# Well-posedness for integro-differential sweeping processes of Volterra type

Emilio Vilches111Instituto de Ciencias de la Ingeniería, Universidad de
O’Higgins and Center for Mathematical Modeling, Universidad de Chile, Chile.
Email<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the well-posedness of integro-differential sweeping
processes of Volterra type. Using new enhanced versions of Gronwall’s
inequality, a reparametrization technique, and a fixed point argument for
history-dependent operators, we obtain the existence of solutions and provide
a fully continuous dependence result for the integro-differential sweeping
process. The paper ends with an application to projected dynamical systems.

##  1 Introduction

Given a real Hilbert space $\mathcal{H}$, the Moreau’s sweeping process is a
differential inclusion involving the normal cone to a family of closed moving
sets. Since its introduction by J.J. Moreau in [20, 21], the sweeping process
has been an important mathematical framework for the modeling for several
problems in contact mechanics, electrical circuits, and crowd motion, among
others (see, e.g., [1, 8, 19]). Moreover, so far, there is a well-developed
existence theory for extensive classes of nonconvex sets.

In this paper, we are interested in a new variant of the sweeping process
called the integro-differential sweeping process of Volterra type. In its
general form, it is the following differential inclusion:

$\left\\{\begin{aligned} \dot{x}(t)&\in-
N_{C(t)}(x(t))+f(t,x(t))+\int_{T_{0}}^{t}g(t,s,x(s))ds&\textrm{ a.e. }t\in
I,\\\ x(T_{0})&=x_{0}\in C(T_{0}),\end{aligned}\right.$ (1)

where $C(t)$ is a nonempty, closed and $\rho$-uniformly prox-regular set for
all $t\in I$ and $f\colon I\times\mathcal{H}\to\mathcal{H}$ and $g\colon
I\times I\to\mathcal{H}$ are functions satisfying assumptions
$(\mathcal{H}^{f})$ and $(\mathcal{H}^{g})$ below (see Section 4). The
integro-differential sweeping process (1) is a natural extension of the
sweeping process. It was introduced by Brenier, Gangbo, Savaré and
Westdickenberg [6] to describe a one-dimensional compressible fluid under the
influence of an endogenous force field. Then, Colombo and Kozaily [13] obtain
the existence of solutions for a particular version of (1) by applying the
Moreau-Yosida regularization. A great step was given by Bouach, Haddad and
Thibault [4, 5], who develop a complete existence theory for the integro-
differential sweeping process using an appropriate catching-up algorithm.

In this paper, we show that the well-posedness for the integro-differential
sweeping process can be obtained through a reparametrization technique and a
fixed point argument for history-dependent operators. Moreover, we provide a
fully continuous dependence result with respect to the data of the problem.
Our approach is based in some enhanced version of the Gronwall’s inequality.

The paper is organized as follows. After some mathematical preliminaries, in
Section 3, we prove two new versions of Gronwall’s inequality, which are used,
in Section 4, to obtain the existence of solutions for the integro-
differential sweeping process. Our approach is based on a fixed point
argument. In Section 5, we provide a fully continuous dependence result for
the integro-differential sweeping process. The paper ends with an application
to projected dynamical systems.

##  2 Mathematical Preliminaries

Let $\mathcal{H}$ be a real Hilbert space endowed with an inner product
$\langle\cdot,\cdot\rangle$ and associated norm $\|\cdot\|$. As usual,
$\mathbb{R}$ will denote the set of real numbers and $\mathbb{R}+$ the set of
nonnegative reals, that is, $\mathbb{R}_{+}:=[0,+\infty[$. It will be
convenient to use the notation $I_{=}[T_{0},T]$ for $-\infty<T_{0}<T<+\infty$.

Given a set $S\subset\mathcal{H}$ and $x\in S$. We say that $v$ belongs to the
_proximal normal cone_ $N_{S}^{P}(x)$ if there exists $\sigma\geq 0$ such that

$\langle v,y-x\rangle\leq\sigma\|y-x\|^{2}\textrm{ for all }y\in S.$

If $x\notin C$, we set $N_{S}^{P}(x)=\emptyset$. We refer to [11, 24] for more
details and properties.

Now, we recall the concept of uniformly prox-regular sets. Introduced by
Federer in the finite-dimensional setting (see [17]) and later developed by
Rockafellar, Poliquin, and Thibault in [25], the prox-regularity generalizes
and unifies convex sets and nonconvex bodies with $C^{2}$ boundary. We refer
to [14, 27] for a survey.

######  Definition 1.

Let $S$ be a closed subset of $\mathcal{H}$ and $\rho\in]0,+\infty]$. The set
$S$ is called $\rho$-uniformly prox-regular if for all $x\in S$ and $\zeta\in
N^{P}(S;x)$ one has

$\langle\zeta,x^{\prime}-x\rangle\leq\frac{\|\zeta\|}{2\rho}\|x^{\prime}-x\|^{2}\textrm{
for all }x^{\prime}\in S.$

It is important to emphasize that convex sets are $\rho$-uniformly prox-
regular for any $\rho>0$.

The following result provides a characterization of uniformly prox-regular
sets (see, e.g., [14, Theorem 16]).

######  Proposition 2.1.

Let $S$ be a closed subset of $\mathcal{H}$ and $\rho\in]0,+\infty]$. Then,
the following assertions are equivalent.

  * (a)

The set $S$ is $\rho$-uniformly prox-regular.

  * (b)

For any $x_{i}\in S$, $v_{i}\in N^{P}(S;x_{i})\cap\mathbb{B}$ with $i=1,2$ one
has

$\langle
v_{1}-v_{2},x_{1}-x_{2}\rangle\geq-\frac{1}{\rho}\|x_{1}-x_{2}\|^{2},$

that is, the set-valued mapping $N^{P}(S;\cdot)\cap\mathbb{B}$ is
$1/\rho$-hypomonotone.

It is well-known that the proximal normal cone coincides with Fréchet,
Limiting, and Clarke normal cones (see [14]). Hence, we simply denote the
proximal normal cone to a prox-regular set $S$ as $N(S;\cdot)=N_{S}(\cdot)$.

######  Lemma 1.

Let $w_{1},w_{2}\colon I\to\mathcal{H}$ be two absolutely continuous functions
such that

$\frac{1}{2}\frac{d}{dt}\|w_{1}(t)-w_{2}(t)\|^{2}\leq\alpha(t)\|w_{1}(t)-w_{2}(t)\|^{2}+\beta(t)\|w_{1}(t)-w_{2}(t)\|\textrm{
for a.e. }t\in I,$

where $\alpha,\beta\colon I\to\mathbb{R}$ are nonnegative functions. Then, for
a.e. $t\in I$, it holds

$\frac{d}{dt}\|w_{1}(t)-w_{2}(t)\|\leq\alpha(t)\|w_{1}(t)-w_{2}(t)\|+\beta(t).$
(2)

###### Proof.

Let $h(t):=\|w_{1}(t)-w_{2}(t)\|$ and let $\Omega$ be the set of full measure
in which $\dot{h}$ exists. Let us consider the sets

$\Omega_{1}:=\\{t\in\Omega\colon w_{1}(t)\neq w_{2}(t)\\}\textrm{ and
}\Omega_{2}:=\\{t\in\Omega\colon w_{1}(t)=w_{2}(t)\\}.$

On the one hand, for $t\in\Omega_{1}$, one has

$\frac{1}{2}\frac{d}{dt}\|w_{1}(t)-w_{2}(t)\|^{2}=\|w_{1}(t)-w_{2}(t)\|\frac{d}{dt}\|w_{1}(t)-w_{2}(t)\|,$

which implies that (2) holds for $t\in\Omega_{1}$. On the other hand, for any
$t\in\Omega_{2}$, the map $t\mapsto h(t)$ attains a minimum at this point.
Thus, for $t\in\Omega_{2}$, we have

$\frac{d}{dt}\|x_{1}(t)-x_{2}(t)\|=0,$

which implies that (2) holds for $t\in\Omega_{2}$. The proof is then complete.
∎

Given a closed interval $J\subset\mathbb{R}$, we denote by $C(J;\mathcal{H})$
the space of continuous mappings from $J$ to $\mathcal{H}$ equipped with the
supremum norm $\|\cdot\|_{\infty}$. We end this section with a fixed point
theorem for history-dependent operators. We refer to [26, p. 41-45] for a
proof.

######  Proposition 2.2.

Let $\mathcal{F}\colon C([0,T];\mathcal{H})\to C([0,T];\mathcal{H})$ be a
history-dependent operator, i.e., there exists $L_{\mathcal{F}}\geq 0$ such
that

$\|\mathcal{F}(y_{1})(t)-\mathcal{F}(y_{2})(t)\|\leq
L_{\mathcal{F}}\int_{0}^{t}\|y_{1}(s)-y_{2}(s)\|ds\textrm{ for all
}y_{1},y_{2}\in C([0,T];\mathcal{H})\textrm{ and }t\in[0,T].$

Then, $\mathcal{F}$ has a unique fixed point, i.e., there exists a unique
element $y^{\ast}\in C([0,T];\mathcal{H})$ such that
$\mathcal{F}(y^{\ast})=y^{\ast}$.

##  3 Enhanced Gronwall’s inequalities

In this section we present two new versions of Gronwall’s inequality. We will
see later that these two inequalities play a key role in obtaining the
existence of solutions for integro-differential sweeping processes of Volterra
type.

######  Theorem 3.1 (Enhanced Gronwall’s Inequality I).

Let $I:=[T_{0},T]$, and let $\rho\colon I\to\mathbb{R}$ be a nonnegative
absolutely continuous function. Let $K_{1},K_{2},\varepsilon\colon
I\to\mathbb{R}_{+}$, and $K_{3}\colon I\times I\to\mathbb{R}_{+}$ be
nonnegative measurable functions such

$t\mapsto K_{1}(t)\textrm{ and }t\mapsto
K_{2}(t)\int_{T_{0}}^{t}K_{3}(t,s)ds\textrm{ are }integrable.$

Suppose that

$\dot{\rho}(t)\leq\varepsilon(t)+K_{1}(t)\rho(t)+K_{2}(t)\int_{T_{0}}^{t}K_{3}(t,s)\rho(s)ds\quad\textrm{
for a.e. }t\in I.$

Then, one has

$\rho(t)\leq\rho(T_{0})\exp\left(\int_{T_{0}}^{t}\gamma(s)ds\right)+\int_{T_{0}}^{t}\varepsilon(s)\exp\left(\int_{s}^{t}\gamma(\tau)d\tau\right)ds\quad\textrm{
for all }t\in I,$

where $\gamma(t):=K_{1}(t)+K_{2}(t)\int_{T_{0}}^{t}K_{3}(t,s)ds$.

###### Proof.

Let us consider the non-decreasing function:

$\vartheta(t):=\rho(T_{0})+\int_{T_{0}}^{t}\varepsilon(s)ds+\int_{T_{0}}^{t}K_{1}(s)\rho(s)ds+\int_{T_{0}}^{t}K_{2}(s)\int_{T_{0}}^{s}K_{3}(s,\tau)\rho(\tau)d\tau
ds.$

Then, by assumption, $\rho(t)\leq\vartheta(t)$ for all $t\in I$. Therefore,
for a.e. $t\in I$, one has

$\displaystyle\dot{\vartheta}(t)$
$\displaystyle=\varepsilon(t)+K_{1}(t)\rho(t)+K_{2}(t)\int_{T_{0}}^{t}K_{3}(t,s)\rho(s)ds$
$\displaystyle\leq\varepsilon(t)+(K_{1}(t)+K_{2}(t)\int_{T_{0}}^{t}K_{3}(t,s)ds)\vartheta(t)$
$\displaystyle=\varepsilon(t)+\gamma(t)\vartheta(t),$

where we have used that $\rho(t)\leq\vartheta(t)$ for all $t\in I$ and the
function $t\mapsto\vartheta(t)$ is nondecreasing. Hence, by applying the
classical Gronwall’s inequality we get that

$\vartheta(t)\leq\vartheta(T_{0})\exp\left(\int_{T_{0}}^{t}\gamma(s)ds\right)+\int_{T_{0}}^{t}\varepsilon(s)\exp\left(\int_{s}^{t}\gamma(\tau)d\tau\right)ds\quad\textrm{
for all }t\in I.$

Finally, by using that $\vartheta(T_{0})=\rho(T_{0})$ and
$\rho(t)\leq\vartheta(t)$ for all $t\in I$, we obtain the result. ∎

######  Theorem 3.2 (Enhanced Gronwall’s Inequality II).

Let $I:=[T_{0},T]$, and let $\rho\colon I\to\mathbb{R}$ be a nonnegative
absolutely continuous function. Let $\varepsilon,K_{1},K_{2},K_{3},K_{4}\colon
I\to\mathbb{R}_{+}$ be nonnegative measurable functions such

$t\mapsto\varepsilon(t),t\mapsto K_{1}(t),t\mapsto K_{2}(t)\textrm{ and
}t\mapsto K_{3}(t)\int_{T_{0}}^{t}K_{4}(s)ds\textrm{ are }integrable.$

Suppose that for a.e. $t\in I$ one has

$\dot{\rho}(t)\leq\varepsilon(t)+K_{1}(t)\sqrt{\rho(t)}+K_{2}(t)\rho(t)+K_{3}(t)\sqrt{\rho(t)}\int_{T_{0}}^{t}K_{4}(s)\sqrt{\rho(s)}ds.$

Then, the following assertions hold:

  * (a)

For all $t\in I$, one has

$\rho(t)\leq\rho(T_{0})\exp\left(\int_{T_{0}}^{t}(\gamma(s)+K_{1}(s))ds\right)+\int_{T_{0}}^{t}(\varepsilon(s)+K_{1}(s))\exp\left(\int_{s}^{t}(\gamma(\tau)+K_{1}(\tau))d\tau\right)ds,$

where $\gamma(t):=(K_{2}(t)+K_{3}(t)\int_{T_{0}}^{t}K_{4}(s)ds)$.

  * (b)

If, in addition $\varepsilon(t)\equiv 0$, then for all $t\in I$, one has

$\sqrt{\rho(t)}\leq\sqrt{\rho(T_{0})}\exp\left(\frac{1}{2}\int_{T_{0}}^{t}\gamma(s)ds\right)+\frac{1}{2}\int_{T_{0}}^{t}K_{1}(s)\exp\left(\frac{1}{2}\int_{s}^{t}\gamma(\tau)d\tau\right)ds.$

###### Proof.

Let us consider the non-decreasing function $\vartheta(t)$ such that
$\vartheta(0)=\rho(T_{0})$ and

$\dot{\vartheta}(t):=\varepsilon(t)+K_{1}(t)\sqrt{\rho(t)}+K_{2}(t)\rho(t)+K_{3}(t)\sqrt{\rho(t)}\int_{T_{0}}^{t}K_{4}(s)\sqrt{\rho(s)}ds.$

Then, by assumption, $\rho(t)\leq\vartheta(t)$ for all $t\in I$. Therefore,
for a.e. $t\in I$, one has

$\displaystyle\dot{\vartheta}(t)$
$\displaystyle=\varepsilon(t)+K_{1}(t)\sqrt{\rho(t)}+K_{2}(t)\rho(t)+K_{3}(t)\sqrt{\rho(t)}\int_{T_{0}}^{t}K_{4}(s)\sqrt{\rho(s)}ds$
$\displaystyle\leq\varepsilon(t)+K_{1}(t)\sqrt{\vartheta(t)}+(K_{2}(t)+K_{3}(t)\int_{T_{0}}^{t}K_{4}(s)ds)\vartheta(t)$
$\displaystyle=\varepsilon(t)+K_{1}(t)\sqrt{\vartheta(t)}+\gamma(t)\vartheta(t),$

where we have used that $t\mapsto\vartheta(t)$ is nondecreasing. Moreover, by
noting that $\sqrt{\vartheta(t)}\leq\vartheta(t)+1$ for all $t\in I$, we
obtain that

$\displaystyle\dot{\vartheta}(t)\leq(\varepsilon(t)+K_{1}(t))+(\gamma(t)+K_{1}(t))\vartheta(t)\textrm{
a.e. }t\in I,$

which, by the classical Gronwall’s inequality, implies that

$\vartheta(t)\leq\vartheta(T_{0})\exp\left(\int_{T_{0}}^{t}(\gamma(t)+K_{1}(s))ds\right)+\int_{T_{0}}^{t}(\varepsilon(s)+K_{1}(t))\exp\left(\int_{s}^{t}(\gamma(\tau)+K_{1}(\tau))d\tau\right)ds\quad
t\in I.$

Then, by using that $\vartheta(T_{0})=\rho(T_{0})$ and
$\rho(t)\leq\vartheta(t)$ for $t\in I$, the first assertion is proved. To
prove $(b)$, assume that $\varepsilon(t)\equiv 0$ and set
$\sigma(t):=\sqrt{\rho(t)}$. Then, by virtue of Lemma 1, we obtain that

$\dot{\sigma}(t)\leq\frac{1}{2}K_{1}(t)+\frac{1}{2}K_{2}(t)\sigma(t)+\frac{1}{2}K_{3}(t)\int_{T_{0}}^{t}K_{4}(s)\sigma(s)ds.$

Hence, applying Theorem 3.1, we obtain the result. ∎

##  4 Integro-differential sweeping process of Volterra type

In this section, we will use enhanced Gronwall’s inequalities proved in
Section 3 to obtain the existence and uniqueness of solutions for the integro-
differential sweeping process of Volterra type:

$\left\\{\begin{aligned} \dot{x}(t)&\in-
N_{C(t)+z(t)}(x(t))+f(t,x(t))+\int_{T_{0}}^{t}g(t,s,x(s))ds&\textrm{ a.e.
}t\in I,\\\ x(T_{0})&=x_{0}\in C(T_{0}),\end{aligned}\right.$
($\mathcal{SP}(x_{0},z,f,g)$)

where $z\colon I\to\mathcal{H}$ is an absolutely continuous function, $C(t)$
is a nonempty, closed and $\rho$-uniformly prox-regular set for all $t\in I$
and $f\colon I\times\mathcal{H}\to\mathcal{H}$ and $g\colon I\times
I\to\mathcal{H}$ are functions satisfying assumptions $(\mathcal{H}^{f})$ and
$(\mathcal{H}^{g})$ below.

#### Assumptions on the data:

We consider the following assumptions on the data for
($\mathcal{SP}(x_{0},z,f,g)$).

  1. $(\mathcal{H}_{C})$

The set-valued map $C\colon I\rightrightarrows\mathcal{H}$ has nonempty,
closed and $\rho$-uniformly prox-regular values for some constant
$\rho\in]0,+\infty]$. Moreover, there exist an absolutely continuous function
$v\colon I\to\mathbb{R}$ such that

$C(t)\subset C(s)+|v(t)-v(s)|\mathbb{B}\textrm{ for all }s,t\in I.$

  1. $(\mathcal{H}^{f})$

The function $f\colon I\times\mathcal{H}\to\mathcal{H}$ satisfies

    1. (a)

For each $x\in\mathcal{H}$, the map $t\mapsto f(t,x)$ is measurable.

    2. (b)

For all $r>0$, there exists an integrable function $\kappa_{r}\colon
I\to\mathbb{R}_{+}$ such that

$\|f(t,x)-f(t,y)\|\leq\kappa_{r}(t)\|x-y\|\textrm{ for all }t\in I\textrm{ and
}x,y\in r\mathbb{B}.$

    3. (c)

There exists a nonnegative integrable function $\beta\colon I\to\mathbb{R}$
such that

$\|f(t,x)\|\leq\beta(t)(1+\|x\|)\textrm{ for all }t\in I\textrm{ and
}x\in\mathcal{H}.$

  1. $(\mathcal{H}^{g})$

The function $g\colon I\times I\times\mathcal{H}\to\mathcal{H}$ satisfies

    1. (a)

For each $x\in\mathcal{H}$, the map $(t,s)\mapsto g(t,s,x)$ is measurable.

    2. (b)

For all $r>0$, there exists an integrable function $\mu_{r}\colon
I\to\mathbb{R}_{+}$ such that for all $(t,s)\in D$

$\|g(t,s,x)-g(t,s,y)\|\leq\mu_{r}(t)\|x-y\|\textrm{ for all }x,y\in
r\mathbb{B}.$

Here $D:=\\{(t,s)\in I\times I\colon s\leq t\\}$.

    3. (c)

There exists a nonnegative integrable function $\sigma\colon D\to\mathbb{R}$
such that

$\|g(t,s,x)\|\leq\sigma(t,s)(1+\|x\|)\textrm{ for all }(t,s)\in D\textrm{ and
}x\in\mathcal{H}.$

In order to prove the existence of solutions, we will apply a fixed point
argument (see Proposition 2.2) to the classical sweeping process. Hence, the
following result will be needed (see [16, Proposition 1]).

######  Lemma 4.1.

Let $\mathcal{H}$ be a real Hilbert space, and suppose that $C(\cdot)$
satisfies $(\mathcal{H}_{C})$. Let $h\colon I\to\mathcal{H}$ be a single-
valued integrable mapping. Then, for any $x_{0}\in C(T_{0})$ there exists a
unique absolutely continuous solution $x(\cdot)$ for the following
differential inclusion:

$\left\\{\begin{aligned} \dot{x}(t)&\in-N(C(t);x(t))+h(t)\quad\textrm{ a.e.
}t\in I\\\ x(T_{0})&=x_{0}.\end{aligned}\right.$

Moreover, $x(\cdot)$ satisfies the following inequality:

$\|\dot{x}(t)-h(t)\|\leq\|h(t)\|+|\dot{v}(t)|\quad\textrm{ a.e. }t\in I.$

Now, we present the main result of this section, that is, the well-posedness
for the nonconvex integro-differential sweeping process
($\mathcal{SP}(x_{0},z,f,g)$). It is worth emphasizing that the existence of
solutions for ($\mathcal{SP}(x_{0},z,f,g)$) was obtained in [4, 5] using an
approximation scheme and different Gronwall inequalities. In this section, our
contribution is to get the existence of solutions through a fixed point
argument and enhanced versions of Gronwall’s inequality obtained in Section 3.
We refer to [10, Theorem 4.1] for a related fixed point argument for ball
compact moving sets.

######  Theorem 4.1.

Let $z$ be an absolutely continuous function and assume, in addition to
$(\mathcal{H}_{C})$, that $(\mathcal{H}^{f})$ and $(\mathcal{H}^{g})$ hold.
Then, for any $x_{0}\in C(T_{0})$, there exists a unique absolutely continuous
solution $x(\cdot)$ of ($\mathcal{SP}(x_{0},z,f,g)$).
Moreover, the following bounds hold

$\displaystyle\|x(t)\|$ $\displaystyle\leq
r(t):=\|x_{0}\|\exp\left(\int_{T_{0}}^{t}\gamma(s)ds\right)+\int_{T_{0}}^{t}(\gamma(s)+|\dot{v}(s)|+\|\dot{z}(t)\|)\exp\left(\int_{s}^{t}\gamma(\tau)d\tau\right)ds$
$\displaystyle\textrm{ for all }t\in I,$ $\displaystyle\|\dot{x}(t)\|$
$\displaystyle\leq\theta(t):=\gamma(t)(1+r(t))+|\dot{v}(t)|+\|\dot{z}(t)\|$
$\displaystyle\textrm{ for a.e. }t\in I,$

where $\gamma(t):=2\beta(t)+2\int_{T_{0}}^{t}\sigma(t,s)ds$.

###### Proof.

Let us consider the integrable function

$\varphi(t):=\max\\{1,\kappa_{r(T)}(t),\mu_{r(T)}(t),\beta(t),\int_{T_{0}}^{t}\sigma(t,s)ds\\}.$
(3)

The map $t\mapsto\int_{T_{0}}^{t}\varphi(s)ds$ from $I$ to
$J:=[0,\int_{T_{0}}^{T}\varphi(s)ds]$ is continuous and strictly increasing.
Let $\phi$ be its inverse and define

$\tilde{f}(t,x):=\frac{1}{\varphi(\phi(t))}f(\phi(t),x)\textrm{ and
}\tilde{g}(t,s,x)=\frac{1}{\varphi(\phi(t))}\frac{1}{\varphi(\phi(s))}g(\phi(t),\phi(s),x).$

It is clear that $\tilde{f}$ and $\tilde{g}$ satisfy $(\mathcal{H}^{f})$ and
$(\mathcal{H}^{g})$, respectively. Moreover, for all $x,y\in r(T)\mathbb{B}$,
one has

$\displaystyle\|\tilde{f}(t,x)-\tilde{f}(t,y)\|\leq\|x-y\|\textrm{ and
}\|\tilde{g}(t,s,x)-\tilde{g}(t,s,y)\|\leq\|x-y\|.$ (4)

Given $y\in C(J;\mathcal{H})$, let us consider $w(\cdot)$ be the unique
solution (see Lemma 4.1) to the problem:

$\left\\{\begin{aligned} \dot{w}(t)&\in-
N_{C(\phi(t))+z(\phi(t))}(w(t))+h(y)(t)\quad\textrm{ a.e. }t\in J,\\\
w(0)&=x_{0},\end{aligned}\right.$ (5)

where
$h(y)(t):=\tilde{f}(t,\operatorname{proj}_{r(\phi(t))\mathbb{B}}(y(t)))+\int_{0}^{t}\tilde{g}(t,s,\operatorname{proj}_{r(\phi(s))\mathbb{B}}(y(s)))ds$.
According to Lemma 4.1, $(\mathcal{H}^{f})$, and $(\mathcal{H}^{g})$, for a.e.
$t\in J$ one has

$\displaystyle\|\dot{w}(t)-h(y)(t)\|$
$\displaystyle\leq\|h(y)(t)\|+|\dot{v}(\phi(t))|+\|\dot{z}(\phi(t))\|$
$\displaystyle\leq\frac{\beta(\phi(t))}{\varphi(\phi(t))}(1+r(\phi(t)))+\int_{0}^{t}\frac{\sigma(\phi(t),\phi(s))}{\varphi(\phi(t))\varphi(\phi(s))}(1+r(\phi(s)))ds+|\dot{v}(\phi(t))|+\|\dot{z}(\phi(t))\|$
$\displaystyle\leq\left[\frac{\beta(\phi(t))}{\varphi(\phi(t))}+\int_{0}^{t}\frac{\sigma(\phi(t),\phi(s))}{\varphi(\phi(t))\varphi(\phi(s))}ds\right](1+r(\phi(t)))+|\dot{v}(\phi(t))|+\|\dot{z}(\phi(t))\|$
$\displaystyle=\left[\frac{\beta(\phi(t))}{\varphi(\phi(t))}+\frac{\int_{T_{0}}^{\phi(t)}\sigma(\phi(t),s)ds}{{\varphi(\phi(t))}}\right](1+r(\phi(t)))+|\dot{v}(\phi(t))|+\|\dot{z}(\phi(t))\|$
$\displaystyle\leq\eta(t):=2(1+r(\phi(t)))+|\dot{v}(\phi(t))|+\|\dot{z}(\phi(t))\|,$

where we have used that $t\mapsto r(\phi(t))$ is nondecreasing and definition
(3).
Next, we consider the operator $\mathcal{F}\colon C(J;\mathcal{H})\to
C(J;\mathcal{H})$ defined as $\mathcal{F}(y):=w$, where $w(\cdot)$ is the
unique solution of the problem (5).
Claim 1: The operator $\mathcal{F}$ has a unique fixed point.
_Proof of Claim 1_ : Let us consider
$V(t):=\frac{1}{2}\|w_{1}(t)-w_{2}(t)\|^{2}$, where
$w_{i}:=\mathcal{F}(y_{i})$, for $i=1,2$. Then, for a.e. $t\in J$, we obtain
that

$\displaystyle\dot{V}(t)$ $\displaystyle=\langle
w_{1}(t)-w_{2}(t),\dot{w}_{1}(t)-\dot{w}_{2}(t)\rangle$
$\displaystyle=\langle(w_{1}(t)-z(\phi(t)))-(w_{2}(t)-z(\phi(t))),(\dot{w}_{1}(t)-h(y_{1})(t))-(\dot{w}_{2}(t)-h(y_{2})(t))\rangle$
$\displaystyle+\langle w_{1}(t)-w_{2}(t),h(y_{1})(t)-h(y_{2})(t)\rangle$
$\displaystyle\leq\frac{\eta(t)}{\rho}\|w_{1}(t)-w_{2}(t)\|^{2}+\|w_{1}(t)-w_{2}(t)\|\cdot\|h(y_{1})(t)-h(y_{2})(t)\|,$

where we have used Proposition 2.1 and that for $i=1,2,$ one has

$\frac{w_{i}(t)-h(y_{i})(t)}{\eta(t)}\in-
N_{C(\phi(t))}(w_{i}(t)-z(\phi(t)))\cap\mathbb{B}\textrm{ for a.e. }t\in J.$

Next, if we consider $W(t):=\|w_{1}(t)-w_{2}(t)\|$, according to Lemma 1, we
obtain that

$\dot{W}(t)\leq\frac{\eta(t)}{\rho}W(t)+\|h(y_{1})(t)-h(y_{2})(t)\|\textrm{
for a.e. }t\in J.$

On the other hand, by virtue of (4) and the Lipschitzianity of constant 1 of
the map $x\mapsto\operatorname{proj}_{r(t)}(x)$, we obtain that for a.e. $t\in
J$

$\displaystyle\|h(y_{1})(t)-h(y_{2})(t)\|$
$\displaystyle\leq\|\tilde{f}(t,\operatorname{proj}_{r(t)\mathbb{B}}(y_{1}(t)))-\tilde{f}(t,\operatorname{proj}_{r(t)\mathbb{B}}(y_{2}(t)))\|$
$\displaystyle+\int_{0}^{t}\|\tilde{g}(t,s,\operatorname{proj}_{r(s)\mathbb{B}}(y_{1}(s)))-\tilde{g}(t,s,\operatorname{proj}_{r(s)\mathbb{B}}(y_{2}(s)))\|ds$
$\displaystyle\leq\|y_{1}(t)-y_{2}(t)\|+\int_{0}^{t}\|y_{1}(s)-y_{2}(s)\|ds.$

Therefore, for a.e. $t\in J$,

$\displaystyle\frac{d}{dt}\left(W(t)\exp\left(-\int_{0}^{t}\frac{\eta(s)}{\rho}ds\right)\right)$
$\displaystyle\leq\exp\left(-\int_{0}^{t}\frac{\eta(s)}{\rho}ds\right)\left(\|y_{1}(t)-y_{2}(t)\|+\int_{0}^{t}\|y_{1}(s)-y_{2}(s)\|ds\right)$
$\displaystyle\leq
e^{t}\left(\|y_{1}(t)-y_{2}(t)\|+\int_{0}^{t}\|y_{1}(s)-y_{2}(s)\|ds\right)$
$\displaystyle=\frac{d}{dt}\left(e^{t}\int_{0}^{t}\|y_{1}(s)-y_{2}(s)\|ds\right),$

which implies that

$W(t)\leq\exp\left(\int_{0}^{t}\frac{\eta(s)}{\rho}ds+1\right)\int_{0}^{t}\|y_{1}(s)-y_{2}(s)\|ds\quad\textrm{
for all }t\in J.$

Hence, the operator $\mathcal{F}$ is history-dependent. Therefore, by virtue
of Proposition 2.2, the operator $\mathcal{F}$ has a unique fixed point
$\bar{w}\in C(J;\mathcal{H})$, which proves Claim 1.
On the other hand, it is routine to prove that
$x(t):=\bar{w}\left(\int_{T_{0}}^{t}\varphi(s)ds\right)$ solves the problem:

$\left\\{\begin{aligned} \dot{x}(t)&\in-
N_{C(t)+z(t)}(x(t))+f(t,\operatorname{proj}_{r(t)\mathbb{B}}(x(t)))+\int_{T_{0}}^{t}{g}(t,s,\operatorname{proj}_{r(s)\mathbb{B}}(x(s)))ds\quad\textrm{
a.e. }t\in I,\\\ x(T_{0})&=x_{0}.\end{aligned}\right.$ (6)

Claim 2: For all $t\in I$, $\|x(t)\|\leq r(t)$.
_Proof of Claim 2:_ According to Lemma 4.1, for a.e. $t\in I$, one has

$\displaystyle\|\dot{x}(t)\|$ $\displaystyle\leq
2\|f(t,\operatorname{proj}_{r(t)\mathbb{B}}(x(t)))\|+2\int_{T_{0}}^{t}\|g(t,s,\operatorname{proj}_{r(s)\mathbb{B}}(x(s)))\|ds+|\dot{v}(t)|+\|\dot{z}(t)\|$
$\displaystyle\leq
2\beta(t)(1+\|\operatorname{proj}_{r(t)\mathbb{B}}(x(t))\|)+2\int_{T_{0}}^{t}\sigma(t,s)\|\operatorname{proj}_{r(s)\mathbb{B}}(x(s))\|ds+|\dot{v}(t)|+\|\dot{z}(t)\|$
$\displaystyle\leq
2\beta(t)(1+\|x(t)\|)+2\int_{T_{0}}^{t}\sigma(t,s)(1+\|x(s)\|)ds+|\dot{v}(t)|+\|\dot{z}(t)\|,$

where we have used that
$\|\operatorname{proj}_{r(t)\mathbb{B}}(x(t))\|\leq\|x(t)\|$. Therefore,
according to Theorem 3.1, we obtain that $\|x(t)\|\leq r(t)$ for all $t\in I$,
which proves Claim 2.
Claim 3: The trajectory $x$ solves ($\mathcal{SP}(x_{0},z,f,g)$) and for a.e.
$t\in I$, $\|\dot{x}(t)\|\leq\theta(t)$.
_Proof of Claim 3:_ Since $\|x(t)\|\leq r(t)$ for all $t\in I$, it follows
from (6) that $x$ solves ($\mathcal{SP}(x_{0},z,f,g)$). Moreover, from Lemma
4.1, it follows that

$\displaystyle\|\dot{x}(t)\|$ $\displaystyle\leq
2\|f(t,x(t))\|+2\int_{T_{0}}^{t}\|g(t,s,x(s))\|ds+|\dot{v}(t)|+\|\dot{z}(t)\|$
$\displaystyle\leq
2\beta(t)(1+r(t))+2\int_{T_{0}}^{t}\sigma(t,s)(1+r(s))ds+|\dot{v}(t)|+\|\dot{z}(t)\|$
$\displaystyle\leq\gamma(t)(1+r(t))+|\dot{v}(t)|+\|\dot{z}(t)\|,$

which proves Claim 3.
Claim 4: The trajectory $x$ is the unique solution of
($\mathcal{SP}(x_{0},z,f,g)$).
_Proof of Claim 4:_ Let $x_{1}$ and $x_{2}$ be two solutions of
($\mathcal{SP}(x_{0},z,f,g)$). Consider
$V(t):=\frac{1}{2}\|w_{1}(t)-w_{2}(t)\|^{2}$, where
$w_{i}(t):=x_{i}(\phi(t))$, for $i=1,2$. Similarly to the proof of Claim 1, we
can prove the existence of a constant $\mathcal{C}\geq 0$ such that

$\|w_{1}(t)-w_{2}(t)\|\leq\mathcal{C}\int_{0}^{t}\|w_{1}(s)-w_{2}(s)\|ds\textrm{
for all }t\in J.$

Then, by using that $w_{1}(0)=w_{2}(0)$ and the above inequality, we obtain
that $w_{1}\equiv w_{2}$. Therefore, $x_{1}\equiv x_{2}$, which proves Claim
4. ∎

###  4.1 Integro-differential sweeping processes driven by a fixed convex set

In this section, we consider the particular case of $C(t)\equiv C$, where $C$
is a nonempty, closed and convex set of a Hilbert space $\mathcal{H}$. Hence,
we are interested in the problem

$\left\\{\begin{aligned} \dot{x}(t)&\in-
N_{C}(x(t))+f(t,x(t))+\int_{T_{0}}^{t}g(t,s,x(s))ds&\textrm{ a.e. }t\in I,\\\
x(T_{0})&=x_{0}\in C.\end{aligned}\right.$ (7)

In the present case, the operator $x\mapsto N_{C}(x)$ is maximal monotone,
making possible to obtain more information about the behavior of the
solutions. We recall the concept of slow solution introduced in [15] and
widely used in differential inclusions governed by maximal monotone operators
(see, e.g., [7]).

######  Definition 2 (Slow solution).

We say that $x(\cdot)$ is a _slow_ solution of (7) if $x(T_{0})=x_{0}$ and

$\dot{x}(t)=-\bar{\nu}(t)+f(t,x(t))+\int_{T_{0}}^{t}g(t,s,x(s))ds\quad\textrm{
a.e. }t\in I,$

where

$\bar{\nu}(t):=\operatorname{argmin}_{\nu\in
N_{C}(x(t))}\|f(t,x(t))+\int_{T_{0}}^{t}g(t,s,x(s))ds-\nu\|.$

The next result asserts that the unique solution of (7) is slow.

######  Theorem 4.2.

Assume, in addition to $(\mathcal{H}_{C})$, that $(\mathcal{H}^{f})$ and
$(\mathcal{H}^{g})$ hold. Then, for any $x_{0}\in C$, there exists a unique
absolutely continuous solution $x(\cdot)$ of (7). Moreover, $x(\cdot)$ is a
slow solution of (7) and the following bounds hold

$\displaystyle\|x(t)\|$ $\displaystyle\leq
r(t):=\|x_{0}\|\exp\left(\int_{T_{0}}^{t}\gamma(s)ds\right)+\int_{T_{0}}^{t}\gamma(s)\exp\left(\int_{s}^{t}\gamma(\tau)d\tau\right)ds$
$\displaystyle\textrm{ for all }t\in I,$ $\displaystyle\|\dot{x}(t)\|$
$\displaystyle\leq\theta(t):=\gamma(t)(1+r(t))$ $\displaystyle\textrm{ for
a.e. }t\in I,$

where $\gamma(t):=2\beta(t)+2\int_{T_{0}}^{t}\sigma(t,s)ds$.

###### Proof.

It follows from [7, Proposition 3.4] and Theorem 4.1. ∎

##  5 Continuous dependence of solutions for the integro-differential sweeping
process

In this section, we prove the continuous dependence of solutions for the
integro-differential sweeping process of Volterra type. Our approach is based
on enhanced Gronwall inequalities established in Section 3.
Given a nonnegative integrable function $R$, we consider the set

$\mathcal{B}_{R}:=\\{z\in AC(I;\mathcal{H})\colon\|\dot{z}(t)\|\leq
R(t)\textrm{ a.e. }t\in I\\}.$

In what follows, given $r>0$, we denote

$d_{r}(f_{1},f_{2})(t):=\sup_{x\in
r\mathbb{B}}\|f_{1}(t,x)-f_{2}(t,x)\|\textrm{ and
}D_{r}(g_{1},g_{2})(t,s):=\sup_{x\in
r\mathbb{B}}\|g_{1}(t,s,x)-g_{2}(t,s,x)\|.$

######  Theorem 5.1.

Assume that $(\mathcal{H}_{C})$ holds. Let $x_{0}^{1},x_{0}^{2}\in C(T_{0})$,
$z_{1},z_{2}\in\mathcal{B}_{R}$, and assume that $f_{1}$, $f_{2}$, and
$g_{1}$, $g_{2}$ satisfy $(\mathcal{H}^{f})$ and $(\mathcal{H}^{g})$,
respectively. Let $x_{i}$ be the unique solution of
$\mathcal{SP}(x_{0}^{i},z_{i},f_{i},g_{i})$ for $i=1,2$. Then, for all $t\in
I$, one has

$\displaystyle\|x_{1}(t)-x_{2}(t)\|^{2}\leq\|x_{0}^{1}-x_{0}^{2}\|^{2}\exp\left(\int_{T_{0}}^{t}(\delta(s)+\Delta(s))ds\right)+2\int_{T_{0}}^{t}(\varepsilon(s)+\Delta(s))\exp\left(\int_{s}^{t}(\delta(\tau)+\Delta(\tau))d\tau\right)ds,$

where

$\displaystyle\Delta(t)$
$\displaystyle:=\sqrt{2}d_{r(T)}(f_{1},f_{2})(t)+\sqrt{2}\int_{T_{0}}^{t}D_{r(T)}(g_{1},g_{2})(t,s)ds$
$\displaystyle\delta(t)$
$\displaystyle:=2\frac{\nu(t)}{\rho}+2\kappa_{r(T)}+2\int_{T_{0}}^{t}\mu_{r(T)}(s)ds$
$\displaystyle\varepsilon(t)$
$\displaystyle:=2\nu(t)\|z_{1}(t)-z_{2}(t)\|\textrm{ and
}\nu(t):=\frac{\gamma(t)}{2}(1+r(t))+|\dot{v}(t)|+R(t).$

###### Proof.

For $i=1,2$, let us consider

$h_{i}(t):=f_{i}(t,x_{i}(t))+\int_{T_{0}}^{t}g_{i}(t,s,x_{i}(s))ds.$

Then, according to Theorem 4.1, for $i=1,2$, one has

$\|\dot{x}_{i}(t)-h_{i}(t)\|\leq\nu(t):=\frac{\gamma(t)}{2}(1+r(t))+|\dot{v}(t)|+R(t),$
(8)

where

$r(t):=\|x_{0}\|\exp\left(\int_{T_{0}}^{t}\gamma(s)ds\right)+\int_{T_{0}}^{t}(\gamma(s)+|\dot{v}(s)|+R(t))\exp\left(\int_{s}^{t}\gamma(\tau)d\tau\right)ds.$

Hence, for $i=1,2$, one has

$\frac{\dot{x}_{i}(t)-h_{i}(t)}{\nu(t)}\in-
N_{C(t)}(x_{i}(t)-z_{i}(t))\cap\mathbb{B}\textrm{ for a.e. }t\in I.$

Next, let us consider $V(t):=\frac{1}{2}\|x_{1}(t)-x_{2}(t)\|^{2}$. Then, for
a.e. $t\in I$, we obtain that

$\displaystyle\dot{V}(t)$ $\displaystyle=\langle
x_{1}(t)-x_{2}(t),\dot{x}_{1}(t)-\dot{x}_{2}(t)\rangle$
$\displaystyle=\langle(x_{1}(t)-z_{1}(t))-(x_{2}(t)-z_{2}(t)),(\dot{x}_{1}(t)-h_{1}(t))-(\dot{x}_{2}(t)-h_{2}(t))\rangle$
$\displaystyle+\langle
z_{1}(t)-z_{2}(t),(\dot{x}_{1}(t)-h_{1}(t))-(\dot{x}_{2}(t)-h_{2}(t))\rangle$
$\displaystyle+\langle x_{1}(t)-x_{2}(t),h_{1}(t)-h_{2}(t)\rangle$
$\displaystyle\leq\frac{\nu(t)}{\rho}\|x_{1}(t)-x_{2}(t)\|^{2}+2\nu(t)\|z_{1}(t)-z_{2}(t)\|+\|x_{1}(t)-x_{2}(t)\|\cdot\|h_{1}(t)-h_{2}(t)\|$
$\displaystyle=\frac{2\nu(t)}{\rho}V(t)+2\nu(t)\|z_{1}(t)-z_{2}(t)\|+\sqrt{2V(t)}\cdot\|h_{1}(t)-h_{2}(t)\|,$

where we have used Proposition 2.1 and inequality (8). On the other hand, for
all $t\in I$, one has

$\displaystyle\|h_{1}(t)-h_{2}(t)\|$
$\displaystyle\leq\|f_{1}(t,x_{1}(t))-f_{2}(t,x_{2}(t))\|+\int_{T_{0}}^{t}\|g_{1}(t,s,x_{1}(s))-g_{2}(t,s,x_{2}(s))\|ds$
$\displaystyle\leq
d_{r(T)}(f_{1},f_{2})(t)+\kappa_{r(T)}(t)\|x_{1}(t)-x_{2}(t)\|$
$\displaystyle+\int_{T_{0}}^{t}D_{r(T)}(g_{1},g_{2})(t,s)ds+\int_{T_{0}}^{t}\mu_{r(T)}(s)\|x_{1}(s)-x_{2}(s)\|ds$
$\displaystyle=d_{r(T)}(f_{1},f_{2})(t)+\kappa_{r(T)}(t)\sqrt{2V(t)}+\int_{T_{0}}^{t}D_{r(T)}(g_{1},g_{2})(t,s)ds+\int_{T_{0}}^{t}\mu_{r(T)}(s)\sqrt{2V(s)}ds$

Hence, by using the above inequalities, we obtain that for a.e. $t\in I$, one
has

$\displaystyle\dot{V}(t)$ $\displaystyle\leq
2\nu(t)\|z_{1}(t)-z_{2}(t)\|+\sqrt{2}\left(d_{r(T)}(f_{1},f_{2})(t)+\int_{T_{0}}^{t}D_{r(T)}(g_{1},g_{2})(t,s)ds\right)\sqrt{V(t)}$
(9)
$\displaystyle+2\left(\frac{\nu(t)}{\rho}+\kappa_{r(T)}\right)V(t)+2\sqrt{V(t)}\int_{T_{0}}^{t}\mu_{r(T)}(s)\sqrt{V(s)}ds.$

Therefore, by the enhanced Gronwall’s inequality II (Theorem 3.2), we obtain
that for all $t\in I$, one has

$\displaystyle V(t)\leq
V(T_{0})\exp\left(\int_{T_{0}}^{t}(\delta(s)+\Delta(s))ds\right)+\int_{T_{0}}^{t}(\varepsilon(s)+\Delta(s))\exp\left(\int_{s}^{t}(\delta(\tau)+\Delta(\tau))d\tau\right)ds,$

which proves the result. ∎

It is worth noting that when $z_{1}$ and $z_{2}$ are equal, the conclusions of
the above theorem can be improved.

######  Theorem 5.2.

Assume that $(\mathcal{H}_{C})$ holds. Let $x_{0}^{1},x_{0}^{2}\in C(T_{0})$
and $z\in\mathcal{B}_{R}$. Suppose that $f_{1}$, $f_{2}$, and $g_{1}$, $g_{2}$
satisfy $(\mathcal{H}^{f})$ and $(\mathcal{H}^{g})$, respectively. Let $x_{i}$
be the unique solution of $\mathcal{SP}(x_{0}^{i},z,f_{i},g_{i})$ for $i=1,2$.
Then, for all $t\in I$, one has

$\displaystyle\|x_{1}(t)-x_{2}(t)\|\leq\|x_{0}^{1}-x_{0}^{2}\|\exp\left(\int_{T_{0}}^{t}\delta(s)ds\right)+\int_{T_{0}}^{t}\Delta(s)\exp\left(\int_{s}^{t}\delta(\tau)d\tau\right)ds,$

where

$\displaystyle\Delta(t)$
$\displaystyle:=\sqrt{2}d_{r(T)}(f_{1},f_{2})(t)+\sqrt{2}\int_{T_{0}}^{t}D_{r(T)}(g_{1},g_{2})(t,s)ds,$
$\displaystyle\delta(t)$
$\displaystyle:=2\frac{\nu(t)}{\rho}+2\kappa_{r(T)}(t)+2\int_{T_{0}}^{t}\mu_{r(T)}(s)ds,$
$\displaystyle\nu(t)$
$\displaystyle:=\frac{\gamma(t)}{2}(1+r(t))+|\dot{v}(t)|+R(t).$

###### Proof.

From (9), we obtain that

$\displaystyle\dot{V}(t)$
$\displaystyle\leq\sqrt{2}\left(d_{r(T)}(f_{1},f_{2})(t)+\int_{T_{0}}^{t}D_{r(T)}(g_{1},g_{2})(t,s)ds\right)\sqrt{V(t)}$
$\displaystyle+2\left(\frac{\nu(t)}{\rho}+\kappa_{r(T)}\right)V(t)+2\sqrt{V(t)}\int_{T_{0}}^{t}\mu_{r(T)}(s)\sqrt{V(s)}ds.$

Hence, by virtue of Lemma 1, if we consider $W(t):=\sqrt{V(t)}$, then one has

$\displaystyle\dot{W}(t)$
$\displaystyle\leq\sqrt{2}\left(d_{r(T)}(f_{1},f_{2})(t)+\int_{T_{0}}^{t}D_{r(T)}(g_{1},g_{2})(t,s)ds\right)$
$\displaystyle+2\left(\frac{\nu(t)}{\rho}+\kappa_{r(T)}\right)W(t)+2\int_{T_{0}}^{t}\mu_{r(T)}(s)W(s)ds.$

Therefore, by virtue of the enhanced Gronwall’s inequality I (Theorem 3.1 ),
we obtain that

$W(t)\leq
W(T_{0})\exp\left(\int_{T_{0}}^{t}\delta(s)ds\right)+\int_{T_{0}}^{t}\Delta(s)\exp\left(\int_{s}^{t}\delta(\tau)d\tau\right)ds,$

which proves the result. ∎

##  6 Integro-differential projected dynamical systems

Let $\mathcal{H}$ be a real Hilbert space and consider the following integro-
differential equation:

$\left\\{\begin{aligned}
\dot{x}(t)&=f(t,x(t))+\int_{T_{0}}^{t}g(t,s,x(s))ds&\textrm{ a.e. }t\in I,\\\
x(T_{0})&=x_{0},\end{aligned}\right.$ (10)

where $f\colon I\times\mathcal{H}\to\mathcal{H}$ and $g\colon I\times
I\times\mathcal{H}\to\mathcal{H}$ are functions satisfying $(\mathcal{H}^{f})$
and $(\mathcal{H}^{g})$, respectively. According to Theorem 4.1, the above
problem has a unique absolutely continuous solution $x\colon I\to\mathcal{H}$.
Moreover, Theorem 5.1 provides a quantified continuous dependence result for
(10). Hence, the problem (10) is well-posed.

Given a nonempty and closed set $C\subset\mathcal{H}$ and $x_{0}\in C$, the
viability problem for (10) related to $C$ consists in providing necessary and
sufficient conditions on $C$, $f$ and $g$ in order that the unique solution
$x(\cdot)$ of (10) stays on the set $C$, that is, $x(t)\in C$ for all $t\in
I$. For classical differential equations ($g\equiv 0$), the first result in
this sense goes back to Nagumo [22] who proved that a necessary and sufficient
condition for viability is that the right-hand side of the differential
equation belongs to the tangent cone of the viability set $C$. Since then, a
well-developed viability theory has been developed for differential
inclusions, including the case of differential equations. We refer to [3, 2]
for more details. However, viability theory for integro-differential equations
seems inadequate due to the memory effects induced by the integral term on
(10). Indeed, roughly speaking, a tangential condition for the viability
result would require that the right-hand side of (10) belongs to the tangent
cone of $C$, involving the solutions of the dynamical system, which is
impractical from a useful point of view. One way to overcome this difficulty
is to consider a related dynamical system whose solutions naturally stay
within the given set $C$. Although this changes the system under study, it can
still provide important information about the phenomenon of interest.

In what follows, we assume that $C$ is a nonempty, closed and convex set and
$x_{0}\in C$. We follow the developments of the theory of _projected dynamical
systems_ (see, e.g., [12, 9]) and consider the following integro-differential
projected dynamical system:

$\left\\{\begin{aligned}
\dot{x}(t)&=\operatorname{proj}_{T_{C}(x(t))}\left(f(t,x(t))+\int_{T_{0}}^{t}g(t,s,x(s))ds\right)&\textrm{
a.e. }t\in I,\\\ x(T_{0})&=x_{0}\in C,\end{aligned}\right.$ (11)

where $T_{C}(x(t))$ denotes the tangent cone of the set $C$ at $x(t)$. When
$g\equiv 0$ the above system is called projected dynamical system and have
been used to model several equilibrium problems (see [12] and the references
therein). Before establishing the well-posedness for the above dynamical
system, we provide the equivalence of (11) with the following integro-
differential sweeping process of Volterra type:

$\left\\{\begin{aligned} \dot{x}(t)&\in-
N_{C}(x(t))+f(t,x(t))+\int_{T_{0}}^{t}g(t,s,x(s))ds&\textrm{ a.e. }t\in I,\\\
x(T_{0})&=x_{0}\in C.\end{aligned}\right.$ (12)

The following result follows in the same way as [9, Corollary 2]

######  Proposition 6.1.

Let $C$ be a nonempty, closed and convex set. Then, any solution of (11) is a
solution of (12). Reciprocally, any slow solution of (12) is a solution of
(11).

It is worth emphasizing that due to the presence of the normal cone, any
solution of 12 satisfies the constraint $x(t)\in C$. Now, we are in position
to prove the existence of solutions for (11).

######  Proposition 6.2.

Let $C$ be a nonempty, closed and convex set and assume that
$(\mathcal{H}^{f})$ and $(\mathcal{H}^{g})$ hold. Then, for any $x_{0}\in C$,
there exists a unique absolutely continuous solution $x(\cdot)$ of (11).
Moreover, $x(\cdot)$ is a slow solution of (7) and the following bounds hold

$\displaystyle\|x(t)\|$ $\displaystyle\leq
r(t):=\|x_{0}\|\exp\left(\int_{T_{0}}^{t}\gamma(s)ds\right)+\int_{T_{0}}^{t}\gamma(s)\exp\left(\int_{s}^{t}\gamma(\tau)d\tau\right)ds$
$\displaystyle\textrm{ for all }t\in I,$ $\displaystyle\|\dot{x}(t)\|$
$\displaystyle\leq\theta(t):=\gamma(t)(1+r(t))$ $\displaystyle\textrm{ for
a.e. }t\in I,$

where $\gamma(t):=2\beta(t)+2\int_{T_{0}}^{t}\sigma(t,s)ds$.

###### Proof.

It follows directly from Theorem 4.2 and Proposition 6.1. ∎

##  7 Concluding remarks

In this paper, we have studied a variant of Moreau’s sweeping process known as
the integro-differential sweeping process of Volterra type. After establishing
new enhanced versions of Gronwall’s inequality, we provided an existence
result for integro-differential sweeping process of Volterra type. Our
approach is based on a reparametrization technique and a fixed point argument
for history-dependent operators. We observe that the main result from Section
4 (Theorem 4.1) is based on an existence result for the sweeping process due
to Edmond and Thibault [16], where the prox-regularity and a priori bound on
the derivative of the trajectory is provided. Hence, by using the same fixed
point argument with a different existence result for Moreau’s sweeping process
(for example, with truncated bounded retraction), more general results for the
integro-differential sweeping process of Volterra type can be obtained.
Another key point of this work is the fully continuous dependence result
obtained in Section 5. Indeed, we provide estimates of the dependence of the
solutions with respect to all the data of the problem. Again, our analysis is
based on new enhanced versions of Gronwall’s inequality.

Several open questions remain. On the one hand, a natural continuation of this
work is to consider integro-differential state-dependent sweeping process of
Volterra type with nonconvex sets, e.g., submooth moving sets (see, e.g.,
[18]). Degenerate versions of integro-differential state-dependent sweeping
processes could also be explored (see, e.g., [23]).

#### Acknowledgements

This work was supported by the Center for Mathematical Modeling (CMM) and
ANID-Chile under BASAL funds for Center of Excellence FB210005, Fondecyt
Regular 1200283, Fondecyt Regular 1240120 and Fondecyt Exploración 13220097.
We also acknowledge ECOS-Anid Project PC23E11 and MathAmsud Project
23-MATH-17.

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