# Second order splitting dynamics with vanishing damping for additively
structured monotone inclusions ††thanks: Research partially supported by the
Doctoral Programme Vienna Graduate School on Computational Optimization
(VGSCO) which is funded by FWF (Austrian Science Fund), project W1260-N35.

Radu Ioan Boţ<EMAIL_ADDRESS>David Alexander Hulett
<EMAIL_ADDRESS>
###### Abstract

In the framework of a real Hilbert space, we address the problem of finding
the zeros of the sum of a maximally monotone operator $A$ and a cocoercive
operator $B$. We study the asymptotic behaviour of the trajectories generated
by a second order equation with vanishing damping, attached to this problem,
and governed by a time-dependent forward-backward-type operator. This is a
splitting system, as it only requires forward evaluations of $B$ and backward
evaluations of $A$. A proper tuning of the system parameters ensures the weak
convergence of the trajectories to the set of zeros of $A+B$, as well as fast
convergence of the velocities towards zero. A particular case of our system
allows to derive fast convergence rates for the problem of minimizing the sum
of a proper, convex and lower semicontinuous function and a smooth and convex
function with Lipschitz continuous gradient. We illustrate the theoretical
outcomes by numerical experiments.

Keywords Asymptotic stabilization $\cdot$ Damped inertial dynamics $\cdot$
Lyapunov analysis $\cdot$ Vanishing viscosity $\cdot$ Splitting system $\cdot$
Monotone inclusions

Mathematics Subject Classification (2020) 37N40 $\cdot$ 46N10 $\cdot$ 65K05
$\cdot$ 65K10 $\cdot$ 90B50 $\cdot$ 90C25

##  1 Introduction

###  1.1 Problem formulation and a continuous time splitting scheme with
vanishing damping

Let $\mathcal{H}$ be a real Hilbert, $A:\mathcal{H}\to 2^{\mathcal{H}}$ a
maximally monotone operator and $B:\mathcal{H}\to\mathcal{H}$ a
$\beta$-cocoercive operator for some $\beta>0$ such that
$\operatorname{zer}(A+B)\neq\emptyset$. Devising fast convergent continuous
and discrete time dynamics for solving monotone inclusions of the type

$\mbox{find}\ x\in\mathcal{H}\ \mbox{such that}\ 0\in(A+B)(x)$ (1)

is of great importance in many fields, including, but not limited to,
optimization, equilibrium theory, economics and game theory, partial
differential equations, and statistics. One of our main motivations comes from
the fact that solving the convex optimization problem

$\min_{x\in\mathcal{H}}f(x)+g(x),$

where $f:\mathcal{H}\to\mathbb{R}\cup\\{+\infty\\}$ is proper, convex and
lower semicontinuous and $g:\mathcal{H}\to\mathbb{R}$ is convex and Fréchet
differentiable with a Lipschitz continuous gradient, is equivalent to solving
the monotone inclusion

$0\in(\partial f+\nabla g)(x).$

We want to exploit the additive structure of $(\ref{monotoneinclusion})$ and
approach $A$ and $B$ separately, in the spirit of the splitting paradigm.

For $t\geq t_{0}>0$, $\alpha>1,\xi\geq 0$, and functions
$\lambda,\gamma:[t_{0},+\infty)\to(0,+\infty)$, we will study the asymptotic
behaviour of the trajectories of the second order differential equation

$\text{(Split-DIN-
AVD)}\quad\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\left(\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))\right)+T_{\lambda(t),\gamma(t)}(x(t))=0,$
(2)

where, for $\lambda,\gamma>0$, the operator
$T_{\lambda,\gamma}:\mathcal{H}\to\mathcal{H}$ is given by

$T_{\lambda,\gamma}=\frac{1}{\lambda}\Big{[}\operatorname{Id}-J_{\gamma
A}\circ(\operatorname{Id}-\gamma B)\Big{]}.$

The sets of zeros of $A+B$ and of $T_{\lambda,\gamma}$, for
$\lambda,\gamma>0$, coincide. The nomenclature (Split-DIN-AVD) comes from the
splitting feature of the continuous time scheme, as well as the link with the
(DIN-AVD) system developed by Attouch and László in [9] (Dynamic Inertial
Newton - Asymptotic Vanishing Damping), which we will emphasize later. We will
discuss the existence and uniqueness of the trajectories generated (Split-DIN-
AVD), and also show their weak convergence to the set of zeros of $A+B$ as
well as the fast convergence of the velocities to zero, and convergence rates
for $T_{\lambda(t),\gamma(t)}(x(t))$ and
$\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))$ as $t\to+\infty$.

For the particular case $B=0$, we are left with the monotone inclusion problem

$\mbox{find}\ x\in\mathcal{H}\ \mbox{such that}\ 0\in A(x),$

and the attached system

$\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\left(\frac{d}{dt}A_{\lambda(t),\gamma(t)}(x(t))\right)+A_{\lambda(t),\gamma(t)}(x(t))=0,$

where, for $\lambda,\gamma>0$, the operator
$A_{\lambda,\gamma}:\mathcal{H}\to\mathcal{H}$ can be seen as a generalized
Moreau envelope of the operator $A$, i.e.,

$A_{\lambda,\gamma}=\frac{1}{\lambda}\Big{[}\operatorname{Id}-J_{\gamma
A}\Big{]}.$

In particular, we will be able to set $\gamma(t)=\lambda(t)$ for every $t\geq
t_{0}$. Since for $\lambda>0$, $A_{\lambda,\lambda}=A_{\lambda}$, this allows
us to recover the (DIN-AVD) system

$\text{(DIN-
AVD)}\quad\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\left(\frac{d}{dt}A_{\lambda(t)}(x(t))\right)+A_{\lambda(t)}(x(t))=0,$

addressed by Attouch and László in [9].

If $A=0$, and after properly redefining some parameters, we obtain the
following system

$\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\left(\frac{d}{dt}\frac{1}{\eta(t)}Bx(t)\right)+\frac{1}{\eta(t)}Bx(t)=0,$

with $\eta:[t_{0},+\infty)\to(0,+\infty)$, which addresses the monotone
equation

$\mbox{find}\ x\in\mathcal{H}\ \mbox{such that}\ B(x)=0.$

This dynamical system approaches the cocoercive operator $B$ directly through
a forward evaluation, which is more natural, instead of having to resort to
its Moreau envelope, as in (DIN-AVD).

###  1.2 Notation and preliminaries

In this subsection, we will explain the notions which were mentioned in the
previous subsection, and we will introduce some definitions and preliminary
results that will be required later. Throughout the paper, we will be working
in a real Hilbert space $\mathcal{H}$ with inner product
$\langle\cdot,\cdot\rangle$ and corresponding norm
$\|\cdot\|=\sqrt{\langle\cdot,\cdot\rangle}$.

Let $A:\mathcal{H}\to 2^{\mathcal{H}}$ be a set-valued operator, that is, $Ax$
is a subset of $\mathcal{H}$ for every $x\in\mathcal{H}$. The operator $A$ is
totally characterized by its graph
$\operatorname{gra}A=\\{(x,u)\in\mathcal{H}\times\mathcal{H}:u\in Ax\\}$. The
inverse of $A$ is the operator $A^{-1}:\mathcal{H}\to 2^{\mathcal{H}}$ well-
defined through the equivalence $x\in A^{-1}u$ if and only if $u\in Ax$. The
set of zeros of $A$ is the set $\operatorname{zer}A=\\{x\in\mathcal{H}:0\in
Ax\\}$. For a subset $C\subseteq\mathcal{H}$, we say that $A(C)=\cup_{x\in
C}Ax$. The range of $A$ is the set $\operatorname{ran}A=A(\mathcal{H})$.

A set-valued operator $A$ is said to be monotone if $\langle
v-u,y-x\rangle\geq 0$ whenever $(x,u),(y,v)\in\operatorname{gra}A$, and
maximally monotone if it is monotone and the following implication holds:

$\widetilde{A}\>\>\text{is
monotone},\>\>\operatorname{gra}A\subseteq\operatorname{gra}\widetilde{A}\>\Longrightarrow\>A=\widetilde{A}.$

Let $\lambda>0$. The resolvent of index $\lambda$ of $A$ is the operator
$J_{\lambda A}:\mathcal{H}\to 2^{\mathcal{H}}$ given by

$J_{\lambda A}=(\operatorname{Id}+\lambda A)^{-1},$

and the Moreau envelope (or Yosida approximation or Yosida regularization) of
index $\lambda$ of $A$ is the operator $A_{\lambda}:\mathcal{H}\to
2^{\mathcal{H}}$ given by

$A_{\lambda}=\frac{1}{\lambda}(\operatorname{Id}-J_{\lambda A}),$

where $\operatorname{Id}:\mathcal{H}\to\mathcal{H}$, defined by
$\operatorname{Id}(x)=x$ for every $x\in\mathcal{H}$, is the identity operator
of $\mathcal{H}$. For $\lambda_{1},\lambda_{2}>0$, it holds
$(A_{\lambda_{1}})_{\lambda_{2}}=A_{\lambda_{1}+\lambda_{2}}$.

A single-valued operator $B:\mathcal{H}\to\mathcal{H}$ is said to be
$\beta$-cocoercive for some $\beta>0$ if for every $x,y\in\mathcal{H}$ we have

$\beta\|Bx-By\|^{2}\leq\langle Bx-By,x-y\rangle.$

In this case, $B$ is $\frac{1}{\beta}$-Lipschitz continuous, namely, for every
$x,y\in\mathcal{H}$ we have

$\|Bx-By\|\leq\frac{1}{\beta}\|x-y\|.$

We say $B$ is nonexpansive if it is $1$-Lipschitz continuous, and firmly
nonexpansive if it is $1$-cocoercive. For $\alpha\in(0,1)$, we say $B$ is
$\alpha$-averaged if there exists a nonexpansive operator
$R:\mathcal{H}\to\mathcal{H}$ such that

$B=(1-\alpha)\operatorname{Id}+\alpha R.$

Let $\lambda>0$ and $A:\mathcal{H}\to 2^{\mathcal{H}}$. According to Minty’s
Theorem, $A$ is maximally monotone if and only if
$\operatorname{ran}(\operatorname{Id}+\lambda A)=\mathcal{H}$. In this case
$J_{\lambda A}$ is single-valued and firmly nonexpansive, $A_{\lambda}$ is
single-valued, $\lambda$-cocoercive, and for every $x\in\mathcal{H}$ and every
$\lambda_{1},\lambda_{2}>0$ we have

$\|J_{\lambda_{1}A}(x)-J_{\lambda_{2}A}(x)\|\leq|\lambda_{1}-\lambda_{2}|\|A_{\lambda_{1}}(x)\|.$

Let $B:\mathcal{H}\to\mathcal{H}$ be a single-valued operator. If $B$ is
$\alpha$-averaged for some $\alpha\in(0,1)$, then $\operatorname{Id}-B$ is
$\frac{1}{2\alpha}$-cocoercive. If $B$ is monotone and continuous, then it is
maximally monotone.

The following concepts and results show the strong interplay between the
theory of monotone operators and the convex analysis.

Let $f:\mathcal{H}\to\mathbb{R}\cup\\{+\infty\\}$ be a proper, convex and
lower semicontinuous function. We denote the infimum of $f$ over $\mathcal{H}$
by $\min_{\mathcal{H}}f$ and the set of global minimizers of $f$ by
$\operatorname{argmin}_{\mathcal{H}}f$. The subdifferential of $f$ is the
operator $\partial f:\mathcal{H}\to 2^{\mathcal{H}}$ defined, for every
$x\in\mathcal{H}$, by

$\partial f(x)=\\{x^{*}\in\mathcal{H}:\langle x^{*},y-x\rangle+f(x)\leq
f(y)\>\>\forall y\in\mathcal{H}\\}.$

The subdifferential operator of $f$ is maximally monotone and
$\overline{x}\in\operatorname{zer}\partial f$ $\Leftrightarrow$ $\overline{x}$
is a global minimizer of $f$.

Let $\lambda>0$. The proximal operator of $f$ of index $\lambda$ is the
operator $\operatorname{prox}_{\lambda f}:\mathcal{H}\to\mathcal{H}$ defined,
for every $x\in\mathcal{H}$, by

$\operatorname{prox}_{\lambda f}(x)=J_{\lambda\partial
f}(x)=\operatorname{argmin}_{y\in\mathcal{H}}\left[f(y)+\frac{1}{2\lambda}\|x-y\|^{2}\right],$

which also means that $\operatorname{prox}_{\lambda f}$ is firmly
nonexpansive. The Moreau envelope of $f$ of index $\lambda$ is the function
$f_{\lambda}:\mathcal{H}\to\mathbb{R}$ given, for every $x\in\mathcal{H}$, by

$f_{\lambda}(x)=f\left(\operatorname{prox}_{\lambda
f}(x)\right)+\frac{1}{2\lambda}\|x-\operatorname{prox}_{\lambda f}(x)\|^{2}.$

The function $f_{\lambda}$ is Fréchet differentiable and

$\nabla f_{\lambda}(x)=\frac{1}{\lambda}\left(x-\operatorname{prox}_{\lambda
f}(x)\right)=(\partial f)_{\lambda}(x)\quad\forall x\in\mathcal{H}.$

Finally, if $f:\mathcal{H}\to\mathbb{R}$ has full domain and is Fréchet
differentiable with $\frac{1}{\beta}$-Lipschitz continuous gradient, for
$\beta>0$, then, according to Baillon-Haddad’s Theorem, $\nabla f$ is
$\beta$-cocoercive.

###  1.3 A brief history of inertial systems attached to optimization problems
and monotone inclusions

In the last years there have been many advances in the study of continuous
time inertial systems with vanishing damping attached to monotone inclusion
problems. We briefly visit them in the following paragraphs.

####  1.3.1 The Heavy Ball Method with friction

Consider a convex and continuously differentiable function
$f:\mathcal{H}\to\mathbb{R}$ with at least one minimizer. The heavy ball with
friction system

$\text{(HBF)}\qquad\ddot{x}(t)+\mu\dot{x}(t)+\nabla f(x(t))=0$ (3)

was introduced by Álvarez in [2] as a suitable continuous time scheme to
approach the minimization of the function $f$. This system can be seen as the
equation of the horizontal position $x(t)$ of an object that moves, under the
force of gravity, along the graph of the function $f$, subject to a kinetic
friction represented by the term $\mu\dot{x}(t)$ (a nice derivation can be
seen in the work done by Attouch-Goudou-Redont in [8]). It is known that, if
$x$ is a solution of (HBF), then $x$ converges weakly to a minimizer of $f$
and $f(x(t))-\min_{\mathcal{H}}f=\mathcal{O}\left(\frac{1}{t}\right)$ as
$t\to+\infty$.

In recent times, the question was raised whether the damping coefficient $\mu$
could be chosen to be time-dependent. An important contribution was made by
Su-Boyd-Candés (in [20]) who studied the case of an Asymptotic Vanishing
Damping coefficient $\mu(t)=\frac{\alpha}{t}$, namely,

$\text{(AVD)}\qquad\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\nabla f(x(t))=0,$
(4)

and proved when $\alpha\geq 3$ the rate of convergence for the functional
values $f(x(t))-\min_{\mathcal{H}}f=O\left(\frac{1}{t^{2}}\right)$ as
$t\to+\infty$. This second order system can be seen as a continuous
counterpart to Nesterov’s accelerated gradient method from [19]. Weak
convergence of the trajectories generated by (AVD) when $\alpha>3$ has been
shown by Attouch-Chbani-Peypouquet-Redont [6] and May [18], with the improved
rate of convergence for the functional values
$f(x(t))-\min_{\mathcal{H}}f=o\left(\frac{1}{t^{2}}\right)$ as $t\to+\infty$.
For $\alpha=3$, the convergence of the trajectories remains an open question,
except for the one dimensional case (see [7]). In the subcritical case
$\alpha\leq 3$, it has been shown by Apidopoulos-Aujol-Dossal [5] and Attouch-
Chbani-Riahi [7] that the objective values converge at a rate
$\mathcal{O}(t^{-\frac{2\alpha}{3}})$ as $t\to+\infty$.

####  1.3.2 Heavy Ball dynamics and cocoercive operators

If $f:\mathcal{H}\to\mathbb{R}\cup\\{+\infty\\}$ is a proper, convex and lower
semicontinuous function which is not necessarily differentiable, then we
cannot make direct use of $(\ref{HBF})$. However, since for $\lambda>0$ we
have $\operatorname{argmin}f=\operatorname{argmin}f_{\lambda}$, we can replace
$f$ by its Moreau envelope $f_{\lambda}$, and the system now becomes

$\ddot{x}(t)+\mu\dot{x}(t)+\nabla f_{\lambda}(x(t))=0.$

In line with this idea, and in analogy with $(\ref{HBF})$, Álvarez and Attouch
[3] and Attouch and Maingé [11] studied the dynamics

$\ddot{x}(t)+\mu\dot{x}(t)+B(x(t))=0,$ (5)

where $B:\mathcal{H}\to\mathcal{H}$ is a $\beta$-cocoercive operator. They
were able to prove that the solutions of this system weakly converge to
elements of $\operatorname{zer}B$ provided that the cocoercitivity parameter
$\beta$ and the damping coefficient $\mu$ satisfy $\beta\mu^{2}>1$. For a
maximally monotone operator $A:\mathcal{H}\to 2^{\mathcal{H}}$, we know that
its Moreau envelope is $\lambda$-cocoercive and thus, under the condition
$\lambda\mu^{2}>1$, the trajectories of

$\ddot{x}(t)+\mu\dot{x}(t)+A_{\lambda}(x(t))=0$

converge weakly to elements of
$\operatorname{zer}A_{\lambda}=\operatorname{zer}A$.

Also related to $(\ref{cocoercive sin vanishing damping})$, Boţ-Csetnek [16]
considered the system

$\ddot{x}(t)+\mu(t)\dot{x}(t)+\nu(t)Bx(t)=0,$ (6)

where $B:\mathcal{H}\to\mathcal{H}$ is again $\beta$-cocoercive. Under the
assumption that $\mu$ and $\nu$ are locally absolutely continuous,
$\dot{\mu}(t)\leq 0\leq\dot{\nu}(t)$ for almost every $t\in[0,+\infty)$ and
$\inf_{t\geq 0}\frac{\mu^{2}(t)}{\nu(t)}>\frac{1}{\beta}$, the authors were
able to prove that the solutions to this system converge weakly to zeros of
$B$.

In [12], Attouch and Peypouquet addressed the system

$\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+A_{\lambda(t)}(x(t))=0,$ (7)

where $\alpha>1$ and the time-dependent regularizing parameter $\lambda(t)$
satisfies $\lambda(t)\frac{\alpha^{2}}{t^{2}}>1$ for every $t\geq t_{0}>0$. As
well as ensuring the weak convergence of the trajectories towards elements of
$\operatorname{zer}A$, choosing the regularizing parameter in such a fashion
allowed the authors to obtain fast convergence of the velocities and
accelerations towards zero.

####  1.3.3 Inertial dynamics with Hessian damping

Let us return briefly to the (AVD) system $(\ref{AVD})$. In addition to the
viscous vanishing damping term $\frac{\alpha}{t}\dot{x}(t)$, the following
system with Hessian-driven damping was considered by Attouch-Peypouquet-Redont
in [13]

$\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\nabla^{2}f(x(t))\dot{x}(t)+\nabla
f(x(t))=0,$

where $\xi\geq 0$. While preserving the fast convergence properties of the
Nesterov accelerated method, the Hessian-driven damping term reduces the
oscillatory aspect of the trajectories. In [9], Attouch and László studied a
version of $(\ref{DINAVD no damping})$ with an added Hessian-driven damping
term:

$\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\left(\frac{d}{dt}A_{\lambda(t)}(x(t))\right)+A_{\lambda(t)}(x(t))=0.$

While preserving the convergence results of $(\ref{DINAVD no damping})$, the
main benefit of the introduction of this damping term is the fast convergence
rates that can be obtained for $A_{\lambda(t)}(x(t))$ and
$\frac{d}{dt}A_{\lambda(t)}(x(t))$ as $t\to+\infty$. The regularizing
parameter $\lambda(t)$ is again chosen to be time-dependent; in the general
case, the authors take $\lambda(t)=\lambda t^{2}$, and in [12] it is shown
that taking $\lambda(t)$ this way is critical. However, in the case where
$A=\partial f$ for a proper, convex and lower semicontinuous function $f$, it
is also allowed to take $\lambda(t)=\lambda t^{r}$ with $r\geq 0$.

###  1.4 Layout of the paper

In Section 2, we give the proof for the existence and uniqueness of strong
global solutions to (Split-DIN-AVD) by means of a Cauchy-Lipschitz-Picard
argument. In Section 3 we state the main theorem of this work, and we show the
weak convergence of the solutions of $(\ref{mainsystem})$ to elements of
$\operatorname{zer}(A+B)$, as well as the fast convergence of the velocities
and accelerations to zero. We also provide convergence rates for
$T_{\lambda(t),\gamma(t)}(x(t))$ and
$\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))$ as $t\to+\infty$. We explore the
particular cases $A=0$ and $B=0$, and show improvements with respect to
previous works. In Section 4, we address the convex minimization case, namely,
when $A=\partial f$ and $B=\nabla g$, where
$f:\mathcal{H}\to\mathbb{R}\cup\\{+\infty\\}$ is a proper, convex and lower
semicontinuous function and $g:\mathcal{H}\to\mathbb{R}$ is a convex and
Fréchet differentiable function with Lipschitz continuous gradient, and
derive, in addition, a fast convergence rate for the function values. In
Section 5, we illustrate the theoretical results by numerical experiments. In
Section 5, we provide an algorithm that arises from a time discretization of
(Split-DIN-AVD) and discuss its convergence properties.

##  2 Existence and uniqueness of trajectories

In this section, we show the existence and uniqueness of strong global
solutions to (Split-DIN-AVD). For the sake of clarity, first we state the
definition of a strong global solution.

######  Definition 2.1.

We say that $x:[t_{0},+\infty)\to\mathcal{H}$ is a strong global solution of
(Split-DIN-AVD) with Cauchy data
$(x_{0},u_{0})\in\mathcal{H}\times\mathcal{H}$ if

  1. (i)

$x,\dot{x}:[t_{0},+\infty)\to\mathcal{H}$ are locally absolutely continuous;

  2. (ii)

$\ddot{x}(t)+\frac{\alpha}{t}\dot{x}+\xi\left(\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))\right)+T_{\lambda(t),\gamma(t)}(x(t))=0$
for almost every $t\in[t_{0},+\infty)$;

  3. (iii)

$x(t_{0})=x_{0}$, $\dot{x}(t_{0})=u_{0}$.

A classic solution is just a strong global solution which is
$\mathcal{C}^{2}$. Sometimes we will mention the terms strong global solution
or classic global solution without explicit mention of the Cauchy data.

The following lemma will be used to prove the existence of strong global
solutions of our system, and we will need it in the proof of the main theorem
as well.

######  Lemma 2.2.

Let $A:\mathcal{H}\to 2^{\mathcal{H}}$ be a maximally monotone operator and
$B:\mathcal{H}\to\mathcal{H}$ a $\beta$-cocoercive operator for some
$\beta>0$. Then, the following statements hold:

  1. (i)

For $\lambda>0$ and $\gamma\in(0,2\beta)$, $T_{\lambda,\gamma}$ is a
$\lambda\frac{4\beta-\gamma}{4\beta}$-cocoercive operator. In particular, this
also implies that $T_{\lambda,\gamma}$ is $\frac{\lambda}{2}$-cocoercive.

  2. (ii)

Choose $\lambda_{1},\lambda_{2}>0$, $\gamma_{1},\gamma_{2}\in(0,2\beta)$ and
$x,y\in\mathcal{H}$. Then, for $\overline{x}\in\operatorname{zer}(A+B)$ it
holds

$\displaystyle\|\lambda_{1}T_{\lambda_{1},\gamma_{1}}(x)-\lambda_{2}T_{\lambda_{2},\gamma_{2}}(y)\|\leq\>$
$\displaystyle
4\|x-y\|+\frac{4\beta|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|B(x)\|+\frac{2|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|x-\overline{x}\|,$
$\displaystyle\left\|T_{\lambda_{1},\gamma_{1}}(x)-T_{\lambda_{2},\gamma_{2}}(y)\right\|\leq\>$
$\displaystyle\frac{1}{\lambda_{1}}\left[4\|x-y\|+4\beta\frac{|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|Bx\|+2\frac{|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|x-\overline{x}\|\right]$
$\displaystyle+2\frac{|\lambda_{2}-\lambda_{1}|}{\lambda_{1}\lambda_{2}}\|y-\overline{x}\|.$

  3. (iii)

If $x$ is a classic global solution to $(\ref{mainsystem})$ and
$\overline{x}\in\operatorname{zer}(A+B)$, then, for every $t\geq t_{0}$, we
have

$\left\|\frac{d}{dt}\left(\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right)\right\|\leq
4\|\dot{x}(t)\|+4\beta\frac{|\dot{\gamma}(t)|}{\gamma(t)}\|B(x(t))\|+2\frac{|\dot{\gamma}(t)|}{\gamma(t)}\|x(t)-\overline{x}\|.$

###### Proof.

(i) From [14, Proposition 26.1(iv)(d)] we know that the operator $J_{\gamma
A}\circ(\operatorname{Id}-\gamma B)$ is
$\alpha=\frac{2\beta}{4\beta-\gamma}$-averaged. From [14, Proposition 4.39],
we obtain that $\operatorname{Id}-J_{\gamma A}\circ(\operatorname{Id}-\gamma
B)$ is $\frac{1}{2\alpha}$-cocoercive, namely, it is
$\frac{4\beta-\gamma}{4\beta}$-cocoercive. Since $\gamma\in(0,2\beta)$, we
have $\frac{4\beta-\gamma}{4\beta}>\frac{2\beta}{4\beta}=\frac{1}{2}$, which
implies that $\operatorname{Id}-J_{\gamma A}\circ(\operatorname{Id}-\gamma B)$
is $\frac{1}{2}$-cocoercive and thus

$T_{\lambda,\gamma}\>\>\>\text{is}\>\>\>\lambda\frac{4\beta-\gamma}{4\beta}\text{-cocoercive}\>\>\>\text{and}\>\>\>T_{\lambda,\gamma}\>\>\>\text{is}\>\>\>\frac{\lambda}{2}\text{-cocoercive}.$

(ii) We have

$\displaystyle\|\lambda_{1}T_{\lambda_{1},\gamma_{1}}(x)-\lambda_{2}T_{\lambda_{2},\gamma_{2}}(y)\|\leq$
$\displaystyle\
\|x-y\|+\|J_{\gamma_{1}A}(x-\gamma_{1}B(x))-J_{\gamma_{2}A}(y-\gamma_{2}B(y))\|$
$\displaystyle\leq$ $\displaystyle\
\|x-y\|+\|J_{\gamma_{1}A}(x-\gamma_{1}B(x))-J_{\gamma_{2}A}(x-\gamma_{1}B(x))\|$
$\displaystyle+\|J_{\gamma_{2}A}(x-\gamma_{1}B(x))-J_{\gamma_{2}A}(y-\gamma_{2}B(y))\|$
$\displaystyle\leq$ $\displaystyle\
2\|x-y\|+|\gamma_{1}-\gamma_{2}|\|A_{\gamma_{1}}(x-\gamma_{1}B(x))\|+\|\gamma_{1}B(x)-\gamma_{2}B(y)\|$
$\displaystyle\leq$ $\displaystyle\
2\|x-y\|+|\gamma_{1}-\gamma_{2}|\|A_{\gamma_{1}}(x-\gamma_{1}B(x))\|$
$\displaystyle+\|\gamma_{1}B(x)-\gamma_{2}B(x)\|+\|\gamma_{2}B(x)-\gamma_{2}B(y)\|$
$\displaystyle=$ $\displaystyle\
2\|x-y\|+|\gamma_{1}-\gamma_{2}|\|A_{\gamma_{1}}(x-\gamma_{1}B(x))\|$
$\displaystyle\ +|\gamma_{1}-\gamma_{2}|\|B(x)\|+\gamma_{2}\|B(x)-B(y)\|.$

Now, notice that

$\displaystyle A_{\gamma_{1}}(x-\gamma_{1}B(x))$
$\displaystyle=\frac{1}{\gamma_{1}}(\operatorname{Id}-J_{\gamma_{1}A})(x-\gamma_{1}B(x))=-B(x)+\frac{1}{\gamma_{1}}(x-J_{\gamma_{1}A}(x-\gamma_{1}B(x)))$
$\displaystyle=-B(x)+T_{\gamma_{1},\gamma_{1}}(x),$

so using (i) and the fact that $T_{\gamma_{1},\gamma_{2}}(\overline{x})=0$, we
obtain

$\begin{split}\|A_{\gamma_{1}}(x-\gamma_{1}B(x))\|&=\|-B(x)+T_{\gamma_{1},\gamma_{2}}(x)\|\leq\|B(x)\|+\|T_{\gamma_{1},\gamma_{2}}(x)-T_{\gamma_{1},\gamma_{2}}(\overline{x})\|\\\
&\leq\|B(x)\|+\frac{2}{\gamma_{1}}\|x-\overline{x}\|.\end{split}$ (8)

Altogether, plugging $(\ref{aux3})$ into our initial inequality yields

$\displaystyle\|\lambda_{1}T_{\lambda_{1},\gamma_{2}}(x)-\lambda_{2}T_{\lambda_{2},\gamma_{2}}(y)\|\leq\>$
$\displaystyle
2\|x-y\|+2|\gamma_{1}-\gamma_{2}|\|B(x)\|+\frac{2|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|x-\overline{x}\|+\gamma_{2}\|B(x)-B(y)\|$
$\displaystyle\leq\>$ $\displaystyle
2\|x-y\|+\frac{4\beta|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|B(x)\|+\frac{2|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|x-\overline{x}\|+2\beta\left(\frac{1}{\beta}\right)\|x-y\|.$

To show the second inequality, we use the previous one. We have

$\displaystyle\left\|T_{\lambda_{1},\gamma_{1}}(x)-T_{\lambda_{2},\gamma_{2}}(y)\right\|=$
$\displaystyle\
\frac{1}{\lambda_{1}}\left\|\lambda_{1}T_{\lambda_{1},\gamma_{1}}(x)-\lambda_{2}T_{\lambda_{2},\gamma_{2}}(y)+(\lambda_{2}-\lambda_{1})T_{\lambda_{2},\gamma_{2}}(y)\right\|$
$\displaystyle\leq$ $\displaystyle\
\frac{1}{\lambda_{1}}\left[4\|x-y\|+4\beta\frac{|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|Bx\|+2\frac{|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|x-\overline{x}\|\right]+\frac{|\lambda_{2}-\lambda_{1}|}{\lambda_{1}}\left\|T_{\lambda_{2},\gamma_{2}}(y)\right\|$
$\displaystyle\leq$ $\displaystyle\
\frac{1}{\lambda_{1}}\left[4\|x-y\|+4\beta\frac{|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|Bx\|+2\frac{|\gamma_{1}-\gamma_{2}|}{\gamma_{1}}\|x-\overline{x}\|\right]+2\frac{|\lambda_{2}-\lambda_{1}|}{\lambda_{1}\lambda_{2}}\left\|y-\overline{x}\right\|,$

where the last line is a consequence of $T_{\lambda_{2},\gamma_{2}}$ being
$\frac{\lambda_{2}}{2}$-cocoercive, and hence
$\frac{2}{\lambda_{2}}$-Lipschitz continuous (see (i)).

(iii) For $t,s\geq t_{0}$ set

$x=x(t),\>\>y=x(s),\>\>\lambda_{1}=\lambda(t),\>\>\gamma_{1}=\gamma(t),\>\>\lambda_{2}=\lambda(s),\>\>\gamma_{2}=\gamma(s)$

and use (ii) to obtain, for every $t\geq t_{0}$,

$\displaystyle\
\frac{\|\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))-\lambda(s)T_{\lambda(s),\gamma(s)}(x(s))\|}{|t-s|}$
$\displaystyle\leq$ $\displaystyle\
4\frac{\|x(t)-x(s)\|}{|t-s|}+\frac{4\beta}{\gamma(t)}\frac{|\gamma(t)-\gamma(s)|}{|t-s|}\|B(x(t))\|+\>\frac{2}{\gamma(t)}\frac{|\gamma(t)-\gamma(s)|}{|t-s|}\|x(t)-\overline{x}\|.$

Hence, by taking the limit as $s\to t$ we get, for any $t\geq t_{0}$,

$\displaystyle\left\|\frac{d}{dt}\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|\leq
4\|\dot{x}(t)\|+4\beta\frac{|\dot{\gamma}(t)|}{\gamma(t)}\|B(x(t))\|+2\frac{|\dot{\gamma}(t)|}{\gamma(t)}\|x(t)-\overline{x}\|.$

∎

The next theorem concerns the existence and uniqueness of strong global
solutions to (Split-DIN-AVD).

######  Theorem 2.3.

Assume that $\lambda,\gamma:[t_{0},+\infty)\to(0,+\infty)$ are Lebesgue
measurable functions and that $\inf_{t\geq t_{0}}\lambda(t)>0$. Then, for any
$(x_{0},u_{0})\in\mathcal{H}\times\mathcal{H}$ there exists a unique strong
global solution $x:[t_{0},+\infty)\to\mathcal{H}$ of the system
$(\ref{mainsystem})$ that satisfies $x(t_{0})=x_{0}$ and
$\dot{x}(t_{0})=u_{0}$.

###### Proof.

We will rely on [17, Proposition 6.2.1] and distinguish between the cases
$\xi>0$ and $\xi=0$. For each chase, we will check that the conditions of the
afforementioned proposition are fulfilled. We will be working in the real
Hilbert space $\mathcal{H}\times\mathcal{H}$ endowed with the norm
$\|(x,y)\|=\|x\|+\|y\|$. Let $\overline{x}\in\operatorname{zer}(A+B)$ be
fixed.

The case $\xi>0$. First, it can be easily checked (see also [4, 9, 13]) that
for all $t\geq t_{0}$ the following dynamical systems are equivalent

  * $*$

$\displaystyle\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\left(\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))\right)+T_{\lambda(t),\gamma(t)}(x(t))=0$.

  * $*$

$\displaystyle\begin{cases}\dot{x}(t)+\xi
T_{\lambda(t),\gamma(t)}(x(t))-\left(\frac{1}{\xi}-\frac{\alpha}{t}\right)x(t)+\frac{1}{\xi}y(t)=0,\\\
\dot{y}(t)-\left(\frac{1}{\xi}-\frac{\alpha}{t}+\frac{\alpha\xi}{t^{2}}\right)x(t)+\frac{1}{\xi}y(t)=0.\end{cases}$

In other words, $(\ref{mainsystem})$ with Cauchy data
$(x_{0},u_{0})=(x(t_{0}),\dot{x}(t_{0}))$ is equivalent to the first order
system

$\begin{cases}\dot{z}(t)=F(t,z(t)),\\\ z(t_{0})=(x_{0},y_{0}),\end{cases}$

where $z(t)=(x(t),y(t))$, $F$ is given, for every $t\geq t_{0}$, by

$F(t,(x,y))=\left[-\xi
T_{\lambda(t),\gamma(t)}(x)+\left(\frac{1}{\xi}-\frac{\alpha}{t}\right)x-\frac{1}{\xi}y,\>\left(\frac{1}{\xi}-\frac{\alpha}{t}+\frac{\alpha\xi}{t^{2}}\right)x-\frac{1}{\xi}y\right]$

and the Cauchy data is $x_{0}=x(t_{0})$, $y_{0}=-\xi\left(u_{0}+\xi
T_{\lambda(t_{0}),\gamma(t_{0})}(x_{0})-\left(\frac{1}{\xi}-\frac{\alpha}{t_{0}}\right)x_{0}\right)$.

(i) Let $t\in[t_{0},+\infty)$ be fixed. We need to verify the Lipschitz
continuity of $F$ on the $z$ variable. Set $z=(x,y)$, $w=(u,v)$. We have

$\displaystyle\|F(t,z)-F(t,w)\|=\>$
$\displaystyle\left\|-\xi\left(T_{\lambda(t),\gamma(t)}(x)-T_{\lambda(t),\gamma(t)}(u)+\left(\frac{1}{\xi}-\frac{\alpha}{t}\right)(x-u)-\frac{1}{\xi}(y-v)\right)\right\|$
$\displaystyle+\left\|\left(\frac{1}{\xi}-\frac{\alpha}{t}+\frac{\alpha\xi}{t^{2}}\right)(x-u)-\frac{1}{\xi}(y-v)\right\|.$

Set $\underline{\lambda}:=\inf_{t\geq t_{0}}\lambda(t)>0$. According to Lemma
2.2(i), the term involving the operator $T_{\lambda(t),\gamma(t)}$ satisfies

$\left\|T_{\lambda(t),\gamma(t)}(x)-T_{\lambda(t),\gamma(t)}(u)\right\|\leq\frac{2}{\lambda(t)}\|x-u\|\leq\frac{2}{\underline{\lambda}}\|x-u\|.$

It follows that, if we take

$K(t):=\max\left\\{\frac{2\xi}{\underline{\lambda}}+\left|\frac{1}{\xi}-\frac{\alpha}{t}\right|+\left|\frac{1}{\xi}-\frac{\alpha}{t}+\frac{\alpha\xi}{t^{2}}\right|,\>\frac{2}{\xi}\right\\}\quad\forall
t\geq t_{0},$

then we have $K\in L_{\text{loc}}^{1}([t_{0},+\infty),\mathbb{R})$ and

$\|F(t,z)-F(t,w)\|\leq K(t)\|z-w\|\quad\forall t\geq t_{0}.$

(ii) Now, we claim that $F$ fulfills a boundedness condition. For
$t\in[t_{0},+\infty)$ and $z=(x,y)\in\mathcal{H}\times\mathcal{H}$ we have

$\displaystyle\|F(t,z)\|=\left\|-\xi
T_{\lambda(t),\gamma(t)}(x)+\left(\frac{1}{\xi}-\frac{\alpha}{t}\right)x-\frac{1}{\xi}y\right\|+\left\|\left(\frac{1}{\xi}-\frac{\alpha}{t}+\frac{\alpha\xi}{t^{2}}\right)x-\frac{1}{\xi}y\right\|.$

By Lemma 2.2(i), we have, for every $t\geq t_{0}$,

$\left\|T_{\lambda(t),\gamma(t)}(x)\right\|=\left\|T_{\lambda(t),\gamma(t)}(x)-T_{\lambda(t),\gamma(t)}(\overline{x})\right\|\leq\frac{2}{\lambda(t)}\|x-\overline{x}\|.$

Hence, if we take

$P(t)=\max\left\\{\frac{2\xi}{\lambda(t)}+\left|\frac{1}{\xi}-\frac{\alpha}{t}\right|+\left|\frac{1}{\xi}-\frac{\alpha}{t}+\frac{\alpha\xi}{t^{2}}\right|,\frac{2\xi}{\lambda(t)},\>\frac{2}{\xi}\right\\}\quad\forall
t\geq t_{0},$

then we have $P\in L_{\text{loc}}^{1}([t_{0},+\infty),\mathbb{R})$ and

$\|F(t,z)\|\leq P(t)(1+\|z\|).$

We have checked that the conditions of [17, Proposition 6.2.1] hold.
Therefore, there exists a unique locally absolutely continuous solution
$t\mapsto x(t)$ of $(\ref{mainsystem})$ that satisfies $x(t_{0})=x_{0}$ and
$\dot{x}(t_{0})=u_{0}$.

The case $\xi=0$. Now, $(\ref{mainsystem})$ is easily seen to be equivalent to

$\begin{cases}\dot{z}(t)=F(t,z(t)),\\\ z(t_{0})=(x_{0},u_{0}),\end{cases},$

where $z(t)=(x(t),y(t))$ and $F$ is given, for every $t\geq t_{0}$, by

$F(t,(x,y))=\left[y,\>-\frac{\alpha}{t}y-T_{\lambda(t),\gamma(t)}(x)\right].$

Showing that $F$ fulfills the required properties is starightforward. ∎

##  3 The convergence properties of the trajectories

In this section, we will study the asymptotic behaviour of the trajectories of
the system

$\text{(Split-DIN-
AVD)}\quad\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\frac{d}{dt}\left(T_{\lambda(t),\gamma(t)}(x(t))\right)+T_{\lambda(t),\gamma(t)}(x(t))=0,$

where

$T_{\lambda,\gamma}(x)=\frac{1}{\lambda}\big{[}\operatorname{Id}-J_{\gamma
A}\circ(\operatorname{Id}-\gamma B)\big{]}.$

We will show weak convergence of the trajectories generated by
$(\ref{mainsystem})$ to elements of $\operatorname{zer}(A+B)$, as well as the
fast convergence of the velocities and accelerations to zero. Additionally, we
will provide convergence rates for $T_{\lambda(t),\gamma(t)}(x(t))$ and
$\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))$ as $t\to+\infty$. To avoid
repetition of the statement “for almost every $t$”, in the following theorem
we will assume we are working with a classic global solution of our system.

######  Theorem 3.1.

Let $A:\mathcal{H}\to 2^{\mathcal{H}}$ be a maximally monotone operator and
$B:\mathcal{H}\to\mathcal{H}$ a $\beta$-cocoercive operator for some $\beta>0$
such that $\operatorname{zer}(A+B)\neq\emptyset$. Assume that $\alpha>1$,
$\xi\geq 0$, $\lambda(t)=\lambda t^{2}$ for $\lambda>\frac{2}{(\alpha-1)^{2}}$
and all $t\geq t_{0}$, and that $\gamma:[t_{0},+\infty)\to(0,2\beta)$ is a
differentiable function that satisfies
$\frac{\dot{\gamma}(t)}{\gamma(t)}=\mathcal{O}\left(\frac{1}{t}\right)$ as
$t\to+\infty$. Then, for a solution $x:[t_{0},+\infty)\to\mathcal{H}$ to
(Split-DIN-AVD), the following statements hold:

  1. (i)

$x$ is bounded.

  2. (ii)

We have the estimates

$\displaystyle\int_{t_{0}}^{+\infty}t\|\dot{x}(t)\|^{2}dt<+\infty,\quad\int_{t_{0}}^{+\infty}t^{3}\|\ddot{x}(t)\|^{2}dt<+\infty,$
$\displaystyle\int_{t_{0}}^{+\infty}\frac{\gamma^{2}(t)}{t}\left\|A_{\gamma(t)}\Big{[}x(t)-\gamma(t)Bx(t)\Big{]}+Bx(t)\right\|^{2}dt<+\infty.$

  3. (iii)

We have the convergence rates

$\displaystyle\|\dot{x}(t)\|=o\left(\frac{1}{t}\right),\|\ddot{x}(t)\|=\mathcal{O}\left(\frac{1}{t^{2}}\right),$
$\displaystyle\left\|A_{\gamma(t)}\Big{[}x(t)-\gamma(t)Bx(t)\Big{]}+Bx(t)\right\|=o\left(\frac{1}{\gamma(t)}\right),$
$\displaystyle\left\|\frac{d}{dt}\left(A_{\gamma(t)}\Big{[}x(t)-\gamma(t)Bx(t)\Big{]}+Bx(t)\right)\right\|=\mathcal{O}\left(\frac{1}{t\gamma(t)}\right)+o\left(\frac{t^{2}\left|\frac{d}{dt}\frac{\gamma(t)}{\lambda(t)}\right|}{\gamma^{2}(t)}\right)$

as $t\to+\infty$.

  4. (iv)

If $0<\inf_{t\geq t_{0}}\gamma(t)\leq\sup_{t\geq t_{0}}\gamma(t)<2\beta$, then
$x(t)$ converges weakly to an element of $\operatorname{zer}(A+B)$ as
$t\rightarrow+\infty$.

###### Proof.

Integral estimates and rates. To develop the analysis, we will fix
$\overline{x}\in\operatorname{zer}(A+B)$ and make of use of the Lyapunov
function $\mathcal{E}:[t_{0},+\infty)\to\mathbb{R}\cup\\{+\infty\\}$ given by

$\mathcal{E}(t):=\frac{1}{2}\left\|\frac{\alpha-1}{2}(x(t)-\overline{x})+t(\dot{x}(t)+\xi\>T_{\lambda(t),\gamma(t)}(x(t)))\right\|^{2}+\frac{(\alpha-1)^{2}}{8}\|x(t)-\overline{x}\|^{2}.$
(9)

Differentiation of $\mathcal{E}$ with respect to time yields, for every $t\geq
t_{0}$,

$\begin{split}\dot{\mathcal{E}}(t)=&\
\left\langle\frac{\alpha-1}{2}(x(t)-\overline{x})+t\left(\dot{x}(t)+\xi\>T_{\lambda(t),\lambda(t)}(x(t))\right)\right.,\\\
&\ \ \
\left.\frac{\alpha+1}{2}\dot{x}(t)+\xi\>T_{\lambda(t),\gamma(t)}(x(t))+t\left(\ddot{x}(t)+\xi\frac{d}{dt}\left(T_{\lambda(t),\gamma(t)}(x(t))\right)\right)\right\rangle\\\
&\ +\frac{(\alpha-1)^{2}}{4}\langle
x(t)-\overline{x},\dot{x}(t)\rangle.\end{split}$

After reduction and employing $(\ref{mainsystem})$, we get, for every $t\geq
t_{0}$,

$\begin{split}\dot{\mathcal{E}}(t)=&\ \frac{(\alpha-1)(\xi-t)}{2}\left\langle
x(t)-\overline{x},T_{\lambda(t),\gamma(t)}(x(t))\right\rangle+\frac{(1-\alpha)t}{2}\|\dot{x}(t)\|^{2}\\\
&\ +\left(-t^{2}+\frac{\xi(3-\alpha)t}{2}\right)\left\langle
T_{\lambda(t),\gamma(t)}(x(t)),\dot{x}(t)\right\rangle+\xi(\xi-t)t\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}.\end{split}$

Now, by Lemma 2.2(i), we know that $T_{\lambda(t),\gamma(t)}$ is
$\frac{\lambda(t)}{2}$-cocoercive for every $t\geq t_{0}$. Using this on the
first summand of the right hand side of the previous inequality yields, for
$t\geq t_{1}=\max\\{\xi,t_{0}\\}$,

$\begin{split}\dot{\mathcal{E}}(t)\leq&\
\frac{(1-\alpha)t}{2}\|\dot{x}(t)\|^{2}+\left(-t^{2}+\frac{\xi(3-\alpha)t}{2}\right)\left\langle
T_{\lambda(t),\gamma(t)}(x(t)),\dot{x}(t)\right\rangle\\\ &\
+\left(\frac{(\alpha-1)(\xi-t)\lambda(t)}{4}+\xi(\xi-t)t\right)\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}.\end{split}$
(10)

Now, since $\lambda>\frac{2}{(\alpha-1)^{2}}$, we can choose $\epsilon>0$ such
that

$0<\epsilon<\alpha-1-\sqrt{\frac{2}{\lambda}}<\alpha-1.$ (11)

From $(\ref{aux5})$ we get, for every $t\geq t_{1}$,

$\begin{split}&\
\dot{\mathcal{E}}(t)\,+\,\frac{\epsilon}{2}t\|\dot{x}(t)\|^{2}+\frac{\epsilon}{4}t\lambda(t)\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}\\\
\leq&\
\left(\frac{1-\alpha}{2}+\frac{\epsilon}{2}\right)t\|\dot{x}(t)\|^{2}+\left(-t^{2}+\frac{\xi(3-\alpha)t}{2}\right)\left\langle
T_{\lambda(t),\gamma(t)}(x(t)),\dot{x}(t)\right\rangle\\\ &\
+\left(\left(\frac{(\alpha-1)(\xi-t)}{2}+\frac{\epsilon}{2}t\right)\frac{\lambda(t)}{2}+\xi(\xi-t)t\right)\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}.\end{split}$
(12)

By $(\ref{aux6})$ and the definition of $\lambda(t)$, we know that
$\frac{1-\alpha}{2}+\frac{\epsilon}{2}<0$, and

$\left(\left(\frac{(\alpha-1)(\xi-t)}{2}+\frac{\epsilon}{2}t\right)\frac{\lambda(t)}{2}+\xi(\xi-t)t\right)=\underbrace{\left(\frac{1-\alpha}{2}+\frac{\epsilon}{2}\right)}_{<0}\frac{\lambda}{2}t^{3}+\mathcal{O}(t^{2}),$

so we can find $t_{2}\geq t_{1}$ such that for every $t\geq t_{2}$ the
previous expression becomes nonpositive. According to Lemma A.2, the right
hand side of $(\ref{aux7})$ is nonpositive whenever

$R(t):=\left(-t^{2}+\frac{\xi(3-\alpha)t}{2}\right)^{2}-4\left(\frac{1-\alpha}{2}+\frac{\epsilon}{2}\right)t\left(\left(\frac{(\alpha-1)(\xi-t)}{2}+\frac{\epsilon}{2}t\right)\frac{\lambda(t)}{2}+\xi(\xi-t)t\right)\leq
0.$

This quantity can be rewritten as

$R(t)=\left(1+4\left(\frac{1-\alpha}{2}+\frac{\epsilon}{2}\right)\left(\frac{\alpha-1}{2}-\frac{\epsilon}{2}\right)\frac{\lambda}{2}\right)t^{4}+\mathcal{O}(t^{3})\quad\text{as}\quad
t\to+\infty.$

Since $\epsilon<\alpha-1-\sqrt{\frac{2}{\lambda}}$, we have
$\frac{\lambda}{2}>\frac{1}{(\alpha-1-\epsilon)^{2}}$. Hence,

$1+4\left(\frac{1-\alpha}{2}+\frac{\epsilon}{2}\right)\left(\frac{\alpha-1}{2}-\frac{\epsilon}{2}\right)\frac{\lambda}{2}=1-(\alpha-1-\epsilon)^{2}\frac{\lambda}{2}<0.$

This means we can find $t_{3}\geq t_{2}$ such that for every $t\geq t_{3}$ we
have $R(t)\leq 0$, that is, for every $t\geq t_{3}$ we have

$\dot{\mathcal{E}}(t)+\frac{\epsilon}{2}t\|\dot{x}(t)\|^{2}+\frac{\epsilon}{4}t\lambda(t)\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}\leq
0.$ (13)

Now, integrating $(\ref{aux8})$ from $t_{3}$ to $t$ we obtain

$\mathcal{E}(t)+\frac{\epsilon}{2}\int_{t_{3}}^{t}s\|\dot{x}(s)\|^{2}ds+\frac{\epsilon}{4}\lambda\int_{t_{3}}^{t}s^{3}\left\|T_{\lambda(s),\gamma(s)}(x(s))\right\|^{2}ds\leq\mathcal{E}(t_{3}).$
(14)

From $(\ref{aux8})$ and the form of $\mathcal{E}$ we immediately obtain

$\displaystyle t\mapsto\|x(t)-\overline{x}\|\>\>\text{is bounded},$ (15)
$\displaystyle\int_{t_{0}}^{+\infty}t\|\dot{x}(t)\|^{2}dt<+\infty,$ (16)
$\displaystyle\int_{t_{0}}^{+\infty}t^{3}\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}dt<+\infty,$
(17) $\displaystyle\sup_{t\geq
t_{0}}\left\|\left(\frac{\alpha-1}{2}\right)(x(t)-\overline{x})+t\left(\dot{x}(t)+\xi\>T_{\lambda(t),\gamma(t)}(x(t))\right)\right\|<+\infty.$
(18)

From Lemma 2.2(i), we know that for every $t\geq t_{0}$ the operator
$T_{\lambda(t),\gamma(t)}$ is $\frac{2}{\lambda(t)}$-Lipschitz continuous,
which gives, for every $t\geq t_{0}$,

$\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|=\left\|T_{\lambda(t),\gamma(t)}(x(t))-T_{\lambda(t),\gamma(t)}(\overline{x})\right\|\leq\frac{2}{\lambda(t)}\|x(t)-\overline{x}\|.$

Thus, from $(\ref{aux10})$ and recalling that $\lambda(t)=\lambda t^{2}$ we
arrive at

$\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|=\mathcal{O}\left(\frac{1}{t^{2}}\right)\quad\text{as}\quad
t\to+\infty.$ (19)

By combining $(\ref{aux10})$, $(\ref{aux13})$ and $(\ref{aux14})$ we obtain
$\sup_{t\geq t_{0}}t\|\dot{x}(t)\|<+\infty$ and therefore

$\|\dot{x}(t)\|=\mathcal{O}\left(\frac{1}{t}\right)\quad\text{as}\quad
t\to+\infty.$ (20)

From Lemma 2.2, $(\ref{aux10})$, $(\ref{aux15})$ and the fact that $B$ is
$\frac{1}{\beta}$-Lipschitz continuous we deduce that, as $t\to+\infty$,

$\left\|\frac{d}{dt}\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|\leq
4\|\dot{x}(t)\|+4\beta\frac{|\dot{\gamma}(t)|}{\gamma(t)}\|B(x(t))\|+2\frac{|\dot{\gamma}(t)|}{\gamma(t)}\|x(t)-\overline{x}\|=\mathcal{O}\left(\frac{1}{t}\right).$
(21)

On the other hand, for every $t\geq t_{0}$ we have

$\left\|\frac{d}{dt}\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|=\left\|\dot{\lambda}(t)T_{\lambda(t),\gamma(t)}(x(t))+\lambda(t)\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))\right\|,$
(22)

so by combining $(\ref{aux14})$, $(\ref{aux16})$, $(\ref{aux17})$ and the fact
that $\dot{\lambda}(t)=2\lambda t$ we arrive at

$\left\|\lambda(t)\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))\right\|\leq\underbrace{\left\|\frac{d}{dt}\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|}_{\mathcal{O}\left(\frac{1}{t}\right)}+\dot{\lambda}(t)\underbrace{\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|}_{\mathcal{O}\left(\frac{1}{t^{2}}\right)}=\mathcal{O}\left(\frac{1}{t}\right)\quad\text{as}\quad
t\to+\infty,$

which yields

$\left\|\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))\right\|=\frac{1}{\lambda(t)}\mathcal{O}\left(\frac{1}{t}\right)=\mathcal{O}\left(\frac{1}{t^{3}}\right)\quad\text{as}\quad
t\to+\infty.$ (23)

Let us now improve $(\ref{aux14})$ and show that

$\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|=o\left(\frac{1}{t^{2}}\right)\quad\text{as}\quad
t\to+\infty.$ (24)

According to $(\ref{aux14})$ and $(\ref{aux16})$ there exists a constant $K>0$
such that for every $t\geq t_{0}$ it holds

$\displaystyle\left|\frac{d}{dt}\left\|\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|^{4}\right|$
$\displaystyle=\left|4\left\|\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}\left\langle\lambda(t)T_{\lambda(t),\gamma(t)}(x(t)),\frac{d}{dt}\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\rangle\right|$
$\displaystyle\leq
4\left\|\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}\left\|\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|\left\|\frac{d}{dt}\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|$
$\displaystyle\leq\frac{4K}{t}\left\|\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}.$

By $(\ref{aux12})$, the right hand side belongs to
$L^{1}([t_{0},+\infty),\mathbb{R})$, so we get

$\frac{d}{dt}\left\|\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|^{4}\in
L^{1}([t_{0},+\infty),\mathbb{R}),$

hence the limit

$\lim_{t\to+\infty}\left\|\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|^{4}$

exists. Obviously, this implies the existence of
$L:=\lim_{t\to+\infty}\left\|\lambda(t)T_{\lambda(t),\gamma(t))}(x(t))\right\|^{2}$.
By using $(\ref{aux12})$ again we come to

$\int_{t_{0}}^{+\infty}\frac{1}{t}\left\|\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}dt=\lambda^{2}\int_{t_{0}}^{+\infty}t^{3}\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}dt<+\infty,$

and so we must have $L=0$, which gives

$\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|=o\left(\frac{1}{t^{2}}\right)\quad\text{as}\quad
t\to+\infty.$ (25)

By combining $(\ref{mainsystem})$, $(\ref{aux14})$, $(\ref{aux15})$ and
$(\ref{aux18})$ we obtain, as $t\to+\infty$,

$\displaystyle\|\ddot{x}(t)\|$
$\displaystyle=\left\|-\frac{\alpha}{t}\dot{x}(t)-\xi\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))-T_{\lambda(t),\gamma(t)}(x(t))\right\|$
$\displaystyle\leq\frac{\alpha}{t}\underbrace{\|\dot{x}(t)\|}_{\mathcal{O}\left(\frac{1}{t}\right)}+\xi\underbrace{\left\|\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))\right\|}_{\mathcal{O}\left(\frac{1}{t^{3}}\right)}+\underbrace{\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|}_{\mathcal{O}\left(\frac{1}{t^{2}}\right)}=\mathcal{O}\left(\frac{1}{t^{2}}\right).$

Moreover, by using the well-known inequality $\|a+b+c\|^{2}\leq
3\|a\|^{2}+3\|b\|^{2}+3\|c\|^{2}$ for every $a,b,c\in\mathcal{H}$, for every
$t\geq t_{0}$ it holds

$\displaystyle t^{3}\|\ddot{x}(t)\|^{2}$ $\displaystyle\leq
t^{3}\left\|-\frac{\alpha}{t}\dot{x}(t)-\xi\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))-T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}$
$\displaystyle\leq 3\alpha
t\|\dot{x}(t)\|^{2}+3\xi^{2}t^{3}\left\|\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}+3t^{3}\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|^{2}.$

From $(\ref{aux11})$, $(\ref{aux18})$ and $(\ref{aux12})$ it follows

$\int_{t_{0}}^{+\infty}t^{3}\|\ddot{x}(t)\|^{2}dt<+\infty.$ (26)

To see that $\|\dot{x}(t)\|=o\left(\frac{1}{t}\right)$ as $t\to+\infty$, we
write, for every $t\geq t_{0}$,

$\displaystyle\frac{d}{dt}\left(t^{2}\|\dot{x}(t)\|^{2}\right)=2t\|\dot{x}(t)\|^{2}+2t^{2}\langle\dot{x}(t),\ddot{x}(t)\rangle\leq
3t\|\dot{x}(t)\|^{2}+t^{3}\|\ddot{x}(t)\|^{2}.$

From $(\ref{aux11})$ and $(\ref{aux21})$ we deduce that the left hand side
belongs to $L^{1}([t_{0},+\infty),\mathbb{R})$, from which we infer that the
limit $\lim_{t\to+\infty}t^{2}\|\dot{x}(t)\|^{2}$ exists. Using
$(\ref{aux11})$ again, we get

$\int_{t_{0}}^{+\infty}\frac{1}{t}\left(t^{2}\|\dot{x}(t)\|^{2}\right)dt=\int_{t_{0}}^{+\infty}t\|\dot{x}(t)\|^{2}dt<+\infty,$

from which we finally deduce $\lim_{t\to+\infty}t^{2}\|\dot{x}(t)\|^{2}=0$,
therefore

$\|\dot{x}(t)\|=o\left(\frac{1}{t}\right)\quad\text{as}\quad t\to+\infty.$
(27)

Notice that we can write for every $t\geq t_{0}$

$T_{\lambda(t),\gamma(t)}=\frac{1}{\lambda(t)}\Big{[}\operatorname{Id}-J_{\gamma(t)A}(\operatorname{Id}-\gamma(t)B)\Big{]}=\frac{\gamma(t)}{\lambda(t)}\left(A_{\gamma(t)}\Big{[}x(t)-\gamma(t)Bx(t)\Big{]}+Bx(t)\right).$

Hence, multiplying both sides of $(\ref{aux20})$ by
$\frac{\lambda(t)}{\gamma(t)}$ and remembering the definition of $\lambda(t)$
we obtain

$\left\|A_{\gamma(t)}\Big{[}x(t)-\gamma(t)Bx(t)\Big{]}+Bx(t)\right\|=o\left(\frac{1}{\gamma(t)}\right)\quad\text{as}\quad
t\to+\infty.$ (28)

For every $t\geq t_{0}$, we have

$\displaystyle\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))=$
$\displaystyle\>\frac{d}{dt}\left(\frac{\gamma(t)}{\lambda(t)}\right)\left(A_{\gamma(t)}\Big{[}x(t)-\gamma(t)Bx(t)\Big{]}+Bx(t)\right)$
$\displaystyle+\frac{\gamma(t)}{\lambda(t)}\frac{d}{dt}\left(A_{\gamma(t)}\Big{[}x(t)-\gamma(t)Bx(t)\Big{]}+Bx(t)\right).$

Therefore, by using $(\ref{aux18})$ and $(\ref{aux65})$, and recalling that
$\lambda(t)=\lambda t^{2}$, we obtain

$\left\|\frac{d}{dt}\left(A_{\gamma(t)}\Big{[}x(t)-\gamma(t)Bx(t)\Big{]}+Bx(t)\right)\right\|=\mathcal{O}\left(\frac{1}{t\gamma(t)}\right)+o\left(\frac{t^{2}\left|\frac{d}{dt}\frac{\gamma(t)}{\lambda(t)}\right|}{\gamma^{2}(t)}\right)\quad\text{as}\quad
t\to+\infty.$

The fact that $\|\ddot{x}(t)\|=\mathcal{O}\left(\frac{1}{t^{2}}\right)$ as
$t\to+\infty$ comes from $(\ref{mainsystem})$, $(\ref{fast decay of x'})$,
$(\ref{aux18})$ and $(\ref{fast decay of T})$.

Weak convergence of the trajectories. Let
$\overline{x}\in\operatorname{zer}(A+B)$. We will work with the energy
function $h:[t_{0},+\infty)\to\mathbb{R}$ given by

$h(t):=\frac{1}{2}\|x(t)-\overline{x}\|^{2}.$

For every $t\geq t_{0}$, we have

$\dot{h}(t)=\langle
x(t)-\overline{x},\dot{x}(t)\rangle,\quad\ddot{h}(t)=\langle
x(t)-\overline{x},\ddot{x}(t)\rangle+\|\dot{x}(t)\|^{2}.$ (29)

Combining $(\ref{mainsystem})$ and $(\ref{aux22})$ gives us, for every $t\geq
t_{0}$,

$\ddot{h}(t)+\frac{\alpha}{t}\dot{h}(t)+\left\langle
T_{\lambda(t),\gamma(t)}(x(t)),x(t)-\overline{x}\right\rangle=\|\dot{x}(t)\|^{2}+\left\langle-\xi\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t)),x(t)-\overline{x}\right\rangle.$

By using the $\frac{\lambda(t)}{2}$-cocoercitivity of
$T_{\lambda(t),\gamma(t)}$ on the left hand side, Cauchy-Schwarz on the right
hand side and multiplying both sides by $t$, the previous inequality entails,
for every $t\geq t_{0}$,

$t\ddot{h}(t)+\alpha\dot{h}(t)+t\frac{\lambda(t)}{2}\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|\leq
t\|\dot{x}(t)\|^{2}+\xi
t\left\|\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))\right\|\|x(t)-\overline{x}\|\quad\forall
t\geq t_{0}.$

Now, putting $(\ref{aux10})$, $(\ref{aux11})$ and $(\ref{aux18})$ together
results in

$k(t):=t\|\dot{x}(t)\|^{2}+\xi
t\left\|\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))\right\|\|x(t)-\overline{x}\|\in
L^{1}([t_{0},+\infty),\mathbb{R}).$

Now apply Lemma A.1 with
$\theta(t):=t\frac{\lambda(t)}{2}\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|$
for every $t\geq t_{0}$ to deduce that the limit

$\lim_{t\to+\infty}h(t)$

exists, which fulfills the first condition of Opial’s Lemma A.3.

Let us now move on to the second condition. Suppose $\widehat{x}$ is a weak
sequential cluster point of $t\mapsto x(t)$, that is, there exists a sequence
$(t_{n})_{n\in\mathbb{N}}\subseteq[t_{0},+\infty)$ such that $t_{n}\to+\infty$
and $x_{n}:=x(t_{n})$ converges weakly to $\widehat{x}$ as $n\to+\infty$.
Define

$U_{\gamma}:=\operatorname{Id}-J_{\gamma A}\circ(\operatorname{Id}-\gamma B).$

According to $(\ref{aux20})$, we have
$U_{\gamma(t)}(x(t))=\lambda(t)T_{\lambda(t),\gamma(t)}(x(t))\to 0$ as
$t\to+\infty$. Now, since $\gamma(t)\in[\delta,2\beta-\delta]$ for all $t\geq
t_{0}$ for some $\delta>0$, we can extract a subsequence
$(\gamma(t_{n_{k}}))_{k\in\mathbb{N}}$ such that
$\gamma(t_{n_{k}})\to\overline{\gamma}\in(0,2\beta)$ as $k\to+\infty$. We may
assume without loss of generality then that
$\gamma_{n}:=\gamma(t_{n})\to\overline{\gamma}$ as $n\to+\infty$. We now have
for every $n\in\mathbb{N}$

$\displaystyle\|U_{\gamma_{n}}(x_{n})-U_{\overline{\gamma}}(x_{n})\|=$
$\displaystyle\
\|J_{\gamma_{n}A}(x_{n}-\gamma_{n}B(x_{n}))-J_{\overline{\gamma}A}(x_{n}-\overline{\gamma}B(x_{n}))\|$
$\displaystyle=$ $\displaystyle\
\|J_{\gamma_{n}A}(x_{n}-\gamma_{n}B(x_{n}))-J_{\gamma_{n}A}(x_{n}-\overline{\gamma}B(x_{n}))\|$
$\displaystyle+\|J_{\gamma_{n}A}(x_{n}-\overline{\gamma}B(x_{n}))-J_{\overline{\gamma}A}(x_{n}-\overline{\gamma}B(x_{n}))\|$
$\displaystyle\leq$ $\displaystyle\
|\overline{\gamma}-\gamma_{n}|\|B(x_{n})\|+|\overline{\gamma}-\gamma_{n}|\|A_{\overline{\gamma}}(x_{n}-\overline{\gamma}B(x_{n}))\|.$

Now, since every weakly convergent sequence is bounded and the operators $B$
and $A_{\overline{\gamma}}$ are Lipschitz-continuous we deduce that the right-
hand side of the previous inequality approaches zero as $n\to+\infty$,
therefore getting

$U_{\overline{\gamma}}(x_{n})=U_{\gamma_{n}}(x_{n})+\big{(}U_{\overline{\gamma}}(x_{n})-U_{\gamma_{n}}(x_{n})\big{)}\to
0$

as $n\to+\infty$. Now, from the proof of part (i) of Lemma 2.2, we know that
$U_{\overline{\gamma}}$ is
$\frac{4\beta-\overline{\gamma}}{4\beta}$-cocoercive, thus monotone and
Lipschitz continuous and therefore maximally monotone. Summarizing, we have

  1. 1.

$U_{\overline{\gamma}}$ is maximally monotone and thus its graph is closed in
the weak$\times$strong topology of $\mathcal{H}\times\mathcal{H}$ (see [14,
Proposition 20.38(ii)]),

  2. 2.

$x_{n}$ converges weakly to $\widehat{x}$ and $U_{\overline{\gamma}}(x_{n})\to
0$ as $n\to+\infty$,

which allows us to conclude that $U_{\overline{\gamma}}(\widehat{x})=0$, and
gives finally $\widehat{x}\in\operatorname{zer}(A+B)$. Now we just invoke
Opial’s Lemma to achieve that $x(t)$ converges weakly to $\overline{x}$ as
$t\to+\infty$ for some $\overline{x}\in\operatorname{zer}(A+B)$. ∎

In the following subsections, we explore the particular cases $B=0$ and $A=0$,
and we will show improvements with respect to previous results from the
literature addressing continuous time approaches to monotone inclusions.

###  3.1 The case $B=0$

If we let $B=0$ in the (Split-DIN-AVD) system $(\ref{mainsystem})$, then,
attached to the monotone inclusion problem

$\mbox{find}\ x\in\mathcal{H}\ \mbox{such that}\ 0\in A(x),$

we obtain the dynamics

$\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\frac{d}{dt}\left(A_{\lambda(t),\gamma(t)}(x(t)\right)+A_{\lambda(t),\gamma(t)}(x(t))=0,$
(30)

where

$A_{\lambda,\gamma}(x)=\frac{1}{\lambda}(\operatorname{Id}-J_{\gamma A}).$

We can state the following theorem.

######  Theorem 3.2.

Let $A:\mathcal{H}\to 2^{\mathcal{H}}$ be a maximally monotone operator such
that $\operatorname{zer}A\neq\emptyset$. Assume that $\alpha>1$, $\xi\geq 0$,
$\lambda(t)=\lambda t^{2}$ for $\lambda>\frac{1}{(\alpha-1)^{2}}$ and all
$t\geq t_{0}$, and that $\gamma:[t_{0},+\infty)\to(0,+\infty)$ is a
differentiable function that satisfies
$\frac{|\dot{\gamma}(t)|}{\gamma(t)}=\mathcal{O}\left(\frac{1}{t}\right)$ as
$t\to+\infty$. Then, for a solution $x:[t_{0},+\infty)\to\mathcal{H}$ to
$(\ref{system3})$, the following statements hold:

  1. (i)

$x$ is bounded.

  2. (ii)

We have the estimates

$\int_{t_{0}}^{+\infty}t\|\dot{x}(t)\|^{2}dt<+\infty,\quad\int_{t_{0}}^{+\infty}t^{3}\|\ddot{x}(t)\|^{2}dt<+\infty,\quad\int_{t_{0}}^{+\infty}\frac{\gamma^{2}(t)}{t}\left\|A_{\gamma(t)}(x(t))\right\|^{2}dt<+\infty.$

  3. (iii)

We have the convergence rates

$\displaystyle\|\dot{x}(t)\|=o\left(\frac{1}{t}\right),\
\|\ddot{x}(t)\|=\mathcal{O}\left(\frac{1}{t^{2}}\right),$
$\displaystyle\left\|A_{\gamma(t)}(x(t))\right\|=o\left(\frac{1}{\gamma(t)}\right),\
\left\|\frac{d}{dt}A_{\gamma(t)}(x(t))\right\|=\mathcal{O}\left(\frac{1}{t\gamma(t)}\right)+o\left(\frac{t^{2}\left|\frac{d}{dt}\frac{\gamma(t)}{\lambda(t)}\right|}{\gamma^{2}(t)}\right)$

as $t\to+\infty$.

  4. (iv)

If $0<\inf_{t\geq t_{0}}\gamma(t)$, then $x(t)$ converges weakly to an element
of $\operatorname{zer}A$ as $t\rightarrow+\infty$.

###### Proof.

The proof proceeds in the exact same way as the proof of Theorem 3.1. However,
a few comments are in order: first of all, now we have
$T_{\lambda,\gamma}=\frac{1}{\lambda}(\operatorname{Id}-J_{\gamma
A})=A_{\lambda,\gamma}$. Since $J_{\lambda A}$ is firmly nonexpansive, by [14,
Proposition 4.4] so is $\operatorname{Id}-J_{\lambda A}$. In other words,
$\operatorname{Id}-J_{\gamma A}$ is $1$-cocoercive, therefore
$A_{\lambda,\gamma}=\frac{1}{\lambda}(\operatorname{Id}-J_{\gamma A})$ is
$\lambda$-cocoercive, so now the condition on $\lambda$ becomes
$\lambda>\frac{1}{(\alpha-1)^{2}}$.

The proof also changes when we verify the second part of the Opial’s Lemma, to
get weak convergence of the trajectories $t\mapsto x(t)$. This is in order to
allow for $\gamma(t)$ not to be necessarily bounded. We do need, however, the
assumption $0<\inf_{t\geq t_{0}}\gamma(t)$. Indeed, from
$\|A_{\lambda(t),\gamma(t)}(x(t))\|=o\left(\frac{1}{t^{2}}\right)$ as
$t\to+\infty$, we obtain

$y(t):=x(t)-J_{\gamma(t)A}x(t)=\lambda(t)A_{\lambda(t),\gamma(t)}(x(t))\to 0$

as $t\to+\infty$. Using the definition of the resolvent, we come to

$J_{\gamma(t)A}x(t)=x(t)-y(t)\Leftrightarrow
y(t)\in\gamma(t)A(x(t)-y(t))\Leftrightarrow\frac{1}{\gamma(t)}y(t)\in
A(x(t)-y(t)).$

for all $t\geq t_{0}$. If $(t_{n})_{n\in\mathbb{N}}\subseteq[t_{0},+\infty)$
is such that $t_{n}\to+\infty$ and $x(t_{n})$ converges weakly to
$\widehat{x}$ as $n\to+\infty$, then the previous inclusion, together with the
assumption on $\gamma$ gives

$x(t_{n})-y(t_{n})\ \mbox{converges weakly to}\
\widehat{x}\quad\text{and}\quad\frac{1}{\gamma(t)}y(t)\to 0\quad\text{as}\quad
n\to+\infty,$

and by the closedness of the graph of $A$ in the weak$\times$strong topology
of $\mathcal{H}\times\mathcal{H}$, we deduce that
$\widehat{x}\in\operatorname{zer}A$. ∎

######  Remark 3.3.

The hypotheses required for $\gamma$ are fulfilled at least by two families of
functions. First, take $r\geq 0$ and set $\gamma(t)=e^{t^{-r}}$. Then, we have

$\frac{\dot{\gamma}(t)}{\gamma(t)}=\frac{-rt^{-(r+1)}e^{t^{-r}}}{e^{t^{-r}}}=-\frac{r}{t^{r+1}}=\mathcal{O}\left(\frac{1}{t}\right)\quad\text{as}\quad
t\to+\infty,$

and

$\gamma(t)=e^{t^{-r}}\geq e^{0}=1\quad\forall t\geq 0.$

If $\gamma$ is a polynomial of degree $n$ for some $n\in\mathbb{N}$, the
conditions are also fulfilled. Assume
$\gamma(t)=a_{n}t^{n}+a_{n-1}t^{n-1}+\cdots+a_{0}$ for all $t\geq t_{0}$, for
some $a_{i}\in\mathbb{R}$ for $i\in\\{0,\ldots,n\\}$ and $a_{n}>0$. Then, we
have

$\displaystyle t\cdot\frac{\dot{\gamma}(t)}{\gamma(t)}$
$\displaystyle=t\cdot\frac{na_{n}t^{n-1}+(n-1)a_{n-1}t^{n-1}+\cdots+a_{1}}{a_{n}t^{n}+a_{n-1}t^{n-1}+\cdots+a_{0}}$
$\displaystyle\to\frac{na_{n}}{a_{n}}=n\quad\text{as}\quad t\to+\infty,$

so $\frac{\dot{\gamma}(t)}{\gamma(t)}=\mathcal{O}\left(\frac{1}{t}\right)$ as
$t\to+\infty$. Since we also have $\gamma(t)\to+\infty$ as $t\to+\infty$, the
condition $\inf_{t\geq t_{0}}\gamma(t)>0$ is fulfilled for large enough
$t_{0}$.

In particular, we can choose $\gamma(t)=\lambda(t)=\lambda t^{2}$, which
fulfills $\gamma(t)\geq\lambda t_{0}^{2}>0$ for any $t\geq t_{0}$ and any
$t_{0}$. Since $A_{\lambda,\lambda}=A_{\lambda}$ for $\lambda>0$, this choice
of $\gamma$ allows us to recover the (DIN-AVD) system studied by Attouch and
László in [9]. Notice the way the convergence rates for $A_{\gamma(t)}(x(t))$
and $\frac{d}{dt}A_{\gamma(t)}(x(t))$ exhibited in part (iii) of Theorem 3.2
depend on $\gamma(t)$. If we set $\gamma(t)=t^{n}$ for every $t\geq t_{0}$ for
any natural number $n>2$, (Split-DIN-AVD) performs from this point of view
better than (DIN-AVD) without increasing the complexity of the governing
operator.

###  3.2 The case $A=0$

Let us return to (Split-DIN-AVD) dynamics $(\ref{mainsystem})$. Set $A=0$, and
for every $t\geq t_{0}$ take $\gamma(t)=\gamma\in(0,2\beta)$ and $\eta(t)=\eta
t^{2}$ with $\eta=\lambda/\gamma$. Then, associated to the problem

$\mbox{find}\ x\in\mathcal{H}\ \mbox{such that}\ B(x)=0,$

we obtain the system

$\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\frac{d}{dt}\left(\frac{1}{\eta(t)}Bx(t)\right)+\frac{1}{\eta(t)}Bx(t)=0.$
(31)

The conditions $\lambda>\frac{2}{(\alpha-1)^{2}}$ and $\gamma\in(0,2\beta)$
imply

$\eta=\frac{\lambda}{\gamma}>\frac{2}{\gamma(\alpha-1)^{2}}>\frac{2}{2\beta(\alpha-1)^{2}}=\frac{1}{\beta(\alpha-1)^{2}}.$

With the previous observation, we are able to state the following theorem.

######  Theorem 3.4.

Let $B:\mathcal{H}\to\mathcal{H}$ be a $\beta$-cocoercive operator for some
$\beta>0$ such that $\operatorname{zer}B\neq\emptyset$. Assume that
$\alpha>1$, $\xi\geq 0$ and $\eta(t)=\eta t^{2}$ for
$\eta>\frac{1}{\beta(\alpha-1)^{2}}$ and all $t\geq t_{0}$. Take
$x:[t_{0},+\infty)\to\mathcal{H}$ a solution to $(\ref{system2})$. Then, the
following hold:

  1. (i)

$x$ is bounded, and $x(t)$ converges weakly to an element of
$\operatorname{zer}B$ as $t\rightarrow+\infty$.

  2. (ii)

We have the estimates

$\int_{t_{0}}^{+\infty}t\|\dot{x}(t)\|^{2}dt<+\infty,\quad\int_{t_{0}}^{+\infty}t^{3}\|\ddot{x}(t)\|^{2}dt<+\infty,\quad\int_{t_{0}}^{+\infty}\frac{1}{t}\left\|Bx(t)\right\|^{2}dt<\infty.$

  3. (iii)

We have the convergence rates

$\|\dot{x}(t)\|=o\left(\frac{1}{t}\right),\quad\|\ddot{x}(t)\|=\mathcal{O}\left(\frac{1}{t^{2}}\right)$

as well as the limit

$\|Bx(t)\|\to 0$

as $t\to+\infty$.

###### Proof.

Since $\eta>\frac{1}{\beta(\alpha-1)^{2}}$, we can find $\epsilon\in(0,\beta)$
such that $\eta>\frac{1}{(\beta-\epsilon)(\alpha-1)^{2}}$, equivalently,
$2(\beta-\epsilon)\eta>\frac{2}{(\alpha-1)^{2}}$. Since $(\ref{system2})$ is
equivalent to (Split-DIN-AVD) with $A=0$ and parameters
$\lambda=2(\beta-\epsilon)\eta>\frac{1}{(\alpha-1)^{2}}$ and $\gamma(t)\equiv
2(\beta-\epsilon)\in(0,2\beta)$, the conclusion follows from Theorem 3.1. ∎

######  Remark 3.5.

(a) As we mentioned in the introduction, the dynamical system (31) provides a
way of finding the zeros of a cocoercive operator directly through forward
evaluations, instead of having to resort to its Moreau envelope when following
the approach in [9].

(b) The dynamics $(\ref{system2})$ bear some resemblance to the system
$(\ref{systemBotCsetnek})$ (see also [16]) with $\mu(t)=\frac{\alpha}{t}$ and
$\nu(t)=\frac{1}{\eta(t)}$, with an additional Hessian-driven damping term. In
our case, since $\eta>\frac{1}{\beta(\alpha-1)^{2}}$, the parameters satisfy

$\dot{\mu}(t)=-\frac{\alpha}{t^{2}}\leq
0,\quad\frac{\mu^{2}(t)}{\nu(t)}=\frac{\alpha^{2}\eta
t^{2}}{t^{2}}=\alpha^{2}\eta>\frac{1}{\beta}\quad\forall t\geq t_{0}.$

However, we have

$\dot{\nu}(t)=-\frac{2}{\lambda t^{3}}\leq 0\quad\forall t\geq t_{0},$

so one of the hypotheses which is needed in $(\ref{systemBotCsetnek})$ is not
fulfilled, which shows that one cannot address the dynamical system
$(\ref{system2})$ as a particular case of it; indeed, for
$(\ref{systemBotCsetnek})$ a vanishing damping is not allowed. With our
system, we obtain convergence rates for $\dot{x}(t)$ and $\ddot{x}(t)$ as
$t\to+\infty$, which are not obtained in [16].

##  4 Structured convex minimization

We can specialize the previous results to the case of convex minimization, and
show additionally the convergence of functional values along the generated
trajectories to the optimal objective value at a rate that will depend on the
choice of $\gamma$. Let $f:\mathcal{H}\to\mathbb{R}\cup\\{+\infty\\}$ be a
proper, convex and lower semicontinuous function, and let
$g:\mathcal{H}\to\mathbb{R}$ be a convex and Fréchet differentiable function
with $L_{\nabla g}$-Lipschitz continuous gradient. Assume that
$\operatorname{argmin}_{\mathcal{H}}(f+g)\neq\emptyset$, and consider the
minimization problem

$\min_{x\in\mathcal{H}}f(x)+g(x).$ (32)

Fermat’s rule tells us that $\overline{x}$ is a global minimum of $f+g$ if and
only if

$0\in\partial(f+g)(\overline{x})=\partial f(\overline{x})+\nabla
g(\overline{x}).$

Therefore, solving $(\ref{ConvexMinimization})$ is equivalent solving the
monotone inclusion $0\in(A+B)(x)$ addressed in the first section, with
$A=\partial f$ and $B=\nabla g$. Moreover, recall that if $\nabla g$ is
$L_{\nabla g}$-Lipschitz then it is $\frac{1}{L_{\nabla g}}$-cocoercive
(Baillon-Haddad’s Theorem, see [14, Corollary 18.17]). Therefore, associated
to the problem $(\ref{ConvexMinimization})$ we have the dynamics

$\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\frac{d}{dt}\left(\frac{\gamma(t)}{\lambda(t)}\left(\nabla
f_{\gamma(t)}(u(t))+\nabla
g(x(t))\right)\right)+\frac{\gamma(t)}{\lambda(t)}\left(\nabla
f_{\gamma(t)}(u(t))+\nabla g(x(t))\right)=0,$ (33)

where we have denoted $u(t)=x(t)-\gamma(t)\nabla g(x(t))$ for all $t\geq
t_{0}$ for convenience.

######  Theorem 4.1.

Let $f:\mathcal{H}\to\mathbb{R}\cup\\{+\infty\\}$ be a proper, convex and
lower semicontinuous function, and let $g:\mathcal{H}\to\mathbb{R}$ be a
convex and Fréchet differentiable function with a $L_{\nabla g}$-Lipschitz
continuous gradient such that
$\operatorname{argmin}_{\mathcal{H}}(f+g)\neq\emptyset$. Assume that
$\alpha>1$, $\xi\geq 0$, $\lambda(t)=\lambda t^{2}$ for
$\lambda>\frac{2}{(\alpha-1)^{2}}$ and all $t\geq t_{0}$, and that
$\gamma:[t_{0},+\infty)\to\left(0,\frac{2}{L_{\nabla g}}\right)$ is a
differentiable function that satisfies
$\frac{\dot{\gamma}(t)}{\gamma(t)}=\mathcal{O}(1/t)$ as $t\to+\infty$. Then,
for a solution $x:[t_{0},+\infty)\to\mathcal{H}$ to $(\ref{systemFunctions})$,
the following statements hold:

  1. (i)

$x$ is bounded.

  2. (ii)

We have the estimates

$\displaystyle\int_{t_{0}}^{+\infty}t\|\dot{x}(t)\|^{2}dt<+\infty,\quad\int_{t_{0}}^{+\infty}t^{3}\|\ddot{x}(t)\|^{2}dt<+\infty,$
$\displaystyle\int_{t_{0}}^{+\infty}\frac{\gamma^{2}(t)}{t}\left\|\nabla
f_{\gamma(t)}\Big{[}x(t)-\gamma(t)\nabla g(x(t))\Big{]}+\nabla
g(x(t))\right\|^{2}dt<+\infty.$

  3. (iii)

We have the convergence rates

$\displaystyle\|\dot{x}(t)\|=o\left(\frac{1}{t}\right),\
\|\ddot{x}(t)\|=\mathcal{O}\left(\frac{1}{t^{2}}\right),$
$\displaystyle\left\|\nabla f_{\gamma(t)}\Big{[}x(t)-\gamma(t)\nabla
g(x(t))\Big{]}+\nabla g(x(t))\right\|=o\left(\frac{1}{\gamma(t)}\right),$
$\displaystyle\left\|\frac{d}{dt}\left(\nabla
f_{\gamma(t)}\Big{[}x(t)-\gamma(t)\nabla g(x(t))\Big{]}+\nabla
g(x(t))\right)\right\|=\mathcal{O}\left(\frac{1}{t\gamma(t)}\right)+o\left(\frac{t^{2}\left|\frac{d}{dt}\frac{\gamma(t)}{\lambda(t)}\right|}{\gamma^{2}(t)}\right)$

as $t\to+\infty$.

  4. (iv)

If $0<\inf_{t\geq t_{0}}\gamma(t)\leq\sup_{t\geq
t_{0}}\gamma(t)<\frac{2}{L_{\nabla g}}$, then $x(t)$ converges converges to a
minimizer of $f+g$ as $t\rightarrow+\infty$.

  5. (v)

Additionally, if $0<\gamma(t)\leq\frac{1}{L_{\nabla g}}$ for every $t\geq
t_{0}$ and we set $u(t):=x(t)-\gamma(t)\nabla g(x(t))$, then

$f\left(\operatorname{prox}_{\gamma(t)f}(u(t))\right)+g\left(\operatorname{prox}_{\gamma(t)f}(u(t))\right)-\min\nolimits_{\mathcal{H}}(f+g)=o\left(\frac{1}{\gamma(t)}\right)$

as $t\to+\infty$. Moreover,
$\left\|\operatorname{prox}_{\gamma(t)f}(u(t))-x(t)\right\|\to 0$ as
$t\to+\infty$.

###### Proof.

Parts (i)-(iv) are a direct consequence of Theorem 3.1. For checking (v),
first notice that for all $t\geq t_{0}$ we have

$\displaystyle T_{\lambda(t),\gamma(t)}(x(t))$
$\displaystyle=\frac{1}{\lambda(t)}\Big{[}\operatorname{Id}-J_{\gamma(t)\partial
f}\circ(\operatorname{Id}-\gamma(t)\nabla
g)\Big{]}(x(t))=\frac{1}{\lambda(t)}\Big{[}x(t)-\operatorname{prox}_{\gamma(t)f}(u(t))\Big{]}.$
(34)

Now, let $\overline{x}\in\operatorname{argmin}_{\mathcal{H}}(f+g)$. According
to [15, Lemma 2.3], for every $t\geq t_{0}$, we have the inequality

$\displaystyle\
f\left(\operatorname{prox}_{\gamma(t)f}(u(t))\right)+g\left(\operatorname{prox}_{\gamma(t)f}(u(t))\right)-\min\nolimits_{\mathcal{H}}(f+g)$
$\displaystyle\leq$ $\displaystyle\
f\left(\operatorname{prox}_{\gamma(t)f}(u(t))\right)+g\left(\operatorname{prox}_{\gamma(t)f}(u(t))\right)-f(\overline{x})-g(\overline{x})$
$\displaystyle\leq$ $\displaystyle\
-\frac{1}{2\gamma(t)}\left\|\operatorname{prox}_{\gamma(t)f}(u(t))-x(t)\right\|^{2}+\frac{1}{\gamma(t)}\left\langle
x(t)-x^{*},x(t)-\operatorname{prox}_{\gamma(t)f}(u(t))\right\rangle.$

After summing the norm squared term and using the Cauchy-Schwarz inequality,
for every $t\geq t_{0}$ we obtain

$\displaystyle\
\frac{1}{2\gamma(t)}\left\|\operatorname{prox}_{\gamma(t)f}(u(t))-x(t)\right\|^{2}$
$\displaystyle\leq$ $\displaystyle\
f\left(\operatorname{prox}_{\gamma(t)f}(u(t))\right)+g\left(\operatorname{prox}_{\gamma(t)f}(u(t))\right)+\frac{1}{2\gamma(t)}\left\|\operatorname{prox}_{\gamma(t)f}(u(t))-x(t)\right\|^{2}-\min\nolimits_{\mathcal{H}}(f+g)$
$\displaystyle\leq$ $\displaystyle\
\left\langle\frac{1}{\gamma(t)}\Big{(}x(t)-\operatorname{prox}_{\gamma(t)f}(u(t))\Big{)},x(t)-\overline{x}\right\rangle\leq\left\|\frac{1}{\gamma(t)}\Big{(}x(t)-\operatorname{prox}_{\gamma(t)f}(u(t))\Big{)}\right\|\|x(t)-\overline{x}\|$
$\displaystyle=$ $\displaystyle\
\frac{\lambda(t)}{\gamma(t)}\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|\|x(t)-\overline{x}\|$
$\displaystyle=$ $\displaystyle\
o\left(\frac{1}{\gamma(t)}\right)\quad\text{as}\quad t\to+\infty,$

which follows as a consequence of $x$ being bounded and
$\left\|T_{\lambda(t),\gamma(t)}(x(t))\right\|=o\left(\frac{1}{t^{2}}\right)$
as $t\to+\infty$. ∎

######  Remark 4.2.

It is also worth mentioning the system we obtain in the case where $g\equiv
0$, since we also get some improved rates for the objective functional values
when we compare (Split-DIN-AVD) to (DIN-AVD) [9]. In this case, we have the
system

$\ddot{x}(t)+\frac{\alpha}{t}+\xi\frac{d}{dt}\left(\frac{\gamma(t)}{\lambda(t)}\nabla
f_{\gamma(t)}(x(t))\right)+\frac{\gamma(t)}{\lambda(t)}\nabla
f_{\gamma(t)}(x(t))=0$ (35)

attached to the convex optimization problem

$\min_{x\in\mathcal{H}}f(x).$

If we assume $\lambda>\frac{1}{(\alpha-1)^{2}}$, allow
$\gamma:[t_{0},+\infty)\to(0,+\infty)$ to be unbounded from above and
otherwise keep the hypotheses of Theorem 4.1, for a solution
$x:[t_{0},+\infty)\to\mathcal{H}$ to $(\ref{system minimizar f})$, the
following statements hold:

  1. (i)

$x$ is bounded,

  2. (ii)

We have the estimates

$\int_{t_{0}}^{+\infty}t\|\dot{x}(t)\|^{2}dt<+\infty,\quad\int_{t_{0}}^{+\infty}t^{3}\|\ddot{x}(t)\|^{2}dt<+\infty,\quad\int_{t_{0}}^{+\infty}\frac{\gamma^{2}(t)}{t}\left\|\nabla
f_{\gamma(t)}(x(t))\right\|^{2}dt<+\infty,$

  3. (iii)

We have the convergence rates

$\displaystyle\|\dot{x}(t)\|=o\left(\frac{1}{t}\right),\
\|\ddot{x}(t)\|=\mathcal{O}\left(\frac{1}{t^{2}}\right),$
$\displaystyle\left\|\nabla
f_{\gamma(t)}(x(t))\right\|=o\left(\frac{1}{\gamma(t)}\right),\
\left\|\frac{d}{dt}\nabla
f_{\gamma(t)}(x(t))\right\|=\mathcal{O}\left(\frac{1}{t\gamma(t)}\right)+o\left(\frac{t^{2}\left|\frac{d}{dt}\frac{\gamma(t)}{\lambda(t)}\right|}{\gamma^{2}(t)}\right)$

as $t\to+\infty$.

  4. (iv)

If $0<\inf_{t\geq t_{0}}\gamma(t)$, then $x(t)$ converges weakly to a
minimizer of $f$ as $t\rightarrow+\infty$.

  5. (v)

We also obtain the rate

$f_{\gamma(t)}(x(t))-\min\nolimits_{\mathcal{H}}f=o\left(\frac{1}{\gamma(t)}\right)\quad\text{as}\quad
t\to+\infty,$

which entails

$f\left(\operatorname{prox}_{\gamma(t)f}(x(t))\right)-\min\nolimits_{\mathcal{H}}f=o\left(\frac{1}{\gamma(t)}\right)\quad\text{and}\quad\left\|\operatorname{prox}_{\gamma(t)f}(x(t))-x(t)\right\|\to
0$

as $t\to+\infty$.

Parts (i)-(iv) are a direct consequence of Theorem 3.2 for the case
$A=\partial f$. For showing part (v), first notice that for $\lambda>0$ and
$u\in\mathcal{H}$ we have, according to the definition of $f_{\lambda}$ and
$\operatorname{prox}_{\lambda f}$,

$\displaystyle f_{\lambda}(u)=f\left(\operatorname{prox}_{\lambda
f}(u)\right)+\frac{1}{2\lambda}\left\|\operatorname{prox}_{\lambda
f}(u)-u\right\|^{2}\leq f(u).$

Let $\overline{x}\in\mathcal{H}$ be a minimizer of $f$. We apply the gradient
inequality to $f_{\gamma(t)}$, from which we obtain, for every $t\geq t_{0}$

$\displaystyle f_{\gamma(t)}(x(t))-\min\nolimits_{\mathcal{H}}f$
$\displaystyle=f_{\gamma(t)}(x(t))-f(\overline{x})\leq
f_{\gamma(t)}(x(t))-f_{\lambda(t)}(\overline{x})$
$\displaystyle\leq\left\langle\nabla
f_{\gamma(t)}(x(t)),x(t)-\overline{x}\right\rangle\leq\left\|\nabla
f_{\gamma(t)}(x(t))\right\|\|x(t)-\overline{x}\|,$

where the last inequality follows from the Cauchy-Schwarz inequality. Since
$\left\|\nabla f_{\gamma(t)}(x(t))\right\|=o\left(\frac{1}{\gamma(t)}\right)$
as $t\to+\infty$ and $x$ is bounded, the previous inequality entails the first
statement of (v). Again recalling the definition of the Moreau envelope of
$f$, this finally gives

$f\left(\operatorname{prox}_{\gamma(t)f}(x(t))\right)+\frac{1}{2\gamma(t)}\left\|\operatorname{prox}_{\gamma(t)f}(x(t))-x(t)\right\|^{2}-\min\nolimits_{\mathcal{H}}f=f_{\gamma(t)}(x(t))-\min\nolimits_{\mathcal{H}}f=o\left(\frac{1}{\gamma(t)}\right)$

as $t\to+\infty$, which implies the last two statements and concludes the
proof.

As pointed out in Remark 3.3, we can choose $\gamma(t)=\lambda t^{2}$ for
every $t\geq t_{0}$ and recover the (DIN-AVD) system for nonsmooth convex
minimization problems studied in [9]. Moreover, we can also set
$\gamma(t)=t^{n}$ for a natural number $n>3$ and all $t\geq t_{0}$. Now, not
only are the convergence rates for $\nabla f_{\gamma(t)}(x(t))$ and
$\frac{d}{dt}\nabla f_{\gamma(t)}(x(t))$ as $t\to+\infty$ improved with
respect to the system in [9], but (Split-DIN-AVD) also provides a better rate
for the convergence of $f_{\gamma(t)}(x(t))$ to $\min_{\mathcal{H}}f$ as
$t\to+\infty$.

##  5 Numerical experiments

In the following paragraphs we describe some numerical experiments that
portray some aspects of the theory.

###  5.1 Minimizing a smooth and convex function

As an example of a continuous time scheme minimizing a convex and Fréchet
differentiable function $g:\mathcal{H}\rightarrow\mathbb{R}$ with $L_{\nabla
g}$-Lipschitz continuous gradient via (Split-DIN-AVD), we consider the system

$\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\xi\frac{d}{dt}\left(\frac{1}{\eta(t)}\nabla
g(x(t))\right)+\frac{1}{\eta(t)}\nabla g(x(t))=0,$ (36)

where for $(x_{1},x_{2})\in\mathbb{R}^{2}$ we set
$g(x_{1},x_{2})=\frac{1}{2}(x_{1}^{2}+100x_{2}^{2})$ and therefore $\nabla
g(x_{1},x_{2})=(x_{1},100x_{2})$. A trajectory generated by $(\ref{system A =
0})$ is a pair $x(t)=(x_{1}(t),x_{2}(t))$. Figure 1 plots both components of
the solution to $(\ref{system A = 0})$ with initial Cauchy data $x_{0}=(1,1)$,
$u_{0}=(1,1)$. Notice that the Lipschitz constant of $\nabla g$ is $L_{\nabla
g}=100$, which means that the cocoercitivity modulus of $\nabla g$ is
$\beta=\frac{1}{L_{\nabla g}}=\frac{1}{100}$. To fulfill
$\eta>\frac{1}{\beta(\alpha-1)^{2}}=\frac{100}{(\alpha-1)^{2}}$, we choose
$\alpha=20$, $\eta=0.278$. Figure 1(a) corresponds to the case with no Hessian
damping, that is, $\xi=0$. Figure 1(b) corresponds to a Hessian damping
parameter $\xi=0.2$.

(a)

(b)

Figure 1: Trajectories of (Split-DIN-AVD) for $B=\nabla g$

Figure 2 depicts the fast convergence of the velocities to zero for the cases
$\xi=0$ (Figure 2(a)) and $\xi=0.2$ (Figure 2(b)). In both figures, notice the
effect of the damping parameter $\xi>0$, which attenuates the oscillations of
the second component of the trajectories, as well as the oscillations present
in the velocities.

(a)

(b)

Figure 2: Fast convergence of the velocities

###  5.2 Minimizing a nonsmooth and convex function

As an example of a continuous time scheme minimizing a proper, convex and
lower semicontinuous function
$f:\mathcal{H}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ via (Split-DIN-AVD), we
consider the system

$\ddot{x}(t)+\frac{\alpha}{t}+\xi\frac{d}{dt}\left(\frac{\gamma(t)}{\lambda(t)}\nabla
f_{\gamma(t)}(x(t))\right)+\frac{\gamma(t)}{\lambda(t)}\nabla
f_{\gamma(t)}(x(t))=0.$ (37)

We will consider three options for $f$ and plot for each of them the
trajectories, the objective function values and the gradients of the Moreau
envelopes as follows:

  * •

$f(x)=\frac{1}{2}x^{2}$ (Figures 3(a) and 4(a)),

  * •

$f(x)=|x|$ (Figures 3(b) and 4(b)),

  * •

$f(x)=|x|+\frac{1}{2}x^{2}$ (Figures 3(c) and 4(c)).

In order to fulfill $\alpha>1$ and $\lambda>\frac{1}{(\alpha-1)^{2}}$, we
choose the parameters $\alpha=2$, $\lambda=1.1$, and we take $\xi=0$ and
$\gamma(t)=t^{8}$. We compare the results given by (DIN-AVD) (that is, when
$\gamma(t)=\lambda t^{2}$) and the ones given by our system (Split-DIN-AVD).
The choice of $\xi$ does not seem to change the plots in a significant way for
the examples we have chosen.

(a)

(b)

(c)

Figure 3: Trajectories and objective function values in the case $A=\partial
f$

(a)

(b)

(c)

Figure 4: Gradients of the Moreau envelopes of $f$

Figure 3 depicts the trajectories $x(t)$ of $(\ref{system B = 0})$ and the
function values $f\left(\operatorname{prox}_{\gamma(t)}(x(t))\right)$ for our
choices of $f$ as $t\rightarrow+\infty$. Figure 4 portrays the fast
convergence to zero of $\|\nabla f_{\gamma(t)}(x(t))\|$ as $t\to+\infty$.
Notice the big improvement over (DIN-AVD) for nonsmooth convex minimization in
[9] when choosing $\gamma(t)=t^{8}$, a result which we already knew
theoretically. Polynomials of high degree seem to be the ones which give the
biggest improvements in terms of rates.

###  5.3 An example with operator splitting

Now we consider the monotone inclusion problem (1) for
$A(x_{1},x_{2})=(-x_{2},x_{1})$ and $B(x_{1},x_{2})=(x_{1},x_{2})$ for every
$(x_{1},x_{2})\in\mathbb{R}^{2}$. For every $(x_{1},x_{2})\in\mathbb{R}^{2}$,
an easy calculation gives

$J_{\gamma A}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}=\begin{bmatrix}\frac{1}{1+\gamma^{2}}&\frac{\gamma}{1+\gamma^{2}}\\\
\frac{-\gamma}{1+\gamma^{2}}&\frac{1}{1+\gamma^{2}}\end{bmatrix}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix},$

and so

$\displaystyle(\operatorname{Id}-J_{\gamma
A}(\operatorname{Id}-\gamma\operatorname{Id}))\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}$ $\displaystyle=\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}-(1-\gamma)\begin{bmatrix}\frac{1}{1+\gamma^{2}}&\frac{\gamma}{1+\gamma^{2}}\\\
\frac{-\gamma}{1+\gamma^{2}}&\frac{1}{1+\gamma^{2}}\end{bmatrix}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}=\begin{bmatrix}\frac{\gamma^{2}+\gamma}{1+\gamma^{2}}&\frac{\gamma-1}{1+\gamma^{2}}\\\
\frac{1-\gamma}{1+\gamma^{2}}&\frac{\gamma^{2}+\gamma}{1+\gamma^{2}}\end{bmatrix}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix},$

and

$T_{\lambda,\gamma}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}=\begin{bmatrix}\frac{\gamma^{2}+\gamma}{\lambda(1+\gamma^{2})}&\frac{\gamma-1}{\lambda(1+\gamma^{2})}\\\
\frac{1-\gamma}{\lambda(1+\gamma^{2})}&\frac{\gamma^{2}+\gamma}{\lambda(1+\gamma^{2})}\end{bmatrix}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}.$

(Split-DIN-AVD) now reads

$\displaystyle\begin{bmatrix}\ddot{x_{1}}(t)\\\
\ddot{x_{2}}(t)\end{bmatrix}+\frac{\alpha}{t}\begin{bmatrix}\dot{x_{1}}(t)\\\
\dot{x_{2}}(t)\end{bmatrix}$
$\displaystyle+\>\xi\frac{d}{dt}\left(\begin{bmatrix}\frac{\gamma^{2}(t)+\gamma(t)}{\lambda(t)(1+\gamma^{2}(t))}&\frac{\gamma(t)-1}{\lambda(t)(1+\gamma^{2}(t))}\\\
\frac{1-\gamma(t)}{\lambda(t)(1+\gamma^{2}(t))}&\frac{\gamma^{2}(t)+\gamma(t)}{\lambda(t)(1+\gamma^{2}(t))}\end{bmatrix}\begin{bmatrix}x_{1}(t)\\\
x_{2}(t)\end{bmatrix}\right)$
$\displaystyle+\begin{bmatrix}\frac{\gamma^{2}(t)+\gamma(t)}{\lambda(t)(1+\gamma(t)^{2})}&\frac{\gamma(t)-1}{\lambda(t)(1+\gamma^{2}(t))}\\\
\frac{1-\gamma(t)}{\lambda(1+\gamma^{2}(t))}&\frac{\gamma^{2}(t)+\gamma(t)}{\lambda(t)(1+\gamma^{2}(t))}\end{bmatrix}\begin{bmatrix}x_{1}(t)\\\
x_{2}(t)\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix}.$

We choose the parameters $\alpha=7$, $\lambda=0.056$, $\gamma(t)\equiv 1.5$,
and the Cauchy data $x_{0}=(1,2)$ and $u_{0}=(-1,-1)$. Figure 5(a) corresponds
to the case $\xi=0$, and Figure 5(b) depicts the trajectory when the Hessian
damping parameter is $\xi=0.8$. Again, notice how, not only for optimization
problems, but also for monotone inclusions which cannot be reduced to the
former, the presence of $\xi$ seems to attenuate the oscillations present in
the trajectories.

(a)

(b)

Figure 5: Trajectories of (Split-DIN-AVD) for finding the zeros of $A+B$

##  6 A numerical algorithm

In the following we will derive via time discretization of (Split-DIN-AVD) a
numerical algorithm for solving the monotone inclusion problem (1). We perform
a discretization of (Split-DIN-AVD) with stepsize $1$ and set, for an integer
$k\geq 1$, $x(k):=x_{k}$, $\lambda(k):=\lambda_{k}$, $\gamma(k):=\gamma_{k}$.
We make the approximations

$\displaystyle\ddot{x}(t)$ $\displaystyle\approx x_{k+1}-2x_{k}+x_{k-1},$
$\displaystyle\quad\frac{\alpha}{t}\dot{x}(t)$
$\displaystyle\approx\frac{\alpha}{k}(x_{k}-x_{k-1}),$
$\displaystyle\frac{d}{dt}T_{\lambda(t),\gamma(t)}(x(t))$
$\displaystyle\approx
T_{\lambda_{k},\gamma_{k}}(x_{k})-T_{\lambda_{k-1},\gamma_{k-1}}(x_{k-1}),$
$\displaystyle\quad T_{\lambda(t),\gamma(t)}(x(t))$ $\displaystyle\approx
T_{\lambda_{k+1},\gamma_{k+1}}(x_{k+1}),$

so we get, for every $k\geq 1$,

$x_{k+1}-2x_{k}-x_{k-1}+\frac{\alpha}{k}(x_{k}-x_{k-1})+\xi\left(T_{\lambda_{k},\gamma_{k}}(x_{k})-T_{\lambda_{k-1},\gamma_{k-1}}(x_{k-1})\right)+T_{\lambda_{k+1},\gamma_{k+1}}(x_{k+1})=0.$
(38)

After rearranging the terms of $(\ref{discretesystem})$, for every $k\geq 1$
we obtain

$x_{k+1}+T_{\lambda_{k+1},\gamma_{k+1}}(x_{k+1})=x_{k}+\left(1-\frac{\alpha}{k}\right)(x_{k}-x_{k-1})-\xi\left(T_{\lambda_{k},\gamma_{k}}(x_{k})-T_{\lambda_{k-1},\gamma_{k-1}}(x_{k-1})\right).$
(39)

In other words, after setting $\alpha_{k}=1-\frac{\alpha}{k}$ and denoting the
right hand side of $(\ref{discretesystem2})$ by $y_{k}$ for every $k\geq 1$,
we obtain the following iterative scheme

$(\forall k\geq 1)\ \left\\{\begin{aligned}
y_{k}&=x_{k}+\alpha_{k}(x_{k}-x_{k-1})-\xi\left(T_{\lambda_{k},\gamma_{k}}(x_{k})-T_{\lambda_{k-1},\gamma_{k-1}}(x_{k-1})\right),\\\
x_{k+1}&=\left(\operatorname{Id}+T_{\lambda_{k+1},\gamma_{k+1}}\right)^{-1}(y_{k}).\end{aligned}\right.$
(40)

Observe that the second step in (40) is always well-defined. Indeed, for
$\lambda,\gamma>0$, $T_{\lambda,\gamma}$ is $\frac{\lambda}{2}$-cocoercive,
hence monotone (see Lemma 2.2(i)). This also implies that $T_{\lambda,\gamma}$
is $\frac{2}{\lambda}$-Lipschitz continuous, and a monotone and continuous
operator is maximally monotone, according to [14, Corollary 20.28]. Hence, by
Minty’s Theorem (see [14, Theorem 21.1]), we know that
$\operatorname{Id}+T_{\lambda,\gamma}:\mathcal{H}\to\mathcal{H}$ is
surjective.

We are in conditions of stating the main theorem concerning our previous
algorithm.

######  Theorem 6.1.

Let $A:\mathcal{H}\to 2^{\mathcal{H}}$ be a maximally monotone operator and
$B:\mathcal{H}\to\mathcal{H}$ a $\beta$-cocoercive operator for some
$\beta\geq 0$ such that $\operatorname{zer}(A+B)\neq\emptyset$. Choose
$x_{0},x_{1}\in\mathcal{H}$ any initial points. Let $\alpha>1$, $\xi\geq 0$,
and $(\lambda_{k})_{k\geq 0}$, $(\gamma_{k})_{k\geq 0}$ sequences of positive
numbers that fulfill

$\displaystyle\lambda_{k}=\lambda k^{2}\>\>\>\forall k\geq
1,\quad\text{with}\quad\lambda>\frac{4\xi+2}{(\alpha-1)^{2}},$ $\displaystyle
0<\inf_{k\geq 0}\gamma_{k}\leq\sup_{k\geq
0}\gamma_{k}<2\beta\quad\text{and}\quad\frac{\gamma_{k}-\gamma_{k-1}}{\gamma_{k}}=\mathcal{O}\left(\frac{1}{k}\right)\quad\text{as}\quad
k\to+\infty.$

Now, consider the sequences $(y_{k})_{k\geq 1}$ and $(x_{k})_{k\geq 0}$
generated by algorithm $(\ref{algorithm})$. The following properties are
satisfied:

  1. (i)

We have the estimates

$\|x_{k+1}-x_{k}\|=\mathcal{O}\left(\frac{1}{k}\right)\quad\text{and}\quad\left\|A_{\gamma_{k}}(x_{k}-\gamma_{k}Bx_{k})+Bx_{k}\right\|=o\left(\frac{1}{\gamma_{k}}\right)\quad\text{as}\quad
k\to+\infty.$

  2. (ii)

The sequence $(x_{k})_{k\geq 0}$ converges weakly to an element of
$\operatorname{zer}(A+B)$.

  3. (iii)

The sequence $(y_{k})_{k\geq 1}$ converges weakly to an element of
$\operatorname{zer}(A+B)$. Precisely, we have
$\|x_{k}-y_{k}\|=\mathcal{O}\left(\frac{1}{k}\right)$ as $k\to+\infty$.

The proof can be done by transposing the techniques used in the continuous
time case to the discrete time case. Algorithm $(\ref{algorithm})$ can be seen
as a splitting version of the (PRINAM) algorithm studied by Attouch and László
in [10].

######  Remark 6.2.

The second step in $(\ref{algorithm})$ can be quite complicated to compute.
However, if $B=0$, we can resort to the fact that
$(A_{\lambda_{1}})_{\lambda_{2}}=A_{\lambda_{1}+\lambda_{2}}$ for
$\lambda_{1},\lambda_{2}>0$. We now have, for $\lambda,\gamma>0$,

$T_{\lambda,\gamma}=\frac{1}{\lambda}\Big{[}\operatorname{Id}-J_{\gamma
A}\Big{]}=\frac{\gamma}{\lambda}A_{\gamma},$

which gives

$\left(\operatorname{Id}+T_{\gamma,\lambda}\right)^{-1}=J_{\frac{\gamma}{\lambda}A_{\gamma}}=-\frac{\gamma}{\lambda}\left(A_{\lambda}\right)_{\frac{\gamma}{\lambda}}+\operatorname{Id}=\operatorname{Id}-\frac{\gamma}{\lambda}A_{\lambda+\frac{\gamma}{\lambda}}.$

It is now possible to write $(\ref{algorithm})$ in terms of the resolvents of
$A$. We have, for every $k\geq 1$,

$\displaystyle
T_{\lambda_{k},\gamma_{k}}(x_{k})-T_{\lambda_{k-1},\gamma_{k-1}}(x_{k-1})=\>$
$\displaystyle\frac{1}{\lambda_{k}}\Big{[}x_{k}-J_{\gamma_{k}A}(x_{k})\Big{]}-\frac{1}{\lambda_{k-1}}\Big{[}x_{k-1}-J_{\gamma_{k-1}A}(x_{k-1})\Big{]}$
$\displaystyle=\>$
$\displaystyle\left(\frac{1}{\lambda_{k}}-\frac{1}{\lambda_{k-1}}\right)x_{k}+\frac{1}{\lambda_{k-1}}(x_{k}-x_{k-1})$
$\displaystyle-\left(\frac{1}{\lambda_{k}}J_{\gamma_{k}A}(x_{k})-\frac{1}{\lambda_{k-1}}J_{\gamma_{k-1}A}(x_{k-1})\right),$
$\displaystyle
y_{k}-\frac{\gamma_{k+1}}{\lambda_{k+1}}A_{\lambda_{k+1}+\frac{\gamma_{k+1}}{\lambda_{k+1}}}(y_{k})=\>$
$\displaystyle
y_{k}-\frac{\gamma_{k+1}}{\lambda_{k+1}}\frac{1}{\frac{\lambda_{k+1}^{2}+\gamma_{k+1}}{\lambda_{k+1}}}\left[y_{k}-J_{\left(\lambda_{k+1}+\frac{\gamma_{k+1}}{\lambda_{k+1}}\right)A}(y_{k})\right]$
$\displaystyle=\>$
$\displaystyle\frac{\lambda_{k+1}^{2}}{\lambda_{k+1}^{2}+\gamma_{k+1}}y_{k}+\frac{\gamma_{k}}{\lambda_{k+1}^{2}+\gamma_{k+1}}J_{\left(\lambda_{k+1}+\frac{\gamma_{k+1}}{\lambda_{k+1}}\right)A}(y_{k}).$

So now $(\ref{algorithm})$ becomes

$(\forall k\geq 1)\ \left\\{\begin{aligned}
y_{k}=\>&\left(1-\xi\left(\frac{1}{\lambda_{k}}-\frac{1}{\lambda_{k-1}}\right)\right)x_{k}+\left(\alpha_{k}-\frac{\xi}{\lambda_{k-1}}\right)(x_{k}-x_{k-1})\\\
&+\xi\left(\frac{1}{\lambda_{k}}J_{\gamma_{k}A}(x_{k})-\frac{1}{\lambda_{k-1}}J_{\gamma_{k-1}A}(x_{k-1})\right),\\\
x_{k+1}=\>&\frac{\lambda_{k+1}^{2}}{\lambda_{k+1}^{2}+\gamma_{k+1}}y_{k}+\frac{\gamma_{k}}{\lambda_{k+1}^{2}+\gamma_{k+1}}J_{\left(\lambda_{k+1}+\frac{\gamma_{k+1}}{\lambda_{k+1}}\right)A}(y_{k}).\end{aligned}\right.$
(41)

Now, if we assume $0<\inf_{k\geq 0}\gamma_{k}$ and
$\lambda>\frac{2\xi+1}{(\alpha-1)^{2}}$ and otherwise keep the hypotheses of
Theorem 6.1, then for the sequences $(x_{k})_{k\geq 0}$ and $(y_{k})_{k\geq
1}$ generated by $(\ref{algorithm4})$, the following statements hold:

  1. (i)

We have the estimates

$\|x_{k+1}-x_{k}\|=\mathcal{O}\left(\frac{1}{k}\right)\quad\text{and}\quad\left\|A_{\gamma_{k}}(x_{k})\right\|=o\left(\frac{1}{\gamma_{k}}\right)\quad\text{as}\quad
k\to+\infty.$

  2. (ii)

The sequence $(x_{k})_{k\geq 0}$ converges weakly to an element of
$\operatorname{zer}A$.

  3. (iii)

The sequence $(y_{k})_{k\geq 1}$ converges weakly to an element of
$\operatorname{zer}A$ as well. Precisely, we have
$\|x_{k}-y_{k}\|=\mathcal{O}\left(\frac{1}{k}\right)$ as $k\to+\infty$.

Notice that the condition required for $(\gamma_{k})_{k\geq 0}$ is fulfilled
in particular for $\gamma_{k}=k^{n}$ for every $k\geq 1$ and a natural number
$n\geq 1$. Thus, by choosing large $n$, we obtain a fast convergence rate for
$A_{\gamma_{k}}(x_{k})$ as $k\to+\infty$.

##  Appendix A Appendix

The following are three auxiliary lemmas that are used in the proof of Theorem
3.1. The proof for Lemma A.1 can be found in [12], while the proof of Lemma
A.2 is straightforward. For the proof of Opial’s Lemma, we refer the reader to
[1, Lemma 1.10].

######  Lemma A.1.

Let $t_{0}>0$, and let $u:[t_{0},+\infty)\to\mathbb{R}$ be a continuously
differentiable function which is bounded from below. Given $\alpha>1$, a
nonnegative function $\theta:[t_{0},+\infty)\to\mathbb{R}$ and a nonnegative
function $k\in L^{1}([t_{0},+\infty),\mathbb{R})$, let us assume that

$t\ddot{u}(t)+\alpha\dot{u}(t)+\theta(t)\leq k(t)$

for almost every $t\geq t_{0}$. Then, the positive part $[\dot{u}]_{+}$ of
$\dot{u}$ belongs to $L^{1}([t_{0},+\infty),\mathbb{R})$ and
$\lim_{t\to+\infty}u(t)$ exists. Moreover, we have
$\int_{t_{0}}^{+\infty}\theta(t)dt<+\infty$.

######  Lemma A.2.

Let $A,B,C\in\mathbb{R}$ and $\mathcal{H}$ a real Hilbert space. Then the
inequality

$A\|X\|^{2}+2C\langle X,Y\rangle+B\|Y\|^{2}\leq 0$

holds for every $X,Y\in\mathcal{H}$ if and only if $A,B\leq 0$ and
$C^{2}-AB\leq 0$.

######  Lemma A.3 (Opial’s Lemma).

Let $S\subseteq\mathcal{H}$ be a nonempty set and
$x:[t_{0},+\infty)\to\mathcal{H}$ a given map, where $t_{0}>0$. Assume that

  1. (i)

for every $x^{*}\in S$, $\lim_{t\to+\infty}\|x(t)-x^{*}\|$ exists;

  2. (ii)

every weak sequential cluster point of the map $x$ belongs to $S$.

Then, there exists $x_{\infty}\in S$ such that $x(t)$ converges weakly to
$x_{\infty}$ as $t\to+\infty$.

## References

  * [1] Abbas, B., Attouch, H. “Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator”. In: Optimization 64(10), pp. 2223–2252 (2015)
  * [2] Álvarez, F. “On the minimizing property of a second-order dissipative system in Hilbert spaces”. In: SIAM Journal on Control Optimization 38(4), pp. 1102–1119 (2020)
  * [3] Álvarez, F., Attouch, H. “An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping”. In: Set-Valued Analysis 9(1-2), pp. 3–11 (2001)
  * [4] Álvarez, F., Attouch, H., Bolte, J., Redont, P. “A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics”. In: Journal de Mathématiques Pures et Apliqueés 81(8) pp. 747–779 (2002)
  * [5] Apidopoulos, V., Aujol, J.-F., Dossal, C. “Convergence rate of inertial forward-backward algorithm beyond Nesterov’s rule”. In: Mathematical Programming 180, pp. 137–156 (2020)
  * [6] Attouch, H., Chbani, Z., Peypouquet, J., Redont, P. “Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity”. In: Mathematical Programming 168, pp. 123–175 (2018)
  * [7] Attouch, H., Chbani, Z., Riahi, H. “Rate of convergence of the Nesterov accelerated gradient method in the subcritical case $\alpha\leq 3$”. In: ESAIM: COCV 25, Article number 2 (2019)
  * [8] Attouch, H., Goudou, X., Redont, P. “ The heavy ball with friction method. The continuous dynamical system, global exploration of the local minima of a real-valued function by asymptotical analysis of a dissipative dynamical system. In: Communications in Contemporary Mathematics 2(1), pp. 1–34 (2000)
  * [9] Attouch, H., László, S.C. “Continuous Newton-like inertial dynamics for monotone inclusions”. In: Set-Valued and Variational Analysis 29, pp. 555–581 (2021)
  * [10] Attouch, H., László, S.C. “Newton-like inertial dynamics and proximal algorithms governed by maximally monotone operators”. In: SIAM Journal on Optimization 30(4), pp. 3252–3283 (2020)
  * [11] Attouch, H., Maingé, P.E. “Asymptotic behavior of second order dissipative evolution equations combining potential with non-potential effects”. In: ESAIM Control Optimization and Calculus of Variations 17(3), pp. 836–857 (2011)
  * [12] Attouch, H., Peypouquet, J. “Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators”. In: Mathematical Programming 174(1-2), pp. 391–432 (2019).
  * [13] Attouch, H., Peypouquet, J., Redont, P. “Fast convex minimization via inertial dynamics with Hessian driven damping”. In: Journal of Differential Equations 210(10), pp. 5734–5783 (2016).
  * [14] Bauschke, H. H., Combettes, P. L. Convex Analysis and Monotone Operator Theory. Second Edition. CMS Books in Mathematics. Springer, 2017
  * [15] Beck, A., Teboulle, M. “A Fast Iterative Shrinkage-Thresholding algorithm for linear inverse problems”. In: SIAM Journal on Imaging Sciences 2(1), pp. 183–202
  * [16] Boţ, R.I., Csetnek, E.R., “Second order forward-backward dynamical systems for monotone inclusion problems”. In: SIAM Journal on Control and Optimization 54, pp. 1423–1443 (2016)
  * [17] Haraux, A. Systémes Dynamiques Dissipatifs et Applications. Masson, 1991
  * [18] May, R. “Asymptotic for a second order evolution equation with convex potential and vanishing damping term”. In: Turkish Journal of Mathematics 41(3), pp. 681–685 (2017)
  * [19] Nesterov, Y. “A method of solving a convex programming problem with convergence rate $\mathcal{O}(1/k^{2})$”. In: Doklady Akademii Nauk SSSR 269(3), pp. 543–547 (1983)
  * [20] Su, W.J., Boyd, S., Candès, E.J. “A differential equation for modeling Nesterov’s accelerated gradient method: theory and insights”. In: Neural Information Processing Systems 27, pp. 2510–2518 (2014)