# Variable metric backward-forward dynamical systems for monotone inclusion
problems

Pankaj Gautam  Department of Mathematical Sciences, Indian Institute of
Technology(BHU), Varanasi 221005, India<EMAIL_ADDRESS>,  D. R. Sahu
Department of Mathematics, Banaras Hindu University, Varanasi 221005, India
<EMAIL_ADDRESS>and  J. C. Yao  Center for General Education,China Medical
University,Taichung 40402, Taiwan  Department of Applied Mathematics, National
Sun Yat-sen University, Kaohsiung 804, Taiwan<EMAIL_ADDRESS>
###### Abstract.

This paper investigates first-order variable metric backward forward dynamical
systems associated with monotone inclusion and convex minimization problems in
real Hilbert space. The operators are chosen so that the backward-forward
dynamical system is closely related to the forward-backward dynamical system
and has the same computational complexity. We show existence, uniqueness, and
weak asymptotic convergence of the generated trajectories and strong
convergence if one of the operators is uniformly monotone. We also establish
that an equilibrium point of the trajectory is globally exponentially stable
and monotone attractor. As a particular case, we explore similar perspectives
of the trajectories generated by a dynamical system related to the
minimization of the sum of a nonsmooth convex and a smooth convex function.
Numerical examples are given to illustrate the convergence of trajectories.

###### Key words and phrases:

Dynamical systems; Monotone inclusion; Backward-forward algorithm; variable
metric; convex optimization problems

##  1\. Introduction

The monotone inclusion problem is

(1.1) $\displaystyle\text{to~{}find}~{}u\in H\textrm{ such that }0\in(A+B)u,$

where $H$ is a real Hilbert space, $A:H\to 2^{H}$ is maximally monotone
operator and $B:H\to H$ is $\beta$-cocoercive operator for $\beta>0$. This
structure is quite central due to the large range of problems involved in
fields such as signal and image processing, partial differential equations,
mechanics, convex optimization, statics, and game theory [14, 7, 17, 19, 28,
22].

The first order dynamical system, which is linked with forward-backward
algorithm to solve the problem (1.1) was studied by Bot et al. [11] as
follows:

(1.4)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{u}(t)=\lambda(t)[J_{\gamma
A}(I-\gamma B)u(t)-u(t)]\\\\[6.0pt] u(0)=u_{0},\end{array}\right.}$

where $u_{0}\in H$, $A:H\to 2^{H}$ is maximal monotone operator, $B:H\to H$ is
$\beta$-cocercive operator for $\beta>0$, $J_{\gamma A}$ is resolvent of
operator $A$ for $\gamma>0$ and $\lambda:[0,\infty)\to[0,\infty)$ is a
Lebesgue measurable function. They studied the convergence of trajectories
when $\gamma\in(0,2\beta)$. Also, Bot et al. [13] studied that the trajectory
generated by dynamical system (1.4) strongly converges with exponential rate
to solution of problem (1.1), when $A:H\to 2^{H}$ is maximal monotone and
$B:H\to H$ is monotone, $\frac{1}{\beta}$-Lipschitz for $\beta>0$ such that
sum of both the operators is $\rho$-strongly monotone for $\rho>0$.

To determine the zero of $\partial\Phi+B$, Abbas et al. [1] studied the
dynamical system of the form:

(1.6)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{u}(t)+u(t)=\operatorname{prox_{\gamma\Phi}}(I-\gamma
B)u(t)\\\\[6.0pt] u(0)=u_{0},\end{array}\right.}$

where $\Phi:H\to\mathbb{R}_{\infty}$ is a proper, lower semicontinuous and
convex function, $\partial\Phi$ is subdifferential of $\Phi$, $B:H\to H$ is a
cocoercive operator and $\operatorname{prox_{\gamma\Phi}}$ denotes the
proximal point operator of $\gamma\Phi$.

Bot et al. [13] also studied the dynamical system which is linked with the
minimization of sum of smooth and nonsmooth function, which is as follows:

(1.8)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{u}(t)=\lambda(t)[\operatorname{prox_{\gamma
f}}(I-\gamma\nabla g)u(t)-u(t)]\\\\[6.0pt] u(0)=u_{0},\end{array}\right.}$

where $f:H\to\mathbb{R}_{\infty}$ is proper, lower semicontinuous, convex
function and $g:H\to\mathbb{R}$ is convex function , $1/\beta$-Lipschitz
continuous gradient for $\beta>0$ and Fr$\acute{e}$chet differentiable.

To minimize a smooth convex function $f:H\to\mathbb{R}$ over the nonempty,
closed, convex set $C\subseteq H$, Antipin [3] and Bolte [8] discussed the
convergence of the trajectories governed by

(1.11)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{u}(t)+u(t)=P_{C}(I-\gamma\nabla
f)u(t)\\\\[6.0pt] u(0)=u_{0},\end{array}\right.}$

where $\gamma>0$ and $P_{C}$ is the projection operator on the set $C$.
Antipin [3] also obtained the exponential convergence rate of the trajectory
for dynamical system (1.11).

Bot et al. [10] studied existence, uniqueness, weak and strong convergence of
the trajectories achieved by second-order dynamical systems linked with the
problem to find zeros of the sum of two operators, in which one is a maximally
monotone operator, and another one is cocoercive. Some implicit type dynamical
systems have been already studied in the literature ( see [9, 12, 2, 1, 6,
4]).

The backward-forward algorithm was studied by Attouch et al. [5] to solve the
monotone inclusion problem (1.1). Operators are chosen so that they are
closely associated with a forward-backward algorithm to solve the problem
(1.1). The forward-backward algorithms with a symmetric positive definite
operator $M$, called a variable metric, were studied by [15, 18, 21]. Raguet
et al. [26] studied generalized variable metric forward-backward algorithm by
taking the operator $M$ strongly positive.

In this manuscript, we investigate the first order dynamical system, which is
associated with the variable metric backward-forward method to solve
structured monotone inclusion problem of the form:

$\displaystyle\text{find}~{}u\in H:0\in(A+B)u,$

where $A:H\to 2^{H}$ is maximal $(\gamma-\alpha)$-cohypomonotone for
$\gamma\in\mathbb{R},\alpha>0$, $B:H\to H$ is a $\beta$-cocoercive for
$\beta>0$ and $H$ is a real Hilbert space. We study first-order variable
metric backward-forward dynamical system of the form:

(1.14)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{u}(t)=\lambda(t)[(I-\gamma
M^{-1}B)J_{\gamma A}^{M}u(t)-u(t)]\\\\[6.0pt] u(0)=u_{0},\end{array}\right.}$

where $\gamma\neq 0$, $u_{0}\in H$, $\lambda:[0,\infty)\to[0,\infty)$ is a
Lebesgue measurable function and $J_{\gamma A}^{M}:H\to 2^{H}$ is an operator
defined by $J_{\gamma A}^{M}:=(I+\gamma M^{-1}A)^{-1}$ and $M:H\to H$ is a
strongly positive operator. It is shown that the equilibrium point is
exponentially stable and monotone attractor, whenever $B_{-\gamma}$ is
$\rho$-strongly monotone for $\rho>0$.

We study the convergence behaviour of the trajectories generated by forward-
backward dynamical system in variable metric setting:

(1.17)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{x}(t)=\lambda(t)[J_{\gamma
A}^{M}(I-\gamma M^{-1}B)x(t)-x(t)]\\\\[6.0pt] x(0)=x_{0},\end{array}\right.}$

where $x_{0}\in H$, $\lambda:[0,\infty)\to[0,\infty)$ is a Lebesgue measurable
function and operators $A$, $B$ and $M$ satisfy the same conditions as in
dynamical system (1.14).

We also examine the first order dynamical system generated by optimization
problem of the form:

(1.18) $\displaystyle\min_{u\in H}f(u)+g(u),$

where $f:H\to\mathbb{R}\cup\\{\infty\\}$ is proper, convex and lower
semicontinuous function and $g:H\to\mathbb{R}$ is differentiable such that its
gradient $\nabla g$ is $\beta$-cocercive for $\beta>0$.

The remaining parts of this paper are organized as follows: some lemmas and
definitions required for proving the main results are presented in Section 2.
Existence, uniqueness, and convergence of the trajectories generated by the
first-order backward-forward dynamical system (1.14) and forward-backward
dynamical system (1.17) in the variable metric environment are studied in
Section 3. In this section, we also study the convergence behavior of the
dynamical system’s trajectories, which is associated with minimizing the sum
of a smooth and nonsmooth function. Finally, Section 4 is devoted to numerical
experiments to illustrate the convergence of the trajectories of the dynamical
system (1.14).

##  2\. Preliminaries

Throughout this paper $H$ denotes a real Hilbert space with inner product
$\langle\cdot,\cdot\rangle$, corresponding norm
$\|\cdot\|=\sqrt{\langle\cdot,\cdot\rangle}$ and $I$ denote the identity
operator.

Let $T:H\to 2^{H}$ be a set-valued operator. $T^{-1}:H\to 2^{H}$ is inverse of
$T$ which is defined by the relation $y\in Tx\Leftrightarrow x\in T^{-1}y$.
The graph of $T$ is the set $\operatorname{gra}(T):=\\{(x,y)\in H\times H:y\in
Tx\\}$. The resolvent of an operator $T$ of index $\gamma$ is defined by
$J_{\gamma T}:=(I+\gamma T)^{-1}$, where
$\gamma\in\mathbb{R}\setminus\\{0\\}$. The Yosida approximation of $T$ of
index $\gamma$ is defined by $T_{\gamma}=(T^{-1}+\gamma I)^{-1}$,
$\gamma\in\mathbb{R}$. For $\gamma,\delta\in\mathbb{R}$, we have
$(T_{\gamma})_{\delta}=T_{\gamma+\delta}$; in particular,
$(T_{-\gamma})_{\gamma}=T$ (see [5]).

######  Definition 2.1.

[7] A set-valued operator $T:H\to 2^{H}$ is said to be

  1. (i)

monotone if

$\left\langle x-y,u-v\right\rangle\geq
0~{}\forall(x,u),(y,v)\in\operatorname{gra}(T);$

  2. (ii)

maximal monotone if there exist no monotone operator $S:H\rightarrow 2^{H}$
such that $\operatorname{gra}(S)$ properly contains $\operatorname{gra}(T)$;

  3. (iii)

maximally $\rho$-cohypomonotone if $T_{\rho}=(T^{-1}+\rho I)^{-1}$ is
maximally monotone, where $\rho\in\mathbb{R}$;

  4. (iv)

uniformly monotone with modulus function $\phi:[0,\infty)\to[0,\infty)$, if
$\phi$ is increasing, vanishes only at $0$, and

$\langle
x-y,u-v\rangle\geq\phi(\|x-y\|)~{}\forall(x,u),(y,v)\in\operatorname{gra}(T);$

  5. (v)

strongly monotone with constant $\rho\in[0,\infty)$ if $T-\rho I$ is monotone,
i.e.,

$\langle
x-y,u-v\rangle\geq\rho\|x-y\|^{2}~{}\forall(x,u),(y,v)\in\operatorname{gra}(T).$

######  Remark 2.2.

On cohypomonotonicity, its useful and related notions see [16, 27, 20].

######  Example 2.3.

(i) Let $T:H\to 2^{H}$ be a maximally monotone operator. Then $T$ is
$\rho$-cohypomonotone for all $\rho\geq 0$ ([5]).

(ii) Consider a bounded, linear and symmetric operator $N:H\to H$ whose
spectrum $\sigma(N)$ has negative points. Define the set-valued operator
$T:H\to 2^{H}$ by $T=N^{-1}$, which is not monotone. Then $T_{\rho}=(T+\rho
I)^{-1}$ is maximal monotone operator for $\rho>-\min\sigma(N)$. Hence $T$ is
maximally $\rho$-cohypomonotone ([5]).

######  Definition 2.4.

[7] An operator $T:H\to H$ is said to be

  1. (i)

$\beta$-cocoercive for $\beta>0$ if

$\langle Tx-Ty,x-y\rangle\geq\beta\|Tx-Ty\|^{2}~{}for~{}every~{}x,y\in H;$

  2. (ii)

non-expansive if

$\|Tx-Ty\|\leq\|x-y\|~{}for~{}every~{}x,y\in H;$

  3. (iii)

$\alpha$-averaged for $\alpha\in(0,1)$ if there exists a nonexpansive operator
$R:H\to H$ such that $T=(1-\alpha)I+\alpha R$.

######  Remark 2.5.

If $T$ is a nonexpansive operator, then operator defined by $B=I-T$ is
$\frac{1}{2}$-cocoercive.

######  Lemma 2.6.

[7] Let $\beta>0$, $\gamma\in(0,2\beta)$ and $T:H\to H$ be $\beta$-cocoercive.
Then $I-\gamma T$ is $\frac{\gamma}{2\beta}$-averaged.

######  Lemma 2.7.

[24] Let $T_{i}:H\to H$ be $\alpha_{i}$-averaged operators for some
$\alpha_{i}\in[0,1)$, where $i=1,2$. Then
$\frac{\alpha_{1}+\alpha_{2}-2\alpha_{1}\alpha_{2}}{1-\alpha_{1}\alpha_{2}}\in[0,1)$
and $T_{1}T_{2}$ is
$\frac{\alpha_{1}+\alpha_{2}-2\alpha_{1}\alpha_{2}}{1-\alpha_{1}\alpha_{2}}$-averaged.

######  Lemma 2.8.

[5] Let $T:H\to 2^{H}$ be a set-valued operator, $\gamma\in\mathbb{R}$ and
$\alpha>0$. Then $T$ is maximally $(\gamma-\alpha)$-cohypomonotone if and only
if $T_{\gamma}$ is defined everywhere, single-valued and $\alpha$-cocoercive.

Let $M:H\to H$ be a bounded, linear, and self-adjoint operator. $M$ is said to
be positive if $\langle Mx,x\rangle\geq 0$, for all $x\in H$, and strongly
positive if $\exists~{}m\in(0,\infty)$ such that $M-mI$ is positive. The
square root and inverse of the strongly positive operator $M$ are denoted by
$\sqrt{M}$ and $M^{-1}$.

The maps $(x,y)\mapsto\langle x,y\rangle_{M}:=\langle Mx,y\rangle$ and
$x\mapsto\|x\|_{M}:=\sqrt{\langle Mx,x\rangle}$ define an inner product and a
norm over $H$, respectively, where $M$ is a strongly positive operator on $H$.
For all $x\in H$,

$m\|x\|^{2}\leq\|x\|_{M}^{2}\leq\|M\|\|x\|^{2}.$

Thus, $\|\cdot\|$ and $\|\cdot\|_{M}$ are equivalent norms and hence induce
the same topology over $H$.

For $\gamma\in\mathbb{R}\setminus\\{0\\}$ and strongly positive operator $M$,
we denote $J_{\gamma T}^{M}:=(I+\gamma M^{-1}T)^{-1}$.

######  Definition 2.9.

[7] Let $f:H\to(-\infty,\infty]$ be a function.

  * (i)

If $f$ is a proper function, then $f$ is said to be uniformly convex with
modulus function $\phi:[0,\infty)\to[0,\infty)$ if

$f(\nu x+(1-\nu)y)+\nu(1-\nu)\phi(\|x-y\|)\leq\nu f(x)+(1-\nu)f(y)$

for all $\nu\in(0,1)$ and $x,y\in domf$, where function $\phi$ is increasing
and vanishes at $0$.

  * (ii)

Let $\gamma>0$. The Moreau envelope of $f$ with parameter $\gamma$ is

$\displaystyle f_{\gamma}(x)=\inf_{y\in
H}\\{f(y)+\frac{1}{2\gamma}\|x-y\|^{2}\\}.$

######  Definition 2.10.

[6, 2] Let $b>0$, $x:[0,b]\to H$ be a function. Then $x$ is absolutely
continuous if one of the following equivalent properties holds:

  * (i)

there exists an integrable function $y:[0,b]\to H$ such that

$\displaystyle x(t)=x(0)+\int_{0}^{t}y(s)ds~{}~{}~{}for~{}all~{}t\in[0,b];$

  * (ii)

$x$ is continuous and its distribution derivative $\dot{x}$ is Lebesgue
integrable on $[0,b]$;

  * (iii)

$\forall~{}\epsilon>0$, $\exists~{}\eta>0$ such that for any finite family of
intervals $I_{k}=(a_{k},b_{k})$ we have the implication

$\displaystyle\left(I_{k}\cap
I_{j}=\emptyset~{}and~{}\sum_{k}|b_{k}-a_{k}|<\eta\right)\Rightarrow\sum_{k}\|x(b_{k})\|-x(a_{k})\|<\epsilon.$

######  Remark 2.11.

$(a)$ From Definition 2.10, one asserts that an absolutely continuous function
is differentiable almost everywhere, its derivative matches with its
distributional derivative almost everywhere, and one can achieve the function
from its derivative $\dot{x}=y$ with the help of integration formula $(i)$.

$(b)$ Given $b>0$, let $x:[0,b]\to H$ be an absolutely continuous function.
Then one can show by using Definition $\ref{def1}(iii)$ that $z=B\circ x$ is
absolutely continuous for $L$-Lipschitz continuous operator $B$. Also, $z$ is
almost everywhere differentiable and $\|\dot{z}(\cdot)\|\leq
L\|\dot{x}(\cdot)\|$ holds almost everywhere.

######  Definition 2.12.

A map $u:[0,\infty)\to H$ is said to be strong solution of (1.14), if the
following properties hold:

  * (i)

$u:[0,\infty)\to H$ is locally absolutely continuous;

  * (ii)

$\dot{u}(t)=\lambda(t)[(I-\gamma M^{-1}B)J_{\gamma A}^{M}u(t)-u(t)]$ for
almost every $t\in[0,\infty)$;

  * (iii)

$u(0)=u_{0}.$

######  Definition 2.13.

A point $u^{*}$ is said to be an equilibrium point of dynamical system (1.14)
if $u^{*}$ satisfies (1.1), i.e., $0\in(A+B)u^{*}$.

######  Definition 2.14.

[23] Let $u(t)$ be the solution of the dynamical system (1.14) and
$u(0)=u_{0}$. Then an equilibrium point $u^{*}$ is said to be

  1. (i)

globally exponentially stable if there are constants $C_{1}>0$ and $C_{2}>0$
such that

$\displaystyle\|u(t)-u^{*}\|\leq C_{1}\|u_{0}-u^{*}\|\exp(-C_{2}t)~{}\forall
t>0;$

  2. (ii)

global monotone attractor if $\|u(t)-u^{*}\|$ is nonincreasing in $t$.

######  Lemma 2.15.

[2] If $F:[0,\infty)\to[0,\infty)$ is locally absolutely continuous function,
for $1\leq p<\infty$, $1\leq r\leq\infty$, $F\in
L^{p}\left([0,\infty)\right)$, $G:[0,\infty)\to\mathbb{R}$, $G\in
L^{r}\left([0,\infty)\right)$ and for almost every $t\in[0,\infty)$

$\frac{d}{d(t)}F(t)\leq G(t),$

then $\lim\limits_{t\to\infty}F(t)=0$.

######  Lemma 2.16.

[10] Let $C$ be a nonempty subset of $H$ and $u:[0,\infty)\to H$ ba a given
map. Suppose that

  * (i)

$\lim\limits_{t\to\infty}\|u(t)-u^{*}\|$ exists, for every $u^{*}\in C$;

  * (ii)

every weak sequemtial cluster point of the map $u$ belongs to $C$.

Then there exists $u_{\infty}\in C$ such that $u(t)\rightharpoonup u_{\infty}$
as $t\to\infty$.

######  Lemma 2.17.

[7] Let $C\subseteq H$ be a nonempty closed convex set and $T:C\to H$ be a
nonexpansive mapping. Let $\\{u_{n}\\}$ be a sequence in $C$ and $u\in H$.
Suppose that $u_{n}\rightharpoonup u$ and that $u_{n}-Tu_{n}\to 0$. Then
$u\in\operatorname{Fix(T)}$.

######  Proposition 2.18.

Let $T:H\to 2^{H}$ be a set-valued operator and
$\gamma\in\mathbb{R}\setminus\\{0\\}$. Then we have the following:

  * (a)

$J_{\gamma T}^{M}=I-\gamma M^{-1}T_{\gamma}$.

  * (b)

$T_{\gamma}$ is defined everywhere and single-valued whenever $J_{\gamma
T}^{M}$ is.

  * (c)

$J_{\gamma(T_{-\gamma})}^{M}=I-\gamma M^{-1}T$.

  * (d)

$T$ is defined everywhere and single-valued whenever
$J_{\gamma(T_{-\gamma})}^{M}$ is.

###### Proof.

(a) Since $\gamma\neq 0$, we obtain

$\displaystyle y\in J_{\gamma T}^{M}x$ $\displaystyle\Leftrightarrow x\in
y+\gamma M^{-1}Ty$ $\displaystyle\Leftrightarrow\frac{x-y}{\gamma}\in
M^{-1}T(x-\gamma\frac{x-y}{\gamma})$ $\displaystyle\Leftrightarrow y\in
x-\gamma M^{-1}T_{\gamma}x.$

(b) It follows from (a).
(c) Replace $T$ by $T_{-\gamma}$ in (a), we get

$J_{\gamma(T_{-\gamma})}^{M}=I-\gamma M^{-1}(T_{-\gamma})_{\gamma}=I-\gamma
M^{-1}T.$

(d) It follows from (c).

∎

######  Proposition 2.19.

Let $A:H\to 2^{H}$ be maximally $(\gamma-\alpha)$-cohypomonotone, $B:H\to H$
be $\beta$-cocoercive and $M:H\to H$ be a strongly positive such that
$\|M^{-1}\|\leq\frac{1}{\kappa}\min\\{\alpha,\beta\\}$, where $\kappa>0$. Then
we have the following:

  * (a)

$M^{-1}B$ is $\kappa$-cocoercive.

  * (b)

$M^{-1}A_{\gamma}$ is $\kappa$-cocoercive.

###### Proof.

(a) Let $x,y\in H$. Since $B$ is $\beta$-cocercive, we have

$\langle Bx-By,x-y\rangle\geq\beta\|Bx-By\|^{2}\\\ \Leftrightarrow\langle
M^{-1}Bx-M^{-1}By,x-y\rangle_{M}\geq\beta\langle Bx-By,Bx-By\rangle.$

Also, $\kappa\|M^{-1}Bx-M^{-1}By\|_{M}=\kappa\langle Bx-By,M^{-1}(Bx-
By)\rangle$. Altogether, denoting $w:=(Bx-By)$, we obtain

(2.1) $\displaystyle\langle
M^{-1}Bx-M^{-1}By,x-y\rangle_{M}-\kappa\|M^{-1}Bx-M^{-1}By\|_{M}\geq\beta\|w\|^{2}-\kappa\langle
w,M^{-1}w\rangle.$

By the Cauchy-Schwarz inequality, we have

$\kappa\langle w,M^{-1}w\rangle\leq\kappa\|M^{-1}\|\|w\|^{2}.$

Hence, from (2.1), we get

$\displaystyle\langle
M^{-1}Bx-M^{-1}By,x-y\rangle_{M}\geq\kappa\|M^{-1}Bx-M^{-1}By\|_{M},$

which implies that $M^{-1}B$ is $\kappa$-cocoercive.

(b) From Lemma 2.8, $A_{\gamma}$ is $\alpha$-cocoercive operator. So, in the
similar manner, one can show $M^{-1}A_{\gamma}$ is $\kappa$-cocoercive. ∎

######  Proposition 2.20.

Let $\gamma\neq 0.$ Let $A:H\to 2^{H}$ be a set-valued operator such that
$J_{\gamma A}^{M}$ is single-valued and everywhere defined and $B:H\to H$ be
an operator. Define $T_{1}:=(I+\gamma M^{-1}A)^{-1}(I-\gamma M^{-1}B)$ and
$T_{2}:=(I+\gamma M^{-1}B_{-\gamma})^{-1}(I-\gamma M^{-1}A_{\gamma})$, where
$M:H\to H$ is a strongly positive operator. Then the following statements
hold:

  * (a)

$x\in\operatorname{Fix}(T_{1})\Leftrightarrow
x\in\operatorname{Zer}(A+B)\Leftrightarrow A_{\gamma}\circ(I-\gamma
M^{-1}B)x+Bx=0$.

  * (b)

$y\in\operatorname{Fix}(T_{2})\Leftrightarrow
y\in\operatorname{Zer}(B_{-\gamma}+A_{\gamma})\Leftrightarrow B\circ J_{\gamma
A}^{M}y+A_{\gamma}y=0$.

  * (c)

$I-\gamma
M^{-1}B:\operatorname{Zer}(A+B)\to\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$
is a bijection with inverse $J_{\gamma A}^{M}$.

###### Proof.

(a) Suppose that $x\in\operatorname{Zer}(A+B)$. Then

$\displaystyle 0\in(A+B)x$ $\displaystyle\Leftrightarrow 0\in\gamma
M^{-1}Ax+\gamma M^{-1}Bx$ $\displaystyle\Leftrightarrow(I-\gamma
M^{-1}B)x\in(I+\gamma M^{-1}A)x$ $\displaystyle\Leftrightarrow x=(I+\gamma
M^{-1}A)^{-1}(I-\gamma M^{-1}B)x$ $\displaystyle\Leftrightarrow
x\in\operatorname{Fix}(T_{1}).$

Also,

$\displaystyle x\in\operatorname{Fix}(T_{1})$ $\displaystyle\Leftrightarrow
x\in\operatorname{Fix}[(I-\gamma M^{-1}A_{\gamma})(I-\gamma M^{-1}B)]$
$\displaystyle\Leftrightarrow x=x-\gamma M^{-1}Bx-\gamma
M^{-1}A_{\gamma}\circ(I-\gamma M^{-1}B)x$ $\displaystyle\Leftrightarrow
A_{\gamma}\circ(I-\gamma M^{-1}B)x+Bx=0.$

In (a) apply $(B_{-\gamma},A_{\gamma})$ in place of $(A,B)$ and using
Proposition 2.18, we have the result.

Let $x\in\operatorname{Zer}(A+B)$. By (a), we have

$\displaystyle A_{\gamma}\circ(I-\gamma M^{-1}B)x+B\circ J_{\gamma
A}^{M}\circ(I-\gamma M^{-1}B)x=0$
$\displaystyle\Leftrightarrow(A_{\gamma}+B\circ J_{\gamma
A}^{M})\circ(I-\gamma M^{-1}B)x=0$ $\displaystyle\Leftrightarrow(I-\gamma
M^{-1}B)x\in\operatorname{Zer}(B_{-\gamma}+A_{\gamma}).$

Also, for $y\in\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$, from (b), we
deduce

$\displaystyle B\circ J_{\gamma A}^{M}y+A_{\gamma}\circ(I-\gamma M^{-1}B)\circ
J_{\gamma A}^{M}y=0$ $\displaystyle\Leftrightarrow(B+A_{\gamma}\circ(I-\gamma
M^{-1}B))\circ J_{\gamma A}^{M}y=0$ $\displaystyle\Leftrightarrow J_{\gamma
A}^{M}y\in\operatorname{Zer}(A+B).$

Finally, $J_{\gamma A}^{M}\circ(I-\gamma M^{-1}B)x=x$, and $(I-\gamma
M^{-1}B)\circ J_{\gamma A}^{M}y=y$ for $x\in\operatorname{Zer}(A+B)$ and
$y\in\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$. ∎

##  3\. Convergence of trajectories generated by first order dynamical systems

###  3.1. Operator Framework

Consider the dynamical system

(3.3)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{u}(t)=\lambda(t)[T(u(t))-u(t)]\\\\[6.0pt]
u(0)=u_{0},\end{array}\right.}$

where $u_{0}\in H$, $T:H\to H$ is an $\alpha$-averaged operator and
$\lambda:[0,\infty)\to[0,\infty)$ is a Lebesgue measurable function satisfying

(3.4) $\displaystyle 0<\underline{\lambda}\leq\inf\limits_{t\geq
0}\lambda(t)\leq\sup\limits_{t\geq 0}\lambda(t)\leq\overline{\lambda},$

where $\underline{\lambda},\overline{\lambda}\in\mathbb{R}$. First we
establish the following result for the dynamical system (3.3).

######  Proposition 3.1.

Let $T:H\to H$ be an $\alpha$-averaged operator for $\alpha\in(0,1)$ with
$\operatorname{Fix}(T)\neq\emptyset$. Let $u:[0,\infty)\to H$ be the unique
strong global solution of the dynamical system (3.3). Then we have the
following:

  * (i)

The trajectory $u$ is bounded and $\dot{u}$, $(I-T)u\in L^{2}([0,\infty);H)$.

  * (ii)

$\lim\limits_{t\to\infty}\dot{u}(t)=\lim\limits_{t\to\infty}(I-T)(u(t))=0$.

  * (iii)

$u(t)\rightharpoonup\bar{u}\in\operatorname{Fix}(T)$ as $t\to\infty$.

###### Proof.

(i) Let $u^{*}\in\operatorname{Fix}(T)$. Define
$k(t):=\frac{1}{2}\|u(t)-u^{*}\|^{2}$, $t\in[0,\infty)$. Then
$\dot{k}(t)=\langle u(t)-u^{*},\dot{u}(t)\rangle$. From (3.3), and the fact
that $(I-T)u^{*}=0$, we have for every $t\in[0,\infty)$

(3.5) $\displaystyle\dot{k}(t)+\lambda(t)\langle
u(t)-u^{*},(I-T)(u(t))-(I-T)u^{*}\rangle=0.$

Since $T$ is $\alpha$-averaged operator, so there exists a nonexpansive
operator $R:H\to H$ such that $T=(1-\alpha)I+\alpha R$ and
$\operatorname{Fix}(T)=\operatorname{Fix}(R)$. From (3.5) we have

(3.6) $\displaystyle\dot{k}(t)+\alpha\lambda(t)\langle
u(t)-u^{*},(I-R)(u(t))-(I-R)u^{*}\rangle=0.$

Remark 2.5 shows that $I-R$ is $\frac{1}{2}$-cocoercive operator. From (3.6)
we obtain

$\displaystyle\dot{k}(t)+\frac{\alpha\lambda(t)}{2}\|(I-R)(u(t))\|^{2}\leq 0,$

which implies that

(3.7)
$\displaystyle\dot{k}(t)+\frac{\lambda(t)}{2\alpha}\|(I-T)(u(t))\|^{2}\leq 0.$

From (3.3) and $(\ref{2.4a})$, we have

$\displaystyle\dot{k}(t)+\frac{1}{2\alpha\lambda(t)}\|\dot{u}(t)\|^{2}\leq
0~{}for~{}all~{}t\in[0,\infty).$

Using condition (3.4), we get

(3.8)
$\displaystyle\dot{k}(t)+\frac{1}{2\alpha\overline{\lambda}}\|\dot{u}(t)\|^{2}\leq
0~{}for~{}every~{}t\in[0,\infty).$

From (3.8), we get that the function $t\mapsto k(t)$ is monotonically
decreasing. Also, the map $t\mapsto k(t)$ is locally absolutely continuous.
Hence there exists $N_{1}\in\mathbb{R}$ such that

$k(t)\leq N_{1}~{}for~{}all~{}t\in[0,\infty).$

Thus, $k$ is bounded, and hence $u$ is bounded.
To integrate the inequality (3.8), we get that there is $N_{2}\in\mathbb{{R}}$
such that

(3.9) $\displaystyle
k(t)+\frac{1}{2\alpha\overline{\lambda}}\int_{0}^{t}\|\dot{u}(t)\|^{2}\leq
N_{2}~{}~{}for~{}all~{}t\in[0,\infty).$

Since $k$ is bounded, so from (3.9), we conclude that $\dot{u}(t)\in
L^{2}([0,\infty);H)$. Hence, from $(\ref{2.10})$ and (3.4), we get that
$(I-T)u\in L^{2}([0,\infty);H)$.

(ii) Using Remark 2.11(b) and cocoercivity of $I-R$, we have

$\displaystyle\frac{d}{dt}\left(\frac{1}{2}\|(I-T)(u(t))\|^{2}\right)$
$\displaystyle=\frac{d}{dt}\alpha^{2}\left(\frac{1}{2}\|(I-R)(u(t))\|^{2}\right)$
$\displaystyle=\alpha^{2}\left\langle(I-R)(u(t)),\frac{d}{dt}((I-R)(u(t)))\right\rangle$
$\displaystyle\leq\frac{\alpha^{2}}{2}\|(I-R)(u(t))\|^{2}+2\alpha^{2}\|\dot{u}(t)\|^{2}$
$\displaystyle=\frac{1}{2}\|(I-T)(u(t))\|^{2}+2\alpha^{2}\|\dot{u}(t)\|^{2}~{}~{}\forall
t\in[0,\infty).$

By using Lemma 2.15 and part (i) we obtain
$\lim\limits_{t\to\infty}(I-T)(u(t))=0$, and from (3.4) and (3.5), we conclude
that $\lim\limits_{t\to\infty}\dot{u}(t)=0.$

(iii) We show that both the assumptions of Lemma 2.16 are satisfied.
As $k$ is bounded and $t\mapsto k(t)$ is monotonically decreasing, we observe
that $\lim\limits_{t\to\infty}k(t)$ exists and belongs to the set of real
numbers. So, $\lim\limits_{t\to\infty}\|u(t)-u^{*}\|$ exists.
Let $\bar{u}$ be a weak sequential cluster point of $u(t)$, i.e, there exists
a sequence $t_{n}\to\infty$ (as $n\to\infty$) such that
$\\{u(t_{n})\\}\rightharpoonup\bar{u}$. Using Lemma 2.17 and part (ii), we
deduce that $\bar{u}\in\operatorname{Fix(R)}=\operatorname{Fix(T)}$ and the
conclusion follows. ∎

In order to study the convergence behaviour of trajectories generated by
dynamical systems (1.14) and (1.17), we need the following assumptions:

  * (A1)

The operator $A:H\to 2^{H}$ is maximally $(\gamma-\alpha)$-cohypomonotone.

  * (A2)

The operator $B:H\to H$ is $\beta$-cocoercive.

  * (A3)

$\operatorname{Zer}(A+B)\neq\emptyset$.

  * (A4)

The operator $M:H\to H$ is strongly positive.

Now we are ready to establish weak and strong convergence of trajectories
generated by backward-forward first order dynamical system (1.14).

######  Theorem 3.2.

Let assumptions (A1), (A2), (A3) and (A4) hold and
$\lambda:[0,\infty)\to[0,\infty)$ be a Lebesgue measurable function satisfying
condition (3.4). Let $u:[0,\infty)\to H$ be the unique strong solution of
(1.14) and $u_{0}\in H$. Let $\gamma\in(0,2\kappa)$, where
$\kappa\in(0,\infty)$ such that

(3.10) $\displaystyle\kappa\leq\|M\|\min\\{\alpha,\beta\\}.$

Set $\delta:=\frac{2\kappa+\gamma}{2\gamma}$. Then the following statements
hold:

  * (i)

The trajectory $u$ is bounded and $\dot{u}$, $(I-(I-\gamma M^{-1}B)\circ
J_{\gamma A}^{M})u\in L^{2}([0,\infty);H)$.

  * (ii)

$\lim\limits_{t\to\infty}\dot{u}(t)=\lim\limits_{t\to\infty}(I-(I-\gamma
M^{-1}B)\circ J_{\gamma A}^{M})(u(t))=0$.

  * (iii)

$u(t)\rightharpoonup u^{*}\in$ $\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$ as
$t\to\infty$.

  * (iv)

If $u^{*}\in\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$, then
$A_{\gamma}(u(\cdot))-A_{\gamma}u^{*}\in L^{2}([0,\infty);H)$ and
$\lim\limits_{t\to\infty}A_{\gamma}(u(t))=A_{\gamma}u^{*}$.

  * (v)

$M^{-1}A_{\gamma}$ is constant on
$\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$.

  * (vi)

$v(t)\rightharpoonup v^{*}\in$ $\operatorname{Zer}(A+B)$ as $t\to\infty$,
where $v(t)=J_{\gamma A}^{M}(u(t))$.

  * (vii)

$\lim\limits_{t\to\infty}B(v(t))=Bv^{*}=-A_{\gamma}u^{*}$.

  * (viii)

If $B_{-\gamma}$ or $A_{\gamma}$ is uniformly monotone, then $u(t)\to
u^{*}\in\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$.

  * (ix)

If $B_{-\gamma}$ is $\rho$-strongly monotone for $\rho>0$ and choose $\eta>0$
fulfilling the condition:

(3.11)
$\displaystyle\frac{1}{2\alpha}+\frac{\eta\|M\|^{2}}{2\gamma^{2}}\leq\rho+\frac{\|M\|\overline{\lambda}}{\gamma}.$

Let $u^{*}$ be an equilibrium point of dynamical system (1.14). Then we have
the following:

    * (a)

If $\frac{1}{\eta\rho}<4$, then $u^{*}$ is globally exponentially stable.

    * (b)

If $\frac{1}{\eta\rho}=4$, then $u^{*}$ is global monotone attractor.

###### Proof.

(i)-(iii) We can write the dynamical system (1.14) in the form

(3.14)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{u}(t)=\lambda(t)[T(u(t))-u(t)]\\\\[6.0pt]
u(0)=u_{0},\end{array}\right.}$

where $T=(I-\gamma M^{-1}B)(I+\gamma M^{-1}A)^{-1}.$ From Proposition 2.18,
$T=(I-\gamma M^{-1}B)(I-\gamma M^{-1}A_{\gamma})$. Since both $M^{-1}B$ and
$M^{-1}A_{\gamma}$ are $\kappa$-cocoercive. From Lemma 2.6, both $I-\gamma
M^{-1}B$ and $I-\gamma M^{-1}A_{\gamma}$ are
$\frac{\gamma}{2\kappa}$-averaged. Hence, by Lemma 2.7, $T$ is
$\frac{1}{\delta}$-averaged. Now, the statements (i)-(iii) follow from
Propositions 3.1 and 2.20, by observing that
$\operatorname{Fix}(T)=\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$.

(iv) Let $u^{*}\in\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$. From (1.14), we
have

$\displaystyle\frac{\dot{u}(t)}{\lambda(t)}+u(t)=(I-\gamma M^{-1}B)(I-\gamma
M^{-1}A_{\gamma})(u(t)),$

which implies that

$\displaystyle-\frac{\dot{u}(t)}{\gamma\lambda(t)}-M^{-1}A_{\gamma}(u(t))=M^{-1}B(I-\gamma
M^{-1}A_{\gamma})(u(t)).$

From Proposition 2.20(b), we have

$\displaystyle-M^{-1}A_{\gamma}u^{*}=M^{-1}B(I-\gamma M^{-1}A_{\gamma})u^{*}.$

Since operators $M^{-1}B$ and $M^{-1}A_{\gamma}$ are $\kappa$-cocoercive, we
deduce for every $t\in[0,\infty)$

$\displaystyle\kappa\left\|-\frac{\dot{u}(t)}{\gamma\lambda(t)}-M^{-1}(A_{\gamma}(u(t))-A_{\gamma}u^{*})\right\|^{2}$
$\displaystyle~{}~{}~{}\leq\left\langle-\frac{\dot{u}(t)}{\gamma\lambda(t)}-M^{-1}(A_{\gamma}(u(t))-A_{\gamma}u^{*}),u(t)-u^{*}-\gamma(M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*})\right\rangle$
$\displaystyle~{}~{}~{}=\left\langle-\frac{\dot{u}(t)}{\gamma\lambda(t)},u(t)-u^{*}-\gamma(M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*})\right\rangle$
$\displaystyle~{}~{}~{}-\langle
M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*},u(t)-u^{*}\rangle+\gamma\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}$
$\displaystyle~{}~{}~{}\leq\left\langle-\frac{\dot{u}(t)}{\gamma\lambda(t)},u(t)-u^{*}\right\rangle+\gamma\left\langle\frac{\dot{u}(t)}{\gamma\lambda(t)},M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\right\rangle$
(3.15)
$\displaystyle~{}~{}~{}-\kappa\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}+\gamma\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}.$

Also,

$\displaystyle\kappa\left\|-\frac{\dot{u}(t)}{\gamma\lambda(t)}-M^{-1}(A_{\gamma}(u(t))-A_{\gamma}u^{*})\right\|^{2}=$
$\displaystyle\kappa\left\|\frac{\dot{u}(t)}{\gamma\lambda(t)}\right\|^{2}+\kappa\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}$
(3.16)
$\displaystyle+2\kappa\left\langle\frac{\dot{u}(t)}{\gamma\lambda(t)},M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\right\rangle.$

From (3.15) and (3.16), we have

$\displaystyle(2\kappa-\gamma)\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}$
$\displaystyle~{}~{}~{}\leq(\gamma-2\kappa)\left\langle\frac{\dot{u}(t)}{\gamma\lambda(t)},M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\right\rangle$
$\displaystyle~{}~{}~{}-\left\langle\frac{\dot{u}(t)}{\gamma\lambda(t)},u(t)-u^{*}\right\rangle-\frac{\kappa}{\gamma^{2}\lambda^{2}(t)}\|\dot{u}(t)\|^{2}$
$\displaystyle~{}~{}~{}\leq\frac{(\gamma-2\kappa)}{2}\left\|\frac{\dot{u}(t)}{\gamma\lambda(t)}\right\|^{2}+\frac{(\gamma-2\kappa)}{2}\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}$
$\displaystyle~{}~{}~{}-\left\langle\frac{\dot{u}(t)}{\gamma\lambda(t)},u(t)-u^{*}\right\rangle$
$\displaystyle~{}~{}~{}\leq\frac{\gamma}{2}\left\|\frac{\dot{u}(t)}{\gamma\lambda(t)}\right\|^{2}+\frac{(\gamma-2\kappa)}{2}\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}-\frac{\kappa}{\gamma^{2}\lambda^{2}(t)}\|\dot{u}(t)\|^{2}$
$\displaystyle~{}~{}~{}-\left\langle\frac{\dot{u}(t)}{\gamma\lambda(t)},u(t)-u^{*}\right\rangle$
$\displaystyle~{}~{}~{}\leq\frac{\gamma}{2}\left\|\frac{\dot{u}(t)}{\gamma\lambda(t)}\right\|^{2}+\frac{(\gamma-2\kappa)}{2}\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}-\left\langle\frac{\dot{u}(t)}{\gamma\lambda(t)},u(t)-u^{*}\right\rangle.$

By using the function $k:[0,\infty)\to\mathbb{R}$,
$k(t)=\frac{1}{2}\|u(t)-u^{*}\|^{2}$ and the fact that $\dot{k}(t)=\langle
u(t)-u^{*},\dot{u}(t)\rangle$, we obtain

$\displaystyle\left((2\kappa-\gamma)-\frac{(2\kappa-\gamma)}{2}\right)\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}+\frac{1}{\gamma\lambda(t)}\dot{k}(t)\leq$
$\displaystyle\frac{1}{2\gamma\lambda^{2}(t)}\left\|\dot{u}(t)\right\|^{2}.$

Taking into accounts the bounds of $\lambda$, we deduce for every
$t\in[0,\infty)$

$\displaystyle\left(\frac{2\kappa-\gamma}{2}\right)\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}+\frac{1}{\gamma\bar{\lambda}}\dot{k}(t)\leq\frac{1}{2\gamma\overline{\lambda}^{2}}\left\|\dot{u}(t)\right\|^{2}.$

Integrating above equation from $0$ to $\tau$, we get that for every
$\tau\in[0,\infty)$

$\displaystyle\left(\frac{2\kappa-\gamma}{2}\right)\int_{0}^{\tau}\|M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\|^{2}dt+\frac{1}{\gamma\bar{\lambda}}(k(\tau)-k(0))$
$\displaystyle\leq\frac{1}{2\gamma\overline{\lambda}^{2}}\int_{0}^{\tau}\|\dot{u}(t)\|^{2}.$

Since $\dot{u}\in L^{2}([0,\infty);H)$, $\gamma\in(0,2\kappa)$, and
$k(\tau)\geq 0$ for every $\tau\in[0,\infty)$, it follows that
$M^{-1}A_{\gamma}(u(t))-M^{-1}A_{\gamma}u^{*}\in L^{2}([0,\infty);H)$ and
hence $A_{\gamma}(u(t))-A_{\gamma}u^{*}\in L^{2}([0,\infty);H)$.
From Remark 2.11, we get

$\displaystyle\frac{d}{dt}\left(\frac{1}{2}\|A_{\gamma}(u(t))-A_{\gamma}u^{*}\|\right)$
$\displaystyle=\left\langle
A_{\gamma}(u(t))-A_{\gamma}u^{*},\frac{d}{dt}\left(A_{\gamma}(u(t))\right)\right\rangle$
$\displaystyle\leq\frac{1}{2}\|A_{\gamma}(u(t))-A_{\gamma}u^{*}\|^{2}+\frac{1}{2\alpha^{2}}\|\dot{u}(t)\|^{2},$

and by Lemma 2.15, we have
$\lim\limits_{t\to\infty}A_{\gamma}(u(t))=A_{\gamma}u^{*}$.

(v) Let $u$ and $v$ are two zeros of $B_{-\gamma}+A_{\gamma}$, by Proposition
2.20, we obtain $-M^{-1}A_{\gamma}u=M^{-1}B(u-\gamma M^{-1}A_{\gamma}u)$ and
$-M^{-1}A_{\gamma}v=M^{-1}B(v-\gamma M^{-1}A_{\gamma}v)$. Since $M^{-1}B$ is
$\kappa$-cocoercive, we have

$\displaystyle\kappa\|-M^{-1}A_{\gamma}v+M^{-1}A_{\gamma}u\|^{2}$
$\displaystyle\leq\langle-M^{-1}A_{\gamma}v+M^{-1}A_{\gamma}u,v-\gamma
M^{-1}A_{\gamma}v-u+\gamma M^{-1}A_{\gamma}u\rangle$
$\displaystyle=\gamma\|-M^{-1}A_{\gamma}v+M^{-1}A_{\gamma}u\|^{2}-\langle-M^{-1}A_{\gamma}v+M^{-1}A_{\gamma}u,-v+u\rangle.$

By using $\kappa$-cocerciveness of $M^{-1}A_{\gamma}$, we obtain

$\displaystyle\kappa\|-M^{-1}A_{\gamma}v+M^{-1}A_{\gamma}u\|^{2}$
$\displaystyle\leq-\kappa\|-M^{-1}A_{\gamma}v+M^{-1}A_{\gamma}u\|^{2}$
$\displaystyle+\gamma\|-M^{-1}A_{\gamma}v+M^{-1}A_{\gamma}u\|^{2}.$

Since $\gamma<2\kappa$, so we get that
$\|M^{-1}A_{\gamma}u-M^{-1}A_{\gamma}v\|^{2}=0$. Hence $M^{-1}A_{\gamma}$ is
constant on $\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$.

(vi) From statements (iii), (iv) and Proposition 2.18, we have

$v(t)=J_{\gamma A}^{M}(u(t))=u(t)-\gamma M^{-1}A_{\gamma}(u(t))$

converges weakly to $v^{*}=u^{*}-\gamma M^{-1}Bu^{*}$. From fixed point
equality, we get

$u^{*}=(I-\gamma M^{-1}B)\circ J_{\gamma A}^{M}u^{*}=(I-\gamma M^{-1}B)v^{*}.$

Enjoying the operator $J_{\gamma A}^{M}$ with the equality $u^{*}=(I-\gamma
M^{-1}B)v^{*}$ gives $v^{*}=J_{\gamma A}^{M}\circ(I-\gamma M^{-1}B)v^{*}$,
which declares that $v^{*}$ is fixed point of $J_{\gamma A}^{M}\circ(I-\gamma
M^{-1}B)$, hence a zero of $A+B$.

(vii) Note,

$\displaystyle M^{-1}B(v(t))$
$\displaystyle=\frac{1}{\gamma}\left(v(t)-J_{\gamma
B_{-\gamma}}^{M}(v(t))\right)$
$\displaystyle=-M^{-1}A_{\gamma}u(t)+\frac{1}{\gamma}\left(v(t)-J_{\gamma
B_{-\gamma}}^{M}(v(t))\right),$

which conclude that $B(v(t))\to-A_{\gamma}u^{*}$ as $t\to\infty$. Finally,
from statement (vi) we deduce that $-A_{\gamma}u^{*}=Bv^{*}$.
(viii) Assume that $B_{-\gamma}$ is uniformly monotone with modulus function
$\phi_{B_{-\gamma}}:[0,\infty)\to[0,\infty)$. Let
$u^{*}\in\operatorname{Zer}(B_{-\gamma}+A_{\gamma})$ be an unique element.
From Proposition 2.18, we have for every $t\in[0,\infty)$

(3.17) $\displaystyle-M\frac{\dot{u}(t)}{\gamma\lambda(t)}-A_{\gamma}(u(t))\in
B_{-\gamma}\left(\frac{\dot{u}(t)}{\lambda(t)}+u(t)\right).$

Combining (3.17) with $-A_{\gamma}u^{*}\in B_{-\gamma}u^{*}$, we have for
every $t\in[0,\infty)$

$\displaystyle\phi_{B_{-\gamma}}\left(\left\|\frac{\dot{u}(t)}{\lambda(t)}+u(t)-u^{*}\right\|\right)$
$\displaystyle\leq\left\langle\frac{\dot{u}(t)}{\lambda(t)}+u(t)-u^{*},A_{\gamma}u^{*}-M\frac{\dot{u}(t)}{\gamma\lambda(t)}-A_{\gamma}(u(t))\right\rangle$
$\displaystyle=\left\langle\frac{\dot{u}(t)}{\lambda(t)},A_{\gamma}u^{*}-M\frac{\dot{u}(t)}{\gamma\lambda(t)}-A_{\gamma}(u(t))\right\rangle+\left\langle
u(t)-u^{*},M\frac{\dot{u}(t)}{\gamma\lambda(t)}\right\rangle$
$\displaystyle-\langle u(t)-u^{*},A_{\gamma}(u(t))-A_{\gamma}u^{*}\rangle,$

which combines with the monotonicity of $A_{\gamma}$ yields

$\displaystyle\phi_{B_{-\gamma}}\left(\left\|\frac{\dot{u}(t)}{\lambda(t)}+u(t)-u^{*}\right\|\right)$
$\displaystyle\leq\left\langle\frac{\dot{u}(t)}{\lambda(t)},A_{\gamma}u^{*}-A_{\gamma}(u(t))\right\rangle-\frac{1}{\lambda^{2}(t)\gamma}\|\dot{u}(t)\|_{M}^{2}+\left\langle
u(t)-u^{*},M\frac{\dot{u}(t)}{\gamma\lambda(t)}\right\rangle$ (3.18)
$\displaystyle\leq\left\langle\frac{\dot{u}(t)}{\lambda(t)},A_{\gamma}u^{*}-A_{\gamma}(u(t))\right\rangle+\left\langle
u(t)-u^{*},M\frac{\dot{u}(t)}{\gamma\lambda(t)}\right\rangle.$

From parts (i)-(iv), (3.18) and the fact that $\lambda$ is bounded by positive
constants, we get

$\displaystyle\lim\limits_{t\to\infty}\phi_{B_{-\gamma}}\left(\left\|\frac{\dot{u}(t)}{\lambda(t)}+u(t)-u^{*}\right\|\right)=0.$

Since the function $\phi_{B_{-\gamma}}$ is increasing vanishes to 0, so we
have

$\left(\frac{\dot{u}(t)}{\lambda(t)}+u(t)-u^{*}\right)\to
0~{}\text{as}~{}t\to\infty.$

Using statement (ii) and the boundedness of $\lambda$ we get that $u(t)$
converges strongly to $u^{*}$ as $t\to\infty$.

(ix) Suppose that $B_{-\gamma}$ is $\rho$-strong monotonicity. Combining
(3.17) with $-A_{\gamma}u^{*}\in B_{-\gamma}u^{*}$, we have

$\displaystyle\rho\left\|\frac{\dot{u}(t)}{\lambda(t)}+u(t)-u^{*}\right\|^{2}$
$\displaystyle\leq\left\langle\frac{\dot{u}(t)}{\lambda(t)}+u(t)-u^{*},A_{\gamma}u^{*}-\frac{M}{\gamma\lambda(t)}\dot{u}(t)-A_{\gamma}(u(t))\right\rangle$
$\displaystyle=\left\langle\frac{\dot{u}(t)}{\lambda(t)},A_{\gamma}u^{*}-\frac{M}{\gamma\lambda(t)}\dot{u}(t)-A_{\gamma}(u(t))\right\rangle+\left\langle
u(t)-u^{*},A_{\gamma}u^{*}-\frac{M}{\gamma\lambda(t)}\dot{u}(t)-A_{\gamma}(u(t))\right\rangle$
$\displaystyle\leq\frac{1}{2\alpha\lambda^{2}(t)}\|\dot{u}(t)\|^{2}+\frac{\alpha}{2}\|A_{\gamma}u^{*}-A_{\gamma}(u(t))\|^{2}-\frac{1}{\gamma\lambda(t)}\|\dot{u}(t)\|_{M}^{2}+\left\langle
u(t)-u^{*},\frac{-M}{\gamma\lambda(t)}\dot{u}(t)\right\rangle$
$\displaystyle-\langle u(t)-u^{*},A_{\gamma}(u(t))-A_{\gamma}u^{*}\rangle.$

Using $\alpha$-cocoercivity of $A_{\gamma}$, we have

$\displaystyle\rho\left\|\frac{\dot{u}(t)}{\lambda(t)}+u(t)-u^{*}\right\|^{2}$
$\displaystyle\leq\frac{1}{2\alpha\lambda^{2}(t)}\|\dot{u}(t)\|^{2}-\frac{1}{\gamma\lambda(t)}\|\dot{u}(t)\|_{M}^{2}+\left\langle
u(t)-u^{*},\frac{-M}{\gamma\lambda(t)}\dot{u}(t)\right\rangle$
$\displaystyle\leq\frac{1}{2\alpha\lambda^{2}(t)}\|\dot{u}(t)\|^{2}-\|M\|\frac{1}{\gamma\lambda(t)}\|\dot{u}(t)\|^{2}+\frac{1}{2\eta}\|u(t)-u^{*}\|^{2}+\frac{\eta\|M\|^{2}}{2\gamma^{2}\lambda^{2}(t)}\|\dot{u}(t)\|^{2}$
(3.19)
$\displaystyle=\left(\frac{1}{2\alpha\lambda^{2}(t)}-\frac{\|M\|}{\gamma\lambda(t)}+\frac{\eta\|M\|^{2}}{2\gamma^{2}\lambda^{2}(t)}\right)\|\dot{u}(t)\|^{2}+\frac{1}{2\eta}k(t).$

Also,

(3.20)
$\displaystyle\rho\left\|\frac{\dot{u}(t)}{\lambda(t)}+u(t)-u^{*}\right\|^{2}$
$\displaystyle=\frac{\rho}{\lambda^{2}(t)}\|\dot{u}(t)\|^{2}+\frac{2\rho}{\lambda(t)}\dot{k}(t)+2\rho
k(t).$

From (3.19) and (3.20), we have for $t\in[0,\infty)$

$\displaystyle\frac{2\rho}{\lambda(t)}\dot{k}(t)+\left(2\rho-\frac{1}{2\eta}\right)k(t)+\left(\frac{\rho}{\lambda^{2}(t)}-\frac{1}{2\alpha\lambda^{2}(t)}+\frac{\|M\|}{\gamma\lambda(t)}-\frac{\eta\|M\|^{2}}{2\gamma^{2}\lambda^{2}(t)}\right)\|\dot{u}(t)\|^{2}\leq
0.$

So, from (3.11) and the fact that $\lambda(t)$ is bounded, we have for every
$t\in[0,\infty)$

$\displaystyle\frac{2\rho}{\underline{\lambda}}\dot{k}(t)+\left(2\rho-\frac{1}{2\eta}\right)k(t)\leq
0,$

which implies that

$\displaystyle\dot{k}(t)+\frac{2\rho-\frac{1}{2\eta}}{\frac{2\rho}{\underline{\lambda}}}k(t)\leq
0.$

Now we have two cases:

  1. (a)

If $\frac{1}{\eta\rho}<4$, then we have

(3.21) $\displaystyle\dot{k}(t)+Ck(t)\leq 0,$

for every $t\in[0,\infty)$, where
$C:=\underline{\lambda}\left(1-\frac{1}{4\eta\rho}\right)>0$. Now, by
multiplying $\exp(Ct)$ with (3.21) and integrating between $0$ and $\tau$,
where $\tau\geq 0$, we have the desired result.

  2. (b)

If $\frac{1}{\eta\rho}=4$, then $\dot{k}(t)\leq 0$, so $u^{*}$ is global
monotone attractor.

∎

We now study the convergence of trajectories generated by forward-backward
first order dynamical system (1.17). Before analyzing the results based on the
dynamical system (1.17), we need the following proposition.

######  Proposition 3.3.

Let $\gamma\neq 0$ and $\lambda:[0,\infty)\to[0,\infty)$ be a function. Let
$A:H\to 2^{H}$ be a set-valued operator such that $J_{\gamma A}^{M}$ is
everywhere defined and single-valued and $B:H\to H$. Then we have the
following:

  * (i)

For each $u_{0}\in H$, dynamical system (1.14) applied to $(A,B)$ is uniquely
defines a function $u(t)$. For $x_{0}\in H$, dynamical system (1.17) applied
to $(B_{-\gamma},A_{\gamma})$ is uniquely defines a function $x(t)$. Moreover,
if $u_{0}=x_{0}$, then $u(t)=x(t)$ for $t\in[0,\infty)$.

  * (ii)

For each $x_{0}\in H$, dynamical system (1.17) applied to $(A,B)$ is uniquely
defines a function $x(t)$. For $u_{0}\in H$, dynamical system (1.14) applied
to $(B_{-\gamma},A_{\gamma})$ is uniquely defines a function $u(t)$. Moreover,
if $u_{0}=x_{0}$, then $u(t)=x(t)$ for $t\in[0,\infty)$.

###### Proof.

(i) The uniqueness of the dynamical system (1.17) follows from the hypothesis
(as $A_{\gamma}$ and $J_{\gamma B_{-\gamma}}^{M}$ are single-valued and
everywhere defined). If we consider $T=B_{-\gamma}$ in Proposition 2.18(ii)
and $T=A_{\gamma}$ in Proposition 2.18(a), then the system (1.14) is uniquely
defined and satisfies the same relation as $(\ref{meq2})$ does.

(ii) It follows from (i), by using $(A,B)$ in place of
$(B_{-\gamma},A_{\gamma})$. ∎

######  Theorem 3.4.

Let assumptions (A1), (A2), (A3) and (A4) hold and
$\lambda:[0,\infty)\to[0,\infty)$ be a Lebesgue measurable function satisfying
condition (3.4). Let $x:[0,\infty)\to H$ be the unique strong solution of
(1.17) and $x_{0}\in H$. Let $\gamma\in(0,2\kappa)$, where
$\kappa\in(0,\infty)$ satisfying (3.10). Set
$\delta:=\frac{2\kappa+\gamma}{2\gamma}$. Then the following statements hold:

  * (i)

The trajectory $x$ is bounded and $\dot{x}$, $(I-J_{\gamma
A}^{M}\circ(I-\gamma M^{-1}B))x\in L^{2}([0,\infty);H)$.

  * (ii)

$\lim\limits_{t\to\infty}\dot{x}(t)=\lim\limits_{t\to\infty}(I-J_{\gamma
A}^{M}\circ(I-\gamma M^{-1}B))(x(t))=0$.

  * (iii)

$x(t)\rightharpoonup x^{*}\in$ $\operatorname{Zer}(A+B)$ as $t\to\infty$.

  * (iv)

If $x^{*}\in\operatorname{Zer}(A+B)$, then $B(x(\cdot))-Bx^{*}\in
L^{2}([0,\infty);H)$, $\lim\limits_{t\to\infty}B(x(t))=Bx^{*}$.

  * (v)

$y(t)\rightharpoonup y^{*}\in$ $\operatorname{Zer}(A+B)$ as $t\to\infty$,
where $y(t)=(I-\gamma M^{-1}B)(x(t))$.

  * (vi)

$\lim\limits_{t\to\infty}A_{\gamma}(y(t))=A_{\gamma}y^{*}=-Bx^{*}$.

  * (vii)

If $A$ or $B$ is uniformly monotone, then $x(t)$ converges strongly to a
unique point in $\operatorname{Zer}(A+B)$ as $t\to\infty$.

  * (viii)

If $A$ is $\rho$-strongly monotone for some $\rho>0$ and $\eta>0$ such that:

$\displaystyle\frac{1}{2\alpha}+\frac{\eta\|M\|^{2}}{2\gamma^{2}}\leq\rho+\frac{\|M\|\overline{\lambda}}{\gamma}.$

Let $x^{*}$ be an equilibrium point of dynamical system (1.17). Then we have
the following:

    * (a)

If $\frac{1}{\eta\rho}<4$, then $x^{*}$ is globally exponentially stable.

    * (b)

If $\frac{1}{\eta\rho}=4$, then $x^{*}$ is global monotone attractor.

###### Proof.

By Proposition 3.3, dynamical system (1.17) applied to
$(B_{-\gamma},A_{\gamma})$ is system (1.14) applied to $(A,B)$. Since the pair
$(A,B)$ satisfies the same assumption as $(B_{-\gamma},A_{\gamma})$, so by
using $(A,B)$ instead of $(B_{-\gamma},A_{\gamma})$ in Theorem 3.2, we have
the desired result. ∎

###  3.2. Function framework

In this section, to study the optimization problem (1.18) we consider the
following first order dynamical system

(3.24)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{u}(t)=\lambda(t)[(I-\gamma
M^{-1}\nabla g)\operatorname{prox}_{\gamma f}^{M}~{}u(t)-u(t)]\\\\[6.0pt]
u(0)=u_{0},\end{array}\right.}$

where $u_{0}\in H$ and $\lambda:[0,\infty)\to[0,\infty)$ is Lebesgue
measurable function.

To study the convergence behavior of trajectories generated by the dynamical
system (3.24), we need the following assumptions:

  * (B1)

$f:H\to\mathbb{R}\cup\\{\infty\\}$ is proper, convex, and lower-semicontinuous
function.

  * (B2)

$g:H\to\mathbb{R}$ is differentiable and its gradient $\nabla g$ is
$\beta$-cocoercive function.

  * (B3)

$\operatorname{Argmin}(f+g)\neq\emptyset.$

######  Remark 3.5.

Considering $\alpha=\gamma$, $A=\partial f$, $B=\nabla g$, assumptions (A1)
and (A2) hold. Since $A$ is maximal monotone, the operator $A_{\gamma}$ is
defined everywhere, single-valued and $\gamma$-cocoercive [7]. Moreover, by
the Moreau-Rockafellar theorem [25], one has $\partial(f+g)=\partial f+\nabla
g$, so $\operatorname{Zer}(A+B)=\operatorname{Zer}(\partial f+\nabla
g)=\operatorname{Zer}(\partial(f+g))=\text{Argmin}(f+g)\neq\emptyset$.

If Moreau envelope of $f$ is denoted by $f_{\gamma}$, then $\nabla
f_{\gamma}=(I-\operatorname{prox}_{\gamma f})/\gamma$ [7]. So, $\nabla
f_{\gamma}=A_{\gamma}$.

The proximity operator of a proper, lower semicontinuous and convex function
$f$ relative to the metric induced by strongly positive operator $M$ is (see
[18])

$\displaystyle\operatorname{prox}_{\gamma f}^{M}(u)=\underset{v\in
H}{\operatorname{Argmin}}\left\\{f(v)+\frac{1}{2\gamma}\|v-u\|^{2}_{M}\right\\}.$

Note that $J_{\gamma\partial f}^{M}=\operatorname{prox}_{\gamma
f}^{M}=(I+\gamma M^{-1}\partial f)^{-1}$.

######  Theorem 3.6.

Let assumptions (B1), (B2), (B3) and (A4) hold,
$\lambda:[0,\infty)\to[0,\infty)$ be a Lebesgue measurable function satisfying
condition (3.4), $u_{0}\in H$ and $u:[0,\infty)\to H$ be the unique strong
global solution of dynamical system (3.24). Let $\gamma\in(0,2\kappa)$, where
$0<\kappa\leq\|M\|\beta$ and $\|M\|\geq\frac{1}{2}$. Set
$\delta:=\frac{2\kappa+\gamma}{2\gamma}$. Then the following statements hold:

  * (i)

The trajectory $u$ is bounded and $\dot{u}$, $(I-(I-\gamma M^{-1}\nabla
g)\operatorname{prox}_{\gamma f}^{M})u\in L^{2}([0,\infty);H)$.

  * (ii)

$\lim\limits_{t\to\infty}\dot{u}(t)=\lim\limits_{t\to\infty}(I-(I-\gamma
M^{-1}\nabla g)\operatorname{prox}_{\gamma f}^{M})(u(t))=0$.

  * (iii)

$u(t)\rightharpoonup u^{*}$.

  * (iv)

$\lim\limits_{t\to\infty}\nabla f_{\gamma}(u(t))=\nabla f_{\gamma}(u^{*})$.

  * (v)

$v(t)\rightharpoonup v^{*}\in\operatorname{Argmin}(f+g)$, where
$v(t)=\operatorname{prox}_{\gamma f}^{M}(u(t))$.

  * (vi)

$\lim\limits_{t\to\infty}\nabla g(v(t))=\nabla g(v^{*})=-\nabla
f_{\gamma}(u^{*})$.

  * (vii)

$(f+g)(v(t))\to(f+g)(v^{*})=\inf(f+g)$, $f(v(t))\to f(v^{*})$ and $g(v(t))\to
g(v^{*})$.

###### Proof.

The statements (i) through (vi) follow from Theorem 3.2 and Remark 3.5, by
taking $A:=\partial f$ and $B:=\nabla g$.

(vii) First, we show that $(f+g)(v(t))\to(f+g)(v^{*})$. Since $(f+g)$ is lower
semicontinuity and $v(t)\rightharpoonup v^{*}$, we have
$(f+g)(v^{*})\leq\liminf(f+g)(v(t))$. Suppose that $w(t)=v(t)-\gamma
M^{-1}\nabla g(v(t))$ and since $v(t)=\operatorname{prox}_{\gamma
f}^{M}(u(t))$, we have

(3.25) $\displaystyle u(t)\in v(t)+\gamma M^{-1}\partial f(v(t))$
$\displaystyle=w(t)+\gamma M^{-1}\nabla g(v(t))+\gamma M^{-1}\partial f(v(t))$
$\displaystyle=w(t)+\gamma M^{-1}\partial(f+g)(v(t)).$

So, from the definition of subgradient of $f+g$ at $v(t)$

$\displaystyle(f+g)(v^{*})\geq(f+g)(v(t))+\langle
v^{*}-v(t),\frac{M}{\gamma}(u(t)-w(t))\rangle. $

In view of (ii), $(f+g)(v^{*})\geq\limsup(f+g)(v(t))$. So, we have
$(f+g)(v(t))\to(f+g)(v^{*})$.
Secondly, we show that $f(v(t))\to f(v^{*})$ as $t\to\infty$. From statement
(v) and the fact that $f$ is lower semicontinuous, we obtain
$f(v^{*})\leq\liminf f(v(t))$. Also, from (3.25) and by the definition of
subgradient of $f$ at point $v(t)$, we have

$\displaystyle f(v^{*})$ $\displaystyle\geq f(v(t))+\langle
v^{*}-v(t),\frac{M}{\gamma}(u(t)-w(t)-\nabla g(v(t)))\rangle$
$\displaystyle\geq f(v(t))+\langle
v^{*}-v(t),\frac{M}{\gamma}(u(t)-w(t)-(\nabla g(v(t))-\nabla
g(v^{*})))\rangle$ $\displaystyle+\langle v^{*}-v(t),-\nabla g(v^{*})\rangle.$

From statements (ii), (v) and (vi), we have $f(v^{*})\geq\limsup f(v(t))$,
which shows our desired result. ∎

Suppose $\gamma\in\mathbb{R}$. Define the function
$l:H\to\mathbb{R}\cup\\{\infty\\}$, by
$l_{\gamma}^{*}=\left(l^{*}+\frac{\gamma}{2}\|\cdot\|^{2}\right)$, where
$l_{\gamma}^{*}$ is the Frenchel conjugate of a function $l_{\gamma}$ [7]. So,
$l_{\gamma}=\left(l^{*}+\frac{\gamma}{2}\|\cdot\|^{2}\right)^{*}$. Since, for
$\gamma>0$ and convex $l$, the function
$\left(l^{*}+\frac{\gamma}{2}\|\cdot\|^{2}\right)^{*}$ is the Moreau envelope
of $l$, so the notation of Frenchel conjugate is compatible with the notation
for the Moreau envelope.

######  Lemma 3.7.

[5] Let $\gamma\in(0,\beta]$ and $g:H\to\mathbb{R}$ be convex and
differentiable. Let $\nabla g$ be $\beta$-cocoercive. Then the following
statements hold:

  * (i)

For $\gamma\geq-\beta$, $\mu\in\mathbb{R}$, we have
$(g_{\gamma})_{\mu}=g_{\gamma+\mu}$. In particular,
$(g_{-\gamma})_{\gamma}=g$.

  * (ii)

$g_{-\gamma}$ is convex, lower semicontinious, proper and
$\operatorname{prox}_{\gamma g_{-\gamma}}^{M}=I-\gamma M^{-1}\nabla g$.

  * (iii)

For all $u\in H$, $g_{-\gamma}(u)=\sup\limits_{\eta\in
H}\left\\{g(\eta)-\frac{1}{2\gamma}\|u-\eta\|^{2}\right\\}$.

######  Lemma 3.8.

Let $\gamma\in(0,\beta]$, and assumptions (B1), (B2), (B3) and (A4) hold. Then

  1. (i)

$I-M^{-1}\gamma\nabla
g:\operatorname{Argmin}(f+g)\to\operatorname{Argmin}(f_{\gamma}+g_{-\gamma})$
is a bijection with inverse $\operatorname{prox}_{\gamma f}^{M}$.

  2. (ii)

$\inf(f+g)=\inf(g_{-\gamma}+f_{\gamma})$.

###### Proof.

Proof follows from [[5], Proposition 4.6], Remark 3.5, Lemma 3.7 and Lemma
2.20. ∎

In Theorem 3.6, we have discussed the asymptotic convergence of the
trajectories of (3.24) under the condition $\gamma\in(0,2\kappa);~{}\kappa>0$.
In convex optimization, the interesting observation is to think about the
choice of step sizes. Now, we choose $\gamma\in(0,\beta]$ and discuss the
dynamical system (3.24) of the optimization problem.

######  Theorem 3.9.

Let assumptions (B1), (B2), (B3) and (A4) hold. Let $\gamma\in(0,\beta]$,
$\lambda:[0,\infty)\to[0,\infty)$ be a Lebesgue measurable function satisfying
condition (3.4), $u_{0}\in H$ and $u:[0,\infty)\to H$ be the unique strong
global solution of dynamical system (3.24). Then the statements of Theorem 3.6
and the following statements are true:

  * (i)

$u(t)$ converges weakly to a minimizer of $g_{-\gamma}+f_{\gamma}$ as
$t\to\infty$.

  * (ii)

If $u^{*}$ is a minimizer of $g_{-\gamma}+f_{\gamma}$, then $\nabla
f_{\gamma}(u(\cdot))-\nabla f_{\gamma}(u^{*})\in L^{2}([0,\infty);H)$,
$\lim\limits_{t\to\infty}\nabla f_{\gamma}(u(t))=\nabla f_{\gamma}(u^{*})$ and
$\nabla f_{\gamma}(u(t))$ is constant on $g_{-\gamma}+f_{\gamma}$.

  * (iii)

If $\partial g_{-\gamma}$ or $\nabla f_{\gamma}$ is uniformly convex, then
$u(t)$ converges strongly to a minimizer of $g_{-\gamma}+f_{\gamma}$ as
$t\to\infty$.

  * (iv)

If $\partial g_{-\gamma}$ is $\rho$-strongly convex for $\rho>0$, and choose
$\eta>0$ fulfilling the condition:

$\displaystyle\frac{1}{2\alpha}+\frac{\eta\|M\|^{2}}{2\gamma^{2}}\leq\rho+\frac{\|M\|\overline{\lambda}}{\gamma}.$

Let $u^{*}$ be an equilibrium point of dynamical system (3.24). Then we have
the following:

    * (a)

If $\frac{1}{\eta\rho}<4$, then $u^{*}$ is globally exponentially stable.

    * (b)

If $\frac{1}{\eta\rho}=4$, then $u^{*}$ is global monotone attractor.

###### Proof.

It follows by applying Lemma 3.7 and Lemma 3.8 to Theorem 3.2. ∎

Now, we discuss the convergence of the trajectories of (3.24) without the
restriction on the choice of the step size
$(\gamma\in(0,2\kappa);~{}\kappa>0)$.

######  Theorem 3.10.

Let assumptions (B1), (B2), (B3) and (A4) hold. Let
$\lambda:[0,\infty)\to[0,\infty)$ be a Lebesgue measurable function satisfying
condition (3.4), $u_{0}\in H$ and $u:[0,\infty)\to H$ be the unique strong
global solution of (3.24). Then we have the following:

  * (i)

The trajectory $u$ is bounded and $\dot{u}$, $(I-(I-\gamma M^{-1}\nabla
g)\operatorname{prox}_{\gamma f}^{M})u\in L^{2}([0,\infty);H)$.

  * (ii)

$\lim\limits_{t\to\infty}\dot{u}(t)=\lim\limits_{t\to\infty}(I-(I-\gamma
M^{-1}\nabla g)\operatorname{prox}_{\gamma f}^{M})(u(t))=0$.

  * (iii)

$u(t)\rightharpoonup u^{*}\in\operatorname{Zer}((\nabla g)_{-\gamma}+\nabla
f_{\gamma})$ as $t\to\infty$.

  * (iv)

If $u^{*}\in\operatorname{Zer}((\nabla g)_{-\gamma}+\nabla f_{\gamma})$, then
$\nabla f_{\gamma}(u(\cdot))-\nabla f_{\gamma}(u^{*})\in L^{2}([0,\infty);H)$.

  * (v)

If $(\nabla g)_{-\gamma}$ or $\nabla f_{\gamma}$ is uniformly convex, then
$u(t)\to u^{*}\in\operatorname{Zer}((\nabla g)_{-\gamma}+\nabla f_{\gamma})$
as $t\to\infty$.

###### Proof.

Let $u^{*}\in\operatorname{Zer}((\nabla g)_{-\gamma}+\nabla f_{\gamma})$. From
the definition of proximal operator and Proposition 2.18, we have for every
$t\in[0,\infty)$

(3.26) $\displaystyle-M\frac{\dot{u}(t)}{\gamma\lambda(t)}-\nabla
f_{\gamma}=(\nabla
g)_{-\gamma}\left(\frac{\dot{u}(t)}{\lambda(t)}+u(t)\right).$

Combining (3.26) with $-\nabla f_{\gamma}(u^{*})\in(\nabla
g)_{-\gamma}(u^{*})$ and using the maximal monotonicity of $(\nabla
g)_{-\gamma}$, we have for every $t\in[0,\infty)$

$\displaystyle\left\langle\frac{\dot{u}(t)}{\lambda(t)}+u(t)-u^{*},\nabla
f_{\gamma}(u^{*})-M\frac{\dot{u}(t)}{\gamma\lambda(t)}-\nabla
f_{\gamma}(u(t))\right\rangle\geq 0.$

Since $\nabla f_{\gamma}$ is $\alpha$-cocoercive, so for every
$t\in[0,\infty)$

$\displaystyle\alpha\|\nabla f_{\gamma}(u(t))-\nabla f_{\gamma}(u^{*})\|$
$\displaystyle\leq\langle u(t)-u^{*},\nabla f_{\gamma}(u(t))-\nabla
f_{\gamma}(u^{*})\rangle$ $\displaystyle\leq\left\langle
\frac{\dot{u}(t)}{\lambda(t)},\nabla
f_{\gamma}-M\frac{\dot{u}(t)}{\gamma\lambda(t)}-\nabla
f_{\gamma}(u(t))\right\rangle$ $\displaystyle+\left\langle
u(t)-u^{*},-M\frac{\dot{u}(t)}{\gamma\lambda(t)}\right\rangle$
$\displaystyle\leq\frac{1}{\lambda(t)}\langle\dot{u}(t),\nabla
f_{\gamma}(u^{*})-\nabla
f_{\gamma}(u(t))\rangle-\frac{1}{\gamma\lambda^{2}(t)}\|\dot{u}(t)\|_{M}^{2}$
$\displaystyle+\frac{1}{\gamma\lambda(t)}\langle
u(t)-u^{*},-M\dot{u}(t)\rangle.$

Define the map $q:[0,\infty)\to\mathbb{R}$,
$q(t)=f_{\gamma}(u(t))-f_{\gamma}(u^{*})-\langle\nabla
f_{\gamma}(u^{*}),u(t)-u^{*}\rangle$.
Since $\nabla f_{\gamma}$ is convex function, so we have

$q(t)\geq 0~{}\forall t\geq 0.$

Also, for any $t\in[0,\infty)$

$\dot{q}(t)=\langle\dot{u}(t),\nabla f_{\gamma}(u(t))-\nabla
f_{\gamma}(u^{*})\rangle.$

Define the function
$h:[0,\infty)\to\mathbb{R},h(t)=\frac{1}{2}\|u(t)-u^{*}\|^{2}_{M}$ and using
the fact that $\dot{h}(t)=\langle u(t)-u^{*},M\dot{u}(t)\rangle$, we obtain

(3.27) $\displaystyle\alpha\lambda(t)\|\nabla f_{\gamma}(u(t))-\nabla
f_{\gamma}(u^{*})\|\leq-\frac{d}{dt}(q(t))-\frac{1}{\gamma\lambda(t)}\|\dot{u}(t)\|_{M}^{2}-\frac{1}{\gamma}\dot{h}(t),$

which implies that

$\displaystyle\alpha\lambda(t)\|\nabla f_{\gamma}(u(t))-\nabla
f_{\gamma}(u^{*})\|+\frac{d}{dt}\left(\frac{1}{\gamma}h+q\right)+\frac{1}{\gamma\lambda(t)}\|\dot{u}(t)\|_{M}^{2}\leq
0.$

So the function $t\mapsto\frac{1}{\gamma}h+q$ is monotonically decreasing.
Keeping in mind the proof of Proposition 3.1, and the fact that $\lambda$ has
positive upper and lower bounds, we obtain that $\frac{1}{\gamma}h+q,h,q,u$
are bounded and $\dot{u},(I-(I-\gamma M^{-1}\nabla
g)\operatorname{prox}_{\gamma f}^{M})u\in L^{2}([0,\infty);H)$. Also,
$\lim\limits_{t\to\infty}\dot{u}(t)=0$. It follows from (3.27) that

$\nabla f_{\gamma}(u(t))-\nabla f_{\gamma}(u^{*})\in L^{2}([0,\infty);H).$

From Lemma 2.15, we conclude

$\lim\limits_{t\to\infty}\nabla f_{\gamma}(u(t))=\nabla f_{\gamma}(u^{*}).$

Therefore, statements (i), (ii) and (iv) are proved.

(iii) First we show that every weak sequential cluster point of $u(\cdot)$ is
in $\operatorname{Zer}((\nabla g)_{-\gamma}+\nabla f_{\gamma})$. Let
$u^{*}\in\operatorname{Zer}((\nabla g)_{-\gamma}+\nabla f_{\gamma})$ and
$t_{n}\to\infty$ (as $n\to\infty$) be such that
$\\{u(t_{n})\\}\rightharpoonup\bar{u}$. Since $(u(t_{n}),\nabla
f_{\gamma}(u(t_{n}))\in\operatorname{gra}(\nabla f_{\gamma})$,
$\lim\limits_{n\to\infty}\nabla f_{\gamma}(u(t_{n}))=\nabla f_{\gamma}(u^{*})$
and $\operatorname{gra}(\nabla f_{\gamma})$ is sequentially closed in the
weak-strong topology, we get $\nabla f_{\gamma}(\bar{u})=\nabla
f_{\gamma}(u^{*})$.

Using the fact that $\operatorname{gra}(\nabla g)$ is sequentially closed in
the weak-strong topology and letting $A=\partial f$, $B=\nabla g$ and
$t=t_{n}$, we have $-\nabla f_{\gamma}(u^{*})\in(\nabla
g)_{-\gamma}(\bar{u})$. So, we get $-\nabla f_{\gamma}(\bar{u})\in(\nabla
g)_{-\gamma}(\bar{u})$, hence $\bar{u}\in\operatorname{Zer}((\nabla
g)_{-\gamma}+\nabla f_{\gamma})$.

Next, we show that $u(\cdot)$ has at most one weak sequential cluster point.
It proves that the trajectory convergence weakly to a zero of $(\nabla
g)_{-\gamma}+\nabla f_{\gamma}$.

Let $u^{*},v^{*}$ be two weak sequential cluster point of $u(\cdot)$. So,
there exist sequences $\\{t_{n}\\}\to\infty$ and
$\\{t^{\prime}_{n}\\}\to\infty$ such that $\\{u(t_{n})\\}\rightharpoonup
u^{*}$ and $\\{u(t^{\prime}_{n})\\}\rightharpoonup v^{*}$. Since
$u^{*},v^{*}\in\underset{u\in
H}{\operatorname{Argmin}}\\{f(u)+g(u)\\}\neq\emptyset$, we have
$\lim\limits_{t\to\infty}\Phi(t,u^{*})\in\mathbb{R}$ and
$\lim\limits_{t\to\infty}\Phi(t,v^{*})\in\mathbb{R}$, hence
$\exists\lim\limits_{t\to\infty}\Phi(t,u^{*})-\lim\limits_{t\to\infty}\Phi(t,v^{*})\in\mathbb{R}$,
where

$\Phi(t,u^{*})=\frac{1}{2\gamma}\|u(t)-u^{*}\|^{2}+f(u(t))-f(u^{*})-\langle\nabla
f_{\gamma}(u^{*}),u(t)-u^{*}\rangle.$

So, we get

(3.28)
$\displaystyle\exists\lim\limits_{t\to\infty}\left(\frac{1}{\gamma}\langle
u(t),v^{*}-u^{*}\rangle+\langle\nabla f_{\gamma}(u^{*})-\nabla
f_{\gamma}(v^{*}),u(t)\rangle\right)\in\mathbb{R}.$

If we express (3.28) by the means of sequences $\\{t_{n}\\}$ and
$\\{t^{\prime}_{n}\\}$, we obtain

$\displaystyle\frac{1}{\gamma}\langle u^{*},v^{*}-u^{*}\rangle+\langle \nabla
f_{\gamma}(v^{*})-\nabla f_{\gamma}(u^{*}),u^{*}\rangle$
$\displaystyle=\frac{1}{\gamma}\langle v^{*},v^{*}-u^{*}\rangle+\langle \nabla
f_{\gamma}(v^{*})-\nabla f_{\gamma}(u^{*}),v^{*}\rangle,$

which implies that

$\displaystyle\frac{1}{\gamma}\|u^{*}-v^{*}\|+\langle \nabla
f_{\gamma}(v^{*})-\nabla f_{\gamma}(u^{*}),v^{*}-u^{*}\rangle=0,$

and by the monotonicity of $\nabla f_{\gamma}$ we conclude that $u^{*}=v^{*}$.

(v) The proof follows by using $A=\partial f$ and $B=\nabla g$ in Theorem
3.2(vii). ∎

##  4\. Numerical Examples

######  Example 4.1.

Let $H=\mathbb{R}$ be a Hilbert space endowed with Euclidean inner product.
Let $A:H\to 2^{H}$ be a set-valued operator defined by

(4.4)
$\displaystyle{A(x)=\left\\{\begin{array}[]{lc@{}c@{}r}0,~{}~{}~{}~{}~{}~{}~{}if~{}x<0\\\\[6.0pt]
[0,1],~{}~{}if~{}x=0\\\\[6.0pt]
1,~{}~{}~{}~{}~{}~{}~{}if~{}x>0.\end{array}\right.}$

Note that $A$ is maximally monotone operator. So, $A$ is $\rho$-cohypomonotone
operator for $\rho\geq 0$.
Let $B:\mathbb{R}\to\mathbb{R}$ be $1$-cocoercive operator defined by

$B(x)=\frac{x}{2}.$

So $\beta=1$. Let $M:\mathbb{R}\to\mathbb{R}$ be a strongly positive operator
defined by $M(x)=3x$. So, from (1.14), we have the dynamical system

(4.10)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{u}(t)={\lambda(t)\left\\{\begin{array}[]{lc@{}c@{}r}\frac{11u(t)}{12}-u(t),~{}~{}~{}~{}~{}~{}~{}t<0\\\\[6.0pt]
-u(t),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}t\in[0,\frac{1}{6}]\\\\[6.0pt]
\frac{11u(t)}{12}-\frac{1}{6}-u(t),~{}~{}t>\frac{1}{6}\end{array}\right.}\\\
u(0)=u_{0}.\end{array}\right.}$

Choose $\rho=\frac{1}{4}=\frac{1}{2}-\frac{1}{4}$. So,
$\gamma=\frac{1}{2},\alpha=\frac{1}{4}$. Let $\kappa=\frac{1}{2}$. Observe
that all the assumptions of Theorem 3.2 are satisfied. Figure 1 shows the
convergence behaviour of the trajectories generated by dynamical system (1.14)
for the Lebesgue measurable function $\lambda:[0,\infty)\to[0,\infty)$ defined
by

(4.13)
$\displaystyle{\lambda(t)=\left\\{\begin{array}[]{lc@{}c@{}r}1,~{}~{}if~{}t\in[0,50]\\\\[6.0pt]
0,~{}~{}otherwise.\end{array}\right.}$

######  Example 4.2.

Let $H=\mathbb{R}^{2}$ be a real Hilbert space endowed with Euclidean inner
product and $N:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be an operator defined by

$N=\begin{bmatrix}-2&0\\\ 0&0\end{bmatrix}.$

Now, consider the multivalued operator $A:=N^{-1}$, so by Example 2.3,
$A_{\rho}=(N+\rho I)^{-1}$ is maximally monotone operator for $\rho>2$, and
hence $A$ is $\rho(>2)$-cohypomonotone operator. Let
$B:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be $1$-cocercive operator defined by

$\displaystyle B=\begin{bmatrix}\frac{1}{2}&0\\\ 0&\frac{1}{3}\end{bmatrix}.$

Here $\beta=1$. Let $M:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a strongly positive
operator defined by

$\displaystyle M=\begin{bmatrix}4&0\\\ 0&8\end{bmatrix}.$

So, from (1.14), we have the dynamical system

(4.16)
$\displaystyle{\left\\{\begin{array}[]{lc@{}c@{}r}\dot{u}(t)=\lambda(t)\left[\begin{bmatrix}\frac{1}{4}&0\\\
0&\frac{35}{48}\end{bmatrix}u(t)-u(t)\right]\par\\\
u(0)=u_{0}.\end{array}\right.}$

Choose $\rho=3=4-1$. So, $\gamma=4,\alpha=1$. Let $\kappa=3$. Observe that all
the assumptions of the Theorem 3.2 are satisfied. Figure 2 shows the
convergence behaviour of the trajectories generated by dynamical system (1.14)
for the Lebesgue measurable function $\lambda:[0,\infty)\to[0,\infty)$ defined
by (4.13).

Figure 1. Trajectories generated by the dynamical system of Example 4.1 for
$u_{0}=0.5$. Figure 2. Trajectories generated by the dynamical system of
Example 4.2 for $u_{0}=(-0.4,0.5).$

Note that discrete version of preconditioned backward-forward dynamical system
is:

(4.17) $\displaystyle u_{n+1}=u_{n}+\lambda_{n}((I-\gamma M^{-1}B)(I-\gamma
M^{-1}A_{\gamma})u_{n}-u_{n}).$

Taking a bounded sequence $\lambda_{n}=\frac{1}{n}$, one can observe by
Example 4.2 and Figure 3 that sequence $\\{\|u_{n}\|\\}$ converges.

Figure 3. Convergence of sequence $\\{\|u_{n}\|\\}$ given by (4.17) for
Example 4.2 and $u_{0}=(3,2).$

##  5\. Conclusions

In this paper, first-order variable metric backward-forward dynamical systems
associated with monotone inclusion, and convex minimization problems have been
studied. Existence, uniqueness, weak and strong convergence of the
trajectories of dynamical systems (1.14), (1.17), and (3.24) have been
studied. We have also established that an equilibrium point of the trajectory
is globally exponentially stable and monotone attractor.

## Acknowledgement

The first author acknowledges the financial support from Ministry of Human
Resource and Development (MHRD), New Delhi, India under Junior Research Fellow
(JRF) scheme.

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