi \bigl (s,{\varvec{J}}_T\tilde{w},v_0;(J \circ P_{c_0})^{-1}\eta ^*\bigr )\bigr |\nonumber \\{} & {} \quad \le \bigl |\varphi \bigl (t,{\varvec{J}}_T\tilde{w},v_0;\hat{\xi }^*(\tilde{w},s,t)\bigr ) -\varphi \bigl (s,{\varvec{J}}_T\tilde{w},v_0;\xi (\tilde{w},s,t)\bigr )\bigr |\nonumber \\{} & {} \qquad +\bigl |\varphi \bigl (t,{\varvec{J}}_T\tilde{w},v_0;\xi ^*(\tilde{w},s,t)\bigr ) -\varphi \bigl (s,{\varvec{J}}_T\tilde{w},v_0;(J \circ P_{c_0})^{-1}\eta ^*\bigr )\bigr |\nonumber \\{} & {} \quad \le C_4 \bigr \Vert \hat{\xi }^*(\tilde{w},s,t)-\xi (\tilde{w},s,t)\bigr \Vert _H\nonumber \\{} & {} \qquad +\left| \varphi \bigl (t,{\varvec{J}}_T\tilde{w},v_0;\xi ^*(\tilde{w},s,t)\bigr ) -\varphi \bigl (s,{\varvec{J}}_T\tilde{w},v_0;(J \circ P_{c_0})^{-1}\eta ^*\bigr )\right| \nonumber \\{} & {} \quad \le C_4\left( \sqrt{|\varphi _{c_0} (s,\tilde{w},v_0;\eta ^*)|}+1 \right) \left| \int _s^t\left\{ \alpha _{c(r)} \bigl ({\varvec{J}}_T\tilde{w};\tau \bigr )+1 \right\} d\tau \right| \nonumber \\{} & {} \qquad +\left( \bigl |\varphi _{c_0} \bigl (s,\tilde{w},v_0;\eta ^*\bigr )\bigr |+1 \right) \left| \int _s^t \beta _{c(r)} \bigl ({\varvec{J}}_T\tilde{w};\tau \bigr )d\tau \right| \nonumber \\{} & {} \quad \le \left( \bigl |\varphi _{c_0} \bigl (s,\tilde{w},v_0;\eta ^*\bigr )\bigr |+1 \right) \nonumber \\{} & {} \qquad \times \left| \int _s^t\left\{ \frac{3C_4}{2} \cdot \alpha _{c(r)} \bigl ({\varvec{J}}_T\tilde{w};\tau \bigr ) +\beta _{c(r)} \bigl ({\varvec{J}}_T\tilde{w};\tau \bigr )+\frac{3C_4}{2}\right\} d\tau \right| . \end{aligned}$$

(77)

Taking

$$\begin{aligned} \left\{ \begin{array}{l} \bar{\alpha }_r (\tilde{w}):=2 \alpha _{c(r)} \bigl ({\varvec{J}}_T\tilde{w}\bigr )+1 \in L^2(0,T),\\ \displaystyle {\bar{\beta }_r (\tilde{w}):=\frac{3C_4}{2} \cdot \alpha _{c(r)} \bigl ({\varvec{J}}_T\tilde{w}\bigr ) +\beta _{c(r)} \bigl ({\varvec{J}}_T\tilde{w}\bigr )+\frac{3C_4}{2}} \in L^1(0,T), \end{array} \right. \end{aligned}$$

(78)

we see from (76) to (78) that this lemma holds. $\square$

Finally, we give Lemmas 16 and 17, which comes from (A7) and (A9).

Lemma 16

There exists a number $\hat{R}_*>0$, which depends on $R^*$ given in (A7), such that for any $R \ge \hat{R}_*$ the following properties are satisfied, where $\bar{\alpha }_r (\tilde{w})$ and $\bar{\beta }_r (\tilde{w})$ are the same functions that are obtained in Lemma 15:

(a) There exists a family $\{M_0(r,R);\,0<r<\infty \}$ such that

$$\begin{aligned} \forall r>0,\quad \sup _{\tilde{w}\, \in \, \mathscr {U}_0(u_0,\,R)} \left( \Vert \bar{\alpha }_r(\tilde{w})\Vert _{L^2(0,T)}+ \Vert \bar{\beta }_r(\tilde{w})\Vert _{L^1(0,T)}\right) \le M_0(r,R). \end{aligned}$$

(b) For any $r>0$ and $\varepsilon >0$, there exists a constant $\hat{\delta }_{r,\varepsilon ,R}>0$ such that

$$\begin{aligned}{} & {} \sup _{ \tilde{w} \, \in \, \mathscr {U}_0(u_0,\,R)} \Biggl \{ \sup _{0\,\le \,t\,\le \,T} \int _t^{\min \{t+\hat{\delta }_{r,\varepsilon ,R},T\}} \left( |\bar{\alpha }_r(\tilde{w};s)|^2+\bar{\beta }_r (\tilde{w};s)\right. \\{} & {} \qquad \left. +\Vert ({\varvec{S}}_0(\tilde{w},v_0,0))'(s)\Vert _X \right) ds \Biggr \} \le \varepsilon . \end{aligned}$$

Proof

We show (a). From (78) in the proof of Lemma 15 we have

$$\begin{aligned}{} & {} \bigl \Vert \bar{\alpha }_r (\tilde{w})\bigr \Vert _{L^2(0,T)} \le 2 \bigl \Vert \alpha _{c(r)} \bigl ({\varvec{J}}_T \tilde{w} \bigr ) \bigr \Vert _{L^2(0,T)} +\sqrt{T}, \end{aligned}$$

(79)

$$\begin{aligned}{} & {} \bigl \Vert \bar{\beta }_r (\tilde{w})\bigr \Vert _{L^1(0,T)} \le \frac{3C_4 \sqrt{T}}{2} \bigl \Vert \alpha _{c(r)} \bigl ({\varvec{J}}_T \tilde{w} \bigr ) \bigr \Vert _{L^2(0,T)}\nonumber \\{} & {} \quad +\bigl \Vert \beta _{c(r)} \bigl ({\varvec{J}}_T \tilde{w} \bigr ) \bigr \Vert _{L^1(0,T)}+\frac{3C_4T}{2}. \end{aligned}$$

(80)

Moreover, from (25) in the proof of Corollary 1 and Lemma 5 we get the following inequality:

$$\begin{aligned}{} & {} \sup _{0\,\le \,t\le \,T}\bigl \Vert (J \circ P_{c_0})^{-1}\tilde{w}(t)\bigr \Vert _{V^*}+\int _0^T \phi ((J \circ P_{c_0})^{-1}\tilde{w}(t))dt\\{} & {} \quad =\sup _{0\,\le \,t\le \,T}\bigl \Vert J^{-1}\tilde{w}(t)+c_0\eta _0\bigr \Vert _{V^*}+\int _0^T \phi ((J \circ P_{c_0})^{-1}\tilde{w}(t))dt\\{} & {} \quad \le C_9\sup _{0\,\le \,t\le \,T}\bigl \Vert \tilde{w}(t)\bigr \Vert _{V_0^*}+\int _0^T \phi _{c_0}(\tilde{w}(t))dt+c_0 \Vert \eta _0\Vert _{V^*}\\{} & {} \quad \le \max \{1,C_9\}R+c_0\Vert \eta _0\Vert _{V^*}=:d(R),\quad \forall \tilde{w} \in \mathscr {U}_0(u_0,R), \end{aligned}$$

which implies

$$\begin{aligned} {\varvec{J}}_T \left( \mathscr {U}_0(u_0,R) \right) \subset \mathscr {U}\left( u_0,d(R)\right) . \end{aligned}$$

(81)

Then, we choose a number $\hat{R}^*>0$ so that

$$\begin{aligned} d\bigl (\hat{R}_*\bigr )=\max \{1,C_9\}\hat{R}^*+c_0\Vert \eta _0\Vert _{V^*}=R_*. \end{aligned}$$

Using (a) in (A7), we see from (79) to (81) that for any $R \ge \hat{R}_*$ the following uniform estimate holds because of $d(R) \ge R_*$:

$$\begin{aligned}{} & {} \sup _{\tilde{w} \, \in \, \mathscr {U}_0(u_0,\,R)} \left( \bigl \Vert \bar{\alpha }_r(\tilde{w})\bigr \Vert _{L^2(0,T)}+ \bigl \Vert \bar{\beta }_r(\tilde{w})\bigr \Vert _{L^1(0,T)} \right) \\{} & {} \quad \le \sqrt{T}+\frac{3C_4T}{2}+\left( 2+\frac{3C_4 \sqrt{T}}{2}\right) \\{} & {} \qquad \times \sup _{\tilde{w} \, \in \, \mathscr {U}_0(u_0,\,R)} \left( \bigl \Vert \alpha _{c(r)} \bigl ({\varvec{J}}_T\tilde{w}\bigr )\bigr \Vert _{L^2(0,T)}+ \bigl \Vert \beta _{c(r)} \bigl ({\varvec{J}}_T\tilde{w}\bigr )\bigr \Vert _{L^1(0,T)} \right) \\{} & {} \quad \le \sqrt{T}+\frac{3C_4T}{2}+\left( 2+\frac{3C_4 \sqrt{T}}{2}\right) \\{} & {} \qquad \times \sup _{\tilde{u} \, \in \, \mathscr {U}(u_0,\,c(R))} \left( \bigl \Vert \alpha _{c(r)} \bigl (\tilde{u}\bigr )\bigr \Vert _{L^2(0,T)}+ \bigl \Vert \beta _{c(r)} \bigl (\tilde{u}\bigr )\bigr \Vert _{L^1(0,T)} \right) \\{} & {} \quad \le \left( 2+\frac{3C_4 \sqrt{T}}{2}\right) \cdot M(c(r),d(R))+\sqrt{T}+\frac{3C_4T}{2}=:M_0(r,R). \end{aligned}$$

Hence, the number $\hat{R}_*>0$ and the family $\{M_0(r,R);\,0<r<\infty \}$ are the desired ones.

Next, we show (b). Using (b) in (A7), we see that for any $r>0,~\varepsilon >0$ and $R \ge \hat{R}_*$ there exists a number $\hat{\delta }_{r,\varepsilon ,R}>0$ such that

$$\begin{aligned}{} & {} \sup _{\tilde{u} \, \in \, \mathscr {U}(u_0,\,d(R))} \Biggl \{ \sup _{0\,\le \,t\,\le \,T} \int _t^{t+\hat{\delta }_{r,\varepsilon ,R}} \left( \bigl | \alpha _{c(r)} (\tilde{u};s)\bigr |^2+\beta _{c(r)} (\tilde{u};s)\right. \nonumber \\{} & {} \quad \left. +\bigl \Vert ({\varvec{S}}(\tilde{u},v_0,0))'(t)\bigr \Vert _X \right) \,ds \Biggr \} \le \frac{\varepsilon }{17}, \end{aligned}$$

(82)

$$\begin{aligned}{} & {} \left( \frac{9C_4^2}{8}+\frac{3C_4}{2}+2\right) \hat{\delta }_{r,\varepsilon ,R} \le \frac{\varepsilon }{2}. \end{aligned}$$

(83)

Using (78), (82), (83) and repeating the argumentation similar to the proof of (a) in this lemma, we get the following uniform estimate:

$$\begin{aligned}{} & {} \sup _{\tilde{w} \, \in \, \mathscr {U}_0(u_0,\,R)} \left\{ \sup _{0\,\le \,t\,\le \,T} \int _t^{t+\hat{\delta }_{r,\varepsilon ,R}} \left( \bigl | \bar{\alpha }_r (\tilde{w};s)\bigr |^2+\bar{\beta }_r (\tilde{w};s) +\bigl \Vert ({\varvec{S}}_0(\tilde{w},v_0,0))'(s)\bigr \Vert _X \right) \,ds \right\} \nonumber \\{} & {} \quad \le \sup _{\tilde{w} \, \in \, \mathscr {U}_0(u_0,\,R)} \left\{ \sup _{0\,\le \,t\,\le \,T} \int _t^{t+\hat{\delta }_{r,\varepsilon ,R}} \left( \frac{17}{2} \cdot \bigl |\alpha _{c(r)} \bigl ({\varvec{J}}_T\tilde{w};s\bigr )\bigr |^2 +\beta _{c(r)} \bigl ({\varvec{J}}_T\tilde{w};s\bigr )\right. \right. \nonumber \\{} & {} \qquad \left. +\bigl \Vert \bigl ({\varvec{S}}({\varvec{J}}_T\tilde{w},v_0,0)\bigr )'(s)\bigr \Vert _X+\frac{9C_4^2}{8} +\frac{3C_4}{2}+2\right) \Biggr \}ds\nonumber \\{} & {} \quad \le \sup _{\tilde{u} \, \in \, \mathscr {U}(u_0,\,d(R))} \left\{ \sup _{0\,\le \,t\,\le \,T} \int _t^{t+\hat{\delta }_{r,\varepsilon ,R}} \left( \frac{17}{2} \cdot \bigl |\alpha _{c(r)} \bigl (\tilde{u};s\bigr )\bigr |^2+\beta _{c(r)} \bigl (\tilde{u};s\bigr )\right. \right. \nonumber \\{} & {} \qquad \left. +\bigl \Vert \bigl ({\varvec{S}}(\tilde{u},v_0,0)\bigr )'(s)\bigr \Vert _X+\frac{9C_4^2}{8} +\frac{3C_4}{2}+2\right) \Biggr \}ds \le \varepsilon . \end{aligned}$$

(84)

Hence, the number $\hat{\delta }_{r,\varepsilon ,R}>0$ and the uniform estimate (84) are desired ones. $\square$

Lemma 17

The following uniform estimate holds:

$$\begin{aligned} \sup _{\tilde{w}\,\in \, \mathscr {U}_0(u_0)} \left( \sup _{0\,\le \,t\,\le \,T} \ell (({\varvec{S}}_0(\tilde{w},v_0,0))(t))+ \int _0^T \Vert ({\varvec{S}}_0(\tilde{w},v_0,0))'(t)\Vert _Xdt \right) \le C_7. \end{aligned}$$

where $C_7$ is the same constant that is given in (A9).

Proof

From (70) and (A9), we get

$$\begin{aligned}{} & {} \sup _{\tilde{w}\,\in \, \mathscr {U}_0(u_0)} \left( \sup _{0\,\le \,t\,\le \,T} \ell (({\varvec{S}}_0(\tilde{w},v_0,0))(t))+ \int _0^T \Vert ({\varvec{S}}_0(\tilde{w},v_0,0))'(t)\Vert _Xdt \right) \\{} & {} \quad =\sup _{\tilde{w}\,\in \, \mathscr {U}_0(u_0)} \left( \sup _{0\,\le \,t\,\le \,T} \ell (({\varvec{S}}({\varvec{J}}_T\tilde{w},v_0,0))(t))+ \int _0^T \bigl \Vert ({\varvec{S}}({\varvec{J}}_T\tilde{w},v_0,0))'(t)\bigr \Vert _Xdt \right) \\{} & {} \quad =\sup _{\tilde{u}\,\in \, \mathscr {U}(u_0)} \left( \sup _{0\,\le \,t\,\le \,T} \ell (({\varvec{S}}(\tilde{u},v_0,0))(t))+ \int _0^T \bigl \Vert ({\varvec{S}}(\tilde{u},v_0,0))'(t)\bigr \Vert _Xdt \right) \le C_7. \end{aligned}$$

Hence, we see that this lemma holds. $\square$

4 An Evolution Inclusion on $V_0^*$

For any triplet $(t,z,\tilde{v}) \in [0,T] \times D(\phi _{c_0}) \times A$, we define a single-valued perturbation $g_0(t,z,\tilde{v}):V_0 \longmapsto \mathbb {R}$ by

$$\begin{aligned} \left( g_0 (t,z,\tilde{v})\right) (\tilde{\eta }):= \left\langle J g \left( t,(J \circ P_{c_0})^{-1}z,\tilde{v}\right) ,\tilde{\eta } \right\rangle _{V_0^*(\tilde{v}),\,V_0(\tilde{v})}, \quad \forall \tilde{\eta } \in V_0. \end{aligned}$$

(85)

Then, we get Lemma 18, which comes from (A4).

Lemma 18

For any triplet $(t,z,\tilde{v}) \in [0,T] \times D(\phi _{c_0}) \times A$ we have $g_0 (t,z,\tilde{v}) \in V_0^*(\tilde{v})$. Moreover, the following properties are satisfied:

• (a) The following inequality holds:

  $$\begin{aligned} \left\| g_0 \left( t,z,\tilde{v}\right) \right\| _{V_0^*} \le C_2 \ell \left( \tilde{v}\right) \sqrt{\left| \phi _{c_0} (z)\right| +C_5}, \quad \forall t \in [0,T],~\forall z \in D(\phi _{c_0}),~\forall \tilde{v} \in A, \end{aligned}$$

  where the constant $C_2$ is the same one that is given in (A1), the function $\ell$ and the constant $C_5$ are same ones that are given in (A4) and the function $\phi _{c_0}$ is the same one that is given in Lemma 12.

• (b) Assume that a sequence $\{\tilde{w}_m\}_{m \in \mathbb {N}} \subset C([0,T];V_0^*)$ and a function $\tilde{w} \in C([0,T];V_0^*)$ satisfy

  $$\begin{aligned} \tilde{w}_m \longrightarrow \tilde{w} \quad \text{ in } \quad C([0,T];V_0^*) \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

  (86)

  Then, for any $\tilde{v} \in A$ we have the following convergence as $m \rightarrow \infty$:

  $$\begin{aligned} {\varvec{G}}_0 \left( \tilde{w}_m,\tilde{v}\right) \longrightarrow {\varvec{G}}_0 \left( \tilde{w},\tilde{v}\right) \quad \text{ weakly } \text{ in } \quad L^2(0,T;V_0^*), \end{aligned}$$

  (87)

  where the operator ${\varvec{G}}_0 (\tilde{w},\tilde{v}) \in L^2(0,T;V_0^*)$ is defined by

  $$\begin{aligned} ({\varvec{G}}_0 (\tilde{w},\tilde{v}))(t):= & {} g_0 \left( t,\tilde{w}(t),S_0(\tilde{w};t,0)\tilde{v} \right) \nonumber \\= & {} Jg \left( t,({\varvec{J}}_T\tilde{w})(t),S({\varvec{J}}_T\tilde{w};t,0)\tilde{v}\right) \quad \text{ in } \quad V_0^*,\quad \forall t \in [0,T].\nonumber \\ \end{aligned}$$

  (88)

Proof

The perturbation $g_0 (t,z,\tilde{v}):V_0^*(\tilde{v}) \longmapsto \mathbb {R}$ is linear. Moreover, from (A1), (b) in (A4) and Lemma 5 we get

$$\begin{aligned} |(g_0 (t,z,\tilde{v}))(\eta )|\le & {} \left\| g \left( t,(J \circ P_{c_0})^{-1}z,\tilde{v}\right) \right\| _{V^*} \Vert \eta \Vert _V\\\le & {} C_2\ell (\tilde{v}) \sqrt{\left| \phi \left( (J \circ P_{c_0})^{-1} z \right) \right| +C_5}\cdot \Vert \eta \Vert _{V(\tilde{v})}\\= & {} C_2\ell (\tilde{v}) \sqrt{\left| \phi _{c_0} (z) \right| +C_5} \cdot \Vert \eta \Vert _{V(\tilde{v})}, \end{aligned}$$

which implies that $g_0 (t,z,\tilde{v})$ is bounded on $V_0^*(\tilde{v})$. Hence, we see that (a) holds.

Next, we show (b). Using Lemma 9, we see from (86) that the following convergence holds:

$$\begin{aligned} {\varvec{J}}_T \tilde{w}_m \longrightarrow {\varvec{J}}_T \tilde{w} \quad \text{ in } \quad C([0,T];V^*) \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

(89)

We see from (37) in the proof of Lemma 5 and (88) that for any $\eta \in L^2(0,T;V_0)$ the following equality holds:

$$\begin{aligned}{} & {} \left\langle {\varvec{G}}_0(\tilde{w}_m,\tilde{v}),\eta \right\rangle _{L^2(0,T;V_0^*),\,L^2(0,T;V_0)}\nonumber \\{} & {} \quad =\int _0^T \left\langle Jg \left( t,({\varvec{J}}_T\tilde{w}_m)(t),S({\varvec{J}}_T\tilde{w};t,0)\tilde{v}\right) , \eta (t) \right\rangle _{V_0^*,\,V_0}dt\nonumber \\{} & {} \quad =\int _0^T \left\langle g \left( t,({\varvec{J}}_T\tilde{w}_m)(t),S({\varvec{J}}_T\tilde{w};t,0)\tilde{v}\right) , \eta (t) \right\rangle _{V^*,\,V}dt\nonumber \\{} & {} \quad =\left\langle {\varvec{G}}({\varvec{J}}_T\tilde{w}_m,\tilde{v}),\eta \right\rangle _{L^2(0,T;V^*),\,L^2(0,T;V)}. \end{aligned}$$

(90)

From (c) in (A4) and (89), (90) that (87) holds. Hence, we see that (b) holds. $\square$

Now we consider the following Cauchy problem of an evolution inclusion on $V_0^*$ with quasi-variational structures, which is denoted by (E):

$$\begin{aligned} \text{(E) }\left\{ \begin{array}{l} w'(t)+\partial _{V_0^*(v(t))} \varphi _{c_0} \left( t,w,v_0;w(t)\right) +g_0(t,w(t),v(t)) \ni 0\\ \quad \text{ in } \quad V_0^* (v(t)), \quad \text{ a.a. }~t \in (0,T),\\ v(t)=S_0(w;t,0)v_0 \quad \text{ in } \quad A,\quad \forall t \in [0,T],\\ w(0)=(J \circ P_{c_0})u_0 \quad \text{ in } \quad V_0^*. \end{array}\right. \end{aligned}$$

Lemmas 7–17 and 18 enable us to apply the general theory, which is obtained in [9]. As its direct result, we derive Theorem 1, which gives the existence of strong solutions to (E) on [0, T].

Theorem 1

The Cauchy problem (E) has at least one strong solution w in the following quasi-variational sense:

• (w1) $w \in W^{1,2}(0,T;V_0^*)$.

• (w2) There exists a function $\eta \in L^2(0,T;V_0)$ such that the following properties hold;

  $$\begin{aligned} F_0(v(t)) \eta (t) \in \partial _{V_0^*(v(t))} \varphi _{c_0} (t,w,v_0\,;w(t)),\quad \text{ a.a. }~t \in (0,T), \end{aligned}$$

  (91)

  and

  $$\begin{aligned} w'(t)+F_0 (v(t)) \eta (t)+g_0(t,w(t),v(t))=0 \quad \text{ in } \quad V_0^*(v(t)),\quad \text{ a.a. }~t \in (0,T).\nonumber \\ \end{aligned}$$

  (92)

• (w3) $v(t)=S_0(w;t,0)v_0~~$in  A for all $t \in [0,T]$.

• (w4) $w(0)=(J \circ P_{c_0}) u_0$ in $V_0^*$.

• (w5) There exists a constant $R_1>0$ such that

  $$\begin{aligned} \Vert w'\Vert _{L^2(0,T;V_0^*)}+\sup _{0\,\le \,t\,\le \,T} \Vert w(t)\Vert _{V_0^*}+ \sup _{0\,\le \,t\,\le \,T} \bigl | \varphi _{c_0} \bigl (t,w,v_0;w(t)\bigr ) \bigr | \le R_1. \end{aligned}$$

5 A System on $V^*$ with Conservative Property

The main purpose of this section is to consider the relation between strong solutions to (M) and those of (E). In order to do this, at first we give the definition of strong solutions to (M) on [0, T].

Definition 1

A function u is called a strong solution to (M) on [0, T] if and only if the function u satisfied the following conditions (u1)–(u5):

• (u1) $u \in W^{1,2}(0,T;V^*)$.

• (u2) There exists a function $\eta \in L^2(0,T;V)$ such that

  $$\begin{aligned}{} & {} \eta (t) \in V \cap V^*(0),\quad \forall t \in [0,T], \end{aligned}$$

  (93)
  $$\begin{aligned}{} & {} F(v(t)) \eta (t) \in \partial _{V^*(v(t))} \varphi (t,u,v_0;u(t)),\quad \text{ a.a. }~t \in (0,T), \end{aligned}$$

  (94)

  and

  $$\begin{aligned}{} & {} u'(t)+\left( P \circ F(v(t))\right) \eta (t)+g(t,u(t),v(t)) =0 \quad \text{ in } \quad V^*(v(t)), \quad \text{ a.a. }~t \in (0,T).\nonumber \\ \end{aligned}$$

  (95)

• (u3) $v(t)=S(u;t,0)v_0~~$in  A  for all $t \in [0,T]$.

• (u4) $u(0)=u_0~~$in$~~V^*$.

• (u5) There exists a constant $R_2>0$ such that

  $$\begin{aligned} \Vert u'\Vert _{L^2(0,T;V^*)}+\sup _{0\,\le \,t\,\le \,T}\Vert u(t)\Vert _{V^*}+\sup _{0\,\le \,t\,\le \,T}|\varphi (t,u,v_0;u(t))| \le R_2. \end{aligned}$$

First of all, we show Lemma 19.

Lemma 19

Let u be a strong solution to (M) on [0, T] and $w:={\varvec{J}}_T^{-1}u$. Assume that elements $\xi ^* \in D(c_0)$ and $\xi \in V$ satisfy

$$\begin{aligned} F(v(t)) \xi \in \partial _{V^*(v(t))} \varphi (t,u,v_0\,;\xi ^*). \end{aligned}$$

(96)

Then, we have

$$\begin{aligned} (F_0(v(t)) \circ P) \xi \in \partial _{V_0^*(v(t))} \varphi _{c_0} \bigl (t,w,v_0\,;(J \circ P_{c_0})\xi ^*\bigr ). \end{aligned}$$

(97)

Proof

Assume that (96) holds. From Lemma 1 and the definition of $\partial _{V^*(v(t))} \varphi (t,\tilde{u},\tilde{v})$, we get the following inequality for all $\tilde{\xi }^* \in V^*$:

$$\begin{aligned} \bigl (F(v(t)) \xi ,\tilde{\xi }^*-\xi ^*\bigr )_{V^*(v(t))}= & {} \bigl \langle \tilde{\xi }^*-\xi ^*,\xi \bigr \rangle _{V^*,\,V}\nonumber \\\le & {} \varphi \bigl (t,u,v_0;\tilde{\xi }^*\bigr )-\varphi \left( t,u,v_0;\xi ^*\right) . \end{aligned}$$

(98)

Using Proposition 1, we see that the following inequality holds for all $\tilde{\xi } \in D(c_0)$:

$$\begin{aligned} \bigl \langle \tilde{\xi }^*-\xi ^*,\xi \bigr \rangle _{V^*,\,V}= & {} \bigl \langle \tilde{\xi }^*,\xi \bigr \rangle _{V^*,\,V} -\bigl \langle \xi ^*,\xi \bigr \rangle _{V^*,\,V}\nonumber \\= & {} \bigl \langle (J \circ P_{c_0}) \tilde{\xi }^*-(J \circ P_{c_0}) \xi ^*,P\xi \bigr \rangle _{V_0^*,\,V_0}. \end{aligned}$$

(99)

Because of $D_0=(J \circ P_{c_0})D(c_0)$ and the definition of $\varphi _{c_0}(t,w,v_0)$, we have $(J \circ P_{c_0})\xi ^* \in D_0$ and see from (98), (99) that the following inequality holds for all $\tilde{\eta }^* \in D_0$:

$$\begin{aligned}{} & {} \bigl ((F_0(v(t)) \circ P) \xi ,\tilde{\eta }^*-(J \circ P_{c_0})\xi ^*\bigr )_{V_0^*(v(t))}\\{} & {} \quad =\bigl \langle \tilde{\eta }^*-(J \circ P_{c_0})\xi ^*,P\xi \bigr \rangle _{V_0^*,\,V_0}= \bigl \langle (J \circ P_{c_0})^{-1} \tilde{\eta }^*-\xi ^*,\xi \bigr \rangle _{V^*,\,V}\\{} & {} \quad \le \varphi \bigl (t,u,v_0;(J \circ P_{c_0})^{-1}\tilde{\eta }^*\bigr )-\varphi \bigl (t,u,v_0;\xi ^*\bigr )\\{} & {} \quad \le \varphi _{c_0} \bigl (t,w,v_0;\tilde{\eta }^*\bigr )-\varphi _{c_0}\bigl (t,w,v_0;(J \circ P_{c_0})\xi ^*\bigr ), \end{aligned}$$

which implies that (97) holds. $\square$

One of the main theorems of this section is Theorem 2, which shows that (M) can be rewritten into the Cauchy problem of an evolution inclusion on $V_0^*$.

Theorem 2

Assume that a function u is a strong solution to (M) on [0, T]. Then, the function $w:={\varvec{J}}_T^{-1}u$ is a strong solution to (E) on [0, T].

Proof

Let a function u be a strong solution to (M) on [0, T]. Taking the duality pair between $V^*$ and V in both sides of (95) with $\eta _0$, we get

$$\begin{aligned} \begin{array}{l} \langle u'(t),\eta _0 \rangle _{V^*,\,V}+\langle (P \circ F(v(t))) \eta (t),\eta _0 \rangle _{V^*,\,V}+ \langle g(t,u(t),v(t)),\eta _0 \rangle _{V^*,\,V}=0,\\ \quad \text{ a.a. }~t \in (0,T), \end{array} \end{aligned}$$

hence, from (A4)

$$\begin{aligned} \langle u'(t),\eta _0 \rangle _{V^*,V}=\frac{\hbox {d}}{\hbox {d}t} (u(t),\eta _0)_H=0,\quad \text{ hence }, \quad u'(t) \in V^*(0),\quad 0 \le \forall t \le T, \end{aligned}$$

which implies that the following conservative property is satisfied:

$$\begin{aligned} \langle u(t),\eta _0 \rangle _{V^*,\,V}=\langle u_0,\eta _0 \rangle _{V^*,\,V}=c_0,\quad \forall t \in [0,T]. \end{aligned}$$

(100)

From (c) in (A3), (100) and (u4), (u5) in Definition 1 we have $u \in \mathscr {W}(u_0,R_2) \subset \mathscr {U}(u_0)$. Hence, we see from Lemma 9 and (70) that the function $w={\varvec{J}}_T^{-1}u \in \mathscr {U}_0(u_0)$ is well defined.

In the following argumentation, we show that the function w satisfies all conditions (w1)–(w5) in Theorem 1. First of all, from (u3), (u4) and (48), (49) we see that (w3) holds and get

$$\begin{aligned} \left( {\varvec{J}}_Tw\right) (0)=\left( J \circ P_{c_0} \right) ^{-1}w(0)=u_0 \quad \text{ in } \quad V^*, \end{aligned}$$

which implies that (w4) holds.

Secondly, from Lemma 5 we get

$$\begin{aligned} \Vert w(t)\Vert _{V_0^*} \le \Vert P_{c_0}u(t)\Vert _{V^*} \le \Vert u(t)\Vert _{V^*}+c_0\Vert \eta _0\Vert _{V^*},\quad 0 \le \forall t \le T, \end{aligned}$$

(101)

and

$$\begin{aligned} \lim _{h \rightarrow 0} \left\| \frac{w(t+h)-w(t)}{h}-Ju'(t) \right\| _{V_0^*}\le & {} \lim _{h \rightarrow 0} \left\| \frac{P_{c_0}u(t+h)-P_{c_0}u(t)}{h}-u'(t) \right\| _{V^*}\\= & {} \lim _{h \rightarrow 0} \left\| \frac{u(t+h)-u(t)}{h}-u'(t) \right\| _{V^*}=0, \end{aligned}$$

which implies

$$\begin{aligned} w'(t)=J u'(t)\quad \text{ in } \quad V_0^*, \qquad \Vert w'(t)\Vert _{V_0^*} \le \Vert u'(t)\Vert _{V^*},\qquad \text{ a.a. }~t \in (0,T). \end{aligned}$$

(102)

On the other hand, we have

$$\begin{aligned} (J \circ P_{c_0})u(t+h)-(J \circ P_{c_0})u(t)= & {} J \left( P_{c_0}u(t+h)-P_{c_0}u(t)\right) \\= & {} J(u(t+h)-u(t)) \quad \text{ in } \quad V_0^*, \end{aligned}$$

which implies

$$\begin{aligned} \left( (J \circ P_{c_0})u\right) '(t)=Ju'(t)\quad \text{ in } \quad V_0^*,\quad 0 \le \forall t \le T. \end{aligned}$$

(103)

Hence, we see from (102) and (103) that the following equality:

$$\begin{aligned} w'(t)=\left( (J \circ P_{c_0})u\right) '(t)=Ju'(t)\quad \text{ in } \quad V_0^*,\quad 0 \le \forall t \le T. \end{aligned}$$

(104)

Using (u5), we see from (50), (101)–(104) and Lemma 5 that the following boundedness holds:

$$\begin{aligned}{} & {} \Vert w'\Vert _{L^2(0,T;V_0^*)}+\sup _{0\,\le \,t\,\le \,T}\Vert w(t)\Vert _{V_0^*}+\sup _{0\,\le \,t\,\le \,T} |\varphi _{c_0}(t,w,v_0;w(t))|\\{} & {} \quad \le \Vert u'\Vert _{L^2(0,T;V^*)}+\sup _{0\,\le \,t\,\le \,T}\Vert u(t)\Vert _{V^*}+\sup _{0\,\le \,t\,\le \,T} |\varphi (t,u,v_0;u(t))|+c_0\Vert \eta _0\Vert _{V^*}, \end{aligned}$$

which implies that (w1) and (w5) in Theorem 1 hold.

Finally, we show (w2). From (95) in (u2) and Lemma 19, we get

$$\begin{aligned} F_0(v(t)) \eta (t) \in \partial _{V_0^*(v(t))} \varphi _{c_0} (t,w,v_0;w(t)),\quad \text{ a.a. }~t \in (0,T), \end{aligned}$$

which implies that (91) in (w2) holds, and the following equality:

$$\begin{aligned}{} & {} w'(t)+(J \circ P \circ F(v(t))) \eta (t)+J g(t,u(t),v(t))=0\\{} & {} \quad \text{ in } \quad V_0^*(v(t)),\quad \text{ a.a. }~t \in (0,T).\nonumber \\ \end{aligned}$$

Using Proposition 2, we see from (85) and (104) that (92) in (w2) holds. Hence, this theorem is proved. $\square$

In the rest of this section, we consider the equivalence between the systems (M) and (E). In order to do this, for any $t \in [0,T]$ and $u \in \mathscr {U}(u_0)$ we need to make the relation between the subdifferentials $\partial _{V^*(v(t))} \varphi (t,u,v_0)$ and $\partial _{V_0^*(v(t))} \varphi _{c_0} (t,w,v_0)$ clear, where the function w is given by $w={\varvec{J}}_T^{-1}u$. Then, we consider conjugate functions $\varphi ^*(t,u,v_0)$ and $\varphi _{c_0}^*(t,w,v_0)$ defined by the following ways:

$$\begin{aligned}{} & {} \varphi ^* (t,u,v_0;\xi ):= \sup \left\{ \left\langle \xi ^*,\xi \right\rangle _{V^*,V}-\varphi (t,u,v_0;\xi ^*);\, \xi ^* \in V^*\right\} ,\quad \forall \xi \in V,\\{} & {} \varphi _{c_0}^* (t,w,v_0;\eta ):= \sup \left\{ \left\langle \eta ^*,\eta \right\rangle _{V_0^*,V_0} -\varphi _{c_0} (t,w,v_0;\eta ^*);\,\eta ^* \in V_0^*\right\} ,\quad \forall \eta \in V_0.\nonumber \end{aligned}$$

(105)

Before giving Theorem 3, we show Propositions 3–5, which give the relation between the conjugate functions $\varphi ^* (t,u,v_0)$ and $\varphi _{c_0}^* (t,w,v_0)$.

Proposition 3

We have

$$\begin{aligned} \varphi _{c_0}^* \bigl (t,w,v_0\,;P\xi \bigr ) \le \varphi ^* \bigl (t,u,v_0\,;\xi \bigr )-c_0(\xi ,\eta _0)_H, \quad \forall \xi \in V. \end{aligned}$$

(106)

Proof

From (105), we have the following equality:

$$\begin{aligned} \bigl \langle \tilde{\xi }^*,\xi \bigr \rangle _{V^*,\,V}-\varphi \bigl (t,u,v_0\,;\tilde{\xi }^*\bigr ) \le \varphi ^* \bigl (t,u,v_0\,;\xi \bigr ), \quad \forall \tilde{\xi }^* \in V^*. \end{aligned}$$

(107)

Using Proposition 1 and Lemma 1, from (107) we get

$$\begin{aligned}{} & {} \bigl \langle (J \circ P_{c_0})\tilde{\xi }^*,P\xi \bigr \rangle _{V_0^*,\,V_0}+c_0(\xi ,\eta _0)_H -\varphi _{c_0} \bigl (t,w,v_0;(J \circ P_{c_0})\tilde{\xi }^*\bigr ) \le \varphi ^* (t,u,v_0;\xi \bigr ),\\{} & {} \quad \forall \tilde{\xi }^* \in D(c_0)=D(\varphi (t,u,v_0)) \cap V^*(c_0), \end{aligned}$$

hence,

$$\begin{aligned} \bigl \langle \tilde{\eta }^*,P\xi \bigr \rangle _{V_0^*,\,V_0}+c_0(\xi ,\eta _0)_H-\varphi _{c_0} \bigl (t,u,v_0;\tilde{\eta }^*\bigr ) \le \varphi ^* \bigl (t,v,v_0;\xi \bigr ), \quad \forall \tilde{\eta }^* \in D_0, \end{aligned}$$

which implies that (106) holds. $\square$

As you see from Proposition 3, it is quite important to find a sufficient condition so that for each $\xi \in V$ the following inequality holds:

$$\begin{aligned} \varphi _{c_0}^* (t,w,v_0\,;P\xi ) \ge \varphi ^* (t,u,v_0\,;\xi )-c_0(\xi ,\eta _0)_H. \end{aligned}$$

(108)

Actually, both Propositions 4 and 5 give sufficient conditions so that (108) holds.

Proposition 4

Assume that $D(c_0)$ is closed in H and $\xi \in V$ satisfies the following condition: there exist sequences $\{\varepsilon _m\}_{m \in \mathbb {N}} \subset (0,1)$ and $\{\xi _m^*\}_{m \in \mathbb {N}} \subset D(\varphi (t,u,v_0))$, which depend on $\xi$, such that the following convergences $(\star )$ are satisfied:

$$\begin{aligned} (\star )\left\{ \begin{array}{l} \displaystyle {\lim _{m \rightarrow \infty } \varepsilon _m = \lim _{m \rightarrow \infty } \bigl \Vert Q\xi _m^*-\xi _m^*\bigr \Vert _H=0},\\ \varphi ^* \left( t,u,v_0;\xi \right) -\varepsilon _m \le \left\langle \xi _m^*,\xi \right\rangle _{V^*,\,V} -\varphi \left( t,u,v_0;\xi _m^*\right) ,\quad \forall m \in \mathbb {N}, \end{array} \right. \end{aligned}$$

where the operator $Q:H \longmapsto D(c_0)$ is a projection operator defined by

$$\begin{aligned} \left\| Q\xi ^*-\xi ^*\right\| _H:=\inf \left\{ \bigl \Vert \tilde{\xi }^*-\xi ^* \bigr \Vert _H;\,\tilde{\xi }^* \in D(c_0) \right\} , \quad \forall \xi ^* \in H. \end{aligned}$$

Then, for this $\xi \in V$ we have (108).

Proof

We consider the sequences $\{\varepsilon _m\}_{m \in \mathbb {N}}$ and $\{\xi _m^*\}_{m \in \mathbb {N}}$ satisfying the property $(\star )$. Using Proposition 1, we see from $Q\xi _m^* \in D(c_0)$ that the following inequality holds for all $m \in \mathbb {N}$:

$$\begin{aligned} \left\langle Q\xi _m^*,\xi \right\rangle _{V^*,\,V}= & {} \left\langle (J \circ P \circ Q) \xi _m^*,P\xi \right\rangle _{V_0^*,\,V_0} +c_0 (\xi ,\eta _0)_H\nonumber \\\le & {} \left\langle (J \circ P \circ Q) \xi _m^*,P\xi \right\rangle _{V_0^*,\,V_0} -\varphi _{c_0} \left( t,w,v_0;(J \circ P \circ Q)\xi _m^*\right) \nonumber \\{} & {} +\varphi \left( t,u,v_0;(J \circ P_{c_0})^{-1}(J \circ P \circ Q)\xi _m^*\right) +c_0(\xi ,\eta _0)_H\nonumber \\\le & {} \varphi _{c_0}^* \left( t,w,v_0;P\xi \right) +\varphi \left( t,u,v_0;Q\xi _m^*\right) +c_0(\xi ,\eta _0)_H. \end{aligned}$$

(109)

From (b) in (A3), (14), (15), (109) and the inequality in $(\star )$ we get

$$\begin{aligned}{} & {} \varphi ^* \left( t,u,v_0;\xi \right) -\varepsilon _m\nonumber \\{} & {} \quad \le \left\langle Q\xi _m^*,\xi \right\rangle _{V^*,\,V}+ \left\langle \xi _m^*-Q\xi _m^*,\xi \right\rangle _{V^*,\,V}-\varphi \left( t,u,v_0;\xi _m^*\right) \nonumber \\{} & {} \quad \le \varphi \left( t,u,v_0;Q\xi _m^*\right) -\varphi \left( t,u,v_0;\xi _m^*\right) + \left\langle \xi _m^*-Q\xi _m^*,\xi \right\rangle _{V^*,\,V}\nonumber \\{} & {} \qquad +\varphi _{c_0}^* \left( t,w,v_0;P\xi \right) +c_0(\xi ,\eta _0)_H\nonumber \\{} & {} \quad \le (C_4+C_6\Vert \xi \Vert _V) \left\| Q\xi _m^*-\xi _m^*\right\| _H+\varphi _{c_0}^* \left( t,w,v_0;P\xi \right) +c_0(\xi ,\eta _0)_H. \end{aligned}$$

(110)

Taking the limit $m \rightarrow \infty$ in both sides of (110) and using $(\star )$, we get (108). $\square$

Proposition 5

Assume that $\xi \in V$ satisfies the following condition $(\star \star )$:

$$\begin{aligned} (\star \star )\left( \begin{array}{l} \text{ There } \text{ exists } \text{ sequences }\, \{\varepsilon _m\}_{m \in \mathbb {N}}, \{\tilde{\xi }_m^*\}_{m \in \mathbb {N}}\, \text{ and } \,\{\bar{\xi }_m^*\}_{m \in \mathbb {N}}\, \text{ such } \text{ that }\\ \text{ the } \text{ following } \text{ properties } \text{ are } \text{ satisfied }:\\ (a)~\varepsilon _m >0\quad \text{ and } \quad \varepsilon _m \longrightarrow 0 \quad \text{ as } \quad m \rightarrow \infty ,\\ (b)~\tilde{\xi }_m^*,~\bar{\xi }_m^* \in D(\varphi (t,u,v_0))~\text{ for } \text{ all }~m \in \mathbb {N},\\ (c)~\bigl \langle \tilde{\xi }_m^*,\xi \bigr \rangle _{V^*\,V} \le c_0 \le \bigl \langle \bar{\xi }_m^*,\xi \bigr \rangle _{V^*\,V}, ~\text{ for } \text{ all }~m \in \mathbb {N},\\ (d)~\text{ The } \text{ following } \text{ inequality } \text{ holds } \text{ for } \text{ all }~m \in \mathbb {N}:\\ \quad \varphi ^*(t,u,v_0;\xi )-\varepsilon _m\\ \quad \le \min \bigl \{ \bigl \langle \tilde{\xi }_\varepsilon ^*,\xi \bigr \rangle _{V^*\,V} -\varphi \bigl (t,u,v_0;\tilde{\xi }_m^*\bigr ),\, \bigl \langle \bar{\xi }_m^*,\xi \bigr \rangle _{V^*\,V} -\varphi \bigl (t,u,v_0;\bar{\xi }_m^*\bigr )\bigr \}. \end{array}\right) \end{aligned}$$

Then, for this $\xi \in V$ we have (108).

Proof

Since the effective domain $D(\varphi (t,u,v_0))$ is convex, we see from (b) and (c) in the condition $(\star \star )$ that for each $m \in \mathbb {N}$ there exists a number $\lambda _m \in [0,1]$ such that

$$\begin{aligned} \lambda _m \tilde{\xi }_\varepsilon ^*+(1-\lambda _m) \bar{\xi }_\varepsilon ^* \in D(c_0) \cap D(\varphi (t,u,v_0)), \end{aligned}$$

hence,

$$\begin{aligned} \bigl \langle \lambda _m \tilde{\xi }_m^*+(1-\lambda _m) \bar{\xi }_m^*,\eta _0 \bigr \rangle _{V^*,\,V}=c_0. \end{aligned}$$

(111)

From (d) in the condition $(\star \star )$ and (111), we get

$$\begin{aligned} \varphi ^* \bigl (t,u,v_0;\xi \bigr )-\varepsilon _m\le & {} \bigl \langle \lambda _m \tilde{\xi }_m^* +(1-\lambda _m) \bar{\xi }_m^*,\xi \bigr \rangle _{V^*,\,V}\\{} & {} -\bigl \{ \lambda _m \varphi \bigl (t,u,v_0;\tilde{\xi }_m^*)+(1-\lambda _m) \varphi \bigl (t,u,v_0;\bar{\xi }_m^*\bigr ) \bigr \}. \end{aligned}$$

Using Proposition 1 and the convexity of $\varphi (t,u,v_0)$, we get the following inequality for all $m \in \mathbb {N}$:

$$\begin{aligned} \varphi ^*(t,u,v_0;\xi )-\varepsilon _m\le & {} \bigl \langle (J \circ P_{c_0}) (\lambda _m \tilde{\xi }_m^* +(1-\lambda _m) \bar{\xi }_m^*),P\xi \bigr \rangle _{V_0^*,\,V_0}+c_0(\xi ,\eta _0)_H\\{} & {} -\varphi (t,u,v_0;\lambda _m \tilde{\xi }_m^* +(1-\lambda _m) \bar{\xi }_m^*)\\\le & {} \bigl \langle (J \circ P_{c_0}) (\lambda _m \tilde{\xi }_m^*+(1-\lambda _m) \bar{\xi }_m^*), P\xi \bigr \rangle _{V_0^*,\,V_0}+c_0(z,\eta _0)_H\\{} & {} -\varphi _{c_0} \bigl (t,{\varvec{J}}_T^{-1}u,v_0;(J \circ P_{c_0})(\lambda _m \tilde{\xi }_m^* +(1-\lambda _m) \bar{\xi }_m^*)\bigr )\\\le & {} \varphi _{c_0}^* (t,w,v_0;P\xi )+c_0(\xi ,\eta _0)_H. \end{aligned}$$

Taking the limit $m \rightarrow \infty$ in the above inequality and using (a) in the condition $(\star \star )$, we see that (108) holds. $\square$

At last, we show Theorem 3.

Theorem 3

Let w be a strong solution to (E) on [0, T], and assume that for a.a. $t \in (0,T)$ either ($\star$) or ($\star \star$) holds for $\eta (t) \in V \cap V^*(0)$, where the function $\eta$ is the same one that is given in (w2) of Theorem 1. Then, the function $u:={\varvec{J}}_Tw$ is a strong solution to (M) on [0, T].

Proof

Throughout this proof, we assume that $(\star$) holds. Since the function w is a strong solution to (E) on [0, T], we see that (u1), (u3), (u4) and (u5) in Definition 1 are satisfied by repeating the argumentation similar to the proof of Theorem 2. Hence, we only show (u2) in this proof. From (29), we have

$$\begin{aligned} \Vert \eta (t)\Vert _V=\Vert \eta (t)\Vert _{V_0},\quad \forall t \in [0,T], \end{aligned}$$

which implies $\eta \in L^2(0,T;V)$. Moreover, from Proposition 2 and (w2) in Theorem 1 we get

$$\begin{aligned} \left( J \circ P \circ F(v(t))\right) \eta \in \partial _{V_0^*(v(t))} \varphi _{c_0} (t,w,v(t);w(t)),\quad \text{ a.a. }~t \in (0,T), \end{aligned}$$

(112)

and

$$\begin{aligned}{} & {} w'(t)+\left( J \circ P \circ F(v(t))\right) \eta (t)+g_0(t,w(t),v(t))=0\nonumber \\{} & {} \quad \text{ in } \quad V_0^*(v(t)),\quad \text{ a.a. }~t \in (0,T), \end{aligned}$$

(113)

From (85), (102) and (113), we get

$$\begin{aligned} \begin{array}{l} u'(t)+(P \circ F(v(t))) \eta (t)+g(t,u(t),v(t))=0\\ \quad \text{ in } \quad V^*(v(t)),\quad \text{ a.a. }~t \in (0,T), \end{array} \end{aligned}$$

which implies that (95) in (u2) holds. Moreover, from (91) we get

$$\begin{aligned} \varphi _{c_0}^*(t,w,v_0;\eta (t))+\varphi _{c_0}(t,w,v_0;w(t))= & {} \left( F_0(v(t)) \eta (t),w(t)\right) _{V_0^* (v(t))}\nonumber \\= & {} \bigl \langle w(t),\eta (t)\bigr \rangle _{V_0^*,\,V_0},\quad \text{ a.a. }~t \in (0,T).\nonumber \\ \end{aligned}$$

(114)

Using Proposition 2, we get

$$\begin{aligned} \left( J^{-1} \circ F_0(v(t))\right) \eta (t)=\left( P \circ F(v(t))\right) \eta (t) \quad \text{ in } \quad V^*,\quad \text{ a.a. }~t \in (0,T). \end{aligned}$$

Applying Propositions 1 and 4, from (106), (108) and (114), we get

$$\begin{aligned}{} & {} \varphi ^*(t,u,v_0;\eta (t))+\varphi (t,u,v_0;u(t))\nonumber \\{} & {} \quad =\varphi _{c_0}^*(t,w,v_0;\eta (t))+\varphi _{c_0}(t,w,v_0;w(t)) =\bigl \langle (J \circ P_{c_0})u(t),\eta (t)\bigr \rangle _{V_0^*,\,V_0}\nonumber \\{} & {} \quad =\bigl \langle (J \circ P)u(t),\eta (t)\bigr \rangle _{V_0^*,\,V_0}= \bigl \langle u(t),\eta (t)\bigr \rangle _{V^*,\,V}=\bigl ( F(v(t))\eta (t),u(t) \bigr )_{V^*(v(t))}.\nonumber \\ \end{aligned}$$

(115)

Equality (115) implies that (94) in (u2) holds. $\square$

Remark 1

As you see from the proof of Theorem 3, for $\xi \in V_0$ it is enough that the equality

$$\begin{aligned} \varphi ^*(t,u,v_0;P\xi )=\varphi _{c_0}^*(t,w,v_0;\xi )+c_0(\xi ,\eta _0)_H \end{aligned}$$

is satisfied, that is,

$$\begin{aligned} \varphi ^*(t,u,v_0;\eta )=\varphi _{c_0}^*(t,w,v_0;\eta ). \end{aligned}$$

6 An Example

In this section, we consider a haptotaxis tumor invasion system (P):={(116) – (120)}, which is originally proposed in [1]:

$$\begin{aligned}{} & {} u_t=\nabla \cdot \{D(v_1)\nabla u^*-u \nabla v_1\}+u^*(t)-\frac{1}{|\varOmega |}\int _\varOmega u^*(t)\,dx,\nonumber \\{} & {} \quad \text{ a.e. } \text{ in } \quad \varOmega \times (0,T), \end{aligned}$$

(116)

$$\begin{aligned}{} & {} u^* \in \partial _{\mathbb {R}} \hat{\beta } (v_1;u),\quad \text{ a.e. } \text{ in } \quad \varOmega \times (0,T), \end{aligned}$$

(117)

$$\begin{aligned}{} & {} v_t=Av+B(u,v), \quad \text{ a.e. } \text{ in } \quad \varOmega \times (0,T), \end{aligned}$$

(118)

where $v:=(v_1,v_2)$ and the operators A and B(u, v) are defined by

$$\begin{aligned} A:=\left( \begin{array}{cc} 0~&{}~0\\ 0~&{}~d_2\varDelta -b \end{array} \right) , \quad B(u,v):=\left( \begin{array}{c} -a v_1 v_2\\ cu \end{array} \right) , \end{aligned}$$

with boundary conditions

$$\begin{aligned} \{D(v_1) \nabla u^*-u\nabla v_1\} \cdot \nu =\nabla v_2 \cdot \nu =0, \quad \text{ a.e. } \text{ on } \quad \varGamma \times (0,T), \end{aligned}$$

(119)

and initial conditions

$$\begin{aligned} (u(0),v(0))=(u_0,v_0),\quad \text{ a.e. } \text{ in } \quad \varOmega . \end{aligned}$$

(120)

Since from (116), (119) and (120), the system (P) has a mass-conservative property;

$$\begin{aligned} \int _\varOmega u(t) \hbox {d}x=\int _\varOmega u_0\, \hbox {d}x,\quad 0 \le \forall t \le T, \end{aligned}$$

(121)

the system (P) is considered as one of the typical examples of (M).

In order to analyze the system (P) by using the abstract results in the previous sections of this paper, we take $H:=L^2(\varOmega )$ and $V:=H^1(\varOmega )$, and consider the dual space $V^*:=(H^1(\varOmega ))^*$ of V. While using the same notation in the previous sections as far as possible, we give assumptions and the results, which have already obtained in [7]. Although most of the proofs of Lemmas below are omitted in this paper and entrusted to [7], we give short proofs for some lemmas with paying attention to the mass-conservative property since the tumor invasion model treated in [7] does not have the mass-conservative property.

• (H1) A function $D:\mathbb {R} \longrightarrow \mathbb {R}$ is Lipschitz continuous, and there exist constants $d_1>0$ and $d_2>0$ such that

  $$\begin{aligned} d_1 \le D(v_1) \le d_2,\quad \forall v_1 \in \mathbb {R}. \end{aligned}$$

• (H2) For each $v_1 \ge 0$, a proper l.s.c. convex function $\hat{\beta }(v_1):\mathbb {R} \longrightarrow \mathbb {R} \cup \{\infty \}$ is given by

  $$\begin{aligned} \hat{\beta }(v_1;r):=\left\{ \begin{array}{ll} r^{v_1+2},\quad &{}\text{ if } \quad r \in [0,1],\\ \infty ,\quad &{}\text{ if } \quad r \in (-\infty ,0) \cup (1,\infty ). \end{array} \right. \end{aligned}$$

• (H3) $a>0,~b>0,~c>0,~d_2>0$ are constants.

• (H4) $u_0 \in L^\infty (\varOmega )$ satisfies the following doble obstacle conditions:

  $$\begin{aligned} 0 \le u_0 \le 1\quad \text{ a.e. } \text{ in } \quad \varOmega . \end{aligned}$$

  (122)

  Hence, we assume that a constant $c_0$ satisfies

  $$\begin{aligned} 0<c_0<\sqrt{|\varOmega |}, \end{aligned}$$

  (123)

  and define a closed subset $D(c_0)$ of H by

  $$\begin{aligned} D(c_0):=\left\{ \tilde{u}_0 \in H;\,\tilde{u}_0\, \text{ satisfies } \text{(122) }~~\text{ and }~~ \frac{1}{\sqrt{|\varOmega |}} \int _\varOmega \tilde{u}_0 \,dx=c_0\right\} . \end{aligned}$$

  Using a function $\eta _0 \in V$ given by

  $$\begin{aligned} \eta _0 (x)=\frac{1}{\sqrt{|\varOmega |}},\quad \forall x \in \varOmega , \end{aligned}$$

  (124)

  which satisfies $\Vert \eta _0\Vert _H=1$, we see that for any $u_0 \in D(c_0)$ the mass-conservative property (121) can be rewritten by

  $$\begin{aligned} (u(t),\eta _0)_H=(u_0,\eta _0)_H=c_0,\quad 0 \le \forall t \le T. \end{aligned}$$

  (125)

• (H5) We choose $X:=L^\infty (\varOmega ) \times H$ and $A:=A_1 \times A_2$, respectively, where $A_1$ and $A_2$ are given by

  $$\begin{aligned} A_1= & {} \{\tilde{v}_1 \in W^{1,\,\infty }(\varOmega );\,0 \le \tilde{v}_1 \le 1~~\text{ a.e. } \text{ in }~~\varOmega \},\\ A_2= & {} \{\tilde{v}_2 \in W^{1,\,\infty }(\varOmega );\,0 \le \tilde{v}_2~~\text{ a.e. } \text{ in }~~\varOmega \}. \end{aligned}$$

  Then, we assume $v_0:=(v_{0,1},v_{0,2}) \in A$.

Now, we give the definition of a strong solution (u, v) to (P) on [0, T].

Definition 2

A pair (u, v) is called a strong solution to (P) on [0, T] if and only if the following properties are satisfied:

• (P1) $u \in W^{1,2}(0,T;V^*) \cap L^\infty (\varOmega \times (0,T))~~$with$~~u(0)=u_0~~$in$~~V^*$.

• (P2) There exists a function $u^* \in L^2(0,T;V)$ such that (116) and (117) are satisfied in the following quasi-variational sense for a.a. $t \in (0,T)$ and all $\xi \in V$:

  $$\begin{aligned}{} & {} \left\langle u'(t),\xi \right\rangle _{V^*,\,V}+\int _\varOmega D(v_1(t)) \nabla u^*(t) \cdot \nabla \xi \,dx+ \int _\varOmega u^*(t)\xi \,dx\nonumber \\{} & {} \quad -\frac{1}{|\varOmega |} \left( \int _\varOmega u^*(t)\,dx\right) \left( \int _\varOmega \xi \,dx\right) -\int _\varOmega u(t) \nabla v_1(t) \cdot \nabla \xi dx=0, \end{aligned}$$

  (126)

  and the constraint condition

  $$\begin{aligned} u^* \in \partial _{\mathbb {R}} \hat{\beta }(v_1;u) \quad \text{ a.e. } \text{ in }~~\varOmega \times (0,T). \end{aligned}$$

  (127)

• (P3) $v:=(v_1,v_2)$ satisfies the following properties:

  • (3a) $v_1 \in C([0,T];C(\overline{\varOmega }) \cap V) \cap W^{1,\infty }(0,T;L^\infty (\varOmega ))$, and it is expressed by

    $$\begin{aligned} v_1(x,t)=v_{1,0}(x)\exp \left( -a\int _0^t v_2(x,s)\, \hbox {d}s\right) ,\quad \text{ a.a. }~(x,t) \in \varOmega \times (0,T). \end{aligned}$$

  • (3b) $v_2 \in W^{1,2}(0,T;H) \cap L^\infty (0,T;W^{1,\infty }(\varOmega ))$, and it is expressed by

    $$\begin{aligned} v_2(t)=e^{t(d_2\varDelta _N-b)}v_{2,0}+c\int _0^t e^{(t-s)(d_2\varDelta _N-b)}u(s)\, \hbox {d}t, \quad 0 \le \forall t \le T, \end{aligned}$$

    where $\varDelta _N$ is a Laplacian with a homogeneous Neumann boundary condition.

In the following argumentation, we check that all conditions (A1)–(A9) are satisfied in order to apply Theorem 1. Firstly, we introduce a quasi-variational structure for inner products of V. For each $\tilde{v}=(\tilde{v}_1,\tilde{v}_2) \in A$, we denote by $V(\tilde{v})$, which is independent of the second component $\tilde{v}_2 \in A_2$, a real Hilbert space $V(\tilde{v})$ whose inner product $(\cdot ,\cdot )_{V(\tilde{v})}$ is defined by

$$\begin{aligned} (\xi _1,\xi _2)_{V(\tilde{v})}:=\int _\varOmega D(\tilde{v}_1) \nabla \xi _1\cdot \nabla \xi _2\, \hbox {d}x +\int _\varOmega \xi _1 \xi _2 \, \hbox {d}x,\quad \forall \xi _1,\xi _2 \in V. \end{aligned}$$

From [7, Lemma 3.1], we get Lemma 20 as a direct consequence of (H1), which implies that (A1) is satisfied.

Lemma 20

(cf. (A1)) The family $\{(\cdot ,\cdot )_{V(\tilde{v})}\,;\,\tilde{v} \in A\}$ of inner products on V is uniformly equivalent to the usual inner product $(\cdot , \cdot )_V$, where $(\cdot , \cdot )_V$ is given by

$$\begin{aligned} (\xi _1,\xi _2)_V:=\int _\varOmega \nabla \xi _1\cdot \nabla \xi _2\, \hbox {d}x +\int _\varOmega \xi _1 \xi _2 \, \hbox {d}x,\quad \forall \xi _1,\xi _2 \in V. \end{aligned}$$

Then, for each $\tilde{v} \in A$ we denote by $V^*(\tilde{v})$ and $F(\tilde{v})$ the dual space of $V(\tilde{v})$ and the duality map from $V(\tilde{v})$ onto $V^*(\tilde{v})$, respectively.

Secondly, in order to define a dynamical system on (X, A), for each $\bar{T} \in [0,T]$ we consider a subset $\mathscr {U}_{\bar{T}}$ of $C([0,\bar{T}];V^*)$ by

$$\begin{aligned} \mathscr {U}_{\bar{T}}:=\{\tilde{u} \in C([0,\bar{T}];V^*);\,0 \le \tilde{u} \le 1~~\text{ a.e. } \text{ in }~~\varOmega \}. \end{aligned}$$

For each $\tilde{v} \in A$, $\tilde{u} \in \mathscr {U}_{\bar{T}}$ and $s,\,t$ with $0 \le s \le t \le \bar{T}$, we consider the following subsystem $\text{(P) }_{\text{ sub }}$ of (P):

$$\begin{aligned} \text{(P) }_{\text{ sub }}\left\{ \begin{array}{ll} \hat{v}_t=A\hat{v}+B(\tilde{u},\hat{v}), \quad &{}\text{ a.e. } \text{ in } \quad \varOmega \times (0,\bar{T}),\\ \nabla \hat{v}_2 \cdot \nu =0, \quad &{}\text{ a.e. } \text{ on } \quad \varGamma \times (s,\bar{T}),\\ \hat{v}(0)=\tilde{v},\quad &{}\text{ a.e. } \text{ in } \quad \varOmega . \end{array} \right. \end{aligned}$$

Since $\text{(P) }_{\text{ sub }}$ has a unique solution $\hat{v}(\bar{T},\tilde{u},\tilde{v},s)$ on $[s,\bar{T}]$ (cf. (P3) in Definition 2), we define a single-valued solution operator $S(\tilde{u};t,s):A \longmapsto X$ by

$$\begin{aligned} S(\tilde{u};t,s)\tilde{v}=(S_1(\tilde{u};t,s)\tilde{v},S_2(\tilde{u};t,s)\tilde{v}_2)= \hat{v}(\bar{T},\tilde{u},\tilde{v},s;t),\quad s \le \forall t \le \bar{T}, \end{aligned}$$

and consider the class $\{\{S(\tilde{u}\,;t,s)\,;\,0 \le s \le t \le \bar{T}\}\,;\,0 \le \bar{T} \le T,~\tilde{u} \in \mathscr {U}_{\bar{T}}\}$. Then, we get Lemma 21, which is originally obtained in [7, Proposition 2.7].

Lemma 21

For each $\bar{T} \in [0,T]$, the class $\{\{S(\tilde{u}\,;t,s)\,;\,0 \le s \le t \le \bar{T}\}\,;\,\tilde{u} \in \mathscr {U}_{\bar{T}}\}$ satisfies the following properties:

• (1) $S(\tilde{u}\,;t,s)A \subset A$ for all $s,\,t,\,\bar{T}$ with $0 \le s \le t \le \bar{T} \le T$ and $\tilde{u} \in \mathscr {U}_{\bar{T}}$, which implies that A is invariant under the operator $S(\tilde{u};t,s)$.

• (2) (cf. (a) in (A2)) Assume that a sequence $\{(\tilde{u}_m,\tilde{v}_m)\}_{m \in \mathbb {N}} \subset \mathscr {U}_{\bar{T}} \times A$ and a pair $(\tilde{u},\tilde{v}) \in \mathscr {U}_{\bar{T}} \times A$ satisfy

  $$\begin{aligned} (\tilde{u}_m,\tilde{v}_m) \longrightarrow (\tilde{u},\tilde{v}) \quad \text{ in } \quad C([0,\bar{T}];V^*) \times X \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

  Then, for any $s \in [0,\bar{T}]$ we have the following convergence as $m \rightarrow \infty$:

  $$\begin{aligned}{} & {} {\varvec{S}}(\tilde{u}_m,\tilde{v}_m,s) \longrightarrow {\varvec{S}}(\tilde{u},\tilde{v},s)\\{} & {} \quad \text{ in } \quad C([s,\bar{T}];L^\infty (\varOmega ) \cap V) \times \left( C([s,\bar{T}];H) \cap L^2(s,\bar{T};V) \right) . \end{aligned}$$

• (3) All properties (b)–(f) in (A2) are satisfied.

Thirdly, we define a class $\mathscr {C}:=\{\phi (\tilde{v})\,;\,\tilde{v} \in A\}$. For each $\tilde{v} \in A$, a proper l.s.c. convex function $\phi (\tilde{v})$ on $V^*(\tilde{v})$ is defined by

$$\begin{aligned} \phi (\tilde{v};\xi ^*):=\left\{ \begin{array}{ll} \displaystyle {\int _\varOmega \hat{\beta }(\tilde{v}_1(x);\xi ^*(x))dx},\quad &{}\text{ if }\quad \xi ^* \in D,\\ \infty ,&{}\text{ if } \quad \xi ^* \in V^*(\tilde{v}) \setminus D, \end{array} \right. \end{aligned}$$

where the effective domain D of $\phi (\tilde{v})$ is given by

$$\begin{aligned} D:=\left\{ \xi ^* \in H\,;\,0 \le \xi ^* \le 1 \quad \text{ a.e. } \text{ in } \quad \varOmega \right\} . \end{aligned}$$

(128)

Then, we get Lemma 22.

Lemma 22

(cf. [7, Sect. 3]) The following properties are satisfied.

• (1) There exists a proper l.s.c. convex function $\phi$ on $V^*$ such that the following properties are satisfied:

  • (1a) (cf. (a1) in (A3)) The following inequality holds for all $\tilde{v} \in A$:

    $$\begin{aligned} \phi (\xi ^*) \le \phi (\tilde{v};\xi ^*),\quad \forall \xi ^* \in V^*. \end{aligned}$$

  • (1b) The property (a2) in (A3) is satisfied.

  • (1c) (cf. (a3) in (A3)) We have $D(\phi ):=\{\xi ^* \in V^*;\,\phi (\xi ^*)<\infty \}=D$ and

    $$\begin{aligned} |\phi (\xi ^*)| \le |\varOmega |,\quad \forall \xi ^* \in D. \end{aligned}$$

  • (1d) The property (a4) in (A3) is satisfied.

• (2) (cf. (b) in (A3)) There exists a constant $C_{12}>0$ such that

  $$\begin{aligned} \left| \phi (\tilde{v};\xi _1^*)-\phi (\tilde{v};\xi _2^*)\right| \le C_{12} \Vert \xi _1^*-\xi _2^*\Vert _H,\quad \forall \tilde{v} \in A,\quad \forall \xi _1^*,\,\xi _2^* \in D. \end{aligned}$$

Proof

(1) We define a function $\phi :V^* \longmapsto \mathbb {R} \cup \{\infty \}$ by

$$\begin{aligned} \phi (\xi ^*):=\left\{ \begin{array}{ll} \displaystyle {\int _\varOmega \left\{ \xi ^*(x)^2-1 \right\} \hbox {d}x},\quad &{}\text{ if }\quad \xi ^* \in D,\\ \infty ,&{}\text{ if } \quad \xi ^* \in V^* \setminus D. \end{array}\right. \end{aligned}$$

Then, (1a) and (1c) are obtained as direct consequences of the following inequality:

$$\begin{aligned} -1 \le r^2-1 \le 0 \le r^{v_1+2} \le 1,\quad 0 \le \forall r \le 1,~0 \le \forall v_1. \end{aligned}$$

Next, we show (1b). Since we have

$$\begin{aligned} |\phi (\xi ^*)|=|\varOmega |-\Vert \xi ^*\Vert _H^2,\quad \forall \xi ^* \in D, \end{aligned}$$

for any $r \ge 0$ we get

$$\begin{aligned} \left\{ \xi ^* \in V^*;\,\Vert \xi ^*\Vert _{V^*} \le r,~|\phi (\xi ^*)| \le r\right\} \subset \left\{ \xi ^* \in D;\, |\varOmega |-r \le \Vert \xi ^*\Vert _H^2 \le |\varOmega |\right\} , \end{aligned}$$

hence, (1b) is satisfied.

Thirdly, we show (1d). We have the following inequality for all $\xi _1^*,\,\xi _2^* \in D$:

$$\begin{aligned} |\phi (\xi _1^*)-\phi (\xi _2^*)|\le & {} \int _\varOmega \left| \{\xi _1^*(x)\}^2-\{\xi _2^*(x)\}^2\right| \, \hbox {d}x\\\le & {} 2 \int _\varOmega |\xi _1^*(x)-\xi _2^*(x)|\, \hbox {d}x \le 2\sqrt{|\varOmega |} \cdot \Vert \xi _1^*-\xi _2^*\Vert _H, \end{aligned}$$

which implies that $\phi$ is continuous on D with respect to the strong topology of H.

(2) Using the mean value theorem, we have

$$\begin{aligned} \bigl |r_1^{v_1+2}-r_2^{v_1+2}\bigr | \le 2^{v_1+1}(v_1+2)|r_1-r_2|,\quad \forall r_1,\,r_2 \in [0,1],\quad \forall v_1 \ge 0. \end{aligned}$$

Hence, from (128) we get the following inequality for all $\tilde{v} \in A$ and $\xi _1^*,\,\xi _2^* \in D$:

$$\begin{aligned} |\phi (\tilde{v};\xi _1^*)-\phi (\tilde{v};\xi _2^*)|\le & {} \int _\varOmega 2^{\tilde{v}_1(x)+1} (\tilde{v}_1(x)+2) |\xi _1^*(x)-\xi _2^*(x)| \hbox {d}x\\\le & {} \sqrt{|\varOmega |} \cdot 2^{\Vert \tilde{v}_1\Vert _{L^\infty }+1} \cdot (\Vert \tilde{v}_1\Vert _{L^\infty }+2) \cdot \Vert \xi _1^*-\xi _2^*\Vert _H, \end{aligned}$$

which implies that (2) is satisfied because of $v_1 \in A_1$ with the definition of $A_1$. $\square$

Since the class $\mathscr {C}$ is independent of $\mathscr {U}_T(u_0)$, we see that (d) in (A3) is expressed in other words as in Lemma 23 whose proof is omitted in this paper.

Lemma 23

The following properties are satisfied:

• (1) (cf. (d) in (A3)) Assume that a sequence $\{\tilde{v}_m\}_{m \in \mathbb {N}} \subset A$ and $\tilde{v} \in A$ satisfy the following convergence as $m \rightarrow \infty$:

  $$\begin{aligned} \tilde{v}_m:=(\tilde{v}_{1,\,m},\tilde{v}_{2,\,m}) \longrightarrow \tilde{u}:=(\tilde{v}_1,\tilde{v}_2)\quad \text{ in } \quad \left( L^\infty (\varOmega ) \cap V\right) \times H. \end{aligned}$$

  Then, we have

  $$\begin{aligned} \phi (\tilde{v}_m) \longrightarrow \phi (\tilde{v}) \quad \text{ on } \quad V^*(\tilde{v}) \end{aligned}$$

  in the following Mosco sense:

  • (2a) For any $\xi ^* \in D$, we have

    $$\begin{aligned} \lim _{m \rightarrow \infty } \phi (\tilde{v}_m;\xi ^*)=\phi (\tilde{v};\xi ^*). \end{aligned}$$

  • (2b) For any subsequence $\{\tilde{v}_{m_k}\}_{k \in \mathbb {N}}$ of $\{\tilde{v}_m\}_{m \in \mathbb {N}}$, we have

    $$\begin{aligned} \phi (\tilde{v};\xi ^*) \le \liminf _{k \rightarrow \infty } \phi (\tilde{v}_{m_k};\xi _k^*), \end{aligned}$$

    whenever a sequence $\{\xi _k^*\}_{k \in \mathbb {N}} \subset V^*$ and an element $\xi ^* \in V^*$ satisfy

    $$\begin{aligned} \xi _m^* \longrightarrow \xi ^* \quad \text{ weakly } \text{ in } \quad V^*(\tilde{v}) \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

Using $\{\{S(\tilde{u}\,;t,s)\,;\,0 \le s \le t \le T\}\,;\,\tilde{u} \in \mathscr {U}_T(u_0)\}$ and $\mathscr {C}$, we define a class $\mathscr {K}(u_0,\tilde{v})$, which gives a quasi-variational structure for convex functions on $V^*$, by

$$\begin{aligned} \begin{array}{l} \mathscr {K}(u_0,\tilde{v}):=\left\{ \{\varphi (t,\tilde{u},\tilde{v});\,0 \le t \le T\};\, \tilde{u} \in \mathscr {U}_T(u_0),~\tilde{v} \in A\right\} ,\\ \varphi (t,\tilde{u},\tilde{v}):=\phi (S_1(\tilde{u};t,0)\tilde{v}),\quad 0 \le \forall t \le T,~ \forall \tilde{u} \in \mathscr {U}(u_0),~\forall \tilde{v} \in A. \end{array} \end{aligned}$$

However, we have already succeeded in showing Lemma 24 below in [7, Proposition 4.2] without using Lemma 23, which does not seem so important to verify properties of the class $\mathscr {K}(u_0,\tilde{v})$. Hence, we omit its proof in this paper and entrust it to [7].

Lemma 24

(cf. [7, Proposition 4.2]) The following properties are satisfied:

• (1) (cf. (c) in (A3)) For any $\tilde{u}_1,\,\tilde{u}_2 \in \mathscr {U}_T(u_0)$ we assume that there exists $T_0 \in [0,T]$ such that

  $$\begin{aligned} \tilde{u}_1(t)=\tilde{u}_2(t) \quad \text{ in } \quad V^*,\quad 0 \le \forall t \le T_0. \end{aligned}$$

  Then, we have

  $$\begin{aligned} \varphi (t,\tilde{u}_1,\tilde{v})=\varphi (t,\tilde{u}_2,\tilde{v})\quad \text{ on } \quad V^*,\quad 0 \le \forall t \le T_0,~\forall \tilde{v} \in A. \end{aligned}$$

• (2) (cf. (d) in (A3)) Assume that a sequence $\{(\tilde{u}_m,\tilde{v}_m)\}_{m \in \mathbb {N}} \subset \mathscr {U}_T(u_0) \times A$ and a pair $(\tilde{u},\tilde{v}) \in \mathscr {U}_T(u_0) \times A$ satisfy the following convergence as $m \rightarrow \infty$:

  $$\begin{aligned} \begin{array}{c} (\tilde{u}_m,\tilde{v}_m):=(\tilde{u}_m,\tilde{v}_{1,\,m},\tilde{v}_{2,\,m}) \longrightarrow (\tilde{u},\tilde{v}):=(\tilde{u},\tilde{v}_1,\tilde{v}_2)\\ \text{ in } \quad C([0,T];V^*) \times \left( L^\infty (\varOmega ) \cap V\right) \times H. \end{array} \end{aligned}$$

  Then, for each $t \in [0,T]$ we have

  $$\begin{aligned} \varphi (t,\tilde{u}_m,\tilde{v}_m) \longrightarrow \varphi (t,\tilde{u},\tilde{v}) \quad \text{ on } \quad V^*(\tilde{v}) \end{aligned}$$

  in the following Mosco sense:

  • (2a) For any $\xi ^* \in D$ we have

    $$\begin{aligned} \lim _{m \rightarrow \infty } \varphi (t,\tilde{u}_m,\tilde{v}_m;\xi ^*)=\varphi (t,\tilde{u},\tilde{v};\xi ^*). \end{aligned}$$

  • (2b) For any subsequence $\{(\tilde{u}_{m_k},\tilde{v}_{m_k})\}_{k \in \mathbb {N}}$ of $\{(\tilde{u}_m,\tilde{v}_m)\}_{m \in \mathbb {N}}$ we have

    $$\begin{aligned} \varphi (t,\tilde{u},\tilde{v};\xi ^*) \le \liminf _{k \rightarrow \infty } \varphi (t,\tilde{u}_{m_k},\tilde{v}_{m_k};\xi _k^*), \end{aligned}$$

    whenever a sequence $\{\xi _k^*\}_{k \in \mathbb {N}} \subset V^*$ and an element $\xi ^* \in V^*$ satisfy

    $$\begin{aligned} \xi _m^* \longrightarrow \xi ^* \quad \text{ weakly } \text{ in } \quad V^*(\tilde{v}) \quad \text{ as } \quad m \rightarrow \infty . \end{aligned}$$

Moreover, we get Lemma 25.

Lemma 25

The following properties are satisfied:

• (1) (cf. (A6) and (A7)) There exists a constant $C_{13}>0$, which is independent of the choice of $\tilde{u} \in \mathscr {U}_T$, such that the following property is satisfied:

  $$\begin{aligned} \left( \begin{array}{l} \text{ for } \text{ any }\, s,\,t \in [0,T]\, \text{ and }\, \xi ^* \in D\, \text{ there } \text{ exists }\, \xi ^*(\tilde{u},s,t) \in D\, \text{ such } \text{ that }\\ |\varphi (t,\tilde{u},v_0;\xi ^*(\tilde{u},s,t))-\varphi (s,\tilde{u},v_0;\xi ^*)|+ \Vert \xi ^*(\tilde{u},s,t)-\xi ^*\Vert _{L^\infty (\varOmega )} \le C_{13}|t-s|. \end{array} \right) \qquad \quad \end{aligned}$$

• (2) The property (A8) is satisfied.

Proof

From [7, Lemma 4.3], we have already gotten (1), which directly implies that not only (A6) but also all conditions in (A7) are satisfied. Hence, we omit their proofs in this paper and entrust them to [7].

In the rest of this proof, we show (2). We consider a function $h \in L^\infty (\varOmega )$, which is defined by

$$\begin{aligned} h(x):=c_0 \eta _0 (x)=\frac{c_0}{\sqrt{|\varOmega |}},\quad \forall x \in \varOmega . \end{aligned}$$

(129)

Then, from (124) to (129) we get

$$\begin{aligned} \left\langle h,\eta _0\right\rangle _{V^*,\,V}=c_0, \end{aligned}$$

and from (123)

$$\begin{aligned} 0<h(x)<1,\quad \forall x \in \varOmega , \end{aligned}$$

which implies $h \in D$ and

$$\begin{aligned} \varphi (t,\tilde{u},v_0;h)=\int _\varOmega \{h(x)\}^{(S_1(\tilde{u}\,;\,\,t,\,0)v_0)(x)+2}dx \le |\varOmega |. \end{aligned}$$

Hence, we see from Lemma 22 that this lemma holds. $\square$

Fourthly, in order to treat the haptotaxis term in (116), we define a single-valued perturbation $g:D \times A_1 \longmapsto V^*$ by

$$\begin{aligned} \left\langle g(z,\tilde{v}_1),\xi \right\rangle _{V^*,\,V}:=-\int _\varOmega z(x) \nabla \tilde{v}_1(x) \cdot \nabla \xi (x)\,dx, \quad \forall \xi \in V, \end{aligned}$$

(130)

which is independent of $v_2 \in A_2$. Then, we get Lemma 26.

Lemma 26

The perturbation g defined by (130) satisfies the following properties:

• (1) (cf. (a) in (A4)) We have $\left\langle g(z,\tilde{v}_1),\eta _0\right\rangle _{V^*,\,V}=0$ for all $(z,\tilde{v}_1) \in D \times A_1$.

• (2) (cf. (b) in (A4)) There exists a function $\ell :A_1 \longmapsto \mathbb {R}$ such that for any $r \ge 0$ a level set $\{\tilde{v} \in A_1;\,\ell (\tilde{v}_1) \le r\}$ is compact in $L^\infty (\varOmega )$ and

  $$\begin{aligned} \Vert g(z,\tilde{v}_1)\Vert _{V^*} \le \ell (\tilde{v}_1),\quad \forall (z,\tilde{v}_1) \in D \times A_1. \end{aligned}$$

• (3) (cf. (c) in (A4)) Assume that a sequence $\{\tilde{u}_m\}_{m \in \mathbb {N}} \subset \mathscr {U}_T$ and a function $\tilde{u} \in \mathscr {U}_T$ satisfy the following convergence as $m \rightarrow \infty$:

  $$\begin{aligned} \tilde{u}_m \longrightarrow \tilde{u} \quad \text{ in } \quad C([0,T];V^*) \quad \text{ and } \quad *-\text{ weakly } \text{ in } \quad L^\infty (\varOmega \times (0,T)). \end{aligned}$$

  Then, for any $\tilde{v} \in A$ we have

  $$\begin{aligned} {\varvec{G}}(\tilde{u}_m,\tilde{v}) \longrightarrow {\varvec{G}}(\tilde{u},\tilde{v}) \quad \text{ weakly } \text{ in } \quad L^2(0,T;V^*) \quad \text{ as } \quad m \rightarrow \infty , \end{aligned}$$

  where for each $\bar{u} \in \mathscr {U}_T$ the function ${\varvec{G}}(\bar{u},\tilde{v}) \in L^2(0,T;V^*)$ is defined by

  $$\begin{aligned} {\varvec{G}}(\bar{u},\tilde{v};t):=g(\bar{u}(t),S_1(\bar{u};t,0)\tilde{v}) \quad \text{ in } \quad V^*,\quad \forall t \in [0,T]. \end{aligned}$$

• (4) (cf. (A9)) There exists a constant $C_{14}>0$ such that

  $$\begin{aligned} \sup _{\mathscr {U}_T(u_0)} \left( \sup _{0\,\le \,t\,\le \,T} \ell (({\varvec{S}}(\tilde{u},v_0,0))(t))+ \int _0^T \Vert ({\varvec{S}}(\tilde{u},v_0,0))'(t)\Vert _X\,dt \right) \le C_{14}, \end{aligned}$$

  where the function $\ell$ is the same one that is obtained in (2).

Proof

(1) From (124) and (130), we easily get

$$\begin{aligned} \left\langle g(z,\tilde{v}_1),\eta _0 \right\rangle _{V^*,\,V}:=-\int _\varOmega z(x) \nabla \tilde{v}_1(x) \cdot \nabla \eta _0 (x)\,dx=0, \end{aligned}$$

which implies that (1) is satisfied.

(2) From (128) and (130), we get

$$\begin{aligned} \bigl |\left\langle g(z,\tilde{v}_1),\xi \right\rangle _{V^*,\,V}\bigr |\le & {} \int _\varOmega z(x) \Vert \nabla \tilde{v}_1(x)\Vert _{\mathbb {R}^N} \Vert \nabla \xi (x)\Vert _{\mathbb {R}^N}\,dx\\\le & {} \sqrt{|\varOmega |} \cdot \Vert \tilde{v}_1\Vert _{W^{1,\infty }(\varOmega )} \Vert \xi \Vert _V, \end{aligned}$$

which implies

$$\begin{aligned} \Vert g(z,\tilde{v}_1)\Vert _{V^*} \le \sqrt{|\varOmega |} \cdot \Vert \tilde{v}_1\Vert _{W^{1,\infty }(\varOmega )}=:\ell (\tilde{v}_1), \end{aligned}$$

hence we see that (2) is satisfied.

(3) Using the same argumentation in the proof of [7, Lemma 5.5], we can show this property. Hence, its proof is omitted in this paper and entrusted to [7].

(4) This has been obtained in [7, Proposition 2.7]; hence, we also omit its proof in this paper and entrusted to [7]. $\square$

Now, we define $H_0$ and $V_0$ by

$$\begin{aligned} H_0:=\left\{ z \in H;\,\int _\varOmega z\,dx=0\right\} ,\quad V_0:=V \cap H_0.\qquad \text{(cf. } \text{(28) } \text{ and } \text{(29)) }. \end{aligned}$$

and for each $\tilde{v} \in A$

$$\begin{aligned} V_0(\tilde{v}):=V(\tilde{v}) \cap H_0. \end{aligned}$$

We denote by $V_0^*(\tilde{v})$ and $F_0(\tilde{v})$ the dual space of $V_0(\tilde{v})$ and the duality map from $V_0(\tilde{v})$ onto $V_0^*(\tilde{v})$, respectively. Moreover, we define $P_{c_0}$, J, $S_0(w;t,s)$, $\varphi _{c_0}(t,w,v_0)$ and $g_0 (w,v_1)$, which are used in Theorem 4, by the similar way to those in Lemmas 3, 5, (49), (71) and (85). Under these settings, from Lemmas 21 to 26 except Lemma 23 we get Theorem 4 as a direct consequence of Theorem 1.

Theorem 4

For any $(u_0,v_0) \in D(c_0) \times A$, there exists a pair (w, v) such that the following properties are satisfied:

• (1) $w \in W^{1,2}(0,T;V_0^*) \cap L^\infty (0,T;H_0)$.

• (2) There exists a function $\eta \in L^2(0,T;V_0)$ such that

  $$\begin{aligned}{} & {} F_0(v(t)) \eta (t) \in \partial _{V_0^*(v(t))} \varphi _{c_0} (t,w,v_0;w(t)),\quad \text{ a.a. }~t \in (0,T),\\{} & {} w'(t)+F_0 (v(t)) \eta (t)+g_0(w(t),v_1(t))=0 \quad \text{ in } \quad V_0^*(v(t)),\quad \text{ a.a. }~t \in (0,T). \end{aligned}$$

• (3) $v(t)=(v_1(t),v_2(t))=S_0(w;t,0)v_0~~$in  A  for all $t \in [0,T]$.

• (4) $w(0)=(J \circ P_{c_0}) u_0~~$in$~~V_0^*$.

• (5) There exists a constant $R_1>0$ such that

  $$\begin{aligned} \Vert w'\Vert _{L^2(0,T;V_0^*)}+\sup _{0\,\le \,t\,\le \,T} \Vert w(t)\Vert _{V_0^*}+ \sup _{0\,\le \,t\,\le \,T} \bigl | \varphi _{c_0} \bigl (t,w,v_0;w(t)\bigr ) \bigr | \le R_1. \end{aligned}$$

Remark 2

We note that the boundedness (5) of Theorem 4 implies the regularity $w \in L^\infty (0,T;H_0)$ because of the property (1a) in Lemma 22.

Moreover, repeating the similar argumentation to the proof of Theorem 2, we get Theorem 5.

Theorem 5

For each $(u_0,v_0) \in D(c_0) \times A$ we assume that a pair (u, v) is a strong solution to (P) on [0, T]. Then, the pair $(w,v):=({\varvec{J}}_T^{-1}u,v)$ satisfies all properties (1)–(5) in Theorem 4.

Proof

We see from (125) that $w(t):=({\varvec{J}}_T^{-1}u)(t)=(J \circ P_{c_0})u(t)$ is well defined for all $t \in [0,T]$. Since from the definition of the duality map $F(v(t)):V(v(t)) \longmapsto V^*(v(t))$ we have the following equality;

$$\begin{aligned}{} & {} \frac{1}{|\varOmega |}\left( \int _\varOmega u^*(t)\, \hbox {d}x\right) \left( \int _\varOmega \xi \, \hbox {d}x\right) = (u^*(t),\eta _0)_H (\eta _0,\xi )_H\\{} & {} \quad =\left( \int _\varOmega D(v_1(t)) \nabla u^*(t) \cdot \nabla \eta _0 \, \hbox {d}x+\int _\varOmega \eta _0 u^*(t)\, \hbox {d}x \right) (\eta _0,\xi )_H\\{} & {} \quad =\left\langle \left\langle F(v(t))u^*(t),\eta _0\right\rangle _{V^*,\,V} \eta _0,\xi \right\rangle _{V^*,\,V}, \end{aligned}$$

we see that (126) in (2) in Definition 2 can be expressed by

$$\begin{aligned}{} & {} u'(t)+(P \circ F(v(t)))u^*(t)+g(u(t),v_1(t))=0\nonumber \\{} & {} \quad \text{ in } \quad V^*(v(t)),\quad \text{ a.a. }~t \in (0,T). \end{aligned}$$

(131)

Moreover, from [7, Proposition 3.5] we have already known that (127) is rewritten into the following expression:

$$\begin{aligned}{} & {} F(v(t)) u^*(t) \in \partial _{V^*(v(t))} \varphi (t,u(t),v_0;u(t))\nonumber \\{} & {} \quad \text{ in } \quad V^*(v(t)), \quad \text{ a.a. }~t \in (0,T). \end{aligned}$$

(132)

Hence, from (131) and (132) we can repeat the same argumentation to the proof of Theorem 2 and see that this theorem holds. $\square$

Remark 3

Unfortunately, until now the author has not been able to succeed in showing that either the condition $(\star )$ in Proposition 4 or the condition $(\star \star )$ in Proposition 5 is satisfied. As a result, Theorem 3 has not been shown yet. But the author thinks that the method proposed in this paper seems to be one of useful tools when quasi-variational evolution systems with conservative quantities are analyzed.