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This article is cited in 1 scientific paper (total in 1 paper) A class of evolution differential inclusion systems Jing Zhaoa, Zhenhai Liubc, N. S. Papageorgioud a School of Mathematics and Quantitative economics, Guangxi University of Finance and Economics, Nanning, Guangxi, P. R. China b Center for Applied Mathematics of Guangxi, Guangxi Minzu University, Nanning, Guangxi, P. R. China c Center for Applied Mathematics of Guangxi, Yulin Normal University, Yulin, P. R. China d Department of Mathematics, National Technical University, Athens, Greece
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2024, Volume 88, Issue 2, DOI: https://doi.org/10.4213/im9450 
Bibliographic databases:
    § 1. IntroductionFor any Banach space $B$, we denote by $B^*$ the dual space of $B$, by $\langle \,{\cdot}\,,{\cdot}\,\rangle_B$ the duality pairing of $B$ and $B^*$, and by $\|\,{\cdot}\,\|_B$ the norm on $B$. For simplicity, we sometimes omit the subscript $B$ when no confusion arises. The symbols $\rightharpoonup$ and $\to$ denote the weak and strong convergence in a given Banach space, respectively. In this paper, $X$, $Y$, $Z$, and $V$ are real reflexive separable Banach spaces, $H$ a separable Hilbert space which is identified with its dual $H^*$ such that $V\subseteq Z\subseteq H\subseteq Z^*\subseteq V^*$, where all embeddings are dense and continuous and $V$ embeds to $Z$ compactly. We set $I=[0,b]$, where $b>0$. We use the standard Bochner–Lebesgue function spaces $\mathcal{V}=L^2(I,V)$, $\mathcal{Z}=L^2(I,Z)$, $\mathcal{H}=L^2(I,H)$, $\mathcal{V}^{\ast}=L^2(I,V^{\ast})$, and $\mathcal{W}=\{v\in \mathcal{V}\colon v'\in \mathcal{V^{*}}\}$, where the time derivative $v'=\partial v/\partial t$ is understood in the sense of vector-valued distributions. It follows from the reflexivity of $V$ that both $\mathcal{V}$ and its dual space $\mathcal{V^{\ast}}$ are reflexive Banach spaces. The space $\mathcal{W}$ endowed with the graph norm $\|v\|_{\mathcal{W}}=\|v\|_{\mathcal{V}}+ \|v'\|_{\mathcal{V}^*}$ is a separable reflexive Banach space. Identifying $\mathcal{H}$ with its dual, we obtain the continuous embeddings $\mathcal{W}\subset\mathcal{V}\subset\mathcal{Z} \subset\mathcal{H}\subset\mathcal{Z}^*\subset\mathcal{V^{\ast}}$. Moreover, the embedding $\mathcal{W}\subset\mathcal{Z}$ is compact and $\mathcal{W}\subset C(I,H)$ is continuous, $C(I,H)$ being the space of continuous functions on $I$ with values in $H$. The main purpose of this paper is to study the following system of a differential inclusion with a history-dependent operator combined with an evolutionary non-linear inclusion:
$$
\begin{equation}
\begin{cases} Ax(t)+ F(t,x(t),(\mathcal Rx)(t),u(t))\ni x'(t),\ t\in I, \\ u'(t)+K(t,x(t),u(t))+G(t,x(t),u(t))\ni h(t), \ \text{a.e. }t\in I, \\ x(0)=x_0\in X \text{ and } u(0)=u_0\in H. \end{cases}
\end{equation}
\tag{1.1}
$$
Here, $A\colon D(A) \subseteq X \to X$ is the infinitesimal generator of a $C_0$-semigroup $\{T(t)\}_{t\geqslant 0}$ on a reflexive Banach space $X$, $\mathcal R$ is a so-called history-dependent operator, $K$ is a pseudomonotone operator, and $F, G$ are multivalued non-monotone perturbations. The study of system (1.1) is motivated by its relations to a class of multivalued problems and discontinuous dynamical systems, which have been gaining much attention in current literature. Our main goal is to prove existence of solutions to system (1.1). It should also be mentioning that problem (1.1) is a generalized and complicated system which contains several interesting problems as special cases. (a) In a specific setting, for example, $A$ and $K$ are partial differential operators of elliptic type, $F$, $G$ are single-valued, our problem is in the form of a coupled parabolic equations. Parabolic systems have been used to model reversible reaction processes for a set of chemical species (see, for example, [1]), chemotaxis phenomena (in the form of a parabolic-parabolic Keller–Segel system [2]). In particular with chemotaxis systems, there have been extensive studies on global existence and blow-up behaviour (see [3]–[7] and the references cited there). Regarding system (1.1), it can be considered as a non-smooth counterpart of parabolic systems. The multivalued term $F$, $G$ in system (1.1) can be seen as a regularization of a discontinuous non-linearity by means of Filippov [8], and the appearance of multimaps $F=\partial f$, $G=\partial g$ ($\partial$ may stand for subdifferential of a convex function or Clarke generalized subdifferentail of a local Lipschitz function) are due to the lack of regularity of the functions $f$, $g$. (b) If $F$ is single-valued independent of history-dependent operators and without the time derivative item $u'$, the multivalued mapping $G(t,x,{\cdot}\,)$ is a subdifferential $\partial \varphi (t,x,{\cdot}\,)$ of a proper convex lower semicontinuous functional $\varphi (t,x,{\cdot}\,)$, then system (1.1) becomes the differential variational inequality
$$
\begin{equation}
\begin{cases} x'(t)=Ax(t)+ F(t,x(t),u(t)), \ t\in I, \\ K(t,x(t),u(t))+\partial\varphi (t,x(t),u(t))\ni h(t), \text{ a.e. }t\in I, \\ x(0)=x_0. \end{cases}
\end{equation}
\tag{1.2}
$$
Equivalently,
$$
\begin{equation}
\begin{cases} x'(t)=Ax(t)+ F(t,x(t),u(t)), \ t\in I, \\ \langle K(t,x(t),u(t)), v-u(t)\rangle_V+\varphi (t,x(t),v)-\varphi (t,x(t),u(t)) \\ \quad \geqslant \langle h(t),v-u(t)\rangle_V\ \forall\, v\in V, \text{ a.e. }t\in I, \\ x(0)=x_0. \end{cases}
\end{equation}
\tag{1.3}
$$
As is well-known, differential variational inequalities were introduced as a powerful mathematical tool of variational analysis in order to investigate real-life problems coming from operations research, engineering, and physical sciences, which were first systematically discussed by Pang and Stewart [9] in finite-dimensional Banach spaces. Various aspects related to differential variational inequalities have been investigated so far, but in the finite-dimensional framework (see, for instance, [9]–[15] and the references therein). Recently, the theory of differential variational inequalities has been extended to the more general level of infinite-dimensional Banach or Hilbert spaces (see [16]–[20] and the references cited there). (c) If $F$ is single-valued independent of history-dependent operators and without the time derivative item $u'$, the multivalued mapping $G(t,x,{\cdot}\,)$ is a Clarke generalized subdifferential $\partial_{\mathrm{cl}}J (t,x,{\cdot}\,)$ of a local Lipschitz functional $J(t,x,{\cdot}\,)$, then system (1.1) becomes the differential hemivariational inequality
$$
\begin{equation}
\begin{cases} x'(t)=Ax(t)+ F(t,x(t),u(t)), \ t\in I, \\ K(t,x(t),u(t))+\partial_{\mathrm{cl}}J (t,x(t),u(t))\ni h(t), \text{ a.e. }t\in I, \\ x(0)=x_0. \end{cases}
\end{equation}
\tag{1.4}
$$
Equivalently,
$$
\begin{equation}
\begin{cases} x'(t)=Ax(t)+ F(t,x(t),u(t)), \ t\in I, \\ \langle K(t,x(t),u(t)), v-u(t)\rangle_V+J^\circ(t,x(t),v-u(t)) \\ \quad \geqslant \langle h(t),v-u(t)\rangle_V\ \forall\, v\in V, \text{ a.e. }t\in I, \\ x(0)=x_0. \end{cases}
\end{equation}
\tag{1.5}
$$
The notion of differential hemivariational inequalities was first introduced by Liu et al. [21], in which the corresponding energy functionals are non-convex (comparing with the energy functionals being convex). Their derivation is based on properties of the Clarke subgradient defined for locally Lipschitz functions. Liu et al. in [21] studied a differential hemivariational inequality in Banach spaces which is constituted by a non-linear evolution equation and a hemivariational inequality of elliptic type rather than of parabolic type (see also [22]–[25]). (d) If $F$ is single-valued and $F(t,x,u)$ is measurable on $I$ continuous on $X\times H$ and Lipschitz on $X$ independent of history-dependent operators, and the main operator $K$ is independent of the time $t$ and the state $x$, the multivalued mapping $G(t,x,{\cdot}\,)$ is a Clarke generalized subdifferential $\partial_{\mathrm{cl}}J (x,{\cdot}\,)$ of a local Lipschitz functional $J(x,{\cdot}\,)$, then system (1.1) becomes the differential parabolic hemivariational inequality
$$
\begin{equation}
\begin{cases} x'(t)=Ax(t)+ F(t,x(t),u(t)), \ t\in I, \\ u'(t)+K(u(t))+\partial_{\mathrm{cl}}J (x(t),u(t))\ni h(t), \text{ a.e. }t\in I, \\ x(0)=x_0,\ u(0)=u_0. \end{cases}
\end{equation}
\tag{1.6}
$$
Equivalently,
$$
\begin{equation*}
\begin{cases} x'(t)=Ax(t)+ F(t,x(t),u(t)), \ t\in I, \\ u'(t)+\langle K(u(t)), v-u(t)\rangle_V+J^\circ(x(t),u(t),v-u(t)) \\ \quad \geqslant \langle h(t),v-u(t)\rangle_V \ \forall \, v\in V, \text{ a.e. }t\in I, \\ x(0)=x_0,\ u(0)=u_0. \end{cases}
\end{equation*}
\tag{1.7}
$$
Migorski–Zeng [26] was the first to investigate the above abstract system consisting of a hemivariational inequality of parabolic type combined with a non-linear evolution equation in the framework of evolution triples of spaces. Very recently, Migorski [27] has studied a system of coupled non-linear first order history-dependent evolution inclusions in the framework of evolution triples of spaces. The multivalued terms are of the Clarke subgradient or of the convex subdifferential form. Under strong monotone assumptions, using a surjective result for multivalued mappings and a fixed point argument for history-dependent operators, Migorski proved that the system has a unique solution. Anh–Ke [28] studied a class of differential variational inequalities of Parabolic-Parabolic Type, an evolution model formulated by a parabolic differential inclusion and a parabolic variational inequality. They proved the solvability of their problem and showed that the solution set generates an $m$-semiflow. In addition, the existence of a global attractor for the $m$-semiflow was proved by using the technique of measure of non-compactness. Finally, we mention [29], [30], where related problems are studied for more specific cases. Over many years, various publications contributed to the growth of the theoretical field of differential hemivariational inequalities, based on the new results in non-smooth calculus. This latter has proved to be a very good tool to describe the dynamic behaviour of various complex phenomena in the real world (biology, medicine, engineering, mathematics, and physics). Indeed, the above special cases have not been fully studied yet. The main novelties of our present work are three-fold. First, we will discuss an evolutionary non-linear inclusion involving pseudomonotone operators coupled with a first order differential inclusion with a history-dependent operator in Banach spaces. Note that our present problems considered in this paper are more general and more complicated compared with the publications in this area. There is still little information on this kind of problems. A hybrid iterative system will be introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, that is, the Rothe method and a feedback iterative technique. Through a limiting procedure for solutions of the hybrid iterative system, the existence results will be proved. Second, we apply the Rothe method to an abstract complicated system involving a non-linear mapping $F(x,\mathcal Rx,u)\colon X\times Y\times H\to X$, which is a multivalued mapping in comparison with recent works involving single mappings in [26] without the history-dependent item $\mathcal Rx$ and satisfying uniformly a Lipschitz continuous condition for the first argument $x$. Besides, we consider all the non-linear operators explicitly depending on time $t$. Third, under very general assumptions, we can only show the existence of the corresponding nonliear differential inclusion without uniqueness. In this case, we do not impose any convexity assumption on the multivalued mapping $\def\,{\kern 0.4mm} u\,{\to}\, F(t,x,\mathcal Rx,u)$ in comparison with our previous works [16]–[18], [21]. The paper is organized as follows. In § 2, we present the basic hypotheses, some basic definitions and preliminary facts, which will be used throughout the following sections. Section 3 is devoted to the proof of the solvability of the differential inclusions with history-dependent operators and the properties of its solution set. In § 4, we introduce a hybrid iterative system by time-discretization method and show that the hybrid iterative system has at least one solution. Next we give a priori estimates of the approximate solutions and study the strong convergence to a solution of system (1.1). § 2. Assumptions and preliminariesIn what follows, we denote by $\mathcal{P}(B)$ ($\mathcal{P}_{\mathrm{f}}(B)$, $\mathcal{P}_{\mathrm{fc}}(B)$, $\mathcal{P}_{\mathrm{fbc}}(B)$, $\mathcal{P}_{\mathrm{(w)k(c)}}(B)$) the collections of all non-empty (respectively, non-empty closed, non-empty closed and convex, non-empty closed bounded and convex, non-empty (weakly) compact (convex)) subsets of a Banach space $B$. The following results are useful for our later discusses. Lemma 2.1 (see [31], Proposition 2.23). Let $Y_1$, $Y_2$ be Hausdorff topological spaces and $F\colon Y_1\to \mathcal P_f(Y_2)$ be closed and locally compact (that is, for every $y\in Y_1$, there exists a neighbourhood $\mathcal N(y)$ of $y$ such that $\overline{F(\mathcal N(y))}\in \mathcal P_k(Y_2)$), then $F(\,{\cdot}\,)$ is upper semicontinuous. We recall a convergence result for multimaps (see, for example, [32], p. 19, and [33], Proposition 2). Lemma 2.2. Let $X$, $Y$ be two Banach spaces, and $F\colon (0, T )\times X\to 2^Y$ a multimap such that (i) the values of $F$ are non-empty closed convex subsets of $Y$; (ii) for each $x\in X, F(\,{\cdot}\,,x)$ is measurable; (iii) for a.e. $t\in (0,T), F(t,{\cdot}\,)$ is upper semicontinuous from $X$ into $Y_w$. Let $x_n\colon (0,T)\to X$, $y_n\colon (0,T)\to Y$, $n\in \mathbb N$, be measurable maps such that $x_n$ converges a.e. on $(0,T)$ to a map $x\colon (0,T)\to X$ and $y_n$ converges weakly in $L^1(0,T;Y)$ to $y\colon (0,T)\to Y$. If $y_n(t)\in F(t,x_n(t))$ for all $n\in\mathbb N$ and a.e. $t\in (0,T)$, then $y(t)\in F(t,x(t))$ for a.e. $t\in (0,T)$. Definition 2.3. Let $X$ is a reflexive Banach space and let $F\colon X\to 2^{X^{*}}$ be a multimap with the graph $\mathrm{Gr}(F)=\{(x,x^{\ast})\in X\times X^{\ast}|x^{\ast}\in F(x)\}$. We say that $F$ is (i) monotone if, for all $(x,x^{\ast}),(y,y^{\ast})\in \mathrm{Gr}(F)$, (ii) maximal monotone if $F$ is monotone and if the condition $(x,x^{\ast})\in X\times X^{\ast}$,
$$
\begin{equation*}
\langle x^{\ast}-y^{\ast}, x-y\rangle_{X}\geqslant0\quad\forall\, (y,y^{\ast})\in \mathrm{Gr}(F),
\end{equation*}
\notag
$$
implies that $(x,x^{\ast})\in \mathrm{Gr}(F)$; (iii) pseudomonotone if
(iv) generalized pseudomonotone if the above condition $(a)$ is satisfied and, for all sequences $\{x_n\}\subset X$, $\{x^{\ast}_n\}\subset X^{\ast}$ with $x^{\ast}_n\in Fx_n$, $x_n\rightharpoonup x$ in $X$, $x^{\ast}_n\rightharpoonup x^{\ast}$ in $X^{\ast}$ and we have $x^{\ast}\in Fx$ and $\langle x^{\ast}_n, x_n\rangle_{X}\to \langle x^{\ast}, x\rangle_{X}$.Lemma 2.4 (see [34], Proposition 3.58). Let $X$ be a reflexive Banach space and $F\colon X\to 2^{X^*}$ be a multimap. (i) If $F$ is a pseudonmonotone multimap, then $F$ is generalized pseudomonotone. (ii) If $F$ is a generalized pseudomonotone multimap which is bounded (that is, maps bounded sets into bounded ones) and for each $u\in X$, $Fu$ is non-empty, closed and convex subset of $X^*$, then $F$ is pseudomonotone. Pseudomonotone operators exhibit remarkable surjectivity properties. In particular, we need the following result (see, for example, [35], Theorem 1.3.70). Theorem 2.5. Let $V$ be a real, reflexive Banach space, $B\colon V\to 2^{V^*}$ be pseudomonotone and coercive. Then $B$ is surjective, that is, for any $f\in V^*$, there exists $u\in V$ such that $f\in Bu$. In order to show our main results, we need the following assumptions. Assumption $\textrm{H}(A)$. $A$ is an infinitesimal generator of a $C_0$-semigroup $(T(t))_{t\geqslant 0}$ on $X$ and $T(t)$ is compact for $t>0$. Assumption $\textrm{H}(\mathcal R)$. $\mathcal R\colon L^2(I,X)\to L^2(I,Y)$ is a history-dependent operator, that is, there exists a constant $c_{\mathcal R}>0$ such that Obviously, if we denote $\|(\mathcal R0)(t)\|_Y$ by $\varphi(t)$, then, one gets from $\textrm{H}(\mathcal R)$ that $\varphi(t)\in L^2(I,\mathbb R_+)$ and Assumption $\textrm{H}(F)$. $F\colon I\times X\times Y\times H\to \mathcal{P}_{\mathrm{fc}}(X)$ is a multimap such that: (1) for all $(x,y,u)\in X\times Y\times H$, $t\to F(t,x,y,u)$ is measurable; (2) for a.e. $t\in I$, $F(t,{\cdot}\,,{\cdot}\,,{\cdot}\,)$ has a strongly-weakly closed graph, that is, if $x_n\to x$ in $X$, $y_n\to y$ in $Y$, $u_n\to u$ in $H$ and $f_n\rightharpoonup f$ in $X$ with $f_n\in F(t,x_n,y_n,u_n)$, then $f\in F(t,x,y,u)$; (3) there are a function $a_F\in L^2(I,\mathbb{R}^+)$ and a constant $c_F>0$ such that for a.e. $t\in I,$ all $(x,y,u)\in X\times Y\times H$.Assumption $\textrm{H}(K)$. $K\colon I\times X\times V\to V^{\ast}$ is such that: (1) ${K}(\,{\cdot}\,,x,v)\colon I\to V^{\ast}$ is measurable for all $(x,v)\in X\times V$ and there exists a constant $a_{K}\geqslant0$ such that $\|K(t,x,v)\|_{V^{\ast}}\leqslant a_{K}(1+\|x\|_{X}+\|v\|_{V})$ for a.e. $t\in I$, all $x\in X, v\in V$; (2) for a.e. $t\in I$ and all $v\in V$, ${K}(t,{\cdot}\,,v)\colon X\to V^*$ is continuous and there exist two constants $m_{K}>0$, $b_{K}\geqslant0$ such that $\langle K(t,x,v), v\rangle_{V}\geqslant m_{K}\|v\|^2_{V}-b_{K}$ for a.e. $t\in I$, for all $x\in X, v\in V$; (3) for a.e. $t\in I$, ${K}(t,{\cdot}\,,{\cdot}\,)\colon X\times V\to V^{\ast}$ is uniformly generalized pseudomonotone, that is, if $\{x_n\}\subset X$, $\{v_n\}\subset V$, $x_n\to x$ in $X$, $v_n\rightharpoonup v$ in $V$, and then $K(t,x_n,v_n)\rightharpoonup K(t,x,v)$ in $V^*$ and $\langle K(t,x_n,v_n), v_n\rangle_{V}\to \langle K(t,x,v), v\rangle_{V}$.Assumption $\textrm{H}(G)$. $G\colon I\times X\times Z\to \mathcal P_{\mathrm{fc}}(Z^*)$ is a multimap: (1) ${G}(\,{\cdot}\,,x,z)\colon I\to \mathcal P_{\mathrm{fc}}(Z^*)$ is measurable for all $(x,z)\in X\times Z$ and there exists a constant $a_G>0$ such that
$$
\begin{equation*}
\|g\|_{Z^*}\leqslant a_{G}(1+\|x\|_X+\|z\|_Z)\quad \text{for a.e. }t\in I,\quad \forall\, (x,z)\in X\times Z,\quad\forall\, g\in G(t,x,z);
\end{equation*}
\notag
$$
(2) $\mathrm{Gr}(G(t,{\cdot}\,,{\cdot}\,))$ is sequentially closed in $X\times Z\times Z_w^*$ for a.e. $t\in I$, namely, for a.e. $t\in I$, if $x_n\to x$ in $X$, $z_n\to z$ in $Z$, and $g_n\rightharpoonup g$ in $Z^*$ such that $g_n\in G(t,x_n,z_n)$, then $g\in G(t,x,z)$; (3) there exists a constant $b_{G}>0$ with $m_{K}>b_{G}\|i\|^2_{\mathcal L(V,Z)}$ such that $\langle g,z\rangle_Z\geqslant -b_{G}(1+\|x\|^2_X+\|z\|_Z^2)$ for a.e. $t\in I$, for all $g\in G(t,x,z)$ with any $(x,z)\in X\times Z$, where $i$ is the embedding operator from $V$ to $Z$. We now introduce some spaces of functions defined on the interval $I$. Let $\pi$ denote a finite partition of the interval $(0,b)$, say $\pi=\{b_i\}_{i=0}^n$ with $0=b_0< b_1<\dots<b_n=b$, where $n$ depends on the finite partition. Let $\Pi$ denote the family of all such partitions. For a Banach space $V$ and $1\leqslant q<\infty$, we define the space
$$
\begin{equation*}
\mathrm{BV}^q(I,V)=\biggl\{v\colon I\to V\biggm| \sup_{\pi\in\Pi} \biggl\{\sum_{i=1}^n\|v(b_i)-v(b_{i-1})\|_V^q\biggr\}<\infty\biggr\}
\end{equation*}
\notag
$$
and define the seminorm of a vector function $v\colon I\to V$ by
$$
\begin{equation*}
\|v\|_{\mathrm{BV}^q(I,V)}^q=\sup_{\pi\in\Pi}\biggl\{\sum_{i=1}^n\|v(b_{i})-v(b_{i-1})\|_V^q\biggr\}.
\end{equation*}
\notag
$$
Assume that $1\leqslant q<\infty$ and $1\leqslant p\leqslant\infty$, and $V,Z$ are Banach spaces such that $V\subset Z$ with continuous embedding. We introduce the following Banach space: which is endowed with the norm $\|\,{\cdot}\,\|_{L^p(I,V)}+\|\,{\cdot}\,\|_{\mathrm{BV}^q(I,Z)}$. Recall the following useful compactness result (for a proof, see [36], Proposition 2.8). Lemma 2.6. Let $1\,{\leqslant}\, p, q\,{<}\,\infty$, and $X_1\subset X_2\subset X_3$ be Banach spaces such that $X_1$ is reflexive, the embedding $X_1\subset X_2$ is compact, and the embedding $X_2\subset X_3$ is continuous. If a set $S$ is bounded in $M^{p,q}(I, X_1, X_3)$, then $S$ is relatively compact in $L^p(I, X_2)$. Lemma 2.7 (see [37], Lemma 3.3.2). If assumption $\textrm{H}(A)$ is met, then the operator is compact. At the end of this section, we state the following Bohnenblust–Karlin fixed point theorem, which will play an important role in verifying existence of solutions of the abstract system (1.1). Theorem 2.8 (see [38]). Let $\mathfrak{D}$ be a non-empty subset of Banach space $X$, which is bounded closed and convex. Suppose $G\colon \mathfrak{D}\to \mathcal P(X)$ is u.s.c. with closed convex values and such that $G(\mathfrak{D})\subseteq\mathfrak{D}$ and $\overline{G(\mathfrak{D})}$ is compact (that is, $G(\mathfrak{D})$ is relatively compact). Then $G$ has a fixed point. § 3. Existence of solutions to the differential inclusions with history-dependent operatorsIn this section, we are concerned with the following differential inclusion with history-dependent operators:
$$
\begin{equation}
\begin{cases} x'(t)\in Ax(t)+ F(t,x(t),(\mathcal Rx)(t),u(t)),\ \text{a.e. }t\in I, \\ x(0)=x_0 . \end{cases}
\end{equation}
\tag{3.1}
$$
For conveniences, we first define two superposition multimaps as follows:
$$
\begin{equation}
\mathcal N^{\,2}_F \colon C(I,X)\times \mathcal H\to\mathcal P(L^2(I,X)),\qquad \mathcal{N}^{\,2}_G\colon C(I,X)\times \mathcal{Z}\to \mathcal{P}(\mathcal{Z}^*), \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\mathcal N^{\,2}_F(x,u) =\{f\in L^2(I,X)\mid f(t)\in F(t,x(t),(\mathcal Rx)(t),u(t))\text{ for a.e. } t\in I\},
\end{equation}
\tag{3.2}
$$
$$
\begin{equation}
\mathcal{N}^{\,2}_G(x,u) =\{g\in \mathcal{Z}^*\colon g(t)\in G(t,x(t),u(t)), \text{ a.e. } t \in I\}.
\end{equation}
\tag{3.3}
$$
Based on $C_0$-semigroup theory (see [39]), we adopt the following concepts for the mild solutions to (3.1). Definition 3.1. For a given $u\in L^2(I,H)$, $x\in C(I,X)$ is called a mild solution of the integro-differential inclusion (3.1) if there exists $f\in \mathcal{N}^{\,2}_F(x,u)$, where $\mathcal{N}^{\,2}_F$ is defined in (3.2), such that So, we may give out the following concepts of mild solutions for the abstract system (1.1). Definition 3.2. A pair of functions $(x,u)$, with $x\in C([0,T];X)$ and $u\in\mathcal W$, is said to be a mild solution of the abstract system (1.1) if there exist $f\in \mathcal N^{\,2}_F(x,u)$, $g\in \mathcal N^{\,2}_G(x,u)$ such that Next, we present some properties of the two multimaps $\mathcal N^{\,2}_F, \mathcal N^{\,2}_G$, which are useful in what follows. Proposition 3.3. Let assumption $\textrm{H}(F)$ be met. Then the multimap $\mathcal N^{\,2}_F\colon C(I,X)\times \mathcal H\to\mathcal P_{\mathrm{wkc}}(L^2(I,X))$ is bounded. Moreover, $\mathcal{N}_F^{\,2}$ is closed in $C([0,T];X)\times \mathcal H\times L_{w}^2(I,X)$, that is, if $x_n\to x$ in $C(I,X)$, $u_n\to u$ in $\mathcal H$ and $f_n\rightharpoonup f$ in $L^2(I,X)$ with $f_n\in \mathcal N_F^{\,2}(x_n,u_n)$, then $f\in N_F^2(x,u)$. Similarly, if assumption $\textrm{H}(G)$ is met, then the multimap $\mathcal N^{\,2}_G\colon C(I,X)\times \mathcal{Z} \to\mathcal P_{\mathrm{wkc}}(\mathcal{Z}^*)$ is bounded and closed in $C(I,X)\times \mathcal{Z}\times \mathcal{Z}_{w}^*$, where by $\mathcal{Z}^*_w$ we denote the Banach space $\mathcal{Z}^*$ equipped with the weak topology. Proof. We first show that $\mathcal N^{\,2}_F(x,u)\in\mathcal P_{\mathrm{wkc}}(L^2(I,X))$. In fact, from $\textrm{H}(F)$(2), (3), Lemma 2.1 and the reflexivity of the product Banach space $X\times Y\times H$, we easily infer that for a.e. $t\in I$ the multimap $F(t,{\cdot}\,,{\cdot}\,,{\cdot}\,)$ is u.s.c. from $X\times Y\times H$ to $X_w$. So all the conditions of [34], Lemma 5.3, are satisfied by $\textrm{H}(F)$, and hence $\mathcal N^{\,2}_F(x,u)\in\mathcal P_{\mathrm{wkc}}(L^2(I,X))$.
For each $f\in \mathcal{N}^{\,2}_F(x,u)$, we obtain from $\textrm{H}(F)$(3), (2.1) and the Hölder inequality
$$
\begin{equation*}
\begin{aligned} \, &\|f\|^2_{L^2(I,X)}=\int_0^b\|f(s)\|^2_X\, ds \\ &\qquad\leqslant \int_0^b\bigl(a_F(s)+c_F(\|x(s)\|_X+\|(\mathcal Rx)(s)\|_Y+\|u(s)\|_H)\bigr)^2\,ds \\ &\qquad\leqslant 16\int_0^b\bigl(a^2_F(s)+c^2_F(\|x(s)\|^2_X+\|(\mathcal Rx)(s)\|^2_Y +\|u(s)\|^2_H)\bigr)\,ds \\ &\qquad\leqslant 16\biggl(\|a_F\|^2_{L^2(I,\mathbb{R}_+)}+c^2_F\|u\|^2_{\mathcal H} \\ &\qquad\qquad\qquad +c_F^2 \int_0^b\biggl(\|x(s)\|^2_X+\biggl(\varphi(s)+c_{\mathcal R}\int_0^s\|x(\tau)\|_X\,d\tau\biggr)^2\biggr)\,ds\biggr) \\ &\qquad\leqslant 16\biggl(\|a_F\|^2_{L^2(I,\mathbb{R}_+)} +4c^2_F\|\varphi\|^2_{L^2(I,\mathbb{R}_+)} +c^2_F\|u\|^2_{\mathcal H} \\ &\qquad\qquad\qquad +c_F^2\int_0^b(\|x(s)\|^2_X+4c^2_{\mathcal R}b^2\|x(s)\|^2_X)\,ds\biggr) \\ &\qquad\leqslant 16\bigl(\|a_F\|^2_{L^2(I,\mathbb{R}_+)} +4c^2_F\|\varphi\|^2_{L^2(I,\mathbb{R}_+)} +c^2_F\|u\|^2_{\mathcal H} +c_F^2(b+4c^2_{\mathcal R}b^3)\|x\|^2_{C(I,X)}\bigr), \end{aligned}
\end{equation*}
\notag
$$
which implies that $\mathcal N^{\,2}_F\colon C(I,X)\times \mathcal H\to\mathcal P_{\mathrm{wkc}}(L^2(I,X))$ maps bounded subsets into bounded subsets.
In order to prove that the multimap $\mathcal{N}_F^{\,2}$ is closed, let $f_n\rightharpoonup f$ in $L^2(I,X)$ with $f_n\in \mathcal{N}_F^{\,2}(x_n,u_n)$, $x_n\to x$ in $C(I,X)$ and $u_n\to u$ in $\mathcal H$, which imply that $\mathcal Rx_n(t)\to \mathcal Rx(t)$ in $Y$ from H($\mathcal R$) and $u_n(t)\to u(t)$ in $H$ for a.e. $t\in I$ (by passing to a subsequence if necessary). This entails that
$$
\begin{equation*}
\begin{gathered} \, f_n\in L^2(I,X)\quad\text{such that }f_n(t)\in F(t,x_n(t),(\mathcal Rx_n)(t),u_n(t))\text{ for a.e. }t\in I, \\ f_n\rightharpoonup f\quad\text{in }L^2(I,X). \end{gathered}
\end{equation*}
\notag
$$
So, Lemma 2.1 and Lemma 2.2 imply that the multimap $\mathcal{N}_F^{\,2}$ is closed. The remaining conclusion of the proposition can be proved by the same arguments. Proposition 3.3 is proved. For the existence of solutions for the differential inclusion (3.1), the following result holds. Theorem 3.4. If assumptions $\textrm{H}(A)$, $\textrm{H}(\mathcal R)$, and $\textrm{H}(F)$ are met, then, for any $u\in \mathcal H$, the mild solution set $\mathcal S(u)$ of problem (3.1) is non-empty and compact in $C(I,X)$. Proof. By the definition of mild solutions for problem (3.1), for a fixed $u\in \mathcal H$, we define a multimap $\Lambda:C(I,X)\to 2^{C(I,X)}$ by
$$
\begin{equation}
\Lambda(x) =\biggl\{y\in C(I,X)\colon y(t)=T(t)x_0+ \int_0^t T(t-s)f(s)\, ds \ \forall\, t\in I,\ f\in \mathcal{N}^{\,2}_F(x,u)\biggr\}.
\end{equation}
\tag{3.6}
$$
By the definition of $\Lambda$, the problems of establishing the non-emptiness and compactness of the mild solution set $\mathcal S(u)$ of problem (3.1) are equivalent to those of the fixed point set to the multimap $\Lambda$. To this end, we consider the multimap $\Lambda$ on the Banach space $C(I,X)$ with the weighted norm $\|x\|_{\lambda}=\sup_{t\in I}e^{-\lambda t}\|x(t)\|_X$, where $\lambda$ large enough such that
The rest of the proof is in four steps. Step 1. $\Lambda(x)\in \mathcal{P}_{\mathrm{kc}}(C(I,X))$ for each $x\in C(I,X)$. Clearly, for any $x\in C(I,X)$, $\Lambda(x)$ is convex by the convexity of $\mathcal{N}^{\,2}_F$ from Proposition 3.3. It remains us to demonstrate the compactness of $\Lambda(x)$ for each $x\in C(I,X)$. To this end, for any sequence $\{y_n\}_{n\geqslant1}\subset \Lambda(x)$, we have to show that there exists a subsequence, still denoted by $\{y_n\}_{n\geqslant1}$, such that $y_n\to y\in \Lambda (x)$ in $C(I,X)$ as $n \to\infty$. By the definition of $\Lambda$, there exists a sequence $\{f_n\}_{n\geqslant1}\subset\mathcal{N}_F^{\,2}(x,u)$ such that
$$
\begin{equation*}
y_n(t)=T(t)x_0+\int_0^tT(t-s)f_n(s)\, ds,\qquad t\in I.
\end{equation*}
\notag
$$
It follows from Proposition 3.3 that the sequence $\{f_n\}_{n\geqslant1}\subset L^2(I,X)$ is weakly relatively compact. Without loss of generality, we may assume $f_n\rightharpoonup f\in\mathcal{N}^{\,2}_F(x,u)$ in $L^2(I,X)$. On the other hand, the compactness of $T(t)$ for $t>0$ implies by Lemma 2.7 This also means that $y\in \Lambda(x)$, that is, $\Lambda(x)\in\mathcal{P}_{\mathrm{kc}}(C(I,X))$.
Step 2. $\Lambda\colon C(I,X)\to \mathcal P_{\mathrm{kc}}(C(I,X))$ is upper semicontinuous. Let $x_n\to x_*$ in $C(I,X)$ and $y_n\to y_*$ in $C(I,X)$ with $y_n\in \Lambda(x_n)$. Thus, there exists $f_n\in \mathcal{N}^{\,2}_F(x_n,u)$ such that From Proposition 3.3, it is straightforward to show that $\{f_n\}_{n\geqslant1}$ is bounded in $L^2(I,X)$. Hence, passing to a subsequence if necessary, we may suppose that
$$
\begin{equation}
f_n\rightharpoonup f_* \ \text{ in }\ L^2(I,X)\ \text{ and }\ f_*\in \mathcal{N}^{\,2}_F(x,u).
\end{equation}
\tag{3.9}
$$
Another appeal to Lemma 2.7 shows that
$$
\begin{equation}
y_n(t)\to T(t)x_0+\int_0^tT(t-s)f_*(s)\, ds\quad \forall\, t\in I.
\end{equation}
\tag{3.10}
$$
Since $y_n\to y_*$ in $C(I,X)$ and $f_n\in \mathcal{N}^{\,2}_F(x_n,u)$, from (3.10), we obtain $y_*\in \Lambda(x_*),$ which implies that $\Lambda$ is a closed multimap.
If $D\subset C(I,X)$ is a bounded set, then $\mathcal D:=\{f\in L^2(I,X)\mid f\in \mathcal N^{\,2}_F(D,u)\}$ is bounded by Proposition 3.3. So, it is easy to prove from Lemma 2.7 that $\Lambda(D)$ is relatively compact, that is, $\Lambda\colon C(I,X)\to \mathcal P_{\mathrm{kc}}(C(I,X))$ is locally compact. So, $\Lambda\colon C(I,X)\to \mathcal P_{\mathrm{kc}}(C(I,X))$ is upper semicontinuous by Lemma 2.1. Step 3. There exists a constant $R>0$ such that $\Lambda(\overline{B}_R)\subset \overline{B}_R:=\{x\in C(I,X)\colon \|x\|_\lambda\leqslant R\}\subset C(I,X)$. To obtain the conclusion of this step, we choose that
$$
\begin{equation*}
R>2M \|x_0\|_X+2M(\|a_F\|_{L^2(I,\mathbb{R}_+)} +c_F\|\varphi\|_{L^2(I,\mathbb{R}_+)} +c_F\|u\|_{\mathcal H})\sqrt{b}.
\end{equation*}
\notag
$$
Using (2.1), $\textrm{H}(F)$(3), and the Hölder inequality, for any $y\in \Lambda(x)$, there exists $f\in \mathcal{N}^{\,2}_F(x,u)$ such that, for all $t\in I$
$$
\begin{equation}
\begin{aligned} \, \|y(t)\|_X &\leqslant \|T(t)x_0\|_X+\int_0^t\|T(t-s)\|\|f(s)\|_X\, ds \leqslant M \|x_0\|_X+M\int_0^t\|f(s)\|_X\, ds \nonumber \\ &\leqslant M \|x_0\|_X+M\int_0^t\bigl(a_F(s)+c_F(\|x(s)\|_X+\|(\mathcal Rx)(s)\|_Y+\|u(s)\|_H)\bigr)\, ds \nonumber \\ &\leqslant M \|x_0\|_X+M\int_0^t\bigl(a_F(s)+c_F\|u(s)\|_H\bigr)\, ds \nonumber \\ &\qquad+Mc_F\int_0^t\biggl(e^{-\lambda s}e^{\lambda s}\|x(s)\|_X+\varphi(s)+c_{\mathcal R}\int_0^se^{-\lambda \tau}e^{\lambda \tau}\|x(\tau)\|_X\, d\tau \biggr)\, ds \nonumber \\ &\leqslant M \|x_0\|_X+M(\|a_F\|_{L^2(I,\mathbb{R}_+)} +c_F\|\varphi\|_{L^2(I,\mathbb{R}_+)} +c_F\|u\|_{\mathcal H})\sqrt{t} \nonumber \\ &\qquad+\frac{Mc_F}{\lambda}(1+c_{\mathcal R}t)(e^{\lambda t}-1)\|x\|_\lambda, \end{aligned}
\end{equation}
\tag{3.11}
$$
which implies that
$$
\begin{equation}
\begin{aligned} \, \sup_{t\in I}\|y(t)\|_X e^{-\lambda t}&\leqslant M \|x_0\|_X+M(\|a_F\|_{L^2(I,\mathbb{R}_+)} +c_F\|\varphi\|_{L^2(I,\mathbb{R}_+)}+c_F\|u\|_{\mathcal H})\sqrt{b} \nonumber \\ &\qquad+\frac{Mc_F}{\lambda}(1+c_{\mathcal R}b)\|x\|_\lambda<\frac{R}{2}+\frac{R}{2}=R\quad \forall\, x\in \overline{B}_R. \end{aligned}
\end{equation}
\tag{3.12}
$$
Thus, we have verified all the conditions required in Theorem 2.8. Hence, we can conclude that $\Lambda$ has at least a fixed point $x\in C(I,X)$, that is, the mild solution set $\mathcal S(u)$ of problem (3.1) is non-empty.
Step 4. The compactness of the mild solution set $\mathcal S(u)$ of problem (3.1). As seen before, the mild solution set for problem (3.1) coincides with the set of fixed points of the multimap $\Lambda\colon C(I,X)\to \mathcal{P}_{\mathrm{kc}}(C(I,X))$ defined in (3.6). Hence we only need to check the compactness of the set $\operatorname{Fix}\Lambda$. To this end, we note by (3.12) that the mild solution set of problem (3.1) is included in $\overline{B}_R$ with $R>2M \|x_0\|_X+2M(\|a_F\|_{L^2(I,\mathbb{R}_+)}+c_F\|\varphi\|_{L^2(I,\mathbb{R}_+)}+c_F\|u\|_{\mathcal H})\sqrt{b}$. At the end of Step 2, we have shown that $\Lambda\colon C(I,X)\to \mathcal P_{\mathrm{kc}}(C(I,X))$ is a closed multimap which maps bounded subsets into relatively compact subsets. So, the solution set $\mathcal S(u)$ of problem (3.1) is compact in $C(I,X)$. The proof of Theorem 3.4 is complete. § 4. Existence of mild solutionsIn this section, we use the idea of a suitable time discretization method (cf. [36], [26], [40]) combined with a feedback iterative approach (cf. [41]–[43]) to establish the existence of mild solutions to (1.1). We commence by studying the corresponding numerical scheme and its properties. For any $N \in \mathbb N$, let $\tau=b/N$, $t_\tau^k=k\tau$, $k=0,1,\dots,N-1$. We also denote by $\chi_{[t_\tau^k,t_\tau^{k+1}]}$ the characteristic function of the interval $[t_\tau^k,t_\tau^{k+1}]$. For a given Banach space $B$ and any function $h\in L^p(I,B)$ with $1\leqslant p<\infty$, consider the piecewise constant function $h_\tau$ defined by
$$
\begin{equation*}
h_\tau(t)=h_\tau^k=\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}h(s)\, ds
\end{equation*}
\notag
$$
for all $t\in (t_\tau^{k},t_\tau^{k+1})$ and $k=0,\dots,N-1$, we recall the following result (cf. [44], Lemma 1). Lemma 4.1. Assume that $B$ is a Banach space. If $f \in L^p(I,B)$, then $\|f_\tau\|_{L^{p}(I,B)}\,{\leqslant} \|f\|_{L^{p}(I,B)}$ and $f_\tau\to f$ in $L^{p}(I,B)$ as $\tau\to 0$. Now, we are in the position to study the following systems. Hybrid Iterative SystemSetting $u_\tau^0=u_0$, for $k=0,1,\dots,N-1,$ we find $(u_\tau^{k+1},g_\tau^{k+1})\in V\times Z^*$ such that
$$
\begin{equation}
\begin{cases} \dfrac{u_\tau^{k+1}-u_\tau^{k}}{\tau}+K_\tau^k(u_\tau^{k+1})+g_\tau^{k}(u_\tau^{k+1})=h_\tau^k \text{ in } V^*, \\ g_\tau^{k}(u_\tau^{k+1})\in G_\tau^k(u_\tau^{k+1}), \end{cases}
\end{equation}
\tag{4.1}
$$
and $x_\tau\in C([0,t_\tau^{k+1}],X)$ such that
$$
\begin{equation}
x_\tau(t)=T(t)x_0+ \int_0^t T(t-s)f_\tau(s)\, ds \quad \forall\, t\in [0,t_\tau^k],\quad k=0,1,\dots, N-1,
\end{equation}
\tag{4.2}
$$
where
$$
\begin{equation*}
\begin{aligned} \, &K_\tau^k(\,{\cdot}\,)\colon V \to V^*\ \text{by} \ K_\tau^k(u)=\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}K(s,x_\tau(s),u)\, ds, \\ &G_\tau^k(\,{\cdot}\,)\colon Z\to 2^{Z^*}\ \text{by}\ G_\tau^k(u)=\biggl\{ g_\tau^k(u) \biggm| g_\tau^k(u)=\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}\sigma(s)\,ds\ \forall\, \sigma\in\mathcal N^{\,2}_G(x_\tau,u)\biggr\}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{gathered} \, \mathcal N^{\,2}_{G}(x_\tau,u) =\{\sigma\in L^2([t_\tau^k,t_\tau^{k+1}],Z)\mid \sigma(t)\in G(t,x_\tau(t),u),\text{ a.e. }t\in (t_\tau^k,t_\tau^{k+1}) \}, \\ \begin{aligned} \, &f_\tau\in \mathcal N_F^{\,2}(x_\tau, u_\tau) \\ &=\{f\in L^2([t_\tau^k,t_\tau^{k+1}],Z)\mid f_\tau(t)\in F(t,x_\tau(t),(\mathcal Rx_\tau)(t), u_\tau(t)), \text{ a.e. }t\in (t_\tau^k,t_\tau^{k+1}]\}, \end{aligned} \\ u_{\tau}(t)= \begin{cases} {\displaystyle\sum_{k=0}^{n-1}u_\tau^{k}\chi_{(t_\tau^{k},t_\tau^{k+1}]}(t)}, &t\in I, \\ u_0, &t=0. \end{cases} \end{gathered}
\end{equation*}
\notag
$$
Before proceeding further, we present several propositions required in the proof. Proposition 4.2. Let assumption $\textrm{H}(K)$ be met, $x_n{\to}\, x$ in $C(I,X)$, $u_n{\rightharpoonup}\, u$ in $\mathcal{V}$, $u_n(t)\,{\rightharpoonup}\, u(t)$ in $V^*$ for a.e. $t \in I$ as $n{\kern0.8pt}{\to}\,\infty$, and $\limsup\langle K(x_n,u_n),u_n{-}\,u\rangle_{\mathcal{V}}\,{\leqslant}\, 0$. Then Proof. We choose $I_1\,{\subset}\, I$ such that the measure of the set $I\setminus I_1$ is zero, and, for all $t \in I_1$, the operator $K(t,{\cdot}\,,{\cdot}\,)$ on $X\times V$ satisfies all the conditions in $\textrm{H}(K)$(1)–(3). Let $u_n(t)\rightharpoonup u(t)$ in $V^*$ for all $t\in I_1$ as $n\to\infty$. Define $\xi_n(t)=\langle K(t,x_n(t),u_n(t), u_n(t)-u(t))\rangle_{V}$ for all $t\in I_1$. First, we show that, for a.e. $t\in I_1$, Suppose that the assertion is false, that is, for some $t_0\in I_1$ such that which implies that there exists a subsequence, still denoted by $\{\xi_n(t_0)\}$, such that It follows from assumption $\textrm{H}(K)$(2) that
$$
\begin{equation}
\begin{aligned} \, \xi_n(t_0)&=\langle K(t,x_n(t_0),u_n(t_0),u_n(t_0)-u(t_0))\rangle_{V} \nonumber \\ &\geqslant m_K\|u_n(t_0)\|^2_V-c_K-a_K(1+\|x_n(t_0)\|_X+\|u_n(t_0)\|_V)\|u(t_0)\|_V. \end{aligned}
\end{equation}
\tag{4.5}
$$
By (4.4), (4.5) we find that $\{u_n(t_0)\}_{n\geqslant 1}$ is bounded in $V$. Therefore, passing to a subsequence if necessary we may assume that $u_n(t_0)\rightharpoonup \overline u(t_0)$ in $V$ as $n\to\infty$. On the other hand, we already assume that $u_n(t_0)\rightharpoonup u(t_0)$ in $V^*$ from the conditions of the proposition. Since the embedding $V\hookrightarrow V^*$ is dense, we have $\overline u(t_0)=u(t_0)$, and thus $u_n(t_0)\rightharpoonup u(t_0)$ in $V$ as $n\to\infty$. By $x_n(t_0)\to x(t_0)$ in $X$, (4.4) and the uniformly generalized pseudomonotonicity of $K$, we infer that $\lim_{n\to\infty} \xi_n(t_0)=0$, which is a contradiction. Thus (4.3) is true. Consequently, by Fatou’s lemma and the assumption the proposition, which implies that
We next claim that $\xi_n\to 0$ in $L(I,\mathbb R)$ as $n\to\infty$. Let $\xi_n^{\pm}=\max\{\pm\xi_n(t),0\}$. So $|\xi_n(t)|=\xi_n(t)+2\xi_n^-(t)$. By (4.3), we have $\lim_{n\to\infty}\xi_n^-(t)=0$ for all $t\in I_1$. By the dominated the convergence theorem,
$$
\begin{equation*}
\lim_{n\to\infty}\int_0^b|\xi_n(t)|\,dt=\lim_{n\to\infty}\int_0^b[\xi_n(t)+2\xi_n^-(t)]\,dt=0,
\end{equation*}
\notag
$$
which implies that $\xi_n\to 0$ in $L(I_1,\mathbb R)$ as $n\to\infty$. Passing to a subsequence if necessary we deduce that there exists a subset $I_2\subset I_1$ such that the measure $|I\setminus I_2|$ is zero and $\xi_n(t)\to 0$ for all $t\in I_2$. Proceeding as above, we conclude that $u_n(t)\rightharpoonup u(t)$ in $V$ for all $t\in I_2$ as $n\to\infty$.
Since $x_n\to x$ in $C(I,X)$ and using the uniformly generalized pseudomonotonicity of the operator $K$, we find that $K(t,x_n(t),u_n(t))\,{\rightharpoonup}\, K(t,x(t),u(t))$ in $V^*$ and $\langle K(t,x_n(t),u_n(t)), u_n(t)\rangle_{V}\,{\to} \langle K(t,x(t),u(t)), u(t)\rangle_{V}$. Proposition 4.2 is proved. Proposition 4.3. If assumptions $\textrm{H}(A)$, $\textrm{H}(\mathcal R)$, $\textrm{H}(F)$, $\textrm{H}(K)$, and $\textrm{H}(G)$ are met, then, for any $h\in \mathcal{V}^*$, the hybrid iterative system (4.1), (4.2) has at least one solution. Proof. We may rewrite the hybrid iterative system (4.1), (4.2) as the equivalent inclusion
$$
\begin{equation}
\begin{cases} u_\tau^{k+1}+\tau K_\tau^k(u_\tau^{k+1})+\tau G_\tau^k(u_\tau^{k+1})\ni \tau h_\tau^k+u_\tau^{k}, \quad \text{in } V^*,\ k=0,1,\dots, N-1, \\ x_\tau(t)=T(t)x_0+ {\displaystyle\int_0^t T(t-s)f_\tau(s)\, ds} \quad \forall\, t\in [0,t_\tau^k],\ k=0,1,\dots, N-1, \\ u_\tau^0=u_0,\quad f_\tau\in \mathcal N_F^{\,2}(x_\tau, u_\tau). \end{cases}
\end{equation}
\tag{4.6}
$$
Now, we are in a position to construct the sequence $\{u_\tau^{k}\}$ and $f_\tau$ as follows. Firstly, we take $u_\tau^0=u_0$ and define $u_\tau(t)\equiv u_0$ on $[0,t_\tau^1]$. By Theorem 3.4, there exists at least one $x_\tau\in C([0,t_\tau^1],X)$ such that
$$
\begin{equation*}
\begin{gathered} \, x_\tau(t)=T(t)x_0+ \int_0^t T(t-s)f_\tau(s)\, ds \quad \text{for a.e. } t\in [0,t_\tau^1], \\ \begin{aligned} \, &f_\tau\in \mathcal N_F^{\,2}(x_\tau, {u}_\tau)|_{[0,t_\tau^1]} \\ &\quad=\{f\in L^2([0,t_\tau^1],X)v \mid f(t)\in F(t,x_\tau(t),(\mathcal Rx_\tau)(t),{u}_\tau(t))\text{ for a.e. }t\in [0,t_\tau^1]\}. \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
Using $u_\tau(t)=u_0$, $t\in [0,t_\tau^1]$, and $x_\tau\in C([0,t_\tau^1],X)$, we can easily get $K_\tau^0$ and $G_\tau^0$. In order to show that there exists $u_\tau^1\in V$ such that
$$
\begin{equation}
u_\tau^{1}+\tau K_\tau^0(u_\tau^{1})+\tau G_\tau^0(u_\tau^{1})\ni \tau h_\tau^0+u_\tau^{0} \quad \text{in } V^*,
\end{equation}
\tag{4.7}
$$
we need to prove that $S^0(\,{\cdot}\,):=\mathrm{Id}(\,{\cdot}\,)+\tau K_\tau^0(\,{\cdot}\,)+\tau G_\tau^0(\,{\cdot}\,)\colon V\to \mathcal{P}(V^*)$ is pseudomonotone and coercive by Theorem 2.5, where $\mathrm{Id}\colon V \to V^*$ is the canonical isomorphism.
To this end, we first show that $\mathcal{B}_\tau^0(\,{\cdot}\,):=K_\tau^0(\,{\cdot}\,) +G_\tau^0(\,{\cdot}\,)\colon V\to \mathcal{P}(V^*)$ is bounded and generalized pseudomonotone. By Proposition 3.3, $\mathcal N_G^{\,2}(x,u)\in \mathcal P_{\mathrm{wkc}}(\mathcal{Z}^*)$. So, $G_\tau^0(u)\in \mathcal P_{\mathrm{wkc}}(Z^*)$. Since all the embeddings $V\subseteq Z\subseteq H\subseteq Z^*\subseteq V^*$ are dense and continuous and $V\subseteq Z$ compact, we infer that $G_\tau^0|_V(u)\in \mathcal P_{\mathrm{kc}}(V^*)$. Moreover, by assumptions $\textrm{H}(K)$(1) and $\textrm{H}(G)$(1), $\mathcal{B}_\tau^0(\,{\cdot}\,):=K_\tau^0(\,{\cdot}\,)+G_\tau^0(\,{\cdot}\,)\colon V\to \mathcal{P}_{\mathrm{kc}}(V^*)$ is a bounded multimap. Next, we show that it is generalized pseudomonotone. Assume that $u_k\rightharpoonup u$ in $V$, which implies that $u_k\to u$ in $Z$ by the compactness of the embedding $V\subset Z$. Let $g_\tau^0(u_k)\rightharpoonup \overline g$ in $V^*$ with $g_\tau^0(u_k)\in G_\tau^0(u_k)$, where
$$
\begin{equation*}
\begin{gathered} \, g_\tau^0(u_k)=\frac{1}{\tau}\int_0^{t_\tau^{1}}\sigma_k(s)\,ds, \\ \begin{aligned} \, &\sigma_k\in \mathcal N_G^{\,2}(x_\tau,u_k)|_{[0,t_\tau^1]} \\ &\quad=\{\sigma_k\in L^2([0,t_\tau^{1}],Z^*)\mid \sigma_k(t)\in G(t,x_\tau(t),u_k) \text{ for a.e. } t\in (0,t_\tau^{1})\}. \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
Therefore, $\{\sigma_k\}_{k\geqslant 1}\subset L^2([0,t_\tau^{1}],Z^*)$ is bounded. We may assume that $\sigma_k\rightharpoonup \overline{\sigma}$ in $L^2([0,t_\tau^{1}],Z^*)$, which implies
$$
\begin{equation}
g_\tau^0(u_k)=\frac{1}{\tau}\int_0^{t_\tau^{1}}\sigma_k(s)\,ds \rightharpoonup \frac{1}{\tau}\int_0^{t_\tau^{1}}\overline{\sigma}(s)\,ds\quad\text{in } Z^* \text{ as } k\to\infty.
\end{equation}
\tag{4.8}
$$
Hence
$$
\begin{equation*}
\overline{g}=\frac{1}{\tau}\int_0^{t_\tau^{1}}\overline{\sigma}(s)\,ds \quad (\text{recall that } g_\tau^0(u_k)\rightharpoonup \overline{g}\text{ in }V^*).
\end{equation*}
\notag
$$
By Proposition 3.3, $\overline{g}:={g}_\tau^0(u)\in G_\tau^0(u)$,
$$
\begin{equation}
\begin{gathered} \, g_\tau^0(u_k)\rightharpoonup g_\tau^0(u)\in G_\tau^0(u)\quad\text{ in }V^*, \\ \lim_{k\to\infty}\langle g_\tau^0(u_k), u_{k}\rangle_{V}=\lim_{k\to\infty}\langle g_\tau^0(u_k), u_k\rangle_{Z}=\langle {g}_\tau^0(u), u\rangle_{V}. \end{gathered}
\end{equation}
\tag{4.9}
$$
Now we claim that the operator $\mathcal{B}_\tau^0$ is pseudomonotone from $V$ to $V^*$. Let $u_k\rightharpoonup u$ in $V$ and $b_\tau^0(u_k)=K_\tau^0(u_k)+g_\tau^0(u_k)\in \mathcal{B}^0_\tau(u_k)$ with $g_\tau^0(u_k)\in G_\tau^0(u_k)$ such that
$$
\begin{equation}
\limsup_{k\to\infty}\langle b_\tau^0(u_k),u_k-u\rangle_V=\limsup_{k\to\infty}\langle K_\tau^0(u_k)+g_\tau^0(u_k),u_k-u\rangle_V\leqslant 0.
\end{equation}
\tag{4.10}
$$
Consider the following auxiliary sequence
$$
\begin{equation*}
\widehat{u}_k(t)=\begin{cases} u_k, &t\in [0,t_\tau^{1}], \\ 0&\text{otherwise}, \end{cases} \qquad \widehat{u}(t)=\begin{cases} u, &t\in [0,t_\tau^{1}], \\ 0 &\text{otherwise}. \end{cases}
\end{equation*}
\notag
$$
Obviously, $\widehat{u}_k\rightharpoonup \widehat{u}$ in $\mathcal V$. Furthermore, the compact embedding of $V$ into $Z$ implies that $u_k\to u$ in $Z$. Hence, $\widehat{u}_k(t)\to \widehat{u}(t)$ in $Z$ for all $t\in I$ and using (4.10) we have
$$
\begin{equation*}
\begin{aligned} \, &\langle b_\tau^0({u}_k),{u}_k-u\rangle_{V}=\langle K_\tau^0(u_k)+g_\tau^0(u_k),{u}_k-u\rangle_{V} \\ &\qquad=\biggl\langle \frac{1}{\tau}\int_0^{t_\tau^{1}}K(s,x_\tau(s),u_k)\,ds,{u}_k-{u}\biggr\rangle_{V}+\langle g_\tau^0(u_k),{u}_k-u\rangle_{V} \\ &\qquad= \frac{1}{\tau}\int_0^b\langle K(s,x_\tau(s),\widehat {u}_k(s)),\widehat{u}_k(s)-\widehat{u}(s)\rangle_{V}\,ds+\langle g_\tau^0(u_k),{u}_k-u\rangle_{Z} \\ &\qquad=\frac{1}{\tau} \langle K(x_\tau,\widehat{u}_k),\widehat{u}_k-\widehat{u}\rangle_{\mathcal V}\leqslant 0 \quad (\text{recall (4.9) and (4.10)}). \end{aligned}
\end{equation*}
\notag
$$
By Proposition 4.2,
$$
\begin{equation*}
\lim_{k\to\infty}\langle K(x,\widehat u_k),\widehat u_k\rangle_{\mathcal{V}}=\langle K(x,\widehat u),\widehat u\rangle_{\mathcal{V}}\ \ \text{and}\ \ K(x,\widehat u_k)\rightharpoonup K(x,\widehat u)\quad\text{in }\mathcal{V}^*\text{ as }n\to\infty,
\end{equation*}
\notag
$$
that is,
$$
\begin{equation}
\lim_{k\to\infty}\langle K_\tau^0(u_k),u_k\rangle_{V}=\langle K_\tau^0(u),u\rangle_{V}\quad \text{and}\quad K_\tau^0(u_k)\rightharpoonup K_\tau^0(u)\text{ in }V^*\text{ as }k\to\infty.
\end{equation}
\tag{4.11}
$$
Combining (4.9) with (4.11), we have
$$
\begin{equation*}
\begin{gathered} \, K_\tau^0(u_k)+g_\tau^0(u_k)\rightharpoonup K_\tau^0(u)+g_\tau^0(u)\in K_\tau^0(u)+G_\tau^0(u), \\ \lim_{k\to\infty}\langle K_\tau^0(u_k)+g_\tau^0(u_k),u_k\rangle_{V}=\langle K_\tau^0(u)+g_\tau^0(u),u\rangle_{V}. \end{gathered}
\end{equation*}
\notag
$$
Since $\mathcal{B}_{\tau}^0$ is bounded, we have by Lemma 2.4 that the operator $\mathcal{B}_{\tau}^0$ is pseudomonotonic.
Since $\mathrm{Id}\colon V \to V^*$ is bounded linear and non-negative, it is pseudomonotone. Therefore, we conclude that by Proposition 3.59 in [34] that $S^0$ is a pseudomonotone operator. Now we only need to show the coercivity of $S^0$. In fact, for any $u\in V$ and any $g_\tau^0(u)\in G_\tau^0(u)$, using $\textrm{H}(K)$(2) and $\textrm{H}(G)$(3), we have
$$
\begin{equation}
\begin{alignedat}{1} &\langle S^0(u),u\rangle_{V}=\langle \mathrm{Id}(u)+\tau K_\tau^0(u)+\tau g_\tau^0(u),u\rangle_{V} \nonumber \\ &\quad=\langle u,u\rangle_{H}\,{+}\!\int_0^{t_\tau^{1}} \!\! \langle K(s,x_\tau(s),u),u\rangle_{V}\,ds \,{+} \!\int_0^{t_\tau^{1}}\!\!\langle\sigma(s),u\rangle_{V}\,ds \ \ (\text{where }\sigma \in \mathcal N_G^{\,2}(x_\tau,u)) \nonumber \\ &\quad\geqslant \|u\|_H^2+\tau\bigl((m_K-b_G\|i\|^2)\|u\|_V^2-c_K -b_G(1+\|x_\tau\|^2_{C(I,X)})\bigr)\quad \forall\, u\in V. \end{alignedat}
\end{equation}
\tag{4.12}
$$
Taking into account the smallness condition in $\textrm{H}(G)$(3), we conclude that $S^0$ is coercive.
Consequently, we deduce from Theorem 2.5 that there exists at least one $u_\tau^{1}\in V$ such that
$$
\begin{equation}
u_\tau^{1}+\tau K_\tau^0(u_\tau^{1})+\tau G_\tau^0(u_\tau^{1})\ni\tau h_\tau^0+u_\tau^{0}\quad \text{in } V^*.
\end{equation}
\tag{4.13}
$$
Now, we may define
$$
\begin{equation*}
u_{\tau}(t)=u_\tau^{0}\chi_{[t_\tau^{0},t_\tau^{1}]}(t)+u_\tau^{1}\chi_{(t_\tau^{1},t_\tau^2]}(t), \qquad t\in I.
\end{equation*}
\notag
$$
Similarly, by Theorem 3.4 there exists at least one $x_\tau\in C([0,t_\tau^2],X)$ such that
$$
\begin{equation*}
x_\tau(t)=T(t)x_0+ \int_0^t T(t-s)f_\tau(s)\, ds \quad \text{for a.e. } t\in [0,t_\tau^2],\quad f_\tau\in \mathcal N_F^{\,2}(x_\tau, {u}_\tau)|_{[0,t_\tau^2]}.
\end{equation*}
\notag
$$
Furthermore, we get $K_\tau^1$ and $G_\tau^1$.
Inducting and proceeding as above, we easily get $u_\tau^{k}\in V$, $x_\tau\in C([0,t_\tau^{k}],X)$ such that
$$
\begin{equation*}
\begin{cases} u_\tau^{k+1}+\tau K_\tau^{k}(u_\tau^{k+1})+\tau G_\tau^{k}(u_\tau^{k+1})\ni \tau h_\tau^{k}+u_\tau^{k}\quad \text{in } V^*, \\ u_\tau^0=u_0, \\ x_\tau(t)=T(t)x_0+ \int_0^t T(t-s)f_\tau(s)\, ds \quad \text{for all } t\in [0,t_\tau^{k+1}],\ k=0,1,\dots, N-1, \end{cases}
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{gathered} \, f_\tau\in \mathcal N_F^{\,2}(x_\tau, u_\tau), \nonumber \\ f_\tau(t)\in F(t,x_\tau(t),(\mathcal Rx_\tau)(t),u_\tau(t)),\quad \text{for a.e. }t\in (t_\tau^l,t_\tau^{l+1}],\quad l=0,\dots,k, \nonumber \\ u_{\tau}(t)=\begin{cases} {\displaystyle\sum_{k=0}^{N-1}u_\tau^{k}\chi_{(t_\tau^{k},t_\tau^{k+1}]}(t)}, &t>0, \\ u_0, &t=0. \end{cases} \end{gathered}
\end{equation}
\tag{4.14}
$$
This completes the proof of Proposition 4.3. In order to show the main results, we recall a discrete version of the Gronwall inequality, which can be found in Lemma 7.25 of [45] and Lemma 2.32 of [46]. Lemma 4.4. Let $b>0$ be given. For a positive integer $N$, we define $\tau=b/N$. Assume that $\{g_k\}_{k=1}^N$ and $\{e_k\}_{k=1}^N$ are two sequences of nonnegative numbers satisfying for a positive constant $\overline{c}$ independent of $N$ (or $\tau$). Then there exists a positive constant $c$ independent of $N$ (or $\tau$) such that For conveniences, in what follows, we denote by $C$ a positive constant which can vary from line to line but is independent of the time step $\tau$. We first give a priori estimates for the hybrid iterative system (4.1), (4.2). Proposition 4.5. For the hybrid iterative system (4.1), (4.2) if $h \in \mathcal{V}^*$, assumptions $\textrm{H}(A)$, $\textrm{H}(\mathcal R)$, $\textrm{H}(F)$, $\textrm{H}(K)$, and $\textrm{H}(G)$ are met, then there exists $C>0$ independent of $\tau>0$ such that each solution to the hybrid iterative system (4.1), (4.2) satisfies Proof. Let $x_\tau$ be a solution of (4.2). There exists $f_\tau\in \mathcal{N}^{\,2}_F(x_\tau,u_\tau)$ such that, for all $t\in [0,t_\tau^j]$,
$$
\begin{equation*}
\begin{aligned} \, \|x_\tau(t)\|_X &\leqslant \|T(t)x_0\|_X+\int_0^t\|T(t-s)\|\|f_\tau(s)\|_X\, ds \\ &\leqslant M \|x_0\|_X+M(\|a_F\|_{L^2(I,\mathbb{R}_+)} +c_F\|\varphi\|_{L^2(I,\mathbb{R}_+)})\sqrt{t} \\ &\qquad+Mc_F\int_0^t\|u_\tau(s)\|_H\, ds+\frac{Mc_F}{\lambda}(1+c_{\mathcal R}t)e^{\lambda t}\sup_{s\in [0,t]}e^{-\lambda s}\|x_\tau (s)\|_X. \end{aligned}
\end{equation*}
\notag
$$
It follows from formula (3.7) that
$$
\begin{equation*}
\begin{aligned} \, \sup_{s\in [0,t_\tau^{j}]}e^{-\lambda s}\|x_\tau(s)\|_X &\leqslant 2M \|x_0\|_X +2M(\|a_F\|_{L^2(I,\mathbb{R}_+)}+c_F\|\varphi\|_{L^2(I,\mathbb{R}_+)})\sqrt{b} \\ &\qquad +2Mc_F\tau\sum_{k=0}^{j-1}\|u_\tau^k\|_H\quad \forall\, t\in [t_\tau^{j-1},t_\tau^j],\quad j=1,\dots,N, \end{aligned}
\end{equation*}
\notag
$$
which implies that, for all $t\in [0,t_\tau^j]$, $j=1,\dots,N$, where $c_6:=e^{\lambda b}\bigl(2M \|x_0\|_X+2M(\|a_F\|_{L^2(I,\mathbb{R}_+)}+c_F\|\varphi\|_{L^2(I,\mathbb{R}_+)})\sqrt{b}\,\bigr)$, $c_7=2Mc_Fe^{\lambda b}$.
Now, multiplying both sides of (4.1) by a test function $u_\tau^{k+1}$, we obtain
$$
\begin{equation*}
\frac{1}{\tau}\langle u_\tau^{k+1}-u_\tau^{k},u_\tau^{k+1}\rangle_{H}+ \langle K_\tau^k(u_\tau^{k+1})+g_\tau^k(u_\tau^{k+1}),u_\tau^{k+1}\rangle_{V}=\langle h_\tau^k,u_\tau^{k+1}\rangle_{V}.
\end{equation*}
\notag
$$
Taking into account the identity
$$
\begin{equation*}
\langle u-v,u\rangle_{H}=\frac{1}{2}(\|u\|_H^2+\|u-v\|_H^2-\|v\|_H^2)\quad \forall\, u,v\in H,
\end{equation*}
\notag
$$
we have
$$
\begin{equation}
\begin{aligned} \, &\frac{1}{2\tau}(\|u_\tau^{k+1}\|_H^2+\|u_\tau^{k+1}-u_\tau^{k}\|_H^2-\|u_\tau^{k}\|_H^2) \nonumber \\ &\qquad+ \langle K_\tau^k(u_\tau^{k+1})+g_\tau^k(u_\tau^{k+1}),u_\tau^{k+1}\rangle_{V}=\langle h_\tau^k,u_\tau^{k+1}\rangle_{V}. \end{aligned}
\end{equation}
\tag{4.17}
$$
Summing from $k=0$ to $j$ in (4.17) with $j=0,1,\dots,N-1$, we obtain
$$
\begin{equation}
\begin{aligned} \, &\|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2+2\tau\sum_{k=0}^{j}\langle K_\tau^k(u_\tau^{k+1})+g_\tau^k(u_\tau^{k+1}),u_\tau^{k+1}\rangle_{V} \nonumber \\ &\qquad=2\tau\sum_{k=0}^{j}\langle h_\tau^k,u_\tau^{k+1}\rangle_{V}+\|u_0\|_H^2. \end{aligned}
\end{equation}
\tag{4.18}
$$
As in the proof of (4.12), we obtain from the above identity
$$
\begin{equation*}
\begin{aligned} \, &2\tau\sum_{k=0}^{j}\langle h_\tau^k,u_\tau^{k+1}\rangle_{V}+\|u_0\|^2_H \\ &= \|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2+2\tau\sum_{k=0}^{j}\langle K_\tau^k(u_\tau^{k+1})+g_\tau^k(u_\tau^{k+1}),u_\tau^{k+1}\rangle_{V} \\ &\geqslant\|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2+2 \sum_{k=0}^{j}\biggl(\int_{t_\tau^k}^{t_\tau^{k+1}}\langle K(s,x_\tau(s),u_\tau^{k+1}),u_\tau^{k+1}\rangle_{V}\,ds \\ &\qquad+\int_{t_\tau^k}^{t_\tau^{k+1}}\langle\sigma(s),u_\tau^{k+1}\rangle_{V}\,ds\biggr) \quad(\text{where }\sigma\in \mathcal N_\tau^2(x_\tau,u_\tau^{k+1})) \\ &\geqslant\|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2 \\ &\qquad+ 2\tau\sum_{k=0}^{j}\Bigl(m_K\|u_\tau^{k+1}\|_V^2-c_K-b_G\Bigl(1+\sup_{t\in [0,t_\tau^{k+1}]}\|x_\tau(t)\|^2_X+\|u_\tau^{k+1}\|_Z^2\Bigr)\Bigr) \\ &\geqslant \|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2 \\ &\qquad+ 2\tau\sum_{k=0}^{j}\Bigl((m_K-b_G\|i\|^2)\|u_\tau^{k+1}\|_V^2-c_K-b_G(1+\sup_{t\in [0,t_\tau^{k+1}]}\|x_\tau(t)\|^2_X)\Bigr) \\ &\geqslant\|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2+ 2\tau\sum_{k=0}^{j}\biggl((m_K-b_G\|i\|^2)\|u_\tau^{k+1}\|_V^2-c_K-b_G \\ &\quad-b_G\biggl(2c_6^2+2c_7^2(k+1)\tau^2 \sum_{l=0}^{k}\|u_\tau^l\|^2_H\biggr) \biggr)\quad(\text{by H}\unicode{x00F6}\text{lder inequality and (4.16)}) \\ &\geqslant\|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2 \\ &\quad+ 2\tau\sum_{k=0}^{j}\biggl((m_K-b_G\|i\|^2)\|u_\tau^{k+1}\|_V^2-c_K-b_G -b_G(2c_6^2+2bc_7^2\tau \sum_{l=0}^{k}\|u_\tau^l\|^2_H) \biggr) \end{aligned}
\end{equation*}
\notag
$$
(recall for all $k=0,1,\dots, N-1$, $k+1\leqslant N=b/\tau$). So, for $\varepsilon>0$ small enough, by the Young inequality and the above we have
$$
\begin{equation}
\begin{aligned} \, &\|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2+ 2\tau\sum_{k=1}^{j+1}(m_K-b_G\|i\|^2-\varepsilon)\|u_\tau^{k}\|_V^2 \nonumber \\ &\leqslant 2bc_K+2bb_G+4bc_6^2b_G+4c_7^2bb_G \tau\sum_{k=0}^{j}\biggl(\tau \sum_{l=0}^{k}\|u_\tau^l\|^2_H\biggr) \nonumber \\ &\qquad+\tau(2\varepsilon)^{-1}\sum_{k=0}^{j}\|h_\tau^k\|^2_{V^*}+\|u_0\|^2_H, \nonumber \\ &\|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2+ c_8\tau\sum_{k=1}^{j+1}\|u_\tau^{k}\|_V^2 \nonumber \\ &\leqslant c_9+c_{10}\tau\sum_{k=0}^{j}\biggl(\tau \sum_{l=0}^{k}\|u_\tau^l\|^2_H\biggr) +\tau\sum_{k=0}^{j}(2\varepsilon)^{-1}\|h_\tau^k\|^2_{V^*}+\|u_0\|^2_H \nonumber \\ &\leqslant c_9+c_{10}\biggl[(j\,{+}\,1)\tau^2\|u_0\|_H^2+\tau\sum_{k=1}^{j}\biggl(\tau \sum_{l=1}^{k}\|u_\tau^l\|^2_H\biggr)\biggr] +\tau(2\varepsilon)^{-1}\sum_{k=0}^{j}\|h_\tau^k\|^2_{V^*}{+}\,\|u_0\|^2_H, \end{aligned}
\end{equation}
\tag{4.19}
$$
where $c_8=2(m_K-b_G\|i\|^2-\varepsilon)>0,c_9=2bc_K+2bb_G+4bc_6^2b_G, c_{10}=4c_7^2bb_G $.
Let $e_j=\tau\sum_{k=1}^{j}\|u_\tau^{k}\|_V^2$ and $\gamma$ be the embedding operator from $V\subset H$,
$$
\begin{equation}
\begin{aligned} \, &\|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2+ c_8e_{j+1} \nonumber \\ &\leqslant c_9+c_{10}\biggl[b\tau\|u_0\|_H^2+\|\gamma\|^2\tau\sum_{k=1}^{j}e_k\biggr] +\tau(2\varepsilon)^{-1}\sum_{k=0}^{j}\|h_\tau^k\|^2_{V^*}+\|u_0\|^2_H. \end{aligned}
\end{equation}
\tag{4.20}
$$
By Lemma 4.1 and since $h\in \mathcal{V}^*$, we verify that $\tau\sum_{k=0}^{j}\|h_\tau^k\|^2_{V^*}$ is bounded. Therefore, by Lemma 4.4, there exists a positive constant $C$ such that
$$
\begin{equation*}
\|u_\tau^{j+1}\|_H^2+\sum_{k=0}^{j}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2+ \tau\sum_{k=1}^{j+1}\|u_\tau^{k}\|_V^2\leqslant C, \qquad j=0,1,\dots,N-1.
\end{equation*}
\notag
$$
So, from (4.16) and the above inequality, one has
$$
\begin{equation*}
\max_{0\leqslant k\leqslant N}\|u_\tau^{k}\|_H +\sum_{k=1}^{N}\|u_\tau^{k}-u_\tau^{k-1}\|_H +\tau\sum_{k=1}^{N}\|u_\tau^{k}\|_V^2+\|x_\tau\|_{C(I,X)}\leqslant C,
\end{equation*}
\notag
$$
which completes the proof Proposition 4.5. Now, we define a piecewise affine function $\widetilde u_\tau$ and piecewise constant interpolant functions $\overline u$, $K_\tau$, $h_\tau$ and $g_\tau$ as follows:
$$
\begin{equation}
\widetilde u_\tau(t) =\begin{cases} u_\tau^{k+1}+\dfrac{t-t^{k+1}_\tau}{\tau}(u_\tau^{k+1}-u_\tau^{k}), &t\in (t_\tau^{k},t_\tau^{k+1}],\ 0\leqslant k\leqslant N-1, \\ u_0, &t=0, \end{cases}
\end{equation}
\tag{4.21}
$$
$$
\begin{equation}
\overline u_\tau(t) =\begin{cases} u_\tau^{k+1}, &t\in (t_\tau^{k},t_\tau^{k+1}],\ 0\leqslant k\leqslant N-1, \\ u_0, &t=0, \end{cases}
\end{equation}
\tag{4.22}
$$
$$
\begin{equation}
\begin{gathered} \, K_\tau(t)=K_\tau^{k}(u_\tau^{k+1}),\qquad g_\tau(t)=g_\tau^{k}(u_\tau^{k+1}), \qquad h_\tau(t)=h_\tau^{k}, \\ t\in (t_\tau^{k},t_\tau^{k+1}],\quad 0\leqslant k\leqslant N-1. \end{gathered}
\end{equation}
\tag{4.23}
$$
Obviously, equality (4.1) can be rewritten as
$$
\begin{equation}
\frac{d\widetilde{u}_\tau}{dt}+K_\tau(t)+g_\tau(t)=h_\tau(t)\quad \text{for a.e. }t\in I.
\end{equation}
\tag{4.24}
$$
Proposition 4.6. For the hybrid iterative system (4.1), (4.2) if $h\in \mathcal{V}^*$, assumptions $\textrm{H}(A)$, $\textrm{H}(\mathcal R)$, $\textrm{H}(F)$, $\textrm{H}(K)$, and $\textrm{H}(G)$ are met, then there exists a positive constant $C$ independent of $\tau>0$ such that Proof. By (4.21), (4.22) we have
$$
\begin{equation*}
\begin{gathered} \, \|\widetilde{u}_\tau(t)\|_{H}\leqslant\|u_\tau^{k+1}\|_H +\frac{|t-t_\tau^{k+1}|}{\tau}\|u_\tau^{k+1}-u_\tau^{k}\|_H\leqslant 2\|u_\tau^{k+1}\|_H+\|u_\tau^{k}\|_H, \\ \|\overline{u}_\tau(t)\|_{H}=\|u_\tau^{k+1}\|_H \quad\text{for all }t\in (t_\tau^{k},t_\tau^{k+1}],\quad k=0,1,\dots,N-1. \end{gathered}
\end{equation*}
\notag
$$
So, from (4.15), there exists a positive constant $C$ such that
$$
\begin{equation}
\|\widetilde{u}_\tau\|_{C(I,H)}\leqslant C\quad \text{and}\quad \|u_\tau\|_{L^\infty(I,H)}\leqslant C.
\end{equation}
\tag{4.27}
$$
Another appeal to (4.15) shows that
$$
\begin{equation}
\|\overline{u}_\tau\|_{\mathcal{V}}^2 =\int_0^b\|u_\tau(t)\|^2_V\,dt =\tau\sum_{k=0}^{N-1}\|u_\tau^{k+1}\|_V\leqslant C,
\end{equation}
\tag{4.28}
$$
Furthermore, we have
$$
\begin{equation}
\begin{aligned} \, \|K_\tau\|_{\mathcal{V}^*}^2&=\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\|K_\tau(t)\|_{V^*}^2\,dt =\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\|K_\tau^k(u_\tau^{k+1})\|_{V^*}^2\,dt \nonumber \\ &\leqslant\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}} \|K(s,x_\tau(s),u_\tau^{k+1})\|^2_{V^*}\,ds \,dt \nonumber \\ &\leqslant c_{11}\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}(1+\|x_\tau(s)\|_X^2 +\|u_\tau^{k+1}\|_{V}^2)\,ds\stackrel{(4.15)}{\leqslant} C, \end{aligned}
\end{equation}
\tag{4.30}
$$
$c_{11}>0$ by $\textrm{H}(K)$(1),
$$
\begin{equation}
\begin{aligned} \, \|g_\tau\|_{\mathcal{Z}^*}^2 &=\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\|g_\tau(t)\|_{Z^*}^2\,dt =\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\|g_\tau^k(u_\tau^{k+1})\|_{Z^*}^2\,dt \nonumber \\ &\leqslant\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\biggl\|\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}} \sigma(s)\,ds\biggr\|^2_{Z^*} \,dt \leqslant\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}} \|\sigma(s)\|^2_{Z^*}\,ds \,dt \nonumber \\ &\leqslant c_{12}\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}(1+\|x_\tau(s)\|_X^2 +\|u_\tau^{k+1}\|_{V}^2)\,ds \,dt\stackrel{(4.15)}{\leqslant} C, \end{aligned}
\end{equation}
\tag{4.31}
$$
where $\sigma(s)\in G(s,x_\tau(s),u_\tau^{k+1})$, $c_{12}>0$ by $\textrm{H}(G)$(1) and the continuous embedding $V\hookrightarrow Z$.
By (4.24), (4.30), (4.31) and Lemma 4.1, there exists a positive constant $C$ such that
$$
\begin{equation}
\biggl\|\frac{d\widetilde{u}_\tau}{dt}\biggr\|_{\mathcal V^*} =\sum_{k=1}^{N}\biggl\|\frac{{u}^{k}_\tau-{u}^{k-1}_\tau}{\tau}\biggr\|_{ V^*}\leqslant \|K_\tau\|_{\mathcal{V}^*}+\|g_\tau\|_{\mathcal{V}^*}+\|h_\tau\|_{\mathcal{V}^*}\leqslant C.
\end{equation}
\tag{4.32}
$$
In order to show that the boundedness of $\{\overline u_\tau\}$ in $M^{2,2}(I;V,V^*)$, it remains to verify that $\{\overline u_\tau\}$ is bounded in $\mathrm{BV}^2(I,V^*)$. Consider any finite partition $\pi$ of the interval $(0,b)$, say $\pi=\{b_i\}_{i=0}^{k}$ with $0=b_0<b_1<\dots<b_k=b$. For this partition, we construct two sequences $\{k_i\}_{i=0}^{s}$ and $\{m_i\}_{i=0}^{s}$ with $0<s\leqslant k$ as follows.
Choosing $m_1=1$ and $k_1=\max\{i\mid b_i\in [0,\tau]\}$, there exists $m_2> m_1$ such that $b_{k_1+1}\in ((m_2-1)\tau,m_2\tau]$, so we may define $k_2=\max\{k_1+i\mid b_{k_1+i}\in ((m_2- 1)\tau, m_2\tau]\}$. By induction, we may construct $\{k_i\}_{i=1}^{s}$ and $\{m_i\}_{i=1}^{s}$ such that
$$
\begin{equation*}
k_1<k_2<\dots<k_{s-1}<k_s=N\quad\text{and}\quad 1=m_1<m_2<\dots<m_{s-1}<m_s=b
\end{equation*}
\notag
$$
with $b_{k_i}\in ((m_i-1)\tau, m_i\tau]$ for all $i=1,\dots,s$. Hence, $u_\tau(b_{k_i})=u_\tau^{m_i}$ with $m_1=1$, $m_k=N$ and $m_{i+1}>m_i$ for $i=1,2,\dots,s-1$. Hence, we have
$$
\begin{equation*}
\begin{aligned} \, &\|\overline u_\tau\|_{\mathrm{BV}^2(I,V^*)}^2=\sup_{\pi\in\Pi}\sum_{i=1}^{N}\|\overline u_\tau(b_{i})-\overline u_\tau(b_{i-1})\|_{V^*}^2 =\sup_{\pi\in\Pi}\sum_{i=1}^{s}\|u_\tau^{m_i}-u_\tau^{m_{i-1}}\|_{V^*}^2 \\ &\leqslant\sup_{\pi\in\Pi}\sum_{i=1}^{s}\sum_{l=m_{i-1}+1}^{m_i}\|u_\tau^{l}-u_\tau^{l-1}\|_{V^*}^2 \leqslant\sum_{i=1}^{N}\sum_{l=1}^{N}\|u_\tau^{l}-u_\tau^{l-1}\|_{V^*}^2 \\ &\leqslant N\sum_{l=1}^{N}\|u_\tau^{l}-u_\tau^{l-1}\|_{V^*}^2 =b\tau\sum_{l=1}^{N}\biggl\|\frac{u_\tau^{l}-u_\tau^{l-1}}{\tau}\biggr\|_{V^*}^2 =b\|\frac{d\widetilde{u}_\tau}{dt}\|^2_{\mathcal{V}^*}\stackrel{(4.32)}{<} C, \end{aligned}
\end{equation*}
\notag
$$
which completes the proof of Proposition 4.6. Theorem 4.7. If $h\in \mathcal{V}^*$, assumptions $\textrm{H}(A)$, $\textrm{H}(\mathcal R)$, $\textrm{H}(F)$, $\textrm{H}(K)$, and $\textrm{H}(G)$ are met, then there exists at least a mild solution of the abstract system (1.1). Proof. Note that
$$
\begin{equation}
\|\widetilde{u}_\tau-\overline u_\tau\|_{\mathcal{V}^*}^2 \leqslant\sum_{k=0}^{N-1}\int_{t_\tau^{k}}^{t_\tau^{k+1}}(t_\tau^{k+1}-s)^2 \biggl\|\frac{u_\tau^{k+1}-u_\tau^{k}}{\tau}\biggr\|_{V^*}^2\, ds \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
=\frac{\tau^2}{3}\biggl\|\frac{d\widetilde{u}_\tau}{dt}\biggr\|^2_{\mathcal{V}^*} \xrightarrow{(4.32)} 0 \quad\text{as }\tau\to 0,
\end{equation}
\tag{4.33}
$$
$$
\begin{equation}
\|\overline{u}_\tau-u_\tau\|_{\mathcal H}^2 =\tau\sum_{k=0}^{N-1}\|u_\tau^{k+1}-u_\tau^{k}\|_{\mathcal H}^2 \xrightarrow{(4.15)} 0 \quad\text{as }\tau\to 0.
\end{equation}
\tag{4.34}
$$
In view of the reflexivity of $\mathcal{V}, \mathcal{Z},\mathcal H$, Lemma 2.6, estimates (4.25), (4.26), (4.33), (4.34), and by passing to subsequences, if necessary, there exist $u\in \mathcal{V}$, $k\in \mathcal{V}^*$, and $g\in \mathcal{Z}^*$ such that as $\tau\to 0$
$$
\begin{equation}
\overline u_\tau, \widetilde{u}_\tau\rightharpoonup u \quad\text{in }\mathcal V,\qquad u_\tau, \widetilde{u}_\tau, \overline{u}_\tau\to u \quad\text{in }\mathcal Z,
\end{equation}
\tag{4.35}
$$
$$
\begin{equation}
\frac{d\widetilde{u}_\tau}{dt}\rightharpoonup \frac{du}{dt}\quad\text{in } \mathcal V^*,
\end{equation}
\tag{4.36}
$$
$$
\begin{equation}
\overline u_\tau, \widetilde{u}_\tau\rightharpoonup u \quad\text{weakly$^*$ in } L^\infty(I,Z^*),
\end{equation}
\tag{4.37}
$$
$$
\begin{equation}
K_\tau \rightharpoonup k \quad\text{in } \mathcal{V}^*,
\end{equation}
\tag{4.38}
$$
$$
\begin{equation}
g_\tau \rightharpoonup g \quad\text{in } \mathcal{Z}^*.
\end{equation}
\tag{4.39}
$$
In what follows, we are going to show that, for any sequence $\tau\to 0$, there exists a subsequence, still denoted by $\{\tau\}$, such that $x_\tau\to x$ in $C(I,X)$. Since by (4.15) $\{x_\tau\}_{0<\tau\leqslant\tau_0}$ is a bounded subset in $C(I,X)$, in view of Ascoli–Arzelà theorem we only need to show that $\{x_\tau\}_{0<\tau\leqslant\tau_0}$ for some $\tau_0>0$ is equicontinuous and $\{x_\tau(t)\}_{0<\tau\leqslant\tau_0}$ is relatively compact for any $t\in I$ in $X$. In fact, for any $x_\tau\in \mathcal S(u_\tau)$, there is $ f_\tau\in \mathcal N_F^{\,2}(x_\tau,u_\tau)$ such that
$$
\begin{equation}
x_\tau(t)=T(t)x_0+ \int_0^tT(t-s)f_\tau(s)\, ds\quad \forall\, t\in I.
\end{equation}
\tag{4.40}
$$
By (4.35) we have $u_\tau\to u$ in $\mathcal H$ as $\tau\to 0$. It is also easily shown that $f_\tau\in L^2(I,X)$ is bounded from Proposition 3.3 and (4.15). Thus, we may assume that $f_\tau\rightharpoonup f$ in $L^2(I,X)$, which implies that $x_\tau\to x$ in $C(I,X)$ from Lemma 2.7. Thus, making $\tau\to 0$ in (4.40), we obtain
$$
\begin{equation}
x_\tau(t) \to x(t)=T(t)x_0+ \int_0^tT(t-s)f(s)\, ds \quad \forall\, t\in I.
\end{equation}
\tag{4.41}
$$
Since $x_\tau\to x$ in $C(I,X)$, one may assume $\mathcal R x_\tau\to \mathcal R x$ in $L^2(I,Y)$. Passing to a subsequence, if necessary, we may also assume $\mathcal R x_\tau(t)\to \mathcal R x(t)$ in $Y$ and $u_\tau(t)\to u(t)$ in $H$ for a.e. $t\in I$. So from Proposition 3.3, we infer $f\in \mathcal N_F^{\,2}(x,u)$, which implies $x\in \mathcal S(u)$.
By (4.24) and from the above arguments, we deduce that
$$
\begin{equation}
\frac{d u}{dt}+k+g=h \quad \text{in } \mathcal{V}^*.
\end{equation}
\tag{4.42}
$$
In the following, we show that $K(x_\tau,\overline u_\tau)\rightharpoonup k$ in $\mathcal{V}^*$ and $g(x_\tau,\overline u_\tau) \rightharpoonup g$ in $\mathcal{Z}^*$. For any $\zeta\in\mathcal{V}$, we have from (4.23)
$$
\begin{equation*}
\begin{aligned} \, &\langle K_\tau-K(x_\tau,\overline u_\tau), \zeta\rangle_{\mathcal{V}} \\ &=\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\biggl\langle \frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}} K(s,x_\tau(s),u_\tau^{k+1})\,ds-K(t,x_\tau(t),u_\tau^{k+1}), \zeta(t)\biggr\rangle_{V}\,dt \\ &=\sum_{k=0}^{N-1}\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}\int_{t_\tau^k}^{t_\tau^{k+1}} \langle K(s,x_\tau(s),u_\tau^{k+1})-K(t,x_\tau(t),u_\tau^{k+1}), \zeta(t)\rangle_{V}\,ds\,dt \\ &=\sum_{k=0}^{N-1}\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}\int_{t_\tau^k}^{t_\tau^{k+1}} \langle K(s,x_\tau(s),u_\tau^{k+1})-k(s), \zeta(t)\rangle_{V}\,ds\,dt \\ &\qquad+\sum_{k=0}^{N-1}\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}\int_{t_\tau^k}^{t_\tau^{k+1}} \langle k(s)-k(t), \zeta(t)\rangle_{V}\,ds\,dt \\ &\qquad+\sum_{k=0}^{N-1}\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}\int_{t_\tau^k}^{t_\tau^{k+1}} \langle k(t)-K(t,x_\tau(t),u_\tau^{k+1}), \zeta(t)\rangle_{V}\,ds\,dt \\ &=\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}} \biggl\langle K(s,x_\tau(s),u_\tau^{k+1})-k(s), \frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}\zeta(t)\,dt\biggr\rangle_{V}\,ds \\ &\quad \text{(by exchanging the order of integration)} \\ &\qquad+\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\biggl\langle \frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}k(s)\,ds-k(t), \zeta(t)\biggr\rangle_{V}\,ds\,dt \\ &\qquad+\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}} \langle k(t)-K(t,x_\tau(t),u_\tau^{k+1}), \zeta(t)\rangle_{V}\,dt \\ &=\langle K_\tau-k, \zeta_\tau\rangle_{\mathcal{V}}+\langle k_\tau-k, \zeta\rangle_{\mathcal{V}}+\langle k-K_\tau, \zeta\rangle_{\mathcal{V}}, \end{aligned}
\end{equation*}
\notag
$$
where $k_\tau$ and $\zeta_\tau$ are piecewise constant approximation functions of $k$ and $\zeta$ similar to the definition of $f_\tau$ as defined in Lemma 4.1. Thus, by (4.38) and Lemma 4.1,
$$
\begin{equation*}
\langle K_\tau-K(u_\tau), \zeta\rangle_{\mathcal{V}}\to 0\quad\forall\, \zeta\in \mathcal{V},
\end{equation*}
\notag
$$
which implies that $K(x_\tau,\overline u_\tau)\rightharpoonup k$. Similarly, we can easily show that $g(x_\tau,\overline u_\tau)\rightharpoonup g$ and omit the details.
By (4.35) we may assume that $\overline u_\tau\rightharpoonup u$ in $\mathcal{V}$ and $\overline u_\tau\to u$ in $\mathcal H$. Hence, by passing to a subsequence we get $\overline u_\tau(t)\to u(t)$ in $H$ for a.e. $t\in I$. Taking $j=N-1$ in (4.18), we have
$$
\begin{equation}
\begin{aligned} \, &\|u_\tau(b)\|_H^2+\sum_{k=0}^{N-1}\|u_\tau^{k+1}-u_\tau^{k}\|_H^2+2\tau\sum_{k=0}^{N-1}\langle K_\tau^k(u_\tau^{k+1})+g_\tau^k(u_\tau^{k+1}),u_\tau^{k+1}\rangle_{V} \nonumber \\ &\qquad=2\tau\sum_{k=0}^{N-1}\langle h_\tau^k,u_\tau^{k+1}\rangle_{V}+\|u_0\|_H^2 \end{aligned}
\end{equation}
\tag{4.43}
$$
and hence
$$
\begin{equation*}
\begin{gathered} \, \|u_\tau(b)\|_H^2+ 2\tau\sum_{k=0}^{N-1}\langle K_\tau^k(u_\tau^{k+1})+g_\tau^k(u_\tau^{k+1}),u_\tau^{k+1}\rangle_{V}\leqslant 2\tau\sum_{k=0}^{N-1}\langle h_\tau^k,u_\tau^{k+1}\rangle_{V}+\|u_0\|_H^2, \\ \|u_\tau(b)\|_H^2+ 2\langle K_\tau+g_\tau,\overline u_\tau\rangle_{\mathcal{V}}\leqslant 2 \langle h_\tau,\overline u_\tau\rangle_{\mathcal{V}}+\|u_0\|_H^2. \end{gathered}
\end{equation*}
\notag
$$
Therefore, one has
$$
\begin{equation}
\limsup_{\tau\to 0}\langle K_\tau+g_\tau,\overline u_\tau-u\rangle_{\mathcal{V}}\leqslant \langle h-k-g,u\rangle_{\mathcal{V}}+\frac{1}{2}\|u_0\|_H^2-\frac{1}{2}\liminf_{\tau\to 0}\|u_\tau(b)\|_H^2.
\end{equation}
\tag{4.44}
$$
Multiplying both sides of (4.42) by $u$, we get
$$
\begin{equation*}
\int_0^b\biggl\langle \frac{du(t)}{dt},u(t)\biggr\rangle_{V}=\langle h-k-g,u\rangle_{\mathcal{V}}.
\end{equation*}
\notag
$$
Since
$$
\begin{equation*}
\frac{1}{2}\frac{d}{dt}\|u(t)\|_H^2=\biggl\langle\frac{du(t)}{dt},u(t)\biggr\rangle_{V},\quad \text{a.e. } t\in I.
\end{equation*}
\notag
$$
Integrating over $I$, we have, by the above,
$$
\begin{equation*}
\frac{1}{2}\|u(b)\|_H^2-\frac{1}{2}\|u_0\|_H^2=\langle h-k-g,u\rangle_{\mathcal{V}}.
\end{equation*}
\notag
$$
Since $u_\tau(b)\rightharpoonup u(b)$ in $V^*$ and since the norm is the weakly lower semicontinuous, we have
$$
\begin{equation*}
\liminf_{\tau\to 0}\|u_\tau(b)\|_H\geqslant \|u(b)\|_H.
\end{equation*}
\notag
$$
Using (4.44), we have
$$
\begin{equation*}
\limsup_{\tau\to 0}\langle K_\tau+g_\tau,\overline u_\tau-u\rangle_{\mathcal{V}}\leqslant 0.
\end{equation*}
\notag
$$
On the other hand,
$$
\begin{equation*}
\limsup_{\tau\to 0}\langle g_\tau,\overline u_\tau-u\rangle_{\mathcal{V}}=\lim_{\tau\to 0}\langle g_\tau,\overline u_\tau-u\rangle_{\mathcal{Z}}\stackrel{(4.35)}{=} 0.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation}
\limsup_{\tau\to 0}\langle K_\tau,\overline u_\tau-u\rangle_{\mathcal{V}}\leqslant 0.
\end{equation}
\tag{4.45}
$$
Recalling $u_\tau\rightharpoonup u$ in $\mathcal{V}$ and $u_\tau\to u$ in $\mathcal{Z}$, by Proposition 4.2, we have $K(x,u)=k$ and $\lim_{\tau\to 0}\langle K_\tau,\overline u_\tau-u\rangle=0$.
It remains to show that $g\in G(x,u)$. For any $\eta\in \mathcal{Z}$,
$$
\begin{equation*}
\begin{aligned} \, \langle g_\tau, \eta\rangle_{\mathcal{Z}} &=\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\biggl\langle \frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}} \sigma_\tau(s)\,ds, \eta(t)\biggr\rangle_{Z}\,dt \quad(\sigma_\tau(s)\in \mathcal N_G^{\,2}(x_\tau,u_\tau^{k+1})) \\ &=\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\biggl\langle \frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}} \sigma_\tau(s)\,ds, \eta(t)\biggr\rangle_{Z}\,dt \\ &=\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\biggl\langle \sigma_\tau(s), \frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}\eta(t)\,dt\biggr\rangle_{Z}\,ds \\ &=\sum_{k=0}^{N-1}\int_{t_\tau^k}^{t_\tau^{k+1}}\langle \sigma_\tau(s), \eta_\tau(s)\rangle_{Z}\,ds =\int_0^b\langle \sigma_\tau(s), \eta_\tau(s)\rangle_{Z}\,ds =\langle \sigma_\tau, \eta_\tau\rangle_{\mathcal{Z}}\,ds, \end{aligned}
\end{equation*}
\notag
$$
$\sigma_\tau(s)\in \mathcal N_G^{\,2}(x_\tau,\overline u_{\tau})$, where $\eta_\tau$ is the piecewise constant function
$$
\begin{equation*}
\eta_\tau=\eta_\tau^k=\frac{1}{\tau}\int_{t_\tau^k}^{t_\tau^{k+1}}\eta(s)\,ds
\end{equation*}
\notag
$$
for all $t\in (t_\tau^k,t_\tau^{k+1})$ and $k=0,\dots,N-1$. Thus, we may assume that $\sigma_\tau\rightharpoonup \sigma\in \mathcal N_G^{\,2}(x,u)$ by Proposition 3.3. Making $\tau\to 0$ in the above identity, we have, by Lemma 4.1,
$$
\begin{equation*}
\langle g, \eta\rangle_{\mathcal{Z}}=\langle \sigma, \eta\rangle_{\mathcal{Z}}\quad \forall\, \eta\in \mathcal{Z}.
\end{equation*}
\notag
$$
Therefore, $g=\sigma\in \mathcal N_G^{\,2}(x,u)$.
In conclusion, we have
$$
\begin{equation*}
\begin{cases} \displaystyle x(t)=T(t)x_0+ \int_0^tT(t-s)f(s)\, ds\quad \forall\, t\in I, \\ \displaystyle v'(t)+K(t,x(t),u(t))+g(t)=h(t),\quad \text{a.e. }t\in I, \\ \displaystyle u(0)=u_0,\quad f\in \mathcal N^{\,2}_F(x,u), \quad g\in \mathcal N^{\,2}_G(x,u), \end{cases}
\end{equation*}
\notag
$$
which completes the proof. AcknowledgementThe authors are grateful to the anonymous referees for their valuable remarks and criticisms which improved the results and presentation of this article. 
Citation in format AMSBIB
\Bibitem{ZhaLiuPap24} |
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