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This article is cited in 2 scientific papers (total in 2 papers) Comparison theorems for evolution inclusions with maximal monotone operators. A. A. Tolstonogov Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk, Russia
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     § 1. Main resultsLet $T=[0,a]$, $a>0$, be an interval of the positive half-line ${\mathbb R}^+=[0,+\infty)$ with Lebesgue measure $\mu$ and $\sigma$-algebra $\Sigma$ of measurable subsets of $T$, and let $H$ be a separable Hilbert space with inner product $\langle \,\cdot\,{,} \,\cdot\, \rangle$, the induced norm $\| \cdot \|$ and zero element $\Theta $. The aim of our work is the investigation of questions relating to the solvability of an evolution inclusion
$$
\begin{equation}
\begin{gathered} \, -\dot{x}(t)\in A(t) x(t) + f(t), \\ \notag x(0) = x_0 \in D(A(0)), \qquad f(\,\cdot\,)\in L^1(T,H). \end{gathered}
\end{equation}
\tag{1.1}
$$
Here $A(t)\colon D(A(t))\subset H\rightrightarrows H$, $t\in T$, is a family of maximal monotone operators with domains $D(A(t))\subset H$. By a solution of the inclusion (1.1) for fixed $x_0$ and $f(\,\cdot\,)\in L^1(T,H)$ we mean an absolutely continuous function $x_A(f)\colon T\to H$, $x_A(0) = x_0$, whose derivative $\dot{x}_A(f)(t)$ satisfies the inclusion
$$
\begin{equation*}
-\dot{x}_A(f)(t)\in A(t) x_A(f)(t) + f(t) \quad \text{a.e.}
\end{equation*}
\notag
$$
We assume the following hypotheses $H(A)$:
If (1.2) holds, then for each $t\in T$ the set $D(A(t))$ is closed and convex (see [2], Proposition 2.1). Set and consider the sweeping process (see [3])
$$
\begin{equation}
\begin{gathered} \, -\dot{x}(t)\in N(D(t))x(t) + \varphi (t), \\ x(0)=x_0 \in D(0), \qquad \varphi (\,\cdot\,)\in L^1(T,H), \notag \end{gathered}
\end{equation}
\tag{1.4}
$$
where $N(D(t))x$ is the normal cone to $D(t)$ (in the sense of convex analysis) at the point $x\in D(t)$. A solution of the sweeping process (1.4) for fixed $x_0$ and $\varphi (\,\cdot\,)\in L^1(T,H)$ is an absolutely continuous function $x_N(\varphi)\colon T\to H$, $x_N(\varphi)(0) = x_0$, whose derivative $\dot{x}_N(\varphi)(t)$ satisfies (1.4) almost everywhere. In what follows we assume that $x_0 \in D(A(0))$ is fixed. The main result of our paper is as follows. Theorem 1.1. Assume that hypotheses $H(A)$ are fulfilled. If for each $\varphi (\,\cdot\,)\in L^1(T,H)$ the sweeping process (1.4) has a solution $x_N(\varphi)\colon T\to H$, $ x_N(\varphi)(0) = x_0$, then for each $f(\,\cdot\,)\in L^1(T,H)$ the evolution inclusion (1.1) has a unique solution $x_A(f)$: $T\to H$ such that $x_A(f)(0) = x_0$. Furthermore, for each $f(\,\cdot\,)\in L^1(T,H)$ there exists $\varphi (\,\cdot\,)\in L^1(T,H)$ such that If for each $r_N(\,\cdot\,)\in L^1(T,{\mathbb R}^+)$ the set of functions is equicontinuous, then for each $r_A(\,\cdot\,)\in L^1(T,{\mathbb R}^+)$ the set is an equicontinuous subset of the space $C(T,H)$ of continuous functions from $T$ to $H$ with $\sup$-norm. We present a priori estimates for the solutions $x_A(f)(\,\cdot\,)$, $x_A(f)(0)=x_0$, and their derivatives $\dot x_A(f)(\,\cdot\,)$. We also introduce the concept of $\rho$-pseudoexcess between maximal monotone operators, where $\rho \in [0,+\infty]$. For $\rho =+\infty$ this pseudoexcess is the same as the pseudodistance $\operatorname{dis}(\,\cdot\,{,}\,\cdot\,)$ in the sense of Vladimirov (see [4]) between two maximal monotone operators $A_1$ and $A_2$, which is defined by
$$
\begin{equation}
\operatorname{dis}(A_1,A_2)= \sup \biggl\{\frac{\langle x_1-x_2,\,y_2-y_1\rangle}{1+ \| y_1\| +\| y_2\|};\, y_i\in Ax_i,\, x_i \in D(A_i),\, i=1,2\biggr\}.
\end{equation}
\tag{1.6}
$$
In terms of $\rho$-pseudoexcess we state conditions ensuring that for fixed $\varphi (\,\cdot\,)\in L^1(T,H)$ the sweeping process (1.4) has a solution $x_N(\varphi)(\,\cdot\,)$ such that ${x_N(\varphi)(0) \!=\! x_0}$. Here is one such result for $\rho =+\infty $. Corollary 1.1. Assume that (1.2) holds and there exists an absolutely continuous function $b\colon T\to {\mathbb R}$ such that Here $\|\dot{b}(\,\cdot\,)\|_{L^1}$ is the norm of the element $\dot{b}(\,\cdot\,)$ of $L^1(T,{\mathbb R})$, and $\|f(\,\cdot\,)\|_{L^1}$ is the norm of $f(\,\cdot\,)\in L^1(T,H)$. Note that the statement of Corollary 1.1 does not involve hypothesis $H(A)$, 1): it is an automatic consequence of (1.2) and (1.7). We use our results to examine the evolution inclusion
$$
\begin{equation}
\begin{gathered} \, -\dot{x}(t)\in A(t)x(t) + F(t,x(t)), \\ \notag x(0) = x_0 \in D(A(0)), \end{gathered}
\end{equation}
\tag{1.11}
$$
where $F\colon T\times H\rightrightarrows H$ is a set-valued map with closed (not necessarily convex) values. We assume the following hypotheses $H(F)$: 1) the map $t\to F(t,x)$ is measurable; 2) the inequality
$$
\begin{equation}
\operatorname{haus}(F(t,x),F(t,y))<k(t)\| x-y\| \quad \text{a.e.}
\end{equation}
\tag{1.12}
$$
holds, where $k(\,\cdot\,)\in L^1(T,{\mathbb R}^+)$, $k(t)>0$, $t\in T$, $x,y\in H$, and $\operatorname{haus}(\,\cdot\,{,}\,\cdot\,)$ is the Hausdorff distance between two sets; 3) for some $n(\,\cdot\,)\in L^1(T,{\mathbb R}^+)$ such that $n(t)>0$, $t\in T$, the inequality
$$
\begin{equation}
\| F(t,\Theta)\|= \sup \bigl\{\| y\|;\, y\in F(t,\Theta)\bigr\}\leqslant n(t), \qquad t\in T,
\end{equation}
\tag{1.13}
$$
holds; 3$^*$) the inequality
$$
\begin{equation}
d(\Theta,F(t,\Theta))=\inf\bigl\{\|y\|;\, y\in F(t,\Theta)\bigr\} \leqslant n(t), \qquad t\in T_0,
\end{equation}
\tag{1.14}
$$
holds. By a solution of (1.11) we mean a pair $(x(v)(\,\cdot\,),v(\,\cdot\,))$, where $x(v)\colon T\to H$ is an absolutely continuous function $x(v)(0) = x_0$ and $v(\,\cdot\,)\in L^1(T,H)$, such that
$$
\begin{equation}
-\dot{x}(v)(t)\in A(t)x(v)(t) + v(t) \quad \text{a.e.},
\end{equation}
\tag{1.15}
$$
We call the function $x(v)(\,\cdot\,)$ the trajectory of the inclusion (1.11). We denote the set of solutions of (1.11) by $\mathcal R_F(x_0)$, and we denote the set of trajectories by $\mathcal Tr_F(x_0)$. Consider the differential equation which has a solution
$$
\begin{equation}
r(t)=e^{\alpha (t)}\biggl(r_0+\int_0^t e^{-\alpha (\tau)} n(\tau)\,d\tau \biggr),
\end{equation}
\tag{1.18}
$$
where Theorem 1.2. Under the hypotheses $H(F)$, 1), 2) and 3$^*$) assume that for any $x_0 \in D(A(0))$ and $f(\,\cdot\,)\in L^1(T,H)$ inclusion (1.1) has a solution $x(f)(\,\cdot\,)$ such that $x(f)(0)=x_0$. Then the set $\mathcal R(x_0)$ is not empty. If hypotheses $H(F)$, 1)–3) are fulfilled, then each solution $(x(v)(\,\cdot\,),v(\,\cdot\,))\in \mathcal R(x_0)$ satisfies the inequalities and where $r(t)$ is a solution of (1.17) such that $r(0)=\| x_A(\Theta)\|_C$. Moreover, if the set of functions $\{x_A(f)(\,\cdot\,);\, \| f(t)\| \leqslant \dot{r}(t)\}$ is equicontinuous, then the set $\mathcal Tr_F(x_0)$ is too. Here $\| x_A(\Theta)\|_C$ is the norm in $C(T,H)$ of the solution $x_A(\Theta)(t)$ of inclusion (1.15) for $v(t)\equiv \Theta $, $t\in T$. The fact that inclusion (1.1) is solvable follows from the solvability of (1.4). In a similar way we see that inclusion (1.11) has a solution by comparing it with (1.1). For this reason Theorems 1.1 and 1.2 are called comparison theorems. The term $L^2$-theory indicates that we use the Hilbert space $L^2(T,H)$ in the proofs. All results in this paper and the approach we use are new. As consequences of Theorems 1.1 and 1.2, we can obtain the results from the recently published (and so far unique) paper [5] on the existence of an absolutely continuous solution of the inclusion
$$
\begin{equation*}
-\dot{x}(t)\in A(t)x(t) + f(t,x(t)), \qquad x(0) = x_0 \in D(A(0)).
\end{equation*}
\notag
$$
We compare the results in [5] with ours in § 7. The paper consists of seven sections. In § 1 we present a number of main result and in § 2 we present the main notation and definitions. In § 3 we state some facts necessary for the proofs of the main results. In § 4 we present the proof of Theorem 1.1. In § 5 we specify conditions in terms of the properties of the family of maximal monotone operators $A(t)$, $t\in T$, which ensure that (1.4) is solvable. In § 6 we present the proof of Theorem 1.2. In § 7 we examine the results obtained and compare them with the known results. § 2. Main notation and definitionsLet $X$ be a separable Banach space with norm $\| \cdot \|$ and induced metric $d(\,\cdot\,{,}\,\cdot\,)$, let $\Theta$ be zero in $X$, and let $B$ and $\overline{B}$ be the unit open and closed balls with centre $\Theta$. We let $d(y,C)$ denote the distance of the point $y$ to the set $C\subset X$, and let Given a set $C\subset X$, we let $\operatorname{\overline{co}}C$ denote the closed convex hull of $C$. The space $X$ with weak topology is denoted by $\omega$-$X$. The space of continuous functions $x\colon T\to X$ with the topology of uniform convergence on $T$ is denoted by $C(T,X)$, and the norm in it is denoted by $\| x\|_C$.The space $L^1(T,X)$ of Bochner-integrable functions $f\colon T\to X$ is endowed with the norm
$$
\begin{equation*}
\| f(\,\cdot\,)\|_{L^1}=\int_T \| f(\tau)\| \,d\tau .
\end{equation*}
\notag
$$
A set-valued map $F\colon T\rightrightarrows X$ is a map whose values are nonempty subsets of $X$. Let $F\colon T\rightrightarrows X$ be a set-valued map, and let $F^{-1}(U)$, $U\subset X$, denote the set
$$
\begin{equation*}
F^{-1}(U)=\bigl\{t\in T;\, F(t) \cap U \ne \varnothing\bigr\}.
\end{equation*}
\notag
$$
Then $F\colon T\!\rightrightarrows\! X$ is said to be measurable (weakly measurable; see [6]) if ${F^{-1}(U)\!\in\!\Sigma}$ for each closed (open) subset $U$ of $ X$. If the values of a set-valued map $F\colon T\rightrightarrows X$ are closed subsets, then measurability is equivalent to weak measurability (see [6]). Remark 2.1. Note that in the paper [7], results from which are used in what follows, measurability is understood as weak measurability in the sense of [6]. Let $C(t)$, $t\in T$, be a family of operators $C(t)\colon D(C(t))\subset X\rightrightarrows X$ with domains $D(C(t))$. Let $\operatorname{gr}C(t)$ denote the graph of $C(t)$:
$$
\begin{equation*}
\operatorname{gr}C(t)=\bigl\{(x,y)\in X\times X;\, x\in D(C(t)),\,y\in C(t)x\bigr\}.
\end{equation*}
\notag
$$
We say that the family $C(t)\colon D(C(t))\subset X\rightrightarrows X$ is measurable (weakly measurable) if so is the set-valued map $t\to \operatorname{gr}C(t)$. Let $\mathcal T=[0,a)$ be the Sorgenfrey interval (see [8]). The basis of topology on this interval consists of the half-open intervals $[x,r)$, where $x\in [0,a)$, $x<r<a$, and $r$ is a rational number. Each nonempty open subset of the Sorgenfrey interval is a finite or a countable union of half-open intervals $[x,c)$, where $c$ is a real number (see [8]). Let $\rho \in [0,+\infty ]$, and let $C$ and $D$ be nonempty subsets of $X$. We call the quantity
$$
\begin{equation}
\operatorname{exc}_{\rho}(C,D)= \sup \bigl\{d(x,D);\, x\in C\cap \rho \overline{B}\bigr\}
\end{equation}
\tag{2.1}
$$
the $\rho$-excess of $C$ from $D$. Here we set $\operatorname{exc}_{\rho}(C,D)=0$ if $C\cap \rho \overline{B}=\varnothing $. We define the Hausdorff $\rho$-distance (see [9]) between $C$ and $D$ by
$$
\begin{equation}
\operatorname{haus}_{\rho}(C,D)= \max \bigl\{\operatorname{exc}_{\rho}(C,D),\operatorname{exc}_{\rho}(D,C)\bigr\}.
\end{equation}
\tag{2.2}
$$
Because $\rho \overline{B}=X$ for $\rho =+\infty $, we have
$$
\begin{equation}
\begin{gathered} \, \operatorname{exc}_{\infty}(C,D)= \sup \bigl\{d(x,D);\, x\in C\bigr\} =\operatorname{exc}(C,D), \notag \\ \operatorname{haus}_{\infty}(C,D)=\operatorname{haus}(C,D) =\max \bigl\{\operatorname{exc}(C,D),\operatorname{exc}(D,C)\bigr\}, \end{gathered}
\end{equation}
\tag{2.3}
$$
where $\operatorname{exc}(C,D)$ is the excess of $C$ from $D$ and $\operatorname{haus}(C,D)$ is the Hausdorff distance between these sets. In what follows $H=X$ will be a separable Hilbert space with inner product $\langle \,\cdot\,{,}\, \cdot\, \rangle$. An operator $A\colon D(A)\subset H\rightrightarrows H$ is said to be monotone if for any $(x_i,y_i)\in \operatorname{gr}A$, $ i=1,2$, we have $\langle x_1-x_2,\,y_1-y_2\rangle \geqslant 0$. A monotone operator is maximal monotone if its graph is not a proper subset of the graph of another monotone operator. If $A\colon D(A)\subset H\rightrightarrows H$ is a maximal monotone operator, then $Ax$, $x\in D(A)$, is a convex closed subset of the space $H$, its graph $\operatorname{gr}A$ is a closed subset of $H\times H$, and the closure $\overline{D(A)}$ in $H$ of the set $D(A)$ is a convex closed subset of $H$ (see [1]). For each $x \in D(A)$ there exists an element $A^0x \in Ax$ with the minimum norm, that is, A monotone operator $A$ has a maximal monotone extension $\widehat{A}$ such that $D(\widehat{A})\subset \operatorname{\overline{co}} D(A)$ (see [1]).If $A\colon D(A)\subset H\rightrightarrows H$ is a maximal monotone operator, then the single-valued operators of $H$,
$$
\begin{equation}
J_{\lambda}(A)=(I+\lambda A)^{-1}\quad\text{and} \quad A_{\lambda}=\frac{1}{\lambda}(I-J_{\lambda}(A))
\end{equation}
\tag{2.5}
$$
are defined for each $\lambda >0$. They are called the resolvent and Yosida approximation of $A$ with index $\lambda >0$, respectively. (In (2.5), $I\colon H\to H$ is the identity operator.) Here are the main properties (see [1]) of a maximal monotone operator that we use in what follows. Lemma 2.1. Let $A\colon D(A)\subset H\rightrightarrows H$ be a maximal monotone operator. Then 1) the inequality
$$
\begin{equation}
\| J_{\lambda}(A)x - J_{\lambda}(A)y\| \leqslant \| x-y\| , \qquad x,y\in H, \quad \lambda >0,
\end{equation}
\tag{2.6}
$$
holds; 2) the inequality
$$
\begin{equation}
\| A_{\lambda}x - A_{\lambda}y\| \leqslant \frac{1}{\lambda}\| x-y\| , \qquad x,y\in H, \quad \lambda >0,
\end{equation}
\tag{2.7}
$$
holds; 3) the inclusion
$$
\begin{equation}
A_{\lambda}x \in A (J_{\lambda}(A)x), \qquad x\in H, \quad \lambda >0,
\end{equation}
\tag{2.8}
$$
holds; 4) $A_{\lambda}$, $\lambda >0$, is a maximal monotone operator, and
$$
\begin{equation}
\| A_{\lambda}x\| \leqslant \| A^0x\| , \qquad x\in D(A), \quad \lambda >0;
\end{equation}
\tag{2.9}
$$
5) if $(x_n,y_n)\in \operatorname{gr}A$ for $n\geqslant 1$ and the sequence $x_n$, $n\geqslant 1$, converges to $x$ in the space $H$, while the sequence $y_n$, $n\geqslant 1$, converges to $y$ in $\omega$-$H$, then $(x,y)\in \operatorname{gr}A$. Let $C\subset H$ be a convex closed set. By the normal cone $N(C)x$ to $C$ at the point $x\in C$ in the sense of convex analysis we mean the set
$$
\begin{equation}
N(C)x=\bigl\{y\in H;\, \langle y,z-x\rangle \leqslant 0 \ \forall\, z\in C\bigr\}.
\end{equation}
\tag{2.10}
$$
It is known that $N(C)\colon C\subset H\rightrightarrows H$ is a maximal monotone operator. Let $\rho \in [0,+\infty ]$, and let $A_i\colon D(A_i)\subset H\rightrightarrows H$, $i=1,2$, be maximal monotone operators. Then the quantity
$$
\begin{equation}
\begin{aligned} \, \operatorname{exc\,dis}_{\rho}(A_1,A_2) &= \sup \biggl\{\frac{\langle x_1-x_2,\,y_2-y_1\rangle}{1+ \| y_1\| +\| y_2\|}; \notag \\ &\qquad\qquad x_1\in D(A_1)\cap \rho \overline{B},\, y_1\in A_1x_1,\,x_2\in D(A_2), \,y_2\in A_2x_2\biggl\} \end{aligned}
\end{equation}
\tag{2.11}
$$
is called the $\rho$-pseudoexcess of $A_1$ from $A_2$. If $D(A_1)\cap \rho \overline{B}=\varnothing $, then we set $\operatorname{exc\,dis}_{\rho}(A_1,A_2)=0$. The quantity
$$
\begin{equation}
\operatorname{dis}_{\rho}(A_1,A_2)= \max \bigl\{\operatorname{exc\,dis}_{\rho}(A_1,A_2),\operatorname{exc\,dis}_{\rho} (A_2,A_1)\bigr\}
\end{equation}
\tag{2.12}
$$
is called the $\rho$-pseudodistance between $A_1$ and $A_2$. For $\rho =+\infty$ the pseudodistance $\operatorname{dis}_{\rho}(A_1,A_2)$ is the same as the pseudodistance $\operatorname{dis}(A_1,A_2)$ between maximal monotone operators in the sense of Vladimirov (see [4]), which is defined by (1.6). § 3. Preliminary facts and auxiliary resultsIn this section we present a number of preliminary statements and known results which we need in the proof of our main results. In what follows, unless otherwise specified, $X$ is a separable Banach space. Lemma 3.1. A single-valued map $F\colon [0,a)\to X$ is right continuous if and only if it is continuous on the Sorgenfrey interval $[0,a)$. Proof. Assume that $F\colon [0,a)\to X$ is continuous on the Sorgenfrey interval $[0,a)$. Consider some $t_0\in [0,a)$ and a sequence $t_n\in [0,a)$, $t_n\downarrow t_0$, $n\geqslant 1$, which converges to $t_0$. We must show that the sequence $x_n=F(t_n)$, $n\geqslant 1$, converges to $x_0=F(t_0)$.
Set
$$
\begin{equation*}
B(x_0,\varepsilon)=\bigl\{x\in X;\, \| x-x_0\| < \varepsilon \bigr\}, \qquad \varepsilon >0,
\end{equation*}
\notag
$$
and consider an arbitrary sequence $\varepsilon_n\downarrow 0$, $ n\geqslant 1$. Since $x_0\in B(x_0,\varepsilon)$, there exists a sequence $\delta (\varepsilon_n)>0$, $t_0+\delta (\varepsilon_n)< a$, such that
$$
\begin{equation}
F(t)\in B(x_0,\varepsilon_n) \quad \forall\, t\in [t_0,t_0+\delta (\varepsilon_n)).
\end{equation}
\tag{3.1}
$$
It follows from (3.1) directly that $x_n=F(t_n)$ converges to $x_0$.
Now we prove the converse. Let the map $F\colon [0,a)\to X$ be right continuous. Suppose it is not continuous on the Sorgenfrey interval $[0,a)$. Then there exists a point $t_0\in [0,a)$ and $\varepsilon >0$ such that for any $\delta_n>0$ such that $t_0+\delta_n < a$, $n\geqslant 1$, $ \delta_n\downarrow 0$, there exist points $t_n\in [t_0,t_0+\delta_n)$, $n\geqslant 1$, such that Because $t_n\downarrow t_0$, $n\geqslant 1$, the sequence $x_n=F(t_n)$, $n\geqslant 1$, converges to $x_0=F(t_0)$. However, this contradicts (3.2). Hence $F\colon [0,a)\to X$ is continuous on the Sorgenfrey interval. The proof is complete.Lemma 3.2. Let $F\colon [0,a]\to X$ be a single-valued map which is right-continuous on $[0,a)$. Then $F\colon [0,a]\to X$ is a Borel map. Proof. For an arbitrary open set $U\subset X$ consider the set It follows from Lemma 3.1 that $F^{-1}(U)$ is an open subset of the Sorgenfrey interval $[0,a)$. By [8], $F^{-1}(U)$ is a union of a countable number of half-open intervals $[t,t+r)$, where $t\in [0,a)$, $t+r<a$. Hence $F^{-1}(U)$ is a Borel set. As the singleton $\{a\}$ is a Borel set, the set is also Borel. The proof is complete. Lemma 3.3 (see [10], Lemma 2.1). Let $V$ be a Hilbert space with inner product $\langle \,\cdot\,{,}\, \cdot\, \rangle_V$ and norm $\| \cdot \|_V$. Assume that the set $\{v_{\lambda};\, 0<\lambda \leqslant 1\}\subset V$ is bounded and Then the function $\lambda \to \| v_{\lambda}\|_V$ is nondecreasing and $v_{\lambda}$ converges in $V$ as $\lambda \downarrow 0$. Lemma 3.4. Let $A_i\colon D(A_i)\subset H\rightrightarrows H$, $ i=1,2$, be maximal monotone operators. Then Proof. If $D(A_1)\cap \rho \overline{B}=\varnothing $, then $\operatorname{exc}_{\rho}(D(A_1),D(A_2))=0$ and $\operatorname{exc\,dis}_{\rho}(A_1,A_2)=0$ by definition. A fortiori (3.3) holds.
Let $x_1\in D(A_1)\cap \rho \overline{B}\ne \varnothing$ and $y_1\in A_1x_1$. Using (2.5) and (2.8) we see that for each $\lambda >0$ there exists a point
$$
\begin{equation}
x_2^{\lambda}=J_{\lambda}(A_2)x_1, \qquad x_2^{\lambda}\in D(A_2),
\end{equation}
\tag{3.4}
$$
such that
$$
\begin{equation*}
\frac{x_1-x_2^{\lambda}}{\lambda}\in A_2x_2^{\lambda}.
\end{equation*}
\notag
$$
It follows from this inclusion and (2.11) that
$$
\begin{equation}
\biggl\langle x_1-x_2^{\lambda},\frac{x_1-x_2^{\lambda}}{\lambda}-y_1\biggr\rangle \leqslant \operatorname{exc\,dis}_{\rho}(A_1,A_2) \biggl(\biggl\|\frac{x_1-x_2^{\lambda}}{\lambda}\biggr\|+\| y_1\| +1\biggr).
\end{equation}
\tag{3.5}
$$
Let $\operatorname{pr}(x_1,\overline{D(A_2)})$ denote the projection of the point $x_1$ onto the convex closed set $\overline{D(A_2)}$, where overline indicates the closure of the set $D(A_2)$. It is known (see [1]) that $J_{\lambda}(A_2)x_1\to \operatorname{pr}(x_1,\overline{D(A_2)})$ as $\lambda \downarrow 0$. Using (3.4) and taking the limit in (3.5) as $\lambda \downarrow 0$ we obtain
$$
\begin{equation*}
\| x_1-\operatorname{pr}(x_1,\overline{D(A_2)})\| \leqslant \operatorname{exc\,dis}_{\rho}(A_1,A_2).
\end{equation*}
\notag
$$
This inequality and (2.1) imply (3.3) directly. The proof is complete. Corollary 3.1. Let $A_i\colon D(A_i)\subset H\rightrightarrows H$ be maximal monotone operators. Then This follows from Lemma 3.4, (2.2), (2.3) and (2.12), (1.6) for $\rho =+\infty $. Theorem 3.1. Let $\Gamma (t)$, $t\in T$, be a weakly measurable family of monotone operators $\Gamma (t)\colon D(\Gamma (t))\subset H\rightrightarrows H$. Then there exists a unique measurable family $\widehat{\Gamma}(t)$, $t\in T$, of maximal monotone operators $\widehat{\Gamma}(t)\colon D(\widehat{\Gamma}(t))\subset H\rightrightarrows H$ such that for each ${t\in T}$ $\widehat{\Gamma}(t)$ is a maximal monotone extension of the operator $\Gamma(t)$. Taking Remark 1.1 into account, this theorem follows from [7] (Theorem 3.1 on p. 100). Theorem 3.2. Let $A(t)$, $t\in T$, be a family of maximal monotone operators $A(t)$: $D(A(t))\subset H\rightrightarrows H$. Then the following properties are equivalent: 1) the family $A(t)$, $t\in T$, is measurable; 2) for all $\lambda >0$ and $x\in H$ the map $t\to J_{\lambda}(A(t))x$ is measurable. Since the set-valued map $t\to \operatorname{gr}A(t)$ takes values in closed sets, its measurability is equivalent to weak measurability. Now, in view of Remark 2.1 the required result follows from [7] (Theorem 2.1 on p. 83). Let $C\colon T\rightrightarrows H$ be a set-valued map with convex closed values. Consider the sweeping process
$$
\begin{equation}
\begin{gathered} \, -\dot{x}(t)\in N(C(t))x(t) + \varphi (t), \\ \notag x(0)=x_0 \in C(0), \qquad \varphi (\,\cdot\,)\in L^1(T,H). \end{gathered}
\end{equation}
\tag{3.7}
$$
Let $r_C(\,\cdot\,)\in L^1(T,{\mathbb R}^+)$ and
$$
\begin{equation}
S^1(r_C)=\bigl\{\varphi (\,\cdot\,)\in L^1(T,H);\, \| \varphi (t)\| \leqslant r_C(t) \text{ a.e.}\bigr\}.
\end{equation}
\tag{3.8}
$$
Theorem 3.3. Assume that for each $\rho \in [0,+\infty)$ there exists an absolutely continuous function $b_{\rho}\colon T\to {\mathbb R}$ such that for all $s,t\in T$, $s\leqslant t$, the inequality For all $x_0$ and $r_C(\,\cdot\,)\in L^1(T,{\mathbb R}^+)$ there exists $\rho_*>0$ depending on $\| x_0\| $ and $\| r_C\|_{L^1}$ such that for $\varphi (\,\cdot\,)\in S^1(r_C)$ and the set of functions is equicontinuous.Here $x_C(\Theta)(\,\cdot\,)$, $x_C(\Theta)(0)=x_0$, is a solution of the sweeping process (3.7) for $\varphi (t)\equiv \Theta$, $t\in T$. This theorem is a consequence of Corollary 1.1 in [11]. Corollary 3.2. Assume that there exists an absolutely continuous function $b\colon \!{T\!\to\! {\mathbb R}}$ such that This follows from Theorem 3.3, provided that we take the relations $b_{\infty}(t) = b(t)$ and $\operatorname{exc}_{\infty}(C(s)$, $C(t))=\operatorname{exc}(C(s),C(t))$ into account. Remark 3.1. In (3.11) and (3.14), as $r_C(\,\cdot\,)\in L^1(T,{\mathbb R}^+)$ we can take $r_C(t)=\| \varphi (t)\| $. § 4. Main resultIn this section we consider the question of the solvability of inclusion (1.1). We denote a solution of (1.1) by $x_A(f)(\,\cdot\,)$. The next statement is the core theorem in our investigations. Theorem 4.1. Let $A\colon D(A)\subset H\rightrightarrows H$ be a maximal monotone operator. Assume that where $m>0$ and $l\colon {\mathbb R}^+\to {\mathbb R}^+$ is a nondecreasing function. Then $D(A)$ is a convex closed set and Proof. It is known (see [1]) that the closure $\overline{D(A)}$ of $D(A)$ is a convex set. The fact that $D(A)$ is closed when (4.1) holds was proved in [2]. Since the values of the maximal monotone operator $N(D(A))$ are cones, it follows that Hence It is obvious that the operator $N(D(A))+A$ with domain $D(A)$ is monotone. As $A$ is maximal monotone, its graph $\operatorname{gr} A$ cannot be a proper subset of the graph of another monotone operator. Hence (4.2) is a consequence of (4.3). The proof is complete. If $A(t)\colon D(A(t))\subset H\rightrightarrows H$, $t\in T$, is a family of maximal monotone operators, then for each $\lambda >0$ we let $A_{\lambda}(t)$ denote the Yosida approximation of $A(t)$, which is a maximal monotone operator from $H$ to $H$. We assume in what follows that the family of maximal monotone operators $A(t)\colon D(A(t))\subset H\rightrightarrows H$, $t\in T$, satisfies hypotheses $H(A)$. It follows from (1.2) and Theorem 4.1 that for each $t\in T$ the set $D(A(t))$ is convex and closed. Hence we can consider the inclusion
$$
\begin{equation}
\begin{gathered} \, -\dot{x}_{\lambda}(t)\in N(D(t)) x_{\lambda}(t) + A_{\lambda}(t)x_{\lambda}(t)+f(t), \\ \notag x_{\lambda}(0) = x_0 \in D(A(0)), \qquad f(\,\cdot\,)\in L^1(T,H), \end{gathered}
\end{equation}
\tag{4.4}
$$
where $D(t)=D(A(t))$, $t\in T$. Theorem 4.2. Assume that hypotheses $H(A)$ are fulfilled and for each $x_0\in D(0)$ and all $\varphi (\,\cdot\,)\in L^1(T,H)$ the sweeping process (1.4) has a solution $x_N(\varphi)(\,\cdot\,)$, $x_N(\varphi)(0)=x_0$. Then for all $\lambda >0$, $x_0\in D(0)$ and $f(\,\cdot\,)\in L^1(T,H)$ the inclusion (4.4) has a unique solution $x_{\lambda}(f)\colon T\to H$, $x_{\lambda}(f)(0)=x_0$, such that Proof. It is well known that when a solution $x_N(\varphi)(\,\cdot\,)$, $x_N(\varphi)(0)=x_0$, $\varphi (\,\cdot\,)\in L^1(T,H)$, of (1.4) exists, then it is unique, and for any solutions $x_N(\varphi_i)(\,\cdot\,)$, $x_N(\varphi_i)(0)=x_0$, where $\varphi_i(\,\cdot\,)\in L^1(T,H)$, $i=1,2$, we have the inequality
$$
\begin{equation}
\| x_N(\varphi_1)(t) - x_N(\varphi_2)(t)\| \leqslant \int_0^t \| \varphi_1(\tau) -\varphi_2(\tau) \| \,d\tau .
\end{equation}
\tag{4.6}
$$
Let $\mathcal T\colon L^1(T,H)\to C(T,H)$ denote the operator that with each $\varphi (\,\cdot\,)\in L^1(T,H)$ associates the unique solution $x_N(\varphi)(\,\cdot\,)$, $ x_N(\varphi)(0)=x_0$, of (1.4), so that It follows from hypothesis $H(A)$, 1), Theorem 3.2 and (2.5) that for any $\lambda >0$ and $x\in H$ the map $t\to A_{\lambda}(t)x$ is measurable and
$$
\begin{equation}
\| A_{\lambda}(t)x - A_{\lambda}(t)y\| \leqslant \frac{1}{\lambda}\| x-y\| ,\qquad x,y\in H.
\end{equation}
\tag{4.8}
$$
Since $x_N(\Theta)(t)\in D(t)$, $t\in T$, using (4.8), (1.2) and (2.9) we obtain
$$
\begin{equation}
\| A_{\lambda}(t)x\| \leqslant \frac{1}{\lambda}\| x\| + \frac{1}{\lambda}\| x_N(\Theta)\|_C+m(t)(1+l(\| x_N(\Theta)\|_C)).
\end{equation}
\tag{4.9}
$$
It follows from (4.8) that for any continuous functions $x\colon T\to H$ the function $t\to A_{\lambda}(t)x(t)$ is measurable. Then it is an element of $L^1(T,H)$ by (4.9). Hence for fixed $f(\,\cdot\,)\in L^1(T,H)$ we have a well-defined operator $\mathcal A_{\lambda}\colon L^1(T,H)\to L^1(T,H)$ such that
$$
\begin{equation}
\mathcal A_{\lambda}(\varphi)(t)= A_{\lambda}(t) \mathcal T(\varphi)(t)+f(t).
\end{equation}
\tag{4.10}
$$
From (4.6)–(4.8) and (4.10) we obtain
$$
\begin{equation}
\| \mathcal A_{\lambda}(\varphi_1)(t) - \mathcal A_{\lambda}(\varphi_2)(t)\| \leqslant \frac{1}{\lambda} \int_0^t \| \varphi_1(\tau) - \varphi_2(\tau) \| \,d\tau .
\end{equation}
\tag{4.11}
$$
On the space $L^1(T,H)$ consider the norm
$$
\begin{equation}
P(\varphi)=\int_T \exp \biggl(- \frac{2}{\lambda}\, t\biggr) \| \varphi(\tau)\| \,d\tau ,
\end{equation}
\tag{4.12}
$$
which is equivalent to the standard norm in $L^1(T,H)$. We denote $L^1(T,H)$ with norm (4.12) by $L^1_P(T,H)$. From (4.11) and (4.12) we obtain the inequality
$$
\begin{equation*}
P(\mathcal A_{\lambda}(\varphi_1) - \mathcal A_{\lambda}(\varphi_2))\leqslant \int_T \frac{1}{\lambda} \exp \biggl(- \frac{2}{\lambda}\, t\biggr)\biggl(\int_0^t \| \varphi_1(\tau)-\varphi_2(\tau)\| \,d\tau \biggr)\,dt, \qquad t\in T.
\end{equation*}
\notag
$$
Integrating by parts on the right-hand side we obtain
$$
\begin{equation*}
P(\mathcal A_{\lambda}(\varphi_1) - \mathcal A_{\lambda}(\varphi_2))\leqslant \frac{1}{2}P(\varphi_1-\varphi_2).
\end{equation*}
\notag
$$
Therefore, $\mathcal A_{\lambda}\colon L^1_P(T,H)\to L^1_P(T,H)$ is a contracting operator on the Banach space $L^1_P(T,H)$. Hence there exists a fixed point $\varphi_*(\,\cdot\,)\in L^1_P(T,H)$ of this operator, that is,
$$
\begin{equation}
\varphi_*(\,\cdot\,)= \mathcal A_{\lambda}(\varphi_*).
\end{equation}
\tag{4.13}
$$
From (4.10) and (4.13) we obtain
$$
\begin{equation}
\varphi_*(t)= A_{\lambda}(t) \mathcal T(\varphi_*)(t)+f(t).
\end{equation}
\tag{4.14}
$$
Now, by (1.4) and (4.7)
$$
\begin{equation*}
-\dot{x}_N (\varphi_*)(t)\in N(D(t))x_N (\varphi_*)(t) + \varphi_*(t) \quad \text{a.e.}
\end{equation*}
\notag
$$
It follows from this inclusion and (4.14) that
$$
\begin{equation}
-\dot{x}_N (\varphi_*)(t)\in N(D(t))x_N (\varphi_*)(t) + A_{\lambda}(t)x_N (\varphi_*)(t) +f(t).
\end{equation}
\tag{4.15}
$$
Set $x_{\lambda}(f)(t)=x_N (\varphi_*)(t)$. Then by (4.15)
$$
\begin{equation}
-\dot{x}_{\lambda}(f)(t)\in N(D(t))x_{\lambda}(f)(t) + A_{\lambda}(t)x_{\lambda}(f)(t) +f(t).
\end{equation}
\tag{4.16}
$$
Therefore, $x_{\lambda}(f)(\,\cdot\,)$, $x_{\lambda}(f)(0)=x_0$, is a solution of inclusion (4.4).
We show that it is unique. Let $y_{\lambda}(f)(\,\cdot\,)$, $y_{\lambda}(f)(0)=x_0$, be another solution of (4.4). Using (4.4) and the fact that the operators $N(D(t))$ and $A_{\lambda}(t)$, $t\in T$, are monotone we obtain
$$
\begin{equation*}
\langle \dot{x}_{\lambda}(f)(t)-\dot{y}_{\lambda}(f)(t),\,x_{\lambda}(f)(t) -y_{\lambda}(f)(t)\rangle \leqslant 0.
\end{equation*}
\notag
$$
Integrating this inequality we arrive at
$$
\begin{equation*}
\| x_{\lambda}(f)(t)-y_{\lambda}(f)(t)\|^2 \leqslant 0.
\end{equation*}
\notag
$$
Hence $x_{\lambda}(f)(t)=y_{\lambda}(f)(t)$, $t\in T$, which shows the uniqueness of the solutions of (4.4).
Next we prove (4.5). From (1.4) and (4.4), bearing in mind that $N(D(t))$ for $t\in T$ is a monotone operator, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\langle \dot{x}_{\lambda}(f)(t)-\dot{x}_N(\Theta)(t),\,x_{\lambda}(f)(t)-x_N(\Theta)(t)\rangle \\ &\qquad \leqslant \langle - A_{\lambda}(t) x_{\lambda}(f)(t)-f(t),\,x_{\lambda}(f)(t)- x_N(\Theta)(t)\rangle . \end{aligned}
\end{equation*}
\notag
$$
From this inequality, since $A_{\lambda}(t)$, $t\in T$, is monotone, it follows that
$$
\begin{equation*}
\begin{aligned} \, &\langle \dot{x}_{\lambda}(f)(t)-\dot{x}_N(\Theta)(t),\,x_{\lambda}(f)(t)-x_N(\Theta)(t)\rangle \\ &\qquad \leqslant \langle - A_{\lambda}(t) x_N(\Theta)(t)-f(t),\,x_{\lambda}(f)(t)- x_N(\Theta)(t)\rangle . \end{aligned}
\end{equation*}
\notag
$$
Integrating, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\| x_{\lambda}(f)(t)-x_N(\Theta)(t)\|^2 \\ &\qquad\leqslant 2\int_0^t (\| f(\tau)\| + \| A_{\lambda}(t) x_N(\Theta)\|)(\| x_{\lambda}(f)(\tau)-x_N(\Theta)(\tau)\|)\,d\tau , \qquad t\in T. \end{aligned}
\end{equation*}
\notag
$$
Using Lemma A5 in [1], p. 157, we arrive at the inequality
$$
\begin{equation*}
\| x_{\lambda}(f)(t)-x_N(\Theta)(t)\| \leqslant \int_0^t (\| f(\tau)\| + \| A_{\lambda}(t) x_N(\Theta)(\tau)\|)\,d\tau , \qquad t\in T.
\end{equation*}
\notag
$$
Since $x_N (\Theta)(t)\in D(A(t))$, $t\in T$, it follows from (2.9), (1.2) and the last inequality that (4.5) holds. Theorem 4.2 is proved. For $r_A(\,\cdot\,)\in L^1(T,{\mathbb R}^+)$ set
$$
\begin{equation}
S^1(r_A)=\bigl\{f(\,\cdot\,)\in L^1(T,H);\, \| f(t)\| \leqslant r_A(t)\text{ a.e.}\bigr\}.
\end{equation}
\tag{4.17}
$$
Theorem 4.3. Under hypotheses $H(A)$ assume that for all $x_0\in D(A(0))$ and $\varphi (\,\cdot\,)\in L^1(T,H)$ the sweeping process (1.4) has a solution $x_N(\varphi)(\,\cdot\,)$, ${x_N(\varphi)(0)=x_0}$. Then for any $x_0\in D(A(0))$ and $f(\,\cdot\,)\in L^1(T,H)$ inclusion (1.1) has a unique solution $x_A(f)(\,\cdot\,)$, $ x_A(f)(0)=x_0$, such that Furthermore, if $f(\,\cdot\,)\in S^1(r_A)$, then there exists $\varphi (\,\cdot\,)\in L^1(T,H)$ satisfying and such that where If for each $r_N(\,\cdot\,)\in L^1(T,{\mathbb R}^+)$ the set of functions $\{x_N(\varphi)(\,\cdot\,);\, \varphi (\,\cdot\,)\in S^1(r_N)\bigr\}$ is equicontinuous, then for each $r_A(\,\cdot\,) \in L^1(T,{\mathbb R}^+)$ the set $\bigl\{x_A(f)(\,\cdot\,); f(\,\cdot\,) \in S^1(r_A)\bigr\}$ is too. Proof. Let $x_{\lambda}(f)(\,\cdot\,)$, $ x_{\lambda}(f)(0)=x_0$, $\lambda >0$, be a solution of inclusion (4.4), and let Set
$$
\begin{equation}
M=\| x_N(\Theta)\|_C +(1+l(\| x_N(\Theta)\|_C)) \| m(\,\cdot\,)\|_{L^1}+\| f(\,\cdot\,)\|_{L^1}.
\end{equation}
\tag{4.23}
$$
Since $x_{\lambda}(t)\in D(A(t))$, $t\in T$, from (2.9), (4.5) and (4.22), (4.23) we obtain
$$
\begin{equation}
\| v_{\lambda}(t)\| \leqslant m(t)(1+l(M)), \qquad \lambda >0.
\end{equation}
\tag{4.24}
$$
Because $m(\,\cdot\,)\in L^2(T,{\mathbb R}^+)$, it follows from the last inequality that the set $\bigl\{v_{\lambda}(\,\cdot\,); 0<\lambda \leqslant 1\bigr\}$ is bounded in the Hilbert space $L^2(T,H)$. As the operator $N(D(t))$, $t\in T$, is monotone, it follows from (4.4) that
$$
\begin{equation}
\begin{aligned} \, \notag &\langle \dot{x}_{\lambda}(f)(t)-\dot{x}_{\mu}(f)(t),\,x_{\lambda}(f)(t) -x_{\mu}(f)(t)\rangle \\ &\qquad\leqslant \langle - A_{\lambda}(t) x_{\lambda}(f)(t)+A_{\mu}(t)x_{\mu}(f)(t),\,x_{\lambda}(f)(t)- x_{\mu}(f)(t)\rangle . \end{aligned}
\end{equation}
\tag{4.25}
$$
By (2.5) we have
$$
\begin{equation}
x_{\lambda}(f)(t)= \lambda A_{\lambda}(t) x_{\lambda}(f)(t)+J_{\lambda}(A(t))x_{\lambda}(f)(t)
\end{equation}
\tag{4.26}
$$
and
$$
\begin{equation*}
x_{\mu}(f)(t)=\mu A_{\mu}(t)x_{\mu}(f)(t)+J_{\mu}(A(t))x_{\mu}(f)(t).
\end{equation*}
\notag
$$
Since $A(t)$ for $t\in T$ is monotone, these equalities, (2.8) and (4.25) yield the inequality
$$
\begin{equation*}
\begin{aligned} \, &\| x_{\lambda}(f)(t)-x_{\mu}(f)(t)\|^2 \\ &\qquad +2\int_T \langle A_{\lambda}(t) x_{\lambda}(t)-A_{\mu}(t)x_{\mu}(f)(t),\, \lambda A_{\lambda}(t) x_{\lambda}(f)(t)-\mu A_{\mu}(t)x_{\mu}(f)(t)\rangle \,dt\,{\leqslant}\, 0. \end{aligned}
\end{equation*}
\notag
$$
Using (4.22) we obtain
$$
\begin{equation*}
\langle v_{\lambda}(\,\cdot\,)-v_{\mu}(\,\cdot\,),\,\lambda v_{\lambda}(\,\cdot\,)-\mu v_{\mu}(\,\cdot\,)\rangle_{L^2(T,H)}\leqslant 0.
\end{equation*}
\notag
$$
It follows from this and Lemma 3.3, since the set $\bigl\{v_{\lambda}(\,\cdot\,);\, \lambda \in (0,1]\bigr\}\subset L^2(T,H)$ is bounded, that
$$
\begin{equation}
v_{\lambda}(\,\cdot\,)\to v(\,\cdot\,) \quad \text{in } L^2(T,H) \quad \text{for } \lambda \downarrow 0.
\end{equation}
\tag{4.27}
$$
Let $x_A(f)(\,\cdot\,)$, $x_A(f)(0)=x_0$, be a solution of the inclusion Using (4.4), (4.6), (4.22) and (4.28) we obtain
$$
\begin{equation*}
\| x_{\lambda}(f)(t)-x_A(f)(t)\| \leqslant \int_0^t \| v(\tau)-v_{\lambda}(\tau)\| \,d\tau.
\end{equation*}
\notag
$$
It follows from this inequality and (4.27) that
$$
\begin{equation}
x_{\lambda}(f)(\,\cdot\,)\to x_A(f)(\,\cdot\,) \quad \text{in } C(T,H) \quad \text{for } \lambda \downarrow 0.
\end{equation}
\tag{4.29}
$$
Without loss of generality we can assume that
$$
\begin{equation}
v_{\lambda}(t)\to v(t) \quad \text{a.e. for } \lambda \downarrow 0.
\end{equation}
\tag{4.30}
$$
Then from (4.26), (4.22), (4.24), (4.29) and (4.30) we obtain
$$
\begin{equation}
J_{\lambda}(A(t))x_{\lambda}(f)(t)\to x_A(f)(t) \quad \text{as } \lambda \downarrow 0.
\end{equation}
\tag{4.31}
$$
Using (2.8), (4.22), (4.30), (4.31) and property 5) in Lemma 2.1 we arrive at the inclusion Hence, by (4.28)
$$
\begin{equation}
-\dot{x}_A(f)(t)\in N(D(t))x_A(f)(t) + A(t)x_A(f)(t) +f(t).
\end{equation}
\tag{4.33}
$$
Using (4.2) we obtain
$$
\begin{equation*}
-\dot{x}_A(f)(t)\in A(t)x_A(f)(t) + f(t) \quad \text{a.e.}
\end{equation*}
\notag
$$
Therefore, $x_A(f)(\,\cdot\,)$, $x_A(f)(0)=x_0$, is a solution of (1.1). The uniqueness of the solution $x_A(f)(\,\cdot\,)$, $x_A(f)(0)=x_0$, follows from the well-known inequality (see [2], [5] and other papers)
$$
\begin{equation*}
\| x(f_1)(t)-x(f_2)(t)\| \leqslant \int_0^t \| f_1(\tau)-f_2(\tau)\| \,d\tau , \qquad t\in T,
\end{equation*}
\notag
$$
which holds for $f_i(\,\cdot\,)\in L^1(T,H)$ and the solutions $x(f_i)(\,\cdot\,)$, of (1.1) such that $x(f_i)(0)=x_0$. Inequality (4.18) is a direct consequence of (4.5) and (4.29).
Since for fixed $\varphi (\,\cdot\,)\in L^1(T,H)$ inclusion (1.4) has a unique solution $x_N(\varphi)(\,\cdot\,)$, $x_N(\varphi)(0)=x_0$, using (4.28) we obtain the equality for Using (4.23), (4.24), (4.27) and (4.35) we see that (4.20) holds for $\varphi (t)$ satisfying (4.19).Let Then from (4.17) we obtain Now the last assertion of the theorem follows from (4.36) and (4.19), (4.20). Theorem 4.3 is proved, which also completes the proof of Theorem 1.1.§ 5. Concretizing the resultsIn this section we concretize Theorem 4.3 in terms of the properties of the family of maximal monotone operators $A(t)\colon D(A(t))\subset H\rightrightarrows H$. Theorem 5.1. Let the family of maximal monotone operators $A(t)$, $t\in T$, satisfy inequality (1.2). Assume that for each $\rho \in [0,+\infty)$ there exists an absolutely continuous function $b_{\rho}\colon T\to {\mathbb R}$ such that Hence for all $x_0\in D(A(0))$ and $f(\,\cdot\,)\in L^1(T,H)$ inclusion (1.1) has a unique solution $x_A(f)(\,\cdot\,)$, $x_A(f)(0)=x_0$, satisfying (4.18). If $f(\,\cdot\,)\in S^1(r_A)$, then there exists $\varphi (\,\cdot\,)\in L^1(T,H)$ satisfying (4.19) such that equality (4.20) holds. Moreover, there exists $\rho_*>0$ depending on $\| x_0\| $ and $\| r_A(\,\cdot\,)\|_{L^1}$ such that
$$
\begin{equation}
\| \dot{x}_A(f)(t)\| \leqslant |\dot{b}_{\rho_*}(t)| + 2(m(t)(1+l(M_A))+r_A(t)) \quad \textit{a.e.}
\end{equation}
\tag{5.2}
$$
for each $f(\,\cdot\,)\in S^1(r_A)$, where the positive constant $M_A$ is defined by (4.21). Proof. We show that, under the assumptions made, the family of maximal monotone operators $A(t)$, $t\in T$, is measurable.
From (5.1) and (3.3) we obtain
$$
\begin{equation}
\operatorname{exc}_{\rho}(D(A(s)),D(A(t)))\leqslant |b_{\rho}(t)-b_{\rho}(s)|, \qquad s\leqslant t, \quad s,t\in T.
\end{equation}
\tag{5.3}
$$
Using (5.3) we arrive at the inequality
$$
\begin{equation*}
d(x_0,D(A(t)))\leqslant |b_{\| x_0\|}(t)-b_{\| x_0\|}(0)|, \qquad t\in T.
\end{equation*}
\notag
$$
Hence there exists a function $y_*(t)\in D(A(t))$, $t\in T$, such that
$$
\begin{equation*}
\| x_0-y_*(t)\| \leqslant |b_{\| x_0\|}(t)-b_{\| x_0\|}(0)|, \qquad t\in T.
\end{equation*}
\notag
$$
It follows from this inequality that for some $C>0$.
Fix any $\varepsilon >0$. Then there exists a compact set $T_{\varepsilon}\subset [0,a)$ such that $\mu (T\setminus T_{\varepsilon})\leqslant \varepsilon$ and the restriction of $m(t)$ to $T_{\varepsilon}$ is continuous. Let $x\in H$ and $s\in T_{\varepsilon}$ be arbitrary. Using (2.5)–(2.7) we obtain
$$
\begin{equation*}
\| J_{\lambda}(A(s))x\| \leqslant \| x\| + 2 \| y_*(s)\| + \lambda \| A_{\lambda}(s)y_*(s)\| .
\end{equation*}
\notag
$$
It follows from this inequality and (2.9), (1.2) and (5.4) that
$$
\begin{equation*}
\| J_{\lambda}(A(s))x\| \leqslant \| x\| + 2 C + \lambda m(s)(1+l(C)).
\end{equation*}
\notag
$$
Let
$$
\begin{equation}
C_{\varepsilon}=\sup \bigl\{m(\tau);\, \tau\in T_{\varepsilon}\bigr\}.
\end{equation}
\tag{5.5}
$$
Then $\| J_{\lambda}(A(s))x\| \leqslant \rho $, where $\rho = \| x\| + 2 C + \lambda C_{\varepsilon} (1+l(C))$. From (2.11) and (5.3) we obtain
$$
\begin{equation*}
\begin{aligned} \, &\langle J_{\lambda}(A(s))x-J_{\lambda}(A(t))x,\,A_{\lambda}(t)x-A_{\lambda}(s)x\rangle \\ &\qquad \leqslant |b_{\rho}(t)-b_{\rho}(s)|(1+\| A_{\lambda}(t)x\| + \| A_{\lambda}(s)x\|), \qquad s\leqslant t, \quad s,t\in T_{\varepsilon}. \end{aligned}
\end{equation*}
\notag
$$
This and (2.5) yield the inequality
$$
\begin{equation*}
\begin{aligned} \, &\| J_{\lambda}(A(s))x - J_{\lambda}(A(t))x\|^2 \\ &\qquad \leqslant \lambda |b_{\rho}(t)-b_{\rho}(s)|(1+\| A_{\lambda}(t)x\| + \| A_{\lambda}(s)x\|), \qquad s\leqslant t, \quad s,t\in T_{\varepsilon}. \end{aligned}
\end{equation*}
\notag
$$
Now using (2.5), (2.9), (1.2) and (5.5) we obtain
$$
\begin{equation*}
\begin{aligned} \, &\| J_{\lambda}(A(s))x - J_{\lambda}(A(t))x\|^2 \\ &\qquad \leqslant \lambda |b_{\rho}(t)-b_{\rho}(s)|\biggl(1 + \frac{2}{\lambda}\| x\| + 2 C_{\varepsilon}(1 + l(C))\biggr), \qquad s\leqslant t, \quad s,t\in T_{\varepsilon}. \end{aligned}
\end{equation*}
\notag
$$
It follows from this inequality that the function $t\to J_{\lambda}(A(t))x$ is right continuous at $s$. Then by Lemma 3.1 it is continuous at $s$ in the topology induced by the topology of the Sorgenfrey interval. Since $s\in T_{\varepsilon}$ is an arbitrary point, the function $t\to J_{\lambda}(A(t))x$, $t\in T_{\varepsilon}$, is continuous on $T_{\varepsilon}$ in this topology. Hence by Lemma 3.2 the function $t\to J_{\lambda}(A(t))x$ is Borel on $T_{\varepsilon}$, and thus it is measurable on $T$. Now the measurability of the family $A(t)$, $t\in T$, of maximal monotone operators is a consequence of Theorem 3.2.
Inequalities (5.1) and (3.3) ensure (5.3) for $\rho \in [0,+\infty)$. Hence by Theorem 3.3, for all $x_0\in D(A(0))$ and $\varphi (\,\cdot\,)\in L^1(T,H)$ inclusion (1.4) has a unique solution $x_N(\varphi)(\,\cdot\,)$, $x_N(\varphi)(0)=x_0$. Now the existence and uniqueness of a solution $x(f)(\,\cdot\,)$, $x(f)(0)=x_0$, of (1.1) for all $x_0\in D(A(0))$ and $f(\,\cdot\,)\in L^1(T,H)$ follow from Theorem 4.3. This solutions satisfies (4.18), and for each $f(\,\cdot\,)\in S^1(r_A)$ there exists $\varphi (\,\cdot\,)\in L^1(T,H)$ satisfying (4.19) such that (4.20) holds. By Theorem 3.3 there exists $\rho_*>0$ depending on $\| x_0\| $ and $\| r_N(\,\cdot\,)\|_{L^1}$ such that
$$
\begin{equation}
\| \dot{x}_N(\varphi)(t)\| \leqslant |\dot{b}_{\rho_*}(t)| + 2 r_N(t).
\end{equation}
\tag{5.6}
$$
Let Then it follows from (4.19) that for each $f(\,\cdot\,)\in S^1(r_A)$ there exists $\varphi (\,\cdot\,)\in S^1(r_N)$ such that equality (4.20) holds. In combination with (5.6) and (5.7) this yields inequality (5.2). It follows from (5.7) and (4.21) that $\rho_*$ depends on $\| x_0\| $ and $\| r_A(\,\cdot\,)\|_{L^1}$. Theorem 5.1 is proved. Now Corollary 1.1 follows from Theorem 5.1 for $\rho =+\infty$ and Corollary 3.2 in which $b_{\infty}(t)=b(t)$. From (3.11) we obtain the inequality
$$
\begin{equation}
\| x_N(\Theta)\|_C \leqslant \| x_0\| + \| \dot{b}\|_{L^1}.
\end{equation}
\tag{5.8}
$$
Now inequalities (1.8) and (1.9) and equality (1.10) are consequences of (4.18), (5.2), (5.8) and the fact that $l\colon {\mathbb R}^+\to {\mathbb R}^+$ is a monotone function. Thus, Theorem 1.1 and Corollary 1.1 are proved. Theorem 5.2. Under hypotheses $H(A)$ assume that for each $\rho\in [0,+\infty)$ there exists an absolutely continuous function $b_\rho \colon T\to R$ such that for all $s,t\in T$, $s\leqslant t$, This is a consequence of Theorems 3.3 and 4.3. § 6. ApplicationsIn this section we prove Theorem 1.2. Proof of Theorem 1.2. If for all $f(\,\cdot\,)\in L^1(T,H)$ and $x_0\in D(A(0))$ inclusion (1.1) has a solution $x_A(f)(\,\cdot\,)$, $x_A(f)(0)=x_0$, then this solution is unique and for $ f_i(\,\cdot\,)\in L^1(T,H)$ the following inequality holds for any solutions $x_A(f_i)(\,\cdot\,)$, $x_A(f_1)(0)=x_A(f_2)(0)=x_0$:
$$
\begin{equation}
\| x_A(f_1)(t)-x_A(f_2)(t)\| \leqslant \int_0^t |f_1(\tau)-f_2(\tau)|\,d\tau .
\end{equation}
\tag{6.1}
$$
It follows from hypotheses $H(F)$, 1), 2) that for any $x(\,\cdot\,)\in C(T,H)$ the map $t\to F(t,x(t))$ is measurable and its values are closed. If inequality (1.14) holds, then from (1.12) we obtain the inequality
$$
\begin{equation*}
d(\Theta ,F(t,x(t)))< n(t)+k(t)\| x(t)\| \quad \text{a.e.}
\end{equation*}
\notag
$$
It follows from it and the properties of measurable set-valued maps that there exists $f(\,\cdot\,)\in L^1(T,H)$ such that $f(t)\in F(t,x(t))$ almost everywhere and
$$
\begin{equation}
\| f(t)\| \leqslant n(t)+k(t)\| x(t)\| \quad \text{a.e.}
\end{equation}
\tag{6.2}
$$
For each $x(\,\cdot\,)\in C(T,H)$ let $\mathcal F(x)$ denote the set
$$
\begin{equation}
\mathcal F(x)=\bigl\{f(\,\cdot\,)\in L^1(T,H);\, f(t)\in F(t,x(t)) \text{ a.e.}\bigr\}.
\end{equation}
\tag{6.3}
$$
It follows from (6.2) that $\mathcal F(x)$ is a nonempty decomposable closed subset of $L^1(T,H)$. Thus we obtain a set-valued operator $\mathcal F\colon C(T,H)\rightrightarrows L^1(T,H)$, whose values are closed decomposable sets from the space $L^1(T,H)$.
On $L^1(T,H)$ we introduce the norm where it is obviously equivalent to the standard norm in $L^1(T,H)$. We denote $L^1(T,H)$ with norm (6.4) by $L^1_{\Pi}(T,H)$.We denote the Hausdorff distance between closed subsets of $L^1_{\Pi}(T,H)$ by $\operatorname{haus}_{\Pi}(\,\cdot\,{,}\,\cdot\,)$. Let $x_1(\,\cdot\,),x_2(\,\cdot\,)\in C(T,H)$ and $f_1(\,\cdot\,)\in \mathcal F(x_1)$ be arbitrary. Relations (1.12) and (6.3) yield the inequality Hence we see from the properties of measurable set-valued maps that there exists $f_2(\,\cdot\,)\in \mathcal F(x_2)$ such that
$$
\begin{equation*}
\| f_1(t)-f_2(t)\| \leqslant k(t)\| x_1(t)-x_2(t)\| \quad \text{a.e.}
\end{equation*}
\notag
$$
Using this inequality and (6.4) we obtain
$$
\begin{equation}
\Pi(f_1-f_2)\leqslant \int_T e^{-2\alpha (t)}k(t)\| x_1(t)-x_2(t)\| \,dt.
\end{equation}
\tag{6.6}
$$
Therefore,
$$
\begin{equation*}
\operatorname{haus}_\Pi(\mathcal F(x_1),\mathcal F(x_2))\leqslant \int_T e^{-2\alpha (t)}k(t)\| x_1(t)-x_2(t)\| \,dt.
\end{equation*}
\notag
$$
By (6.1)
$$
\begin{equation}
\operatorname{haus}_\Pi(\mathcal F(x_A(f_1)),\mathcal F(x_A(f_2)))\leqslant \int_T e^{-2\alpha (t)}k(t)\int_0^t \| f_1(\tau)-f_2(\tau)\| \,d\tau \,dt.
\end{equation}
\tag{6.7}
$$
Let
$$
\begin{equation}
\Phi (f)= \mathcal F(x_A(f)), \qquad f(\,\cdot\,)\in L^1(T,H).
\end{equation}
\tag{6.8}
$$
We obtain a set-valued operator $\Phi \colon L^1(T,H)\rightrightarrows L^1(T,H)$ with decomposable closed values. It follows from (6.7) and (6.8) that this operator satisfies
$$
\begin{equation*}
\begin{gathered} \, \operatorname{haus}_\Pi (\Phi (f_1),\Phi (f_2))\leqslant \int_T e^{-2\alpha (t)}k(t)\int_0^t \| f_1(\tau)-f_2(\tau)\| \,d\tau \,dt, \\ f_1(\,\cdot\,),f_2(\,\cdot\,)\in L^1(T,H). \end{gathered}
\end{equation*}
\notag
$$
Integrating by parts on the right-hand side we obtain
$$
\begin{equation*}
\operatorname{haus}_\Pi(\Phi (f_1),\Phi (f_2))\leqslant \frac{1}{2}\int_T e^{-2\alpha (t)} \| f_1(t)-f_2(t)\| \,dt.
\end{equation*}
\notag
$$
This inequality and (6.4) yield
$$
\begin{equation*}
\operatorname{haus}_\Pi(\Phi (f_1),\Phi (f_2))\leqslant \frac{1}{2} \Pi(f_1-f_2),
\end{equation*}
\notag
$$
so that $\Phi$ is a contracting operator on the Banach space $L^1_{\Pi}(T,H)$. Then it has a fixed point by Theorem 3.1 in [12], that is, there exists Using (6.8), (6.3), (6.9) and (1.1) we obtain Set $x_A(v)(\,\cdot\,)=x(v)$. Then it follows from (6.10) and (6.11) that $(x(v)(\,\cdot\,),v(\,\cdot\,))\in \mathcal R_F(x_0)$. Thus the set $\mathcal R_F(x_0)$ is not empty.
If (1.13) holds, then for each function $x(\,\cdot\,)\in C(T,H)$ we have the inequality From it and (6.1) we see that for all $(x(v)(\,\cdot\,),v(\,\cdot\,))\in \mathcal R_F(x_0)$ we have
$$
\begin{equation*}
\| x(v)(t)\| \leqslant \| x_A(\Theta)\|_C+\int_0^t \| v(\tau)\| \,d\tau ,\qquad t\in T,
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\| v(t)\| \leqslant n(t)+k(t)\| x(v)(t)\| \quad \text{a.e.}
\end{equation}
\tag{6.12}
$$
Therefore,
$$
\begin{equation*}
\| x(v)(t)\| \leqslant \| x_A(\Theta)\|_C+\int_0^t (n(\tau)+k(\tau)\| x(v)(\tau)\|)\,d\tau .
\end{equation*}
\notag
$$
From this, (1.17) and theorems on differential inequalities it follows that where $r(t)$ is a solution of equation (1.17) for $r(0)=\| x_A(\Theta)\|_C$. From (6.11), (6.12) and (1.17) we obtain This completes the proof of inequalities (1.20) and (1.21).
It follows from (6.13) that $\mathcal Tr_F(x_0)\subset \bigl\{x_A(f);\, \| f(t)\| \leqslant\dot{r}(t)\text{ a.e.}\bigr\}$. Hence, if the set of functions $\bigl\{x_A(f);\, \| f(t)\| \leqslant \dot{r}(t)\text{ a.e.}\bigr\}$ is equicontinuous, then the set $\mathcal Tr_F(x_0)$ is too. Theorem 1.2 is proved. Lemma 6.1. Under the assumptions of Theorem 5.1 the set $\mathcal R_F(x_0)$ is nonempty. For each $(x(v)(\,\cdot\,),v(\,\cdot\,))\in \mathcal R_F(x_0)$ inequalities (1.20) and (1.21) hold for $r(0)$ satisfying the inequality This lemma follows from Theorems 1.2, 4.3 and 5.1, while (6.16) follows from (3.11). Corollary 6.1. Under the assumptions of Corollary 1.1 assertions of Lemma 6.1 hold for $b(t)$ in place of $b_{\rho_*}(t)$ and $b_{\rho}(t)$ in inequalities (6.16) and (6.17). This is a consequence of Corollary 1.1, Theorems 1.2 and 5.1 and Lemma 6.1. § 7. CommentsThe first attempt to prove a theorem on the existence of an absolutely continuous solution of inclusion (1.1) for $f(t)\equiv \Theta $, $t\in T$, was made in Theorem 3 in [13]. It was assumed in that paper that the family $A(t)\colon D(A(t))\subset H\rightrightarrows H$, $t\in T$, of maximal monotone operators has the following properties: 1) the inequality
$$
\begin{equation}
\| A^0(t)x\| \leqslant c(t)(1+\| x\|), \qquad t\in T, \quad x\in D(A(t)),
\end{equation}
\tag{7.1}
$$
holds and 2) the inequality
$$
\begin{equation}
\operatorname{dis}(A(t),A(s))\leqslant |a(t)-a(s)|, \qquad t,s\in T,
\end{equation}
\tag{7.2}
$$
holds, where $c(\,\cdot\,)\in L^1(T,{\mathbb R}^+)$ and $a(\,\cdot\,)\in W^{1,2}(T,{\mathbb R})$. There is no full proof of Theorem 3 in [13], but only a sketch of it. A full proof of the existence theorem for an absolutely continuous solution of inclusion (1.1) for $f(\,\cdot\,)\in L^2(T,H)$ was presented in [5], Theorem 3.1. It was assumed there that the family of maximal monotone operators satisfies inequalities (7.1) and (7.2) for $c(t)\equiv\mathrm{const}$ and $a(\,\cdot\,)\in W^{1,2}(T,{\mathbb R})$ such that $a(t)$ is a nonnegative nondecreasing function on $T$ and $a(0)=0$. The existence and uniqueness of a solution $x(\,\cdot\,)\in W^{1,2}(T,H)$ was established and a priori estimates for $x(\,\cdot\,)$ and $\dot{x}(\,\cdot\,)$ were presented. The so-called catching-up algorithm, developed in [3] for establishing the existence of solutions to a sweeping process, was used in the proof. In Theorem 1 in [14] the existence of an absolutely continuous solution $x(\,\cdot\,)\in W^{1,2}(T,H)$ of the inclusion
$$
\begin{equation*}
\dot{x}(t)\in A(t) x(t) + f(t,x(t)), \qquad x(0) = x_0,
\end{equation*}
\notag
$$
was proved, provided that the family of maximal monotone operators $A(t)\colon H\rightrightarrows H$, $t\in T$, has the following properties $H(A)'$: (a) for all $x\in H$ the map $t\to A(t)x$ is measurable; (b) for all $t\in T$ and $x\in H$, where $\alpha ,\beta \in L^2(T,{\mathbb R}^+)$.The map $f\colon T\times H\to H$ must have properties $H(f)$: (a) for all $x\in H$ the map $t\to f(t,x)$ is measurable; (b) for almost all $t\in T$ and all $x,y\in H$ where $k(\,\cdot\,)\in L^2(T,{\mathbb R}^+)$;(c) for almost all $t\in T$ and all $x,y\in H$ where $\gamma ,\delta \in L^2(T,{\mathbb R}^+)$.In the proof of Theorem 1 a regularization of the family $A(t)\colon H\rightrightarrows H$, $t\in T$, of maximal monotone operators by Yosida approximations was used. Theorem 3.1 in [5] is a consequence of our Corollary 1.1, while Theorem 1 in [14] is a consequence of Theorems 4.3 and 5.1, because for $D(A(t))=H$, $t\in T$, we have the equality $N(D(t))\equiv \Theta $, $t\in T$. Hence in place of inclusion (1.4) we obtain the equality Note the cardinally different growth conditions (1.2), which are distinct from the usual conditions (7.1) and (7.3). In (1.2) the function $l\colon {\mathbb R}^+ \to {\mathbb R}^+$ can have the form $l(r)=r^{\alpha}$, $0<\alpha <+\infty $.Since
$$
\begin{equation*}
\operatorname{dis}_{\rho}(A_1,A_2) \leqslant \operatorname{dis}(A_1,A_2), \qquad \rho \in [0,\infty),
\end{equation*}
\notag
$$
the use of inequality (5.1) is preferable to the use of (1.8). On the other hand inequality (5.9) allows us to extend the class of maximal monotone operators such that (1.8) fails, but inclusion (1.1) is solvable. Here is one example. Let $v\colon T\to H$ and $l\colon T\to R$ be absolutely continuous functions such that $\|v(t)\|=1$, $t\in T$. Consider the set-valued map $C\colon T\rightrightarrows H$,
$$
\begin{equation}
C(t)=\bigl\{x\in H;\, \langle x,v(t)\rangle -l(t)=0\bigr\}, \qquad t\in T,
\end{equation}
\tag{7.4}
$$
whose values are hyperplanes. Let
$$
\begin{equation}
A(t)=\mathcal{N}(C(t)), \quad D(A(t))=C(t), \qquad t\in T,
\end{equation}
\tag{7.5}
$$
be a family of maximal monotone operators. It is known (see [15]) that the distance of a point $x\in H$ to a set $C(t)$ is defined by
$$
\begin{equation}
d(x,C(t)) = |\langle x, v(t)\rangle -l(t)|, \qquad t\in T.
\end{equation}
\tag{7.6}
$$
It follows from (7.6) that for each $x\in H$ the function $t\to d(x,C(t))$ is measurable. Hence the set-valued map $t\to C(t)$ with convex closed values is measurable (see [6]). Therefore, as follows from Lemma 2.7 in [16], the family of maximal monotone operators $A(t)$, $t\in T$, defined by (7.5) is measurable. Since $A^0(t)=\Theta$, $t\in T$, the family $A(t)$, $t\in T$, satisfies hypotheses $H(A)$. We make a change of variables for $s\in T$, $s\ne t$, by setting $y=x-l(s)v(s)$. From (7.4) we obtain the hyperplanes
$$
\begin{equation}
C_1(s)=\bigl\{y\in H:\, \langle y,v(s)\rangle =0\bigr\},
\end{equation}
\tag{7.7}
$$
$$
\begin{equation}
C_1(t)=\bigl\{y\in H:\, \langle y+l(s)v(s),\,v(t)\rangle - l(t)=0\bigr\}.
\end{equation}
\tag{7.8}
$$
It follows from (7.7) that
$$
\begin{equation}
\lambda y \in C_1(s) \quad \text{for all } y\in C(s) \text{ and } \lambda\in R.
\end{equation}
\tag{7.9}
$$
Let $y\in C_1(s)$. Then
$$
\begin{equation}
d(y,C_1(t))=|\langle y+l(s)v(s),\,v(t)\rangle - l(t)|.
\end{equation}
\tag{7.10}
$$
Assume that for $s\ne t$ the vectors $v(s)$ and $v(t)$ are linearly independent. Then it follows from (7.7) that $\langle y,v(t)\rangle \ne 0$. Using (7.9) and (7.10) we obtain
$$
\begin{equation*}
(\operatorname{exc}C_1(s), C_1(t))=+\infty, \qquad s\ne t, \quad s,t\in T.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\operatorname{haus}(C_1(s),C_1(t))= +\infty, \qquad s\ne t.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\operatorname{haus}(C(s),C(t))= +\infty, \qquad s\ne t.
\end{equation}
\tag{7.11}
$$
By Lemma 3.5 in [4]
$$
\begin{equation}
\operatorname{haus}(C(s),C(t))= \operatorname{dis}(N(C(s)),N(C(t))), \qquad s,t\in T.
\end{equation}
\tag{7.12}
$$
Now it follows from (7.11), (7.12) and (7.5) that for the family of maximal monotone operator $A(t)$, $t\in T$, inequality (1.8) fails for any absolutely continuous function $b(t)$. Using (7.6) we obtain
$$
\begin{equation}
|d(x,C(t))-d(x,C(s))| \leqslant \|v(t)-v(s)\|\, \|x\| + |l(t)-l(s)|
\end{equation}
\tag{7.13}
$$
for $s,t \in T$, $s\ne t$. It follows from (7.13) that
$$
\begin{equation}
|d(x,C(t))-d(x,C(s))| \leqslant \int_s^t (\rho\|\dot v(\tau)\|+|\dot b(\tau)|)\,d\tau
\end{equation}
\tag{7.14}
$$
for $x\in H$ such that $\|x\|\leqslant \rho$, $\rho \geqslant 0$. Set
$$
\begin{equation*}
b_\rho(t)=\int_0^t (\rho\|\dot v(\tau)\|+|\dot b(\tau)|)\,d\tau, \qquad \rho \geqslant 0.
\end{equation*}
\notag
$$
Then the functions $b_\rho \colon T\to R$, $\rho \geqslant 0$, are absolutely continuous, and by (7.14) we have the inequality
$$
\begin{equation}
\sup_{\|x\|\leqslant \rho} |d(x,C(t))-d(x,C(s))| \leqslant |b_\rho (t)-b_\rho (s)|.
\end{equation}
\tag{7.15}
$$
Because
$$
\begin{equation*}
\operatorname{exc}_\rho(C(s),C(t)) \leqslant \sup_{\|x\|\leqslant \rho} |d(x,C(s))-d(x,C(t))|,
\end{equation*}
\notag
$$
it follows from (7.15) and (7.5) that
$$
\begin{equation*}
\operatorname{exc}_\rho(D(A(s)),D(A(t))) \leqslant |b_\rho (t)-b_\rho (s)| \quad\text{for } s\leqslant t \text{ and } \rho\in[0,\infty).
\end{equation*}
\notag
$$
This inequality and Theorem 5.2 show that inclusion (1.1) is solvable. Other examples of maximal monotone operators $A(t)=\mathcal{N}(C(t))$, $t\in T$, failing the assumptions of Theorem 3.1 in [5] are provided by operators such that the values of the set-valued map $C\colon T \rightrightarrows H$ are half-spaces (see [17]) or polyhedra (see [18] and [19]). Hence Theorem 5.2, in comparison to Theorem 3.1 in [5], allows us to extend the class of maximal monotone operators such that inclusion (1.1) is solvable. 
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